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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.17117v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S1" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">1 Introduction</a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S2" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">2 Related work</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S2.SS0.SSS0.Px1" title="In 2 Related work ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Subtraction-based methods.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S2.SS0.SSS0.Px2" title="In 2 Related work ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Statistical approaches.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S2.SS0.SSS0.Px3" title="In 2 Related work ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Learning-based approaches.</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S3" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">3 Image formation models</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S3.SS1" title="In 3 Image formation models ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">3.1 Angular Differential Imaging (ADI)</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S3.SS2" title="In 3 Image formation models ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">3.2 Angular Spectral Differential Imaging (ASDI)</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4 Proposed method</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1" title="In 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.1 Convolutional statistical model</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1.SSS0.Px1" title="In 4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Local Gaussian model of speckles.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1.SSS0.Px2" title="In 4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Detection criterion at a given position.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1.SSS0.Px3" title="In 4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Iterative procedure for characterization.</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS2" title="In 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.2 Extensions of the statistical model: mixture of distributions</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS2.SSS0.Px1" title="In 4.2 Extensions of the statistical model: mixture of distributions ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Linear projection.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS2.SSS0.Px2" title="In 4.2 Extensions of the statistical model: mixture of distributions ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Multi-scale approach.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS2.SSS0.Px3" title="In 4.2 Extensions of the statistical model: mixture of distributions ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Leveraging rotational symmetries.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS2.SSS0.Px4" title="In 4.2 Extensions of the statistical model: mixture of distributions ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Joint multi-spectral modeling.</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS3" title="In 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.3 End-to-end trainable approach</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS3.SSS0.Px1" title="In 4.3 End-to-end trainable approach ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Problem statement.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS3.SSS0.Px2" title="In 4.3 End-to-end trainable approach ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Detection by denoising.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS3.SSS0.Px3" title="In 4.3 End-to-end trainable approach ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Training objective.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS3.SSS0.Px4" title="In 4.3 End-to-end trainable approach ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Model ensembling.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS3.SSS0.Px5" title="In 4.3 End-to-end trainable approach ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Calibration.</a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">5 Experiments</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.SS1" title="In 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">5.1 Data, algorithms and evaluation protocol</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.SS1.SSS0.Px1" title="In 5.1 Data, algorithms and evaluation protocol ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Datasets.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.SS1.SSS0.Px2" title="In 5.1 Data, algorithms and evaluation protocol ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Training procedure.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.SS1.SSS0.Px3" title="In 5.1 Data, algorithms and evaluation protocol ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Baselines.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.SS1.SSS0.Px4" title="In 5.1 Data, algorithms and evaluation protocol ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Metrics.</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.SS2" title="In 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">5.2 Quantitative and qualitative evaluations</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.SS2.SSS0.Px1" title="In 5.2 Quantitative and qualitative evaluations ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Statistical model.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.SS2.SSS0.Px2" title="In 5.2 Quantitative and qualitative evaluations ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">End-to-end learnable model.</a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.SS2.SSS0.Px3" title="In 5.2 Quantitative and qualitative evaluations ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Validation on real data.</a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S6" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">6 Conclusion</a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A Technical details</a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS1" title="In Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.1 Estimation of the statistical parameters</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS2" title="In Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.2 Dense detection map computation details</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS3" title="In Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.3 Fast computation of <math alttext="\boldsymbol{b}_{t}" class="ltx_Math" display="inline"><semantics><msub><mi>𝒃</mi><mi>t</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝒃</ci><ci>𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex">\boldsymbol{b}_{t}</annotation><annotation encoding="application/x-llamapun">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{a}" class="ltx_Math" display="inline"><semantics><mi>𝒂</mi><annotation-xml encoding="MathML-Content"><ci>𝒂</ci></annotation-xml><annotation encoding="application/x-tex">\boldsymbol{a}</annotation><annotation encoding="application/x-llamapun">bold_italic_a</annotation></semantics></math></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS4" title="In Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.4 Iterative optimization scheme for astrometry</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS5" title="In Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.5 Prior information on the exoplanet spectrum</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS6" title="In Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.6 Calibration procedure</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS7" title="In Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.7 Implementation details</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A2" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">B Dataset description</a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A3" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">C Comparison of state-of-the-art methods</a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A4" title="In A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">D Additional results</a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations </h1> <div class="ltx_authors"> Théo Bodrito Corresponding author: theo.bodrito@inria.fr Département d’Informatique de l’École normale supérieure (ENS-PSL, CNRS, Inria) Olivier Flasseur Universite Claude Bernard Lyon 1, Centre de Recherche Astrophysique de Lyon UMR 5574, ENS de Lyon, CNRS, Villeurbanne, F-69622, France Julien Mairal Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK Jean Ponce Département d’Informatique de l’École normale supérieure (ENS-PSL, CNRS, Inria) Courant Institute and Center for Data Science, New York University Maud Langlois Universite Claude Bernard Lyon 1, Centre de Recherche Astrophysique de Lyon UMR 5574, ENS de Lyon, CNRS, Villeurbanne, F-69622, France Anne-Marie Lagrange Laboratoire d’Études Spatiales et d’Instrumentation en Astrophysique, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Diderot Université Grenoble Alpes, Institut de Planétologie et d’Astrophysique de Grenoble </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> The search for exoplanets is an active field in astronomy, with direct imaging as one of the most challenging methods due to faint exoplanet signals buried within stronger residual starlight. Successful detection requires advanced image processing to separate the exoplanet signal from this nuisance component. This paper presents a novel statistical model that captures nuisance fluctuations using a multi-scale approach, leveraging problem symmetries and a joint spectral channel representation grounded in physical principles. Our model integrates into an interpretable, end-to-end learnable framework for simultaneous exoplanet detection and flux estimation. The proposed algorithm is evaluated against the state of the art using datasets from the SPHERE instrument operating at the Very Large Telescope (VLT). It significantly improves the precision-recall trade-off, notably on challenging datasets that are otherwise unusable by astronomers. The proposed approach is computationally efficient, robust to varying data quality, and well suited for large-scale observational surveys. 111Code is available at <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/theobdt/exomild" title="">https://github.com/theobdt/exomild</a> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> 1 Introduction</h2> <div class="ltx_para" id="S1.p1"> Direct imaging <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib65" title="">65</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib30" title="">30</a>]</cite> is an astronomical technique to probe the vicinity of young, nearby stars, where exoplanets and circumstellar disks—structures of dust and gas from which exoplanets can form—are found <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib37" title="">37</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib36" title="">36</a>]</cite>. Unlike indirect methods <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib57" title="">57</a>]</cite>, which detect exoplanets via secondary effects like gravitational wobbles or transit dimming, direct imaging captures visual evidence of exoplanets and circumstellar disks by recording a direct image of their emitted flux. Analyzing this light across spectral bands provides insights into exoplanet properties (e.g., temperature, gravity), atmospheric compositions, molecular abundances, and formation processes <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib66" title="">66</a>]</cite>. </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="349" id="S1.F1.g1" src="x1.png" width="428"/> <figcaption class="ltx_caption ltx_centering">Figure A: Left: typical observations <math alttext="\boldsymbol{y}" class="ltx_Math" display="inline" id="S1.F1.3.1.m1.1"><semantics id="S1.F1.3.1.m1.1b"><mi id="S1.F1.3.1.m1.1.1" xref="S1.F1.3.1.m1.1.1.cmml">𝒚</mi><annotation-xml encoding="MathML-Content" id="S1.F1.3.1.m1.1c"><ci id="S1.F1.3.1.m1.1.1.cmml" xref="S1.F1.3.1.m1.1.1">𝒚</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.3.1.m1.1d">\boldsymbol{y}</annotation><annotation encoding="application/x-llamapun" id="S1.F1.3.1.m1.1e">bold_italic_y</annotation></semantics></math> and PSF <math alttext="\boldsymbol{h}" class="ltx_Math" display="inline" id="S1.F1.4.2.m2.1"><semantics id="S1.F1.4.2.m2.1b"><mi id="S1.F1.4.2.m2.1.1" xref="S1.F1.4.2.m2.1.1.cmml">𝒉</mi><annotation-xml encoding="MathML-Content" id="S1.F1.4.2.m2.1c"><ci id="S1.F1.4.2.m2.1.1.cmml" xref="S1.F1.4.2.m2.1.1">𝒉</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.4.2.m2.1d">\boldsymbol{h}</annotation><annotation encoding="application/x-llamapun" id="S1.F1.4.2.m2.1e">bold_italic_h</annotation></semantics></math> from the SPHERE instrument in ASDI mode. The synthetic exoplanet is very bright for illustration purposes. Right: temporal slice along the vertical line.</figcaption> </figure> <div class="ltx_para" id="S1.p2"> While existing technology has enabled imaging of young, giant, gaseous exoplanets, next-generation ground- and space-based telescopes may soon image rocky exoplanets in habitable zones, advancing the search for life beyond our Solar System <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib14" title="">14</a>]</cite>. Imaging exoplanets is challenging due to the high contrast and angular resolution required. The primary difficulty lies in the large brightness ratio (or contrast) between the host star and exoplanets; in the infrared, where gas giant exoplanets’ thermal emission is most detectable, they are typically <math alttext="10^{5}" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><msup id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml"><mn id="S1.p2.1.m1.1.1.2" xref="S1.p2.1.m1.1.1.2.cmml">10</mn><mn id="S1.p2.1.m1.1.1.3" xref="S1.p2.1.m1.1.1.3.cmml">5</mn></msup><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><apply id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p2.1.m1.1.1.1.cmml" xref="S1.p2.1.m1.1.1">superscript</csymbol><cn id="S1.p2.1.m1.1.1.2.cmml" type="integer" xref="S1.p2.1.m1.1.1.2">10</cn><cn id="S1.p2.1.m1.1.1.3.cmml" type="integer" xref="S1.p2.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">10^{5}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT</annotation></semantics></math>-<math alttext="10^{6}" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><msup id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml"><mn id="S1.p2.2.m2.1.1.2" xref="S1.p2.2.m2.1.1.2.cmml">10</mn><mn id="S1.p2.2.m2.1.1.3" xref="S1.p2.2.m2.1.1.3.cmml">6</mn></msup><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><apply id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p2.2.m2.1.1.1.cmml" xref="S1.p2.2.m2.1.1">superscript</csymbol><cn id="S1.p2.2.m2.1.1.2.cmml" type="integer" xref="S1.p2.2.m2.1.1.2">10</cn><cn id="S1.p2.2.m2.1.1.3.cmml" type="integer" xref="S1.p2.2.m2.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">10^{6}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT</annotation></semantics></math> times fainter. Additionally, because exoplanets appear close to their host stars from Earth’s perspective, high spatial resolution is essential to separate their faint signals. To address these challenges, observatories like the VLT are equipped with specialized instruments for direct imaging (e.g., SPHERE <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib4" title="">4</a>]</cite>) and cutting-edge optical technologies. Adaptive optics employs a deformable mirror to obtain sharp images by correcting for atmospheric turbulence in real time <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib20" title="">20</a>]</cite>. Coronagraphs (optical masks) block some starlight, further improving contrast <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib58" title="">58</a>]</cite>. </div> <div class="ltx_para" id="S1.p3"> Despite advanced optical devices, direct imaging remains challenging as exoplanet signals are still dominated by a strong nuisance component with approximately <math alttext="10^{3}" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><msup id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mn id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml">10</mn><mn id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml">3</mn></msup><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.1.cmml" xref="S1.p3.1.m1.1.1">superscript</csymbol><cn id="S1.p3.1.m1.1.1.2.cmml" type="integer" xref="S1.p3.1.m1.1.1.2">10</cn><cn id="S1.p3.1.m1.1.1.3.cmml" type="integer" xref="S1.p3.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">10^{3}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> times higher contrast. This nuisance is mainly composed of structured speckles <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib24" title="">24</a>]</cite>, caused by imperfections in optical corrections that allow residual starlight, unblocked by the coronagraph, to leak into images as a spatially correlated diffraction pattern. The speckle pattern is both spatially and spectrally correlated, exhibiting quasi-static behavior across exposures with minor fluctuations over time. In addition to speckles, other stochastic noise sources–such as thermal background, detector readout, and photon noise–add further contamination. Together, these factors create a non-stationary nuisance that varies in intensity and structure across the field of view, with higher intensity and correlation near the star. This nuisance, which closely resembles the point-like signals expected from exoplanets (instrumental point-spread function off the optical axis), is the primary limitation to direct imaging. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S1.F1" title="Figure A ‣ 1 Introduction ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A</a> illustrates these characteristics with a dataset recorded by SPHERE. </div> <div class="ltx_para" id="S1.p4"> In this context, dedicated processing is crucial to separate exoplanet signals from the nuisance component <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib50" title="">50</a>]</cite>. Observational techniques like angular differential imaging (ADI <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib41" title="">41</a>]</cite>), spectral differential imaging (SDI <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib61" title="">61</a>]</cite>), and combined angular and spectral differential imaging (ASDI <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib66" title="">66</a>]</cite>) introduce diversity that aids in distinguishing exoplanet signals from the nuisance, see Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S3" title="3 Image formation models ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">3</a> for resulting image formation models. ADI takes advantage of Earth’s rotation by keeping the telescope pupil fixed, causing off-axis exoplanets to follow a predictable circular trajectory, while the star remains centered. This apparent motion helps separate exoplanet signals from quasi-static speckles. SDI further improves separation by capturing images across multiple spectral channels, where speckles scale quasi-linearly with wavelength while exoplanet signals stay fixed. ASDI combines ADI and SDI to produce a four-dimensional dataset (spatial, temporal, and spectral dimensions). This diversity in A(S)DI calls for advanced algorithms that can leverage these priors to optimize exoplanet detection and spectral characterization. </div> <div class="ltx_para" id="S1.p5"> We propose a novel hybrid approach that combines the interpretability of a statistical framework with end-to-end learnable components, improving detection sensitivity and characterization accuracy for directly imaged exoplanets. Our main contributions include: <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> • <div class="ltx_para" id="S1.I1.i1.p1"> A learnable architecture based on a statistical model integrating pixel correlations across spatial scales and leveraging the spatial symmetries of the nuisance component, improving both detection and characterization. </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> • <div class="ltx_para" id="S1.I1.i2.p1"> Statistically reliable detection scores and estimates with uncertainty quantification, essential for astrophysics. </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> • <div class="ltx_para" id="S1.I1.i3.p1"> Joint processing of 4-D datasets incorporating the ASDI forward model, shown to significantly boost detection. </div> </li> </ul> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> 2 Related work</h2> <div class="ltx_para" id="S2.p1"> Various methods have been developed to isolate faint planetary signals from stellar nuisance, broadly categorized as subtraction-, statistical-, and learning-based models <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib50" title="">50</a>]</cite>. </div> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Subtraction-based methods.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p1"> Subtraction-based methods are among the earliest and most common techniques for exoplanet detection in direct imaging, aiming to remove quasi-static speckles that obscure faint signals. The cA(S)DI algorithms <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib41" title="">41</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib38" title="">38</a>]</cite> subtract a reference model by averaging frames and stacking residuals, enhancing the rotating exoplanet signal. TLOCI <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib44" title="">44</a>]</cite> and its variants (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib68" title="">68</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib32" title="">32</a>]</cite>) optimize linear combinations of images to model speckles, while KLIP/PCA <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib60" title="">60</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib3" title="">3</a>]</cite> employs principal component analysis for low-rank subspace projection. Other approaches, such as non-negative matrix factorization <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib51" title="">51</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib52" title="">52</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib49" title="">49</a>]</cite> and LLSG <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib34" title="">34</a>]</cite>, decompose data into components to isolate sources, while the RSM algorithm <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib19" title="">19</a>]</cite> combines outputs from multiple methods to reduce individual biases. However, these methods often lack statistically rigorous outputs, such as interpretable detection scores and unbiased flux estimates, as they rely on heuristic image combinations rather than fully end-to-end models. </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Statistical approaches.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px2.p1"> To address the previous limitations, statistical methods model the nuisance using probabilistic frameworks. The ANDROMEDA <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib6" title="">6</a>]</cite> and FMMF <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib54" title="">54</a>]</cite> approaches rely on matched filtering, simplifying the nuisance as uncorrelated Gaussian noise. Similarly, SNAP <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib64" title="">64</a>]</cite> estimates both the nuisance and exoplanet components jointly through maximum likelihood under the same assumption. The PACO algorithm <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib26" title="">26</a>]</cite> improves on this by extending beyond the white noise assumption. It uses a patch-based statistical framework to model the spatial and spectral covariances of the nuisance, effectively capturing the local structure of speckles. This approach draws on techniques from computer vision, such as denoising, restoration, super-resolution, collaborative filtering <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib15" title="">15</a>]</cite>, sparse coding <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib40" title="">40</a>]</cite>, or mixture models <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib72" title="">72</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib71" title="">71</a>]</cite>. </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Learning-based approaches.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px3.p1"> Machine learning methods are increasingly used in high-contrast imaging, inspired by advances in fields like photography and biomedical imaging. Early applications in exoplanet detection, such as the SVM model by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib23" title="">23</a>]</cite>, leverage structured high-contrast data, while the (NA)-SODINN algorithms <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib8" title="">8</a>]</cite> frames detection as a binary classification problem, using KLIP/PCA-processed patches with random forests or CNNs. Despite their effectiveness, SODINN struggles with high false alarm rates and complex hyperparameter tuning <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib7" title="">7</a>]</cite>. Generative adversarial networks (GANs) have also been applied to simulate nuisance patterns for training deep learning models <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib70" title="">70</a>]</cite>. Approaches like TRAP <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib55" title="">55</a>]</cite> and HSR <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib31" title="">31</a>]</cite> handle unmixing via regularized regression, modeling nuisance evolution with signal-free reference data. Deep PACO <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib28" title="">28</a>]</cite> combines statistical modeling with deep learning, leveraging PACO’s nuisance statistical model and a CNN to detect exoplanets and refine residual mismatches. </div> <div class="ltx_para" id="S2.SS0.SSS0.Px3.p2"> All these algorithms are observation-dependent, building nuisance models directly from the dataset of interest. This dependency hinders detection near the host star due to (i) significant temporal fluctuations in the nuisance and (ii) residual self-subtraction, where part of the exoplanet signal is mistakenly removed. Recent observation-independent approaches address these limitations by using archival survey data to model the nuisance. Super-RDI <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib56" title="">56</a>]</cite> extends KLIP/PCA for large observational databases, while ConStruct <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib69" title="">69</a>]</cite> uses an auto-encoder to learn typical speckle patterns, and <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib11" title="">11</a>]</cite> employs a discriminative nuisance model. MODEL&CO <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib5" title="">5</a>]</cite> improves deep PACO with a deep statistically-modeled nuisance framework. Our proposed approach, ExoMILD (Exoplanet imaging by MIxture of Learnable Distributions), belongs to this new category. </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> 3 Image formation models</h2> <div class="ltx_para" id="S3.p1"> In direct imaging, a stellar system is observed through an optical system that includes the atmosphere, telescope, and scientific instrument. The response of this system to a point source (e.g., an exoplanet) is defined by its point-spread function off the optical axis (off-axis PSF), describing how light from the source is distributed across the sensor. However, as noted in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S1" title="1 Introduction ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">1</a>, residual aberrations uncorrected by adaptive optics produce a quasi-static, structured speckle pattern. To mitigate speckles through numerical processing, several observational strategies are used, as detailed next. </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> 3.1 Angular Differential Imaging (ADI)</h3> <div class="ltx_para" id="S3.SS1.p1"> In ADI, the field derotator of the telescope is turned off causing the field of view (including any exoplanets) to rotate around the target star due to Earth’s rotation. Meanwhile, the optical system remains stationary, causing the speckles to remain fixed. This distinction helps separating exoplanet signals from speckle noise. Formally, let <math alttext="\boldsymbol{y}" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mi id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml">𝒚</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><ci id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">𝒚</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">\boldsymbol{y}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">bold_italic_y</annotation></semantics></math> in <math alttext="\mathbb{R}^{T\times H\times W}" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><msup id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.p1.2.m2.1.1.3" xref="S3.SS1.p1.2.m2.1.1.3.cmml"><mi id="S3.SS1.p1.2.m2.1.1.3.2" xref="S3.SS1.p1.2.m2.1.1.3.2.cmml">T</mi><mo id="S3.SS1.p1.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.2.m2.1.1.3.1.cmml">×</mo><mi id="S3.SS1.p1.2.m2.1.1.3.3" xref="S3.SS1.p1.2.m2.1.1.3.3.cmml">H</mi><mo id="S3.SS1.p1.2.m2.1.1.3.1a" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.2.m2.1.1.3.1.cmml">×</mo><mi id="S3.SS1.p1.2.m2.1.1.3.4" xref="S3.SS1.p1.2.m2.1.1.3.4.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">superscript</csymbol><ci id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2">ℝ</ci><apply id="S3.SS1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.p1.2.m2.1.1.3"><times id="S3.SS1.p1.2.m2.1.1.3.1.cmml" xref="S3.SS1.p1.2.m2.1.1.3.1"></times><ci id="S3.SS1.p1.2.m2.1.1.3.2.cmml" xref="S3.SS1.p1.2.m2.1.1.3.2">𝑇</ci><ci id="S3.SS1.p1.2.m2.1.1.3.3.cmml" xref="S3.SS1.p1.2.m2.1.1.3.3">𝐻</ci><ci id="S3.SS1.p1.2.m2.1.1.3.4.cmml" xref="S3.SS1.p1.2.m2.1.1.3.4">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">\mathbb{R}^{T\times H\times W}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_T × italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> denote the sequence of measurements from the observation, where <math alttext="T" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mi id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><ci id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_T</annotation></semantics></math> is the number of exposures and <math alttext="H" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mi id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><ci id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">italic_H</annotation></semantics></math>, <math alttext="W" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.1"><semantics id="S3.SS1.p1.5.m5.1a"><mi id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.1b"><ci id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.1c">W</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.1d">italic_W</annotation></semantics></math> represent the pixel dimensions of each exposure. Each frame <math alttext="\boldsymbol{y}_{t}" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m6.1"><semantics id="S3.SS1.p1.6.m6.1a"><msub id="S3.SS1.p1.6.m6.1.1" xref="S3.SS1.p1.6.m6.1.1.cmml"><mi id="S3.SS1.p1.6.m6.1.1.2" xref="S3.SS1.p1.6.m6.1.1.2.cmml">𝒚</mi><mi id="S3.SS1.p1.6.m6.1.1.3" xref="S3.SS1.p1.6.m6.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m6.1b"><apply id="S3.SS1.p1.6.m6.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m6.1.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.p1.6.m6.1.1.2.cmml" xref="S3.SS1.p1.6.m6.1.1.2">𝒚</ci><ci id="S3.SS1.p1.6.m6.1.1.3.cmml" xref="S3.SS1.p1.6.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m6.1c">\boldsymbol{y}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m6.1d">bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="S3.SS1.p1.7.m7.1"><semantics id="S3.SS1.p1.7.m7.1a"><msup id="S3.SS1.p1.7.m7.1.1" xref="S3.SS1.p1.7.m7.1.1.cmml"><mi id="S3.SS1.p1.7.m7.1.1.2" xref="S3.SS1.p1.7.m7.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.p1.7.m7.1.1.3" xref="S3.SS1.p1.7.m7.1.1.3.cmml"><mi id="S3.SS1.p1.7.m7.1.1.3.2" xref="S3.SS1.p1.7.m7.1.1.3.2.cmml">H</mi><mo id="S3.SS1.p1.7.m7.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.7.m7.1.1.3.1.cmml">×</mo><mi id="S3.SS1.p1.7.m7.1.1.3.3" xref="S3.SS1.p1.7.m7.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m7.1b"><apply id="S3.SS1.p1.7.m7.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.7.m7.1.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1">superscript</csymbol><ci id="S3.SS1.p1.7.m7.1.1.2.cmml" xref="S3.SS1.p1.7.m7.1.1.2">ℝ</ci><apply id="S3.SS1.p1.7.m7.1.1.3.cmml" xref="S3.SS1.p1.7.m7.1.1.3"><times id="S3.SS1.p1.7.m7.1.1.3.1.cmml" xref="S3.SS1.p1.7.m7.1.1.3.1"></times><ci id="S3.SS1.p1.7.m7.1.1.3.2.cmml" xref="S3.SS1.p1.7.m7.1.1.3.2">𝐻</ci><ci id="S3.SS1.p1.7.m7.1.1.3.3.cmml" xref="S3.SS1.p1.7.m7.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m7.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.7.m7.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> can then be represented as: <table class="ltx_equation ltx_eqn_table" id="S3.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\boldsymbol{y}_{t}={\boldsymbol{s}}_{t}+\sum\nolimits_{k=0}^{K-1}\alpha^{(k)}% \,\boldsymbol{h}(\boldsymbol{x}^{(k)}_{t})+\boldsymbol{\epsilon}_{t}\,," class="ltx_Math" display="block" id="S3.E1.m1.3"><semantics id="S3.E1.m1.3a"><mrow 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xref="S3.E1.m1.3.3.1.1.1.1.2.2.3.cmml"><mi id="S3.E1.m1.3.3.1.1.1.1.2.2.3.2" xref="S3.E1.m1.3.3.1.1.1.1.2.2.3.2.cmml">k</mi><mo id="S3.E1.m1.3.3.1.1.1.1.2.2.3.1" xref="S3.E1.m1.3.3.1.1.1.1.2.2.3.1.cmml">=</mo><mn id="S3.E1.m1.3.3.1.1.1.1.2.2.3.3" xref="S3.E1.m1.3.3.1.1.1.1.2.2.3.3.cmml">0</mn></mrow><mrow id="S3.E1.m1.3.3.1.1.1.1.2.3" xref="S3.E1.m1.3.3.1.1.1.1.2.3.cmml"><mi id="S3.E1.m1.3.3.1.1.1.1.2.3.2" xref="S3.E1.m1.3.3.1.1.1.1.2.3.2.cmml">K</mi><mo id="S3.E1.m1.3.3.1.1.1.1.2.3.1" xref="S3.E1.m1.3.3.1.1.1.1.2.3.1.cmml">−</mo><mn id="S3.E1.m1.3.3.1.1.1.1.2.3.3" xref="S3.E1.m1.3.3.1.1.1.1.2.3.3.cmml">1</mn></mrow></msubsup><mrow id="S3.E1.m1.3.3.1.1.1.1.1" xref="S3.E1.m1.3.3.1.1.1.1.1.cmml"><msup id="S3.E1.m1.3.3.1.1.1.1.1.3" xref="S3.E1.m1.3.3.1.1.1.1.1.3.cmml"><mi id="S3.E1.m1.3.3.1.1.1.1.1.3.2" xref="S3.E1.m1.3.3.1.1.1.1.1.3.2.cmml">α</mi><mrow id="S3.E1.m1.1.1.1.3" xref="S3.E1.m1.3.3.1.1.1.1.1.3.cmml"><mo id="S3.E1.m1.1.1.1.3.1" stretchy="false" xref="S3.E1.m1.3.3.1.1.1.1.1.3.cmml">(</mo><mi id="S3.E1.m1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.cmml">k</mi><mo id="S3.E1.m1.1.1.1.3.2" stretchy="false" xref="S3.E1.m1.3.3.1.1.1.1.1.3.cmml">)</mo></mrow></msup><mo id="S3.E1.m1.3.3.1.1.1.1.1.2" xref="S3.E1.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mi id="S3.E1.m1.3.3.1.1.1.1.1.4" xref="S3.E1.m1.3.3.1.1.1.1.1.4.cmml">𝒉</mi><mo id="S3.E1.m1.3.3.1.1.1.1.1.2a" xref="S3.E1.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.3.3.1.1.1.1.1.1.1" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E1.m1.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.E1.m1.3.3.1.1.1.1.1.1.1.1" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.2.2" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.2.2.cmml">𝒙</mi><mi id="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.3" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.3.cmml">t</mi><mrow id="S3.E1.m1.2.2.1.3" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E1.m1.2.2.1.3.1" stretchy="false" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S3.E1.m1.2.2.1.1" xref="S3.E1.m1.2.2.1.1.cmml">k</mi><mo id="S3.E1.m1.2.2.1.3.2" stretchy="false" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.E1.m1.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E1.m1.3.3.1.1.1.2a" xref="S3.E1.m1.3.3.1.1.1.2.cmml">+</mo><msub id="S3.E1.m1.3.3.1.1.1.4" xref="S3.E1.m1.3.3.1.1.1.4.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.E1.m1.3.3.1.1.1.4.2" mathvariant="bold-italic" xref="S3.E1.m1.3.3.1.1.1.4.2.cmml">ϵ</mi><mi id="S3.E1.m1.3.3.1.1.1.4.3" xref="S3.E1.m1.3.3.1.1.1.4.3.cmml">t</mi></msub></mrow></mrow><mo id="S3.E1.m1.3.3.1.2" xref="S3.E1.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E1.m1.3b"><apply id="S3.E1.m1.3.3.1.1.cmml" xref="S3.E1.m1.3.3.1"><eq id="S3.E1.m1.3.3.1.1.2.cmml" xref="S3.E1.m1.3.3.1.1.2"></eq><apply id="S3.E1.m1.3.3.1.1.3.cmml" xref="S3.E1.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S3.E1.m1.3.3.1.1.3.1.cmml" xref="S3.E1.m1.3.3.1.1.3">subscript</csymbol><ci id="S3.E1.m1.3.3.1.1.3.2.cmml" xref="S3.E1.m1.3.3.1.1.3.2">𝒚</ci><ci id="S3.E1.m1.3.3.1.1.3.3.cmml" xref="S3.E1.m1.3.3.1.1.3.3">𝑡</ci></apply><apply id="S3.E1.m1.3.3.1.1.1.cmml" xref="S3.E1.m1.3.3.1.1.1"><plus id="S3.E1.m1.3.3.1.1.1.2.cmml" xref="S3.E1.m1.3.3.1.1.1.2"></plus><apply id="S3.E1.m1.3.3.1.1.1.3.cmml" xref="S3.E1.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S3.E1.m1.3.3.1.1.1.3.1.cmml" xref="S3.E1.m1.3.3.1.1.1.3">subscript</csymbol><ci id="S3.E1.m1.3.3.1.1.1.3.2.cmml" xref="S3.E1.m1.3.3.1.1.1.3.2">𝒔</ci><ci id="S3.E1.m1.3.3.1.1.1.3.3.cmml" xref="S3.E1.m1.3.3.1.1.1.3.3">𝑡</ci></apply><apply id="S3.E1.m1.3.3.1.1.1.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1"><apply id="S3.E1.m1.3.3.1.1.1.1.2.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E1.m1.3.3.1.1.1.1.2.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2">superscript</csymbol><apply id="S3.E1.m1.3.3.1.1.1.1.2.2.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E1.m1.3.3.1.1.1.1.2.2.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2">subscript</csymbol><sum id="S3.E1.m1.3.3.1.1.1.1.2.2.2.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2.2.2"></sum><apply id="S3.E1.m1.3.3.1.1.1.1.2.2.3.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2.2.3"><eq id="S3.E1.m1.3.3.1.1.1.1.2.2.3.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2.2.3.1"></eq><ci id="S3.E1.m1.3.3.1.1.1.1.2.2.3.2.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2.2.3.2">𝑘</ci><cn id="S3.E1.m1.3.3.1.1.1.1.2.2.3.3.cmml" type="integer" xref="S3.E1.m1.3.3.1.1.1.1.2.2.3.3">0</cn></apply></apply><apply id="S3.E1.m1.3.3.1.1.1.1.2.3.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2.3"><minus id="S3.E1.m1.3.3.1.1.1.1.2.3.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2.3.1"></minus><ci id="S3.E1.m1.3.3.1.1.1.1.2.3.2.cmml" xref="S3.E1.m1.3.3.1.1.1.1.2.3.2">𝐾</ci><cn id="S3.E1.m1.3.3.1.1.1.1.2.3.3.cmml" type="integer" xref="S3.E1.m1.3.3.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S3.E1.m1.3.3.1.1.1.1.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1"><times id="S3.E1.m1.3.3.1.1.1.1.1.2.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.2"></times><apply id="S3.E1.m1.3.3.1.1.1.1.1.3.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E1.m1.3.3.1.1.1.1.1.3.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.3">superscript</csymbol><ci id="S3.E1.m1.3.3.1.1.1.1.1.3.2.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.3.2">𝛼</ci><ci id="S3.E1.m1.1.1.1.1.cmml" xref="S3.E1.m1.1.1.1.1">𝑘</ci></apply><ci id="S3.E1.m1.3.3.1.1.1.1.1.4.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.4">𝒉</ci><apply id="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><apply id="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1">superscript</csymbol><ci id="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.2.2">𝒙</ci><ci id="S3.E1.m1.2.2.1.1.cmml" xref="S3.E1.m1.2.2.1.1">𝑘</ci></apply><ci id="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="S3.E1.m1.3.3.1.1.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply><apply id="S3.E1.m1.3.3.1.1.1.4.cmml" xref="S3.E1.m1.3.3.1.1.1.4"><csymbol cd="ambiguous" id="S3.E1.m1.3.3.1.1.1.4.1.cmml" xref="S3.E1.m1.3.3.1.1.1.4">subscript</csymbol><ci id="S3.E1.m1.3.3.1.1.1.4.2.cmml" xref="S3.E1.m1.3.3.1.1.1.4.2">bold-italic-ϵ</ci><ci id="S3.E1.m1.3.3.1.1.1.4.3.cmml" xref="S3.E1.m1.3.3.1.1.1.4.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E1.m1.3c">\boldsymbol{y}_{t}={\boldsymbol{s}}_{t}+\sum\nolimits_{k=0}^{K-1}\alpha^{(k)}% \,\boldsymbol{h}(\boldsymbol{x}^{(k)}_{t})+\boldsymbol{\epsilon}_{t}\,,</annotation><annotation encoding="application/x-llamapun" id="S3.E1.m1.3d">bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = bold_italic_s start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_k = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K - 1 end_POSTSUPERSCRIPT italic_α start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT bold_italic_h ( bold_italic_x start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) + bold_italic_ϵ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(1)</td> </tr></tbody> </table> where <math alttext="\boldsymbol{s}_{t}" class="ltx_Math" display="inline" id="S3.SS1.p1.8.m1.1"><semantics id="S3.SS1.p1.8.m1.1a"><msub id="S3.SS1.p1.8.m1.1.1" xref="S3.SS1.p1.8.m1.1.1.cmml"><mi id="S3.SS1.p1.8.m1.1.1.2" xref="S3.SS1.p1.8.m1.1.1.2.cmml">𝒔</mi><mi id="S3.SS1.p1.8.m1.1.1.3" xref="S3.SS1.p1.8.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.8.m1.1b"><apply id="S3.SS1.p1.8.m1.1.1.cmml" xref="S3.SS1.p1.8.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.8.m1.1.1.1.cmml" xref="S3.SS1.p1.8.m1.1.1">subscript</csymbol><ci id="S3.SS1.p1.8.m1.1.1.2.cmml" xref="S3.SS1.p1.8.m1.1.1.2">𝒔</ci><ci id="S3.SS1.p1.8.m1.1.1.3.cmml" xref="S3.SS1.p1.8.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.8.m1.1c">\boldsymbol{s}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.8.m1.1d">bold_italic_s start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="S3.SS1.p1.9.m2.1"><semantics id="S3.SS1.p1.9.m2.1a"><msup id="S3.SS1.p1.9.m2.1.1" xref="S3.SS1.p1.9.m2.1.1.cmml"><mi id="S3.SS1.p1.9.m2.1.1.2" xref="S3.SS1.p1.9.m2.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.p1.9.m2.1.1.3" xref="S3.SS1.p1.9.m2.1.1.3.cmml"><mi id="S3.SS1.p1.9.m2.1.1.3.2" xref="S3.SS1.p1.9.m2.1.1.3.2.cmml">H</mi><mo id="S3.SS1.p1.9.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.9.m2.1.1.3.1.cmml">×</mo><mi id="S3.SS1.p1.9.m2.1.1.3.3" xref="S3.SS1.p1.9.m2.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.9.m2.1b"><apply id="S3.SS1.p1.9.m2.1.1.cmml" xref="S3.SS1.p1.9.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.9.m2.1.1.1.cmml" xref="S3.SS1.p1.9.m2.1.1">superscript</csymbol><ci id="S3.SS1.p1.9.m2.1.1.2.cmml" xref="S3.SS1.p1.9.m2.1.1.2">ℝ</ci><apply id="S3.SS1.p1.9.m2.1.1.3.cmml" xref="S3.SS1.p1.9.m2.1.1.3"><times id="S3.SS1.p1.9.m2.1.1.3.1.cmml" xref="S3.SS1.p1.9.m2.1.1.3.1"></times><ci id="S3.SS1.p1.9.m2.1.1.3.2.cmml" xref="S3.SS1.p1.9.m2.1.1.3.2">𝐻</ci><ci id="S3.SS1.p1.9.m2.1.1.3.3.cmml" xref="S3.SS1.p1.9.m2.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.9.m2.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.9.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> is the speckles component, <math alttext="K" class="ltx_Math" display="inline" id="S3.SS1.p1.10.m3.1"><semantics id="S3.SS1.p1.10.m3.1a"><mi id="S3.SS1.p1.10.m3.1.1" xref="S3.SS1.p1.10.m3.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.10.m3.1b"><ci id="S3.SS1.p1.10.m3.1.1.cmml" xref="S3.SS1.p1.10.m3.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.10.m3.1c">K</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.10.m3.1d">italic_K</annotation></semantics></math> is the (unknown) number of exoplanets, <math alttext="\boldsymbol{x}^{(k)}_{t}" class="ltx_Math" display="inline" id="S3.SS1.p1.11.m4.1"><semantics id="S3.SS1.p1.11.m4.1a"><msubsup id="S3.SS1.p1.11.m4.1.2" xref="S3.SS1.p1.11.m4.1.2.cmml"><mi id="S3.SS1.p1.11.m4.1.2.2.2" xref="S3.SS1.p1.11.m4.1.2.2.2.cmml">𝒙</mi><mi id="S3.SS1.p1.11.m4.1.2.3" xref="S3.SS1.p1.11.m4.1.2.3.cmml">t</mi><mrow id="S3.SS1.p1.11.m4.1.1.1.3" xref="S3.SS1.p1.11.m4.1.2.cmml"><mo id="S3.SS1.p1.11.m4.1.1.1.3.1" stretchy="false" xref="S3.SS1.p1.11.m4.1.2.cmml">(</mo><mi id="S3.SS1.p1.11.m4.1.1.1.1" xref="S3.SS1.p1.11.m4.1.1.1.1.cmml">k</mi><mo id="S3.SS1.p1.11.m4.1.1.1.3.2" stretchy="false" xref="S3.SS1.p1.11.m4.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.11.m4.1b"><apply id="S3.SS1.p1.11.m4.1.2.cmml" xref="S3.SS1.p1.11.m4.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.11.m4.1.2.1.cmml" xref="S3.SS1.p1.11.m4.1.2">subscript</csymbol><apply id="S3.SS1.p1.11.m4.1.2.2.cmml" xref="S3.SS1.p1.11.m4.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.11.m4.1.2.2.1.cmml" xref="S3.SS1.p1.11.m4.1.2">superscript</csymbol><ci id="S3.SS1.p1.11.m4.1.2.2.2.cmml" xref="S3.SS1.p1.11.m4.1.2.2.2">𝒙</ci><ci id="S3.SS1.p1.11.m4.1.1.1.1.cmml" xref="S3.SS1.p1.11.m4.1.1.1.1">𝑘</ci></apply><ci id="S3.SS1.p1.11.m4.1.2.3.cmml" xref="S3.SS1.p1.11.m4.1.2.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.11.m4.1c">\boldsymbol{x}^{(k)}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.11.m4.1d">bold_italic_x start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is the position of exoplanet <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.p1.12.m5.1"><semantics id="S3.SS1.p1.12.m5.1a"><mi id="S3.SS1.p1.12.m5.1.1" xref="S3.SS1.p1.12.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.12.m5.1b"><ci id="S3.SS1.p1.12.m5.1.1.cmml" xref="S3.SS1.p1.12.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.12.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.12.m5.1d">italic_k</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S3.SS1.p1.13.m6.1"><semantics id="S3.SS1.p1.13.m6.1a"><mi id="S3.SS1.p1.13.m6.1.1" xref="S3.SS1.p1.13.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.13.m6.1b"><ci id="S3.SS1.p1.13.m6.1.1.cmml" xref="S3.SS1.p1.13.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.13.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.13.m6.1d">italic_t</annotation></semantics></math>, <math alttext="\boldsymbol{h}(\boldsymbol{x})" class="ltx_Math" display="inline" id="S3.SS1.p1.14.m7.1"><semantics id="S3.SS1.p1.14.m7.1a"><mrow id="S3.SS1.p1.14.m7.1.2" xref="S3.SS1.p1.14.m7.1.2.cmml"><mi id="S3.SS1.p1.14.m7.1.2.2" xref="S3.SS1.p1.14.m7.1.2.2.cmml">𝒉</mi><mo id="S3.SS1.p1.14.m7.1.2.1" xref="S3.SS1.p1.14.m7.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p1.14.m7.1.2.3.2" xref="S3.SS1.p1.14.m7.1.2.cmml"><mo id="S3.SS1.p1.14.m7.1.2.3.2.1" stretchy="false" xref="S3.SS1.p1.14.m7.1.2.cmml">(</mo><mi id="S3.SS1.p1.14.m7.1.1" xref="S3.SS1.p1.14.m7.1.1.cmml">𝒙</mi><mo id="S3.SS1.p1.14.m7.1.2.3.2.2" stretchy="false" xref="S3.SS1.p1.14.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.14.m7.1b"><apply id="S3.SS1.p1.14.m7.1.2.cmml" xref="S3.SS1.p1.14.m7.1.2"><times id="S3.SS1.p1.14.m7.1.2.1.cmml" xref="S3.SS1.p1.14.m7.1.2.1"></times><ci id="S3.SS1.p1.14.m7.1.2.2.cmml" xref="S3.SS1.p1.14.m7.1.2.2">𝒉</ci><ci id="S3.SS1.p1.14.m7.1.1.cmml" xref="S3.SS1.p1.14.m7.1.1">𝒙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.14.m7.1c">\boldsymbol{h}(\boldsymbol{x})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.14.m7.1d">bold_italic_h ( bold_italic_x )</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="S3.SS1.p1.15.m8.1"><semantics id="S3.SS1.p1.15.m8.1a"><msup id="S3.SS1.p1.15.m8.1.1" xref="S3.SS1.p1.15.m8.1.1.cmml"><mi id="S3.SS1.p1.15.m8.1.1.2" xref="S3.SS1.p1.15.m8.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.p1.15.m8.1.1.3" xref="S3.SS1.p1.15.m8.1.1.3.cmml"><mi id="S3.SS1.p1.15.m8.1.1.3.2" xref="S3.SS1.p1.15.m8.1.1.3.2.cmml">H</mi><mo id="S3.SS1.p1.15.m8.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.15.m8.1.1.3.1.cmml">×</mo><mi id="S3.SS1.p1.15.m8.1.1.3.3" xref="S3.SS1.p1.15.m8.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.15.m8.1b"><apply id="S3.SS1.p1.15.m8.1.1.cmml" xref="S3.SS1.p1.15.m8.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.15.m8.1.1.1.cmml" xref="S3.SS1.p1.15.m8.1.1">superscript</csymbol><ci id="S3.SS1.p1.15.m8.1.1.2.cmml" xref="S3.SS1.p1.15.m8.1.1.2">ℝ</ci><apply id="S3.SS1.p1.15.m8.1.1.3.cmml" xref="S3.SS1.p1.15.m8.1.1.3"><times id="S3.SS1.p1.15.m8.1.1.3.1.cmml" xref="S3.SS1.p1.15.m8.1.1.3.1"></times><ci id="S3.SS1.p1.15.m8.1.1.3.2.cmml" xref="S3.SS1.p1.15.m8.1.1.3.2">𝐻</ci><ci id="S3.SS1.p1.15.m8.1.1.3.3.cmml" xref="S3.SS1.p1.15.m8.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.15.m8.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.15.m8.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> is the PSF model centered on position <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.p1.16.m9.1"><semantics id="S3.SS1.p1.16.m9.1a"><mi id="S3.SS1.p1.16.m9.1.1" xref="S3.SS1.p1.16.m9.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.16.m9.1b"><ci id="S3.SS1.p1.16.m9.1.1.cmml" xref="S3.SS1.p1.16.m9.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.16.m9.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.16.m9.1d">bold_italic_x</annotation></semantics></math> in <math alttext="\mathbb{R}^{2}" class="ltx_Math" display="inline" id="S3.SS1.p1.17.m10.1"><semantics id="S3.SS1.p1.17.m10.1a"><msup id="S3.SS1.p1.17.m10.1.1" xref="S3.SS1.p1.17.m10.1.1.cmml"><mi id="S3.SS1.p1.17.m10.1.1.2" xref="S3.SS1.p1.17.m10.1.1.2.cmml">ℝ</mi><mn id="S3.SS1.p1.17.m10.1.1.3" xref="S3.SS1.p1.17.m10.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.17.m10.1b"><apply id="S3.SS1.p1.17.m10.1.1.cmml" xref="S3.SS1.p1.17.m10.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.17.m10.1.1.1.cmml" xref="S3.SS1.p1.17.m10.1.1">superscript</csymbol><ci id="S3.SS1.p1.17.m10.1.1.2.cmml" xref="S3.SS1.p1.17.m10.1.1.2">ℝ</ci><cn id="S3.SS1.p1.17.m10.1.1.3.cmml" type="integer" xref="S3.SS1.p1.17.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.17.m10.1c">\mathbb{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.17.m10.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\alpha^{(k)}" class="ltx_Math" display="inline" id="S3.SS1.p1.18.m11.1"><semantics id="S3.SS1.p1.18.m11.1a"><msup id="S3.SS1.p1.18.m11.1.2" xref="S3.SS1.p1.18.m11.1.2.cmml"><mi id="S3.SS1.p1.18.m11.1.2.2" xref="S3.SS1.p1.18.m11.1.2.2.cmml">α</mi><mrow id="S3.SS1.p1.18.m11.1.1.1.3" xref="S3.SS1.p1.18.m11.1.2.cmml"><mo id="S3.SS1.p1.18.m11.1.1.1.3.1" stretchy="false" xref="S3.SS1.p1.18.m11.1.2.cmml">(</mo><mi id="S3.SS1.p1.18.m11.1.1.1.1" xref="S3.SS1.p1.18.m11.1.1.1.1.cmml">k</mi><mo id="S3.SS1.p1.18.m11.1.1.1.3.2" stretchy="false" xref="S3.SS1.p1.18.m11.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.18.m11.1b"><apply id="S3.SS1.p1.18.m11.1.2.cmml" xref="S3.SS1.p1.18.m11.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.18.m11.1.2.1.cmml" xref="S3.SS1.p1.18.m11.1.2">superscript</csymbol><ci id="S3.SS1.p1.18.m11.1.2.2.cmml" xref="S3.SS1.p1.18.m11.1.2.2">𝛼</ci><ci id="S3.SS1.p1.18.m11.1.1.1.1.cmml" xref="S3.SS1.p1.18.m11.1.1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.18.m11.1c">\alpha^{(k)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.18.m11.1d">italic_α start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}_{+}" class="ltx_Math" display="inline" id="S3.SS1.p1.19.m12.1"><semantics id="S3.SS1.p1.19.m12.1a"><msub id="S3.SS1.p1.19.m12.1.1" xref="S3.SS1.p1.19.m12.1.1.cmml"><mi id="S3.SS1.p1.19.m12.1.1.2" xref="S3.SS1.p1.19.m12.1.1.2.cmml">ℝ</mi><mo id="S3.SS1.p1.19.m12.1.1.3" xref="S3.SS1.p1.19.m12.1.1.3.cmml">+</mo></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.19.m12.1b"><apply id="S3.SS1.p1.19.m12.1.1.cmml" xref="S3.SS1.p1.19.m12.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.19.m12.1.1.1.cmml" xref="S3.SS1.p1.19.m12.1.1">subscript</csymbol><ci id="S3.SS1.p1.19.m12.1.1.2.cmml" xref="S3.SS1.p1.19.m12.1.1.2">ℝ</ci><plus id="S3.SS1.p1.19.m12.1.1.3.cmml" xref="S3.SS1.p1.19.m12.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.19.m12.1c">\mathbb{R}_{+}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.19.m12.1d">blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> is the flux of the exoplanet, and <math alttext="\boldsymbol{\epsilon}_{t}" class="ltx_Math" display="inline" id="S3.SS1.p1.20.m13.1"><semantics id="S3.SS1.p1.20.m13.1a"><msub id="S3.SS1.p1.20.m13.1.1" xref="S3.SS1.p1.20.m13.1.1.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.SS1.p1.20.m13.1.1.2" mathvariant="bold-italic" xref="S3.SS1.p1.20.m13.1.1.2.cmml">ϵ</mi><mi id="S3.SS1.p1.20.m13.1.1.3" xref="S3.SS1.p1.20.m13.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.20.m13.1b"><apply id="S3.SS1.p1.20.m13.1.1.cmml" xref="S3.SS1.p1.20.m13.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.20.m13.1.1.1.cmml" xref="S3.SS1.p1.20.m13.1.1">subscript</csymbol><ci id="S3.SS1.p1.20.m13.1.1.2.cmml" xref="S3.SS1.p1.20.m13.1.1.2">bold-italic-ϵ</ci><ci id="S3.SS1.p1.20.m13.1.1.3.cmml" xref="S3.SS1.p1.20.m13.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.20.m13.1c">\boldsymbol{\epsilon}_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.20.m13.1d">bold_italic_ϵ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="S3.SS1.p1.21.m14.1"><semantics id="S3.SS1.p1.21.m14.1a"><msup id="S3.SS1.p1.21.m14.1.1" xref="S3.SS1.p1.21.m14.1.1.cmml"><mi id="S3.SS1.p1.21.m14.1.1.2" xref="S3.SS1.p1.21.m14.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.p1.21.m14.1.1.3" xref="S3.SS1.p1.21.m14.1.1.3.cmml"><mi id="S3.SS1.p1.21.m14.1.1.3.2" xref="S3.SS1.p1.21.m14.1.1.3.2.cmml">H</mi><mo id="S3.SS1.p1.21.m14.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.21.m14.1.1.3.1.cmml">×</mo><mi id="S3.SS1.p1.21.m14.1.1.3.3" xref="S3.SS1.p1.21.m14.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.21.m14.1b"><apply id="S3.SS1.p1.21.m14.1.1.cmml" xref="S3.SS1.p1.21.m14.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.21.m14.1.1.1.cmml" xref="S3.SS1.p1.21.m14.1.1">superscript</csymbol><ci id="S3.SS1.p1.21.m14.1.1.2.cmml" xref="S3.SS1.p1.21.m14.1.1.2">ℝ</ci><apply id="S3.SS1.p1.21.m14.1.1.3.cmml" xref="S3.SS1.p1.21.m14.1.1.3"><times id="S3.SS1.p1.21.m14.1.1.3.1.cmml" xref="S3.SS1.p1.21.m14.1.1.3.1"></times><ci id="S3.SS1.p1.21.m14.1.1.3.2.cmml" xref="S3.SS1.p1.21.m14.1.1.3.2">𝐻</ci><ci id="S3.SS1.p1.21.m14.1.1.3.3.cmml" xref="S3.SS1.p1.21.m14.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.21.m14.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.21.m14.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> is stochastic noise. The position of exoplanet <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.p1.22.m15.1"><semantics id="S3.SS1.p1.22.m15.1a"><mi id="S3.SS1.p1.22.m15.1.1" xref="S3.SS1.p1.22.m15.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.22.m15.1b"><ci id="S3.SS1.p1.22.m15.1.1.cmml" xref="S3.SS1.p1.22.m15.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.22.m15.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.22.m15.1d">italic_k</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S3.SS1.p1.23.m16.1"><semantics id="S3.SS1.p1.23.m16.1a"><mi id="S3.SS1.p1.23.m16.1.1" xref="S3.SS1.p1.23.m16.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.23.m16.1b"><ci id="S3.SS1.p1.23.m16.1.1.cmml" xref="S3.SS1.p1.23.m16.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.23.m16.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.23.m16.1d">italic_t</annotation></semantics></math> can be written as <math alttext="\boldsymbol{x}^{(k)}_{t}=r\big{(}\boldsymbol{x}^{(k)}_{0},\phi_{t}\big{)}" class="ltx_Math" display="inline" id="S3.SS1.p1.24.m17.4"><semantics id="S3.SS1.p1.24.m17.4a"><mrow id="S3.SS1.p1.24.m17.4.4" xref="S3.SS1.p1.24.m17.4.4.cmml"><msubsup id="S3.SS1.p1.24.m17.4.4.4" xref="S3.SS1.p1.24.m17.4.4.4.cmml"><mi id="S3.SS1.p1.24.m17.4.4.4.2.2" xref="S3.SS1.p1.24.m17.4.4.4.2.2.cmml">𝒙</mi><mi id="S3.SS1.p1.24.m17.4.4.4.3" xref="S3.SS1.p1.24.m17.4.4.4.3.cmml">t</mi><mrow id="S3.SS1.p1.24.m17.1.1.1.3" xref="S3.SS1.p1.24.m17.4.4.4.cmml"><mo id="S3.SS1.p1.24.m17.1.1.1.3.1" stretchy="false" xref="S3.SS1.p1.24.m17.4.4.4.cmml">(</mo><mi id="S3.SS1.p1.24.m17.1.1.1.1" xref="S3.SS1.p1.24.m17.1.1.1.1.cmml">k</mi><mo id="S3.SS1.p1.24.m17.1.1.1.3.2" stretchy="false" xref="S3.SS1.p1.24.m17.4.4.4.cmml">)</mo></mrow></msubsup><mo id="S3.SS1.p1.24.m17.4.4.3" xref="S3.SS1.p1.24.m17.4.4.3.cmml">=</mo><mrow id="S3.SS1.p1.24.m17.4.4.2" xref="S3.SS1.p1.24.m17.4.4.2.cmml"><mi id="S3.SS1.p1.24.m17.4.4.2.4" xref="S3.SS1.p1.24.m17.4.4.2.4.cmml">r</mi><mo id="S3.SS1.p1.24.m17.4.4.2.3" xref="S3.SS1.p1.24.m17.4.4.2.3.cmml">⁢</mo><mrow id="S3.SS1.p1.24.m17.4.4.2.2.2" xref="S3.SS1.p1.24.m17.4.4.2.2.3.cmml"><mo id="S3.SS1.p1.24.m17.4.4.2.2.2.3" maxsize="120%" minsize="120%" xref="S3.SS1.p1.24.m17.4.4.2.2.3.cmml">(</mo><msubsup id="S3.SS1.p1.24.m17.3.3.1.1.1.1" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1.cmml"><mi id="S3.SS1.p1.24.m17.3.3.1.1.1.1.2.2" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1.2.2.cmml">𝒙</mi><mn id="S3.SS1.p1.24.m17.3.3.1.1.1.1.3" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1.3.cmml">0</mn><mrow id="S3.SS1.p1.24.m17.2.2.1.3" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1.cmml"><mo id="S3.SS1.p1.24.m17.2.2.1.3.1" stretchy="false" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1.cmml">(</mo><mi id="S3.SS1.p1.24.m17.2.2.1.1" xref="S3.SS1.p1.24.m17.2.2.1.1.cmml">k</mi><mo id="S3.SS1.p1.24.m17.2.2.1.3.2" stretchy="false" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.SS1.p1.24.m17.4.4.2.2.2.4" xref="S3.SS1.p1.24.m17.4.4.2.2.3.cmml">,</mo><msub id="S3.SS1.p1.24.m17.4.4.2.2.2.2" xref="S3.SS1.p1.24.m17.4.4.2.2.2.2.cmml"><mi id="S3.SS1.p1.24.m17.4.4.2.2.2.2.2" xref="S3.SS1.p1.24.m17.4.4.2.2.2.2.2.cmml">ϕ</mi><mi id="S3.SS1.p1.24.m17.4.4.2.2.2.2.3" xref="S3.SS1.p1.24.m17.4.4.2.2.2.2.3.cmml">t</mi></msub><mo id="S3.SS1.p1.24.m17.4.4.2.2.2.5" maxsize="120%" minsize="120%" xref="S3.SS1.p1.24.m17.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.24.m17.4b"><apply id="S3.SS1.p1.24.m17.4.4.cmml" xref="S3.SS1.p1.24.m17.4.4"><eq id="S3.SS1.p1.24.m17.4.4.3.cmml" xref="S3.SS1.p1.24.m17.4.4.3"></eq><apply id="S3.SS1.p1.24.m17.4.4.4.cmml" xref="S3.SS1.p1.24.m17.4.4.4"><csymbol cd="ambiguous" id="S3.SS1.p1.24.m17.4.4.4.1.cmml" xref="S3.SS1.p1.24.m17.4.4.4">subscript</csymbol><apply id="S3.SS1.p1.24.m17.4.4.4.2.cmml" xref="S3.SS1.p1.24.m17.4.4.4"><csymbol cd="ambiguous" id="S3.SS1.p1.24.m17.4.4.4.2.1.cmml" xref="S3.SS1.p1.24.m17.4.4.4">superscript</csymbol><ci id="S3.SS1.p1.24.m17.4.4.4.2.2.cmml" xref="S3.SS1.p1.24.m17.4.4.4.2.2">𝒙</ci><ci id="S3.SS1.p1.24.m17.1.1.1.1.cmml" xref="S3.SS1.p1.24.m17.1.1.1.1">𝑘</ci></apply><ci id="S3.SS1.p1.24.m17.4.4.4.3.cmml" xref="S3.SS1.p1.24.m17.4.4.4.3">𝑡</ci></apply><apply id="S3.SS1.p1.24.m17.4.4.2.cmml" xref="S3.SS1.p1.24.m17.4.4.2"><times id="S3.SS1.p1.24.m17.4.4.2.3.cmml" xref="S3.SS1.p1.24.m17.4.4.2.3"></times><ci id="S3.SS1.p1.24.m17.4.4.2.4.cmml" xref="S3.SS1.p1.24.m17.4.4.2.4">𝑟</ci><interval closure="open" id="S3.SS1.p1.24.m17.4.4.2.2.3.cmml" xref="S3.SS1.p1.24.m17.4.4.2.2.2"><apply id="S3.SS1.p1.24.m17.3.3.1.1.1.1.cmml" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.24.m17.3.3.1.1.1.1.1.cmml" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1">subscript</csymbol><apply id="S3.SS1.p1.24.m17.3.3.1.1.1.1.2.cmml" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.24.m17.3.3.1.1.1.1.2.1.cmml" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1">superscript</csymbol><ci id="S3.SS1.p1.24.m17.3.3.1.1.1.1.2.2.cmml" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1.2.2">𝒙</ci><ci id="S3.SS1.p1.24.m17.2.2.1.1.cmml" xref="S3.SS1.p1.24.m17.2.2.1.1">𝑘</ci></apply><cn id="S3.SS1.p1.24.m17.3.3.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.24.m17.3.3.1.1.1.1.3">0</cn></apply><apply id="S3.SS1.p1.24.m17.4.4.2.2.2.2.cmml" xref="S3.SS1.p1.24.m17.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.24.m17.4.4.2.2.2.2.1.cmml" xref="S3.SS1.p1.24.m17.4.4.2.2.2.2">subscript</csymbol><ci id="S3.SS1.p1.24.m17.4.4.2.2.2.2.2.cmml" xref="S3.SS1.p1.24.m17.4.4.2.2.2.2.2">italic-ϕ</ci><ci id="S3.SS1.p1.24.m17.4.4.2.2.2.2.3.cmml" xref="S3.SS1.p1.24.m17.4.4.2.2.2.2.3">𝑡</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.24.m17.4c">\boldsymbol{x}^{(k)}_{t}=r\big{(}\boldsymbol{x}^{(k)}_{0},\phi_{t}\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.24.m17.4d">bold_italic_x start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_r ( bold_italic_x start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math>, where <math alttext="\boldsymbol{x}^{(k)}_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.25.m18.1"><semantics id="S3.SS1.p1.25.m18.1a"><msubsup id="S3.SS1.p1.25.m18.1.2" xref="S3.SS1.p1.25.m18.1.2.cmml"><mi id="S3.SS1.p1.25.m18.1.2.2.2" xref="S3.SS1.p1.25.m18.1.2.2.2.cmml">𝒙</mi><mn id="S3.SS1.p1.25.m18.1.2.3" xref="S3.SS1.p1.25.m18.1.2.3.cmml">0</mn><mrow id="S3.SS1.p1.25.m18.1.1.1.3" xref="S3.SS1.p1.25.m18.1.2.cmml"><mo id="S3.SS1.p1.25.m18.1.1.1.3.1" stretchy="false" xref="S3.SS1.p1.25.m18.1.2.cmml">(</mo><mi id="S3.SS1.p1.25.m18.1.1.1.1" xref="S3.SS1.p1.25.m18.1.1.1.1.cmml">k</mi><mo id="S3.SS1.p1.25.m18.1.1.1.3.2" stretchy="false" xref="S3.SS1.p1.25.m18.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.25.m18.1b"><apply id="S3.SS1.p1.25.m18.1.2.cmml" xref="S3.SS1.p1.25.m18.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.25.m18.1.2.1.cmml" xref="S3.SS1.p1.25.m18.1.2">subscript</csymbol><apply id="S3.SS1.p1.25.m18.1.2.2.cmml" xref="S3.SS1.p1.25.m18.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.25.m18.1.2.2.1.cmml" xref="S3.SS1.p1.25.m18.1.2">superscript</csymbol><ci id="S3.SS1.p1.25.m18.1.2.2.2.cmml" xref="S3.SS1.p1.25.m18.1.2.2.2">𝒙</ci><ci id="S3.SS1.p1.25.m18.1.1.1.1.cmml" xref="S3.SS1.p1.25.m18.1.1.1.1">𝑘</ci></apply><cn id="S3.SS1.p1.25.m18.1.2.3.cmml" type="integer" xref="S3.SS1.p1.25.m18.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.25.m18.1c">\boldsymbol{x}^{(k)}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.25.m18.1d">bold_italic_x start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{2}" class="ltx_Math" display="inline" id="S3.SS1.p1.26.m19.1"><semantics id="S3.SS1.p1.26.m19.1a"><msup id="S3.SS1.p1.26.m19.1.1" xref="S3.SS1.p1.26.m19.1.1.cmml"><mi id="S3.SS1.p1.26.m19.1.1.2" xref="S3.SS1.p1.26.m19.1.1.2.cmml">ℝ</mi><mn id="S3.SS1.p1.26.m19.1.1.3" xref="S3.SS1.p1.26.m19.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.26.m19.1b"><apply id="S3.SS1.p1.26.m19.1.1.cmml" xref="S3.SS1.p1.26.m19.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.26.m19.1.1.1.cmml" xref="S3.SS1.p1.26.m19.1.1">superscript</csymbol><ci id="S3.SS1.p1.26.m19.1.1.2.cmml" xref="S3.SS1.p1.26.m19.1.1.2">ℝ</ci><cn id="S3.SS1.p1.26.m19.1.1.3.cmml" type="integer" xref="S3.SS1.p1.26.m19.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.26.m19.1c">\mathbb{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.26.m19.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is its initial position on frame <math alttext="\boldsymbol{y}_{0}" class="ltx_Math" display="inline" id="S3.SS1.p1.27.m20.1"><semantics id="S3.SS1.p1.27.m20.1a"><msub id="S3.SS1.p1.27.m20.1.1" xref="S3.SS1.p1.27.m20.1.1.cmml"><mi id="S3.SS1.p1.27.m20.1.1.2" xref="S3.SS1.p1.27.m20.1.1.2.cmml">𝒚</mi><mn id="S3.SS1.p1.27.m20.1.1.3" xref="S3.SS1.p1.27.m20.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.27.m20.1b"><apply id="S3.SS1.p1.27.m20.1.1.cmml" xref="S3.SS1.p1.27.m20.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.27.m20.1.1.1.cmml" xref="S3.SS1.p1.27.m20.1.1">subscript</csymbol><ci id="S3.SS1.p1.27.m20.1.1.2.cmml" xref="S3.SS1.p1.27.m20.1.1.2">𝒚</ci><cn id="S3.SS1.p1.27.m20.1.1.3.cmml" type="integer" xref="S3.SS1.p1.27.m20.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.27.m20.1c">\boldsymbol{y}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.27.m20.1d">bold_italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\phi_{t}" class="ltx_Math" display="inline" id="S3.SS1.p1.28.m21.1"><semantics id="S3.SS1.p1.28.m21.1a"><msub id="S3.SS1.p1.28.m21.1.1" xref="S3.SS1.p1.28.m21.1.1.cmml"><mi id="S3.SS1.p1.28.m21.1.1.2" xref="S3.SS1.p1.28.m21.1.1.2.cmml">ϕ</mi><mi id="S3.SS1.p1.28.m21.1.1.3" xref="S3.SS1.p1.28.m21.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.28.m21.1b"><apply id="S3.SS1.p1.28.m21.1.1.cmml" xref="S3.SS1.p1.28.m21.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.28.m21.1.1.1.cmml" xref="S3.SS1.p1.28.m21.1.1">subscript</csymbol><ci id="S3.SS1.p1.28.m21.1.1.2.cmml" xref="S3.SS1.p1.28.m21.1.1.2">italic-ϕ</ci><ci id="S3.SS1.p1.28.m21.1.1.3.cmml" xref="S3.SS1.p1.28.m21.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.28.m21.1c">\phi_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.28.m21.1d">italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS1.p1.29.m22.1"><semantics id="S3.SS1.p1.29.m22.1a"><mi id="S3.SS1.p1.29.m22.1.1" xref="S3.SS1.p1.29.m22.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.29.m22.1b"><ci id="S3.SS1.p1.29.m22.1.1.cmml" xref="S3.SS1.p1.29.m22.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.29.m22.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.29.m22.1d">blackboard_R</annotation></semantics></math> is the predictable parallactic angle at time <math alttext="t" class="ltx_Math" display="inline" id="S3.SS1.p1.30.m23.1"><semantics id="S3.SS1.p1.30.m23.1a"><mi id="S3.SS1.p1.30.m23.1.1" xref="S3.SS1.p1.30.m23.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.30.m23.1b"><ci id="S3.SS1.p1.30.m23.1.1.cmml" xref="S3.SS1.p1.30.m23.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.30.m23.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.30.m23.1d">italic_t</annotation></semantics></math>, defined as the cumulated apparent rotation (induced by Earth’s rotation) since the start of the sequence, and <math alttext="r:\mathbb{R}^{2}\times\mathbb{R}\to\mathbb{R}^{2}" class="ltx_Math" display="inline" id="S3.SS1.p1.31.m24.1"><semantics id="S3.SS1.p1.31.m24.1a"><mrow id="S3.SS1.p1.31.m24.1.1" xref="S3.SS1.p1.31.m24.1.1.cmml"><mi id="S3.SS1.p1.31.m24.1.1.2" xref="S3.SS1.p1.31.m24.1.1.2.cmml">r</mi><mo id="S3.SS1.p1.31.m24.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p1.31.m24.1.1.1.cmml">:</mo><mrow id="S3.SS1.p1.31.m24.1.1.3" xref="S3.SS1.p1.31.m24.1.1.3.cmml"><mrow id="S3.SS1.p1.31.m24.1.1.3.2" xref="S3.SS1.p1.31.m24.1.1.3.2.cmml"><msup id="S3.SS1.p1.31.m24.1.1.3.2.2" xref="S3.SS1.p1.31.m24.1.1.3.2.2.cmml"><mi id="S3.SS1.p1.31.m24.1.1.3.2.2.2" xref="S3.SS1.p1.31.m24.1.1.3.2.2.2.cmml">ℝ</mi><mn id="S3.SS1.p1.31.m24.1.1.3.2.2.3" xref="S3.SS1.p1.31.m24.1.1.3.2.2.3.cmml">2</mn></msup><mo id="S3.SS1.p1.31.m24.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.31.m24.1.1.3.2.1.cmml">×</mo><mi id="S3.SS1.p1.31.m24.1.1.3.2.3" xref="S3.SS1.p1.31.m24.1.1.3.2.3.cmml">ℝ</mi></mrow><mo id="S3.SS1.p1.31.m24.1.1.3.1" stretchy="false" xref="S3.SS1.p1.31.m24.1.1.3.1.cmml">→</mo><msup id="S3.SS1.p1.31.m24.1.1.3.3" xref="S3.SS1.p1.31.m24.1.1.3.3.cmml"><mi id="S3.SS1.p1.31.m24.1.1.3.3.2" xref="S3.SS1.p1.31.m24.1.1.3.3.2.cmml">ℝ</mi><mn id="S3.SS1.p1.31.m24.1.1.3.3.3" xref="S3.SS1.p1.31.m24.1.1.3.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.31.m24.1b"><apply id="S3.SS1.p1.31.m24.1.1.cmml" xref="S3.SS1.p1.31.m24.1.1"><ci id="S3.SS1.p1.31.m24.1.1.1.cmml" xref="S3.SS1.p1.31.m24.1.1.1">:</ci><ci id="S3.SS1.p1.31.m24.1.1.2.cmml" xref="S3.SS1.p1.31.m24.1.1.2">𝑟</ci><apply id="S3.SS1.p1.31.m24.1.1.3.cmml" xref="S3.SS1.p1.31.m24.1.1.3"><ci id="S3.SS1.p1.31.m24.1.1.3.1.cmml" xref="S3.SS1.p1.31.m24.1.1.3.1">→</ci><apply id="S3.SS1.p1.31.m24.1.1.3.2.cmml" xref="S3.SS1.p1.31.m24.1.1.3.2"><times id="S3.SS1.p1.31.m24.1.1.3.2.1.cmml" xref="S3.SS1.p1.31.m24.1.1.3.2.1"></times><apply id="S3.SS1.p1.31.m24.1.1.3.2.2.cmml" xref="S3.SS1.p1.31.m24.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.31.m24.1.1.3.2.2.1.cmml" xref="S3.SS1.p1.31.m24.1.1.3.2.2">superscript</csymbol><ci id="S3.SS1.p1.31.m24.1.1.3.2.2.2.cmml" xref="S3.SS1.p1.31.m24.1.1.3.2.2.2">ℝ</ci><cn id="S3.SS1.p1.31.m24.1.1.3.2.2.3.cmml" type="integer" xref="S3.SS1.p1.31.m24.1.1.3.2.2.3">2</cn></apply><ci id="S3.SS1.p1.31.m24.1.1.3.2.3.cmml" xref="S3.SS1.p1.31.m24.1.1.3.2.3">ℝ</ci></apply><apply id="S3.SS1.p1.31.m24.1.1.3.3.cmml" xref="S3.SS1.p1.31.m24.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.31.m24.1.1.3.3.1.cmml" xref="S3.SS1.p1.31.m24.1.1.3.3">superscript</csymbol><ci id="S3.SS1.p1.31.m24.1.1.3.3.2.cmml" xref="S3.SS1.p1.31.m24.1.1.3.3.2">ℝ</ci><cn id="S3.SS1.p1.31.m24.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.p1.31.m24.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.31.m24.1c">r:\mathbb{R}^{2}\times\mathbb{R}\to\mathbb{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.31.m24.1d">italic_r : blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × blackboard_R → blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> defines a rotation trajectory centered on the star. </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> 3.2 Angular Spectral Differential Imaging (ASDI)</h3> <div class="ltx_para ltx_noindent" id="S3.SS2.p1"> The speckles pattern induced by the star can be understood as the PSF on the optical axis (on-axis PSF) of the optical system. According to diffraction laws, this pattern scales homothetically with wavelength, creating a chromatic effect similar to chromatic aberrations in photography. In ASDI, this spectral scaling helps further disentangle exoplanet signals from speckles. In this case, each observation is denoted as <math alttext="\boldsymbol{y}" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mi id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">𝒚</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><ci id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">𝒚</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">\boldsymbol{y}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">bold_italic_y</annotation></semantics></math> in <math alttext="\mathbb{R}^{C\times T\times H\times W}" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><msup id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml"><mi id="S3.SS2.p1.2.m2.1.1.2" xref="S3.SS2.p1.2.m2.1.1.2.cmml">ℝ</mi><mrow id="S3.SS2.p1.2.m2.1.1.3" xref="S3.SS2.p1.2.m2.1.1.3.cmml"><mi id="S3.SS2.p1.2.m2.1.1.3.2" xref="S3.SS2.p1.2.m2.1.1.3.2.cmml">C</mi><mo id="S3.SS2.p1.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p1.2.m2.1.1.3.1.cmml">×</mo><mi id="S3.SS2.p1.2.m2.1.1.3.3" xref="S3.SS2.p1.2.m2.1.1.3.3.cmml">T</mi><mo id="S3.SS2.p1.2.m2.1.1.3.1a" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p1.2.m2.1.1.3.1.cmml">×</mo><mi id="S3.SS2.p1.2.m2.1.1.3.4" xref="S3.SS2.p1.2.m2.1.1.3.4.cmml">H</mi><mo id="S3.SS2.p1.2.m2.1.1.3.1b" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p1.2.m2.1.1.3.1.cmml">×</mo><mi id="S3.SS2.p1.2.m2.1.1.3.5" xref="S3.SS2.p1.2.m2.1.1.3.5.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.1b"><apply id="S3.SS2.p1.2.m2.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m2.1.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1">superscript</csymbol><ci id="S3.SS2.p1.2.m2.1.1.2.cmml" xref="S3.SS2.p1.2.m2.1.1.2">ℝ</ci><apply id="S3.SS2.p1.2.m2.1.1.3.cmml" xref="S3.SS2.p1.2.m2.1.1.3"><times id="S3.SS2.p1.2.m2.1.1.3.1.cmml" xref="S3.SS2.p1.2.m2.1.1.3.1"></times><ci id="S3.SS2.p1.2.m2.1.1.3.2.cmml" xref="S3.SS2.p1.2.m2.1.1.3.2">𝐶</ci><ci id="S3.SS2.p1.2.m2.1.1.3.3.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3">𝑇</ci><ci id="S3.SS2.p1.2.m2.1.1.3.4.cmml" xref="S3.SS2.p1.2.m2.1.1.3.4">𝐻</ci><ci id="S3.SS2.p1.2.m2.1.1.3.5.cmml" xref="S3.SS2.p1.2.m2.1.1.3.5">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">\mathbb{R}^{C\times T\times H\times W}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_C × italic_T × italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="C" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><mi id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><ci id="S3.SS2.p1.3.m3.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">italic_C</annotation></semantics></math> is the number of channels. The forward model (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S3.E1" title="Equation 1 ‣ 3.1 Angular Differential Imaging (ADI) ‣ 3 Image formation models ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">1</a>) becomes <table class="ltx_equation ltx_eqn_table" id="S3.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\boldsymbol{y}_{c,t}=\beta_{c}\,{\mathbf{D}}_{\lambda_{c}/\lambda_{0}}\ {% \boldsymbol{s}}_{0,t}+\sum\nolimits_{k=0}^{K-1}\alpha^{(k)}_{c}\boldsymbol{h}_% {c}(\boldsymbol{x}^{(k)}_{t})+\boldsymbol{\epsilon}_{c,t}\,," class="ltx_Math" display="block" id="S3.E2.m1.9"><semantics id="S3.E2.m1.9a"><mrow id="S3.E2.m1.9.9.1" xref="S3.E2.m1.9.9.1.1.cmml"><mrow id="S3.E2.m1.9.9.1.1" xref="S3.E2.m1.9.9.1.1.cmml"><msub id="S3.E2.m1.9.9.1.1.3" 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xref="S3.E2.m1.9.9.1.1.1.3.3.2.cmml">𝐃</mi><mrow id="S3.E2.m1.9.9.1.1.1.3.3.3" xref="S3.E2.m1.9.9.1.1.1.3.3.3.cmml"><msub id="S3.E2.m1.9.9.1.1.1.3.3.3.2" xref="S3.E2.m1.9.9.1.1.1.3.3.3.2.cmml"><mi id="S3.E2.m1.9.9.1.1.1.3.3.3.2.2" xref="S3.E2.m1.9.9.1.1.1.3.3.3.2.2.cmml">λ</mi><mi id="S3.E2.m1.9.9.1.1.1.3.3.3.2.3" xref="S3.E2.m1.9.9.1.1.1.3.3.3.2.3.cmml">c</mi></msub><mo id="S3.E2.m1.9.9.1.1.1.3.3.3.1" xref="S3.E2.m1.9.9.1.1.1.3.3.3.1.cmml">/</mo><msub id="S3.E2.m1.9.9.1.1.1.3.3.3.3" xref="S3.E2.m1.9.9.1.1.1.3.3.3.3.cmml"><mi id="S3.E2.m1.9.9.1.1.1.3.3.3.3.2" xref="S3.E2.m1.9.9.1.1.1.3.3.3.3.2.cmml">λ</mi><mn id="S3.E2.m1.9.9.1.1.1.3.3.3.3.3" xref="S3.E2.m1.9.9.1.1.1.3.3.3.3.3.cmml">0</mn></msub></mrow></msub><mo id="S3.E2.m1.9.9.1.1.1.3.1a" xref="S3.E2.m1.9.9.1.1.1.3.1.cmml">⁢</mo><msub id="S3.E2.m1.9.9.1.1.1.3.4" xref="S3.E2.m1.9.9.1.1.1.3.4.cmml"><mi id="S3.E2.m1.9.9.1.1.1.3.4.2" xref="S3.E2.m1.9.9.1.1.1.3.4.2.cmml">𝒔</mi><mrow id="S3.E2.m1.4.4.2.4" xref="S3.E2.m1.4.4.2.3.cmml"><mn 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id="S3.E2.m1.9.9.1.1.1.4.2.cmml" xref="S3.E2.m1.9.9.1.1.1.4.2">bold-italic-ϵ</ci><list id="S3.E2.m1.8.8.2.3.cmml" xref="S3.E2.m1.8.8.2.4"><ci id="S3.E2.m1.7.7.1.1.cmml" xref="S3.E2.m1.7.7.1.1">𝑐</ci><ci id="S3.E2.m1.8.8.2.2.cmml" xref="S3.E2.m1.8.8.2.2">𝑡</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.9c">\boldsymbol{y}_{c,t}=\beta_{c}\,{\mathbf{D}}_{\lambda_{c}/\lambda_{0}}\ {% \boldsymbol{s}}_{0,t}+\sum\nolimits_{k=0}^{K-1}\alpha^{(k)}_{c}\boldsymbol{h}_% {c}(\boldsymbol{x}^{(k)}_{t})+\boldsymbol{\epsilon}_{c,t}\,,</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.9d">bold_italic_y start_POSTSUBSCRIPT italic_c , italic_t end_POSTSUBSCRIPT = italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT bold_D start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT bold_italic_s start_POSTSUBSCRIPT 0 , italic_t end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_k = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K - 1 end_POSTSUPERSCRIPT italic_α start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT bold_italic_h start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) + bold_italic_ϵ start_POSTSUBSCRIPT italic_c , italic_t end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(2)</td> </tr></tbody> </table> where <math alttext="\beta_{c}" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m1.1"><semantics id="S3.SS2.p1.4.m1.1a"><msub id="S3.SS2.p1.4.m1.1.1" xref="S3.SS2.p1.4.m1.1.1.cmml"><mi id="S3.SS2.p1.4.m1.1.1.2" xref="S3.SS2.p1.4.m1.1.1.2.cmml">β</mi><mi id="S3.SS2.p1.4.m1.1.1.3" xref="S3.SS2.p1.4.m1.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m1.1b"><apply id="S3.SS2.p1.4.m1.1.1.cmml" xref="S3.SS2.p1.4.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.4.m1.1.1.1.cmml" xref="S3.SS2.p1.4.m1.1.1">subscript</csymbol><ci id="S3.SS2.p1.4.m1.1.1.2.cmml" xref="S3.SS2.p1.4.m1.1.1.2">𝛽</ci><ci id="S3.SS2.p1.4.m1.1.1.3.cmml" xref="S3.SS2.p1.4.m1.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m1.1c">\beta_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m1.1d">italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS2.p1.5.m2.1"><semantics id="S3.SS2.p1.5.m2.1a"><mi id="S3.SS2.p1.5.m2.1.1" xref="S3.SS2.p1.5.m2.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.5.m2.1b"><ci id="S3.SS2.p1.5.m2.1.1.cmml" xref="S3.SS2.p1.5.m2.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.5.m2.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.5.m2.1d">blackboard_R</annotation></semantics></math> is the amplitude of the speckles in channel <math alttext="c" class="ltx_Math" display="inline" id="S3.SS2.p1.6.m3.1"><semantics id="S3.SS2.p1.6.m3.1a"><mi id="S3.SS2.p1.6.m3.1.1" xref="S3.SS2.p1.6.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.6.m3.1b"><ci id="S3.SS2.p1.6.m3.1.1.cmml" xref="S3.SS2.p1.6.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.6.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.6.m3.1d">italic_c</annotation></semantics></math>, <math alttext="{\mathbf{D}}_{\lambda_{c}/\lambda_{0}}" class="ltx_Math" display="inline" id="S3.SS2.p1.7.m4.1"><semantics id="S3.SS2.p1.7.m4.1a"><msub id="S3.SS2.p1.7.m4.1.1" xref="S3.SS2.p1.7.m4.1.1.cmml"><mi id="S3.SS2.p1.7.m4.1.1.2" xref="S3.SS2.p1.7.m4.1.1.2.cmml">𝐃</mi><mrow id="S3.SS2.p1.7.m4.1.1.3" xref="S3.SS2.p1.7.m4.1.1.3.cmml"><msub id="S3.SS2.p1.7.m4.1.1.3.2" xref="S3.SS2.p1.7.m4.1.1.3.2.cmml"><mi id="S3.SS2.p1.7.m4.1.1.3.2.2" xref="S3.SS2.p1.7.m4.1.1.3.2.2.cmml">λ</mi><mi id="S3.SS2.p1.7.m4.1.1.3.2.3" xref="S3.SS2.p1.7.m4.1.1.3.2.3.cmml">c</mi></msub><mo id="S3.SS2.p1.7.m4.1.1.3.1" xref="S3.SS2.p1.7.m4.1.1.3.1.cmml">/</mo><msub id="S3.SS2.p1.7.m4.1.1.3.3" xref="S3.SS2.p1.7.m4.1.1.3.3.cmml"><mi id="S3.SS2.p1.7.m4.1.1.3.3.2" xref="S3.SS2.p1.7.m4.1.1.3.3.2.cmml">λ</mi><mn id="S3.SS2.p1.7.m4.1.1.3.3.3" xref="S3.SS2.p1.7.m4.1.1.3.3.3.cmml">0</mn></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.7.m4.1b"><apply id="S3.SS2.p1.7.m4.1.1.cmml" xref="S3.SS2.p1.7.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.7.m4.1.1.1.cmml" xref="S3.SS2.p1.7.m4.1.1">subscript</csymbol><ci id="S3.SS2.p1.7.m4.1.1.2.cmml" xref="S3.SS2.p1.7.m4.1.1.2">𝐃</ci><apply id="S3.SS2.p1.7.m4.1.1.3.cmml" xref="S3.SS2.p1.7.m4.1.1.3"><divide id="S3.SS2.p1.7.m4.1.1.3.1.cmml" xref="S3.SS2.p1.7.m4.1.1.3.1"></divide><apply id="S3.SS2.p1.7.m4.1.1.3.2.cmml" xref="S3.SS2.p1.7.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.p1.7.m4.1.1.3.2.1.cmml" xref="S3.SS2.p1.7.m4.1.1.3.2">subscript</csymbol><ci id="S3.SS2.p1.7.m4.1.1.3.2.2.cmml" xref="S3.SS2.p1.7.m4.1.1.3.2.2">𝜆</ci><ci id="S3.SS2.p1.7.m4.1.1.3.2.3.cmml" xref="S3.SS2.p1.7.m4.1.1.3.2.3">𝑐</ci></apply><apply id="S3.SS2.p1.7.m4.1.1.3.3.cmml" xref="S3.SS2.p1.7.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.p1.7.m4.1.1.3.3.1.cmml" xref="S3.SS2.p1.7.m4.1.1.3.3">subscript</csymbol><ci id="S3.SS2.p1.7.m4.1.1.3.3.2.cmml" xref="S3.SS2.p1.7.m4.1.1.3.3.2">𝜆</ci><cn id="S3.SS2.p1.7.m4.1.1.3.3.3.cmml" type="integer" xref="S3.SS2.p1.7.m4.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.7.m4.1c">{\mathbf{D}}_{\lambda_{c}/\lambda_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.7.m4.1d">bold_D start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is the homothety operator aligning channel <math alttext="0" class="ltx_Math" display="inline" id="S3.SS2.p1.8.m5.1"><semantics id="S3.SS2.p1.8.m5.1a"><mn id="S3.SS2.p1.8.m5.1.1" xref="S3.SS2.p1.8.m5.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.8.m5.1b"><cn id="S3.SS2.p1.8.m5.1.1.cmml" type="integer" xref="S3.SS2.p1.8.m5.1.1">0</cn></annotation-xml></semantics></math> on channel <math alttext="c" class="ltx_Math" display="inline" id="S3.SS2.p1.9.m6.1"><semantics id="S3.SS2.p1.9.m6.1a"><mi id="S3.SS2.p1.9.m6.1.1" xref="S3.SS2.p1.9.m6.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.9.m6.1b"><ci id="S3.SS2.p1.9.m6.1.1.cmml" xref="S3.SS2.p1.9.m6.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.9.m6.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.9.m6.1d">italic_c</annotation></semantics></math> with a dilation coefficient defined as the ratio of their wavelengths, <math alttext="\boldsymbol{s}_{0,t}" class="ltx_Math" display="inline" id="S3.SS2.p1.10.m7.2"><semantics id="S3.SS2.p1.10.m7.2a"><msub id="S3.SS2.p1.10.m7.2.3" xref="S3.SS2.p1.10.m7.2.3.cmml"><mi id="S3.SS2.p1.10.m7.2.3.2" xref="S3.SS2.p1.10.m7.2.3.2.cmml">𝒔</mi><mrow id="S3.SS2.p1.10.m7.2.2.2.4" xref="S3.SS2.p1.10.m7.2.2.2.3.cmml"><mn id="S3.SS2.p1.10.m7.1.1.1.1" xref="S3.SS2.p1.10.m7.1.1.1.1.cmml">0</mn><mo id="S3.SS2.p1.10.m7.2.2.2.4.1" xref="S3.SS2.p1.10.m7.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p1.10.m7.2.2.2.2" xref="S3.SS2.p1.10.m7.2.2.2.2.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.10.m7.2b"><apply id="S3.SS2.p1.10.m7.2.3.cmml" xref="S3.SS2.p1.10.m7.2.3"><csymbol cd="ambiguous" id="S3.SS2.p1.10.m7.2.3.1.cmml" xref="S3.SS2.p1.10.m7.2.3">subscript</csymbol><ci id="S3.SS2.p1.10.m7.2.3.2.cmml" xref="S3.SS2.p1.10.m7.2.3.2">𝒔</ci><list id="S3.SS2.p1.10.m7.2.2.2.3.cmml" xref="S3.SS2.p1.10.m7.2.2.2.4"><cn id="S3.SS2.p1.10.m7.1.1.1.1.cmml" type="integer" xref="S3.SS2.p1.10.m7.1.1.1.1">0</cn><ci id="S3.SS2.p1.10.m7.2.2.2.2.cmml" xref="S3.SS2.p1.10.m7.2.2.2.2">𝑡</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.10.m7.2c">\boldsymbol{s}_{0,t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.10.m7.2d">bold_italic_s start_POSTSUBSCRIPT 0 , italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="S3.SS2.p1.11.m8.1"><semantics id="S3.SS2.p1.11.m8.1a"><msup id="S3.SS2.p1.11.m8.1.1" xref="S3.SS2.p1.11.m8.1.1.cmml"><mi id="S3.SS2.p1.11.m8.1.1.2" xref="S3.SS2.p1.11.m8.1.1.2.cmml">ℝ</mi><mrow id="S3.SS2.p1.11.m8.1.1.3" xref="S3.SS2.p1.11.m8.1.1.3.cmml"><mi id="S3.SS2.p1.11.m8.1.1.3.2" xref="S3.SS2.p1.11.m8.1.1.3.2.cmml">H</mi><mo id="S3.SS2.p1.11.m8.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p1.11.m8.1.1.3.1.cmml">×</mo><mi id="S3.SS2.p1.11.m8.1.1.3.3" xref="S3.SS2.p1.11.m8.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.11.m8.1b"><apply id="S3.SS2.p1.11.m8.1.1.cmml" xref="S3.SS2.p1.11.m8.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.11.m8.1.1.1.cmml" xref="S3.SS2.p1.11.m8.1.1">superscript</csymbol><ci id="S3.SS2.p1.11.m8.1.1.2.cmml" xref="S3.SS2.p1.11.m8.1.1.2">ℝ</ci><apply id="S3.SS2.p1.11.m8.1.1.3.cmml" xref="S3.SS2.p1.11.m8.1.1.3"><times id="S3.SS2.p1.11.m8.1.1.3.1.cmml" xref="S3.SS2.p1.11.m8.1.1.3.1"></times><ci id="S3.SS2.p1.11.m8.1.1.3.2.cmml" xref="S3.SS2.p1.11.m8.1.1.3.2">𝐻</ci><ci id="S3.SS2.p1.11.m8.1.1.3.3.cmml" xref="S3.SS2.p1.11.m8.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.11.m8.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.11.m8.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> is speckles pattern at wavelength <math alttext="\lambda_{0}" class="ltx_Math" display="inline" id="S3.SS2.p1.12.m9.1"><semantics id="S3.SS2.p1.12.m9.1a"><msub id="S3.SS2.p1.12.m9.1.1" xref="S3.SS2.p1.12.m9.1.1.cmml"><mi id="S3.SS2.p1.12.m9.1.1.2" xref="S3.SS2.p1.12.m9.1.1.2.cmml">λ</mi><mn id="S3.SS2.p1.12.m9.1.1.3" xref="S3.SS2.p1.12.m9.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.12.m9.1b"><apply id="S3.SS2.p1.12.m9.1.1.cmml" xref="S3.SS2.p1.12.m9.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.12.m9.1.1.1.cmml" xref="S3.SS2.p1.12.m9.1.1">subscript</csymbol><ci id="S3.SS2.p1.12.m9.1.1.2.cmml" xref="S3.SS2.p1.12.m9.1.1.2">𝜆</ci><cn id="S3.SS2.p1.12.m9.1.1.3.cmml" type="integer" xref="S3.SS2.p1.12.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.12.m9.1c">\lambda_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.12.m9.1d">italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\boldsymbol{\epsilon}_{c,t}" class="ltx_Math" display="inline" id="S3.SS2.p1.13.m10.2"><semantics id="S3.SS2.p1.13.m10.2a"><msub id="S3.SS2.p1.13.m10.2.3" xref="S3.SS2.p1.13.m10.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" id="S3.SS2.p1.13.m10.2.3.2" mathvariant="bold-italic" xref="S3.SS2.p1.13.m10.2.3.2.cmml">ϵ</mi><mrow id="S3.SS2.p1.13.m10.2.2.2.4" xref="S3.SS2.p1.13.m10.2.2.2.3.cmml"><mi id="S3.SS2.p1.13.m10.1.1.1.1" xref="S3.SS2.p1.13.m10.1.1.1.1.cmml">c</mi><mo id="S3.SS2.p1.13.m10.2.2.2.4.1" xref="S3.SS2.p1.13.m10.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p1.13.m10.2.2.2.2" xref="S3.SS2.p1.13.m10.2.2.2.2.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.13.m10.2b"><apply id="S3.SS2.p1.13.m10.2.3.cmml" xref="S3.SS2.p1.13.m10.2.3"><csymbol cd="ambiguous" id="S3.SS2.p1.13.m10.2.3.1.cmml" xref="S3.SS2.p1.13.m10.2.3">subscript</csymbol><ci id="S3.SS2.p1.13.m10.2.3.2.cmml" xref="S3.SS2.p1.13.m10.2.3.2">bold-italic-ϵ</ci><list id="S3.SS2.p1.13.m10.2.2.2.3.cmml" xref="S3.SS2.p1.13.m10.2.2.2.4"><ci id="S3.SS2.p1.13.m10.1.1.1.1.cmml" xref="S3.SS2.p1.13.m10.1.1.1.1">𝑐</ci><ci id="S3.SS2.p1.13.m10.2.2.2.2.cmml" xref="S3.SS2.p1.13.m10.2.2.2.2">𝑡</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.13.m10.2c">\boldsymbol{\epsilon}_{c,t}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.13.m10.2d">bold_italic_ϵ start_POSTSUBSCRIPT italic_c , italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" 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In this setting, the exoplanet flux <math alttext="\alpha_{c}" class="ltx_Math" display="inline" id="S3.SS2.p1.15.m12.1"><semantics id="S3.SS2.p1.15.m12.1a"><msub id="S3.SS2.p1.15.m12.1.1" xref="S3.SS2.p1.15.m12.1.1.cmml"><mi id="S3.SS2.p1.15.m12.1.1.2" xref="S3.SS2.p1.15.m12.1.1.2.cmml">α</mi><mi id="S3.SS2.p1.15.m12.1.1.3" xref="S3.SS2.p1.15.m12.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.15.m12.1b"><apply id="S3.SS2.p1.15.m12.1.1.cmml" xref="S3.SS2.p1.15.m12.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.15.m12.1.1.1.cmml" xref="S3.SS2.p1.15.m12.1.1">subscript</csymbol><ci id="S3.SS2.p1.15.m12.1.1.2.cmml" xref="S3.SS2.p1.15.m12.1.1.2">𝛼</ci><ci id="S3.SS2.p1.15.m12.1.1.3.cmml" xref="S3.SS2.p1.15.m12.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.15.m12.1c">\alpha_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.15.m12.1d">italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> and the off-axis PSF <math alttext="\boldsymbol{h}_{c}" class="ltx_Math" display="inline" id="S3.SS2.p1.16.m13.1"><semantics id="S3.SS2.p1.16.m13.1a"><msub id="S3.SS2.p1.16.m13.1.1" xref="S3.SS2.p1.16.m13.1.1.cmml"><mi id="S3.SS2.p1.16.m13.1.1.2" xref="S3.SS2.p1.16.m13.1.1.2.cmml">𝒉</mi><mi id="S3.SS2.p1.16.m13.1.1.3" xref="S3.SS2.p1.16.m13.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.16.m13.1b"><apply id="S3.SS2.p1.16.m13.1.1.cmml" xref="S3.SS2.p1.16.m13.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.16.m13.1.1.1.cmml" xref="S3.SS2.p1.16.m13.1.1">subscript</csymbol><ci id="S3.SS2.p1.16.m13.1.1.2.cmml" xref="S3.SS2.p1.16.m13.1.1.2">𝒉</ci><ci id="S3.SS2.p1.16.m13.1.1.3.cmml" xref="S3.SS2.p1.16.m13.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.16.m13.1c">\boldsymbol{h}_{c}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.16.m13.1d">bold_italic_h start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> now depend on channel <math alttext="c" class="ltx_Math" display="inline" id="S3.SS2.p1.17.m14.1"><semantics id="S3.SS2.p1.17.m14.1a"><mi id="S3.SS2.p1.17.m14.1.1" xref="S3.SS2.p1.17.m14.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.17.m14.1b"><ci id="S3.SS2.p1.17.m14.1.1.cmml" xref="S3.SS2.p1.17.m14.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.17.m14.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.17.m14.1d">italic_c</annotation></semantics></math>. </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> 4 Proposed method</h2> <figure class="ltx_figure" id="S4.F2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="479" id="S4.F2.g1" src="x2.png" width="951"/> <figcaption class="ltx_caption">Figure B: Workflow of the proposed method: it exploits both the spectral behavior of speckles and the apparent motion of exoplanets to disentangle the exoplanet signal from the nuisance component in the observations <math alttext="\boldsymbol{y}" class="ltx_Math" display="inline" id="S4.F2.3.1.m1.1"><semantics id="S4.F2.3.1.m1.1b"><mi id="S4.F2.3.1.m1.1.1" xref="S4.F2.3.1.m1.1.1.cmml">𝒚</mi><annotation-xml encoding="MathML-Content" id="S4.F2.3.1.m1.1c"><ci id="S4.F2.3.1.m1.1.1.cmml" xref="S4.F2.3.1.m1.1.1">𝒚</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.3.1.m1.1d">\boldsymbol{y}</annotation><annotation encoding="application/x-llamapun" id="S4.F2.3.1.m1.1e">bold_italic_y</annotation></semantics></math>. To achieve this, local patches of the nuisance are modeled as Gaussian distributions, leveraging problem symmetries and incorporating multiple scales. These patches are fed to our convolutional statistical model, and combined to form a detection map. Additionally, a learned object prior, represented by a UNet <math alttext="f_{\nu}" class="ltx_Math" display="inline" id="S4.F2.4.2.m2.1"><semantics id="S4.F2.4.2.m2.1b"><msub id="S4.F2.4.2.m2.1.1" xref="S4.F2.4.2.m2.1.1.cmml"><mi id="S4.F2.4.2.m2.1.1.2" xref="S4.F2.4.2.m2.1.1.2.cmml">f</mi><mi id="S4.F2.4.2.m2.1.1.3" xref="S4.F2.4.2.m2.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S4.F2.4.2.m2.1c"><apply id="S4.F2.4.2.m2.1.1.cmml" xref="S4.F2.4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.F2.4.2.m2.1.1.1.cmml" xref="S4.F2.4.2.m2.1.1">subscript</csymbol><ci id="S4.F2.4.2.m2.1.1.2.cmml" xref="S4.F2.4.2.m2.1.1.2">𝑓</ci><ci id="S4.F2.4.2.m2.1.1.3.cmml" xref="S4.F2.4.2.m2.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.4.2.m2.1d">f_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S4.F2.4.2.m2.1e">italic_f start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>, is introduced to denoise this detection map produced by the statistical model. This approach results in an end-to-end learnable architecture.</figcaption> </figure> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> 4.1 Convolutional statistical model</h3> <figure class="ltx_figure" id="S4.F3"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="188" id="S4.F3.g1" src="x3.png" width="447"/> <figcaption class="ltx_caption">Figure C: Proposed convolutional statistical model: spectrally aligned speckles patches, indexed by <math alttext="j" class="ltx_Math" display="inline" id="S4.F3.12.1.m1.1"><semantics id="S4.F3.12.1.m1.1b"><mi id="S4.F3.12.1.m1.1.1" xref="S4.F3.12.1.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.F3.12.1.m1.1c"><ci id="S4.F3.12.1.m1.1.1.cmml" xref="S4.F3.12.1.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.12.1.m1.1d">j</annotation><annotation encoding="application/x-llamapun" id="S4.F3.12.1.m1.1e">italic_j</annotation></semantics></math>, with dimensions <math alttext="Np" class="ltx_Math" display="inline" id="S4.F3.13.2.m2.1"><semantics id="S4.F3.13.2.m2.1b"><mrow id="S4.F3.13.2.m2.1.1" xref="S4.F3.13.2.m2.1.1.cmml"><mi id="S4.F3.13.2.m2.1.1.2" xref="S4.F3.13.2.m2.1.1.2.cmml">N</mi><mo id="S4.F3.13.2.m2.1.1.1" xref="S4.F3.13.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.F3.13.2.m2.1.1.3" xref="S4.F3.13.2.m2.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.13.2.m2.1c"><apply id="S4.F3.13.2.m2.1.1.cmml" xref="S4.F3.13.2.m2.1.1"><times id="S4.F3.13.2.m2.1.1.1.cmml" xref="S4.F3.13.2.m2.1.1.1"></times><ci id="S4.F3.13.2.m2.1.1.2.cmml" xref="S4.F3.13.2.m2.1.1.2">𝑁</ci><ci id="S4.F3.13.2.m2.1.1.3.cmml" xref="S4.F3.13.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.13.2.m2.1d">Np</annotation><annotation encoding="application/x-llamapun" id="S4.F3.13.2.m2.1e">italic_N italic_p</annotation></semantics></math> and <math alttext="CT" class="ltx_Math" display="inline" id="S4.F3.14.3.m3.1"><semantics id="S4.F3.14.3.m3.1b"><mrow id="S4.F3.14.3.m3.1.1" xref="S4.F3.14.3.m3.1.1.cmml"><mi id="S4.F3.14.3.m3.1.1.2" xref="S4.F3.14.3.m3.1.1.2.cmml">C</mi><mo id="S4.F3.14.3.m3.1.1.1" xref="S4.F3.14.3.m3.1.1.1.cmml">⁢</mo><mi id="S4.F3.14.3.m3.1.1.3" xref="S4.F3.14.3.m3.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.14.3.m3.1c"><apply id="S4.F3.14.3.m3.1.1.cmml" xref="S4.F3.14.3.m3.1.1"><times id="S4.F3.14.3.m3.1.1.1.cmml" xref="S4.F3.14.3.m3.1.1.1"></times><ci id="S4.F3.14.3.m3.1.1.2.cmml" xref="S4.F3.14.3.m3.1.1.2">𝐶</ci><ci id="S4.F3.14.3.m3.1.1.3.cmml" xref="S4.F3.14.3.m3.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.14.3.m3.1d">CT</annotation><annotation encoding="application/x-llamapun" id="S4.F3.14.3.m3.1e">italic_C italic_T</annotation></semantics></math> samples, are first linearly projected into a lower-dimensional space of size <math alttext="m" class="ltx_Math" display="inline" id="S4.F3.15.4.m4.1"><semantics id="S4.F3.15.4.m4.1b"><mi id="S4.F3.15.4.m4.1.1" xref="S4.F3.15.4.m4.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S4.F3.15.4.m4.1c"><ci id="S4.F3.15.4.m4.1.1.cmml" xref="S4.F3.15.4.m4.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.15.4.m4.1d">m</annotation><annotation encoding="application/x-llamapun" id="S4.F3.15.4.m4.1e">italic_m</annotation></semantics></math>. 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As detailed in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS3" title="A.3 Fast computation of 𝒃_𝑡 and 𝒂 ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.3</a>, the efficient computation of <math alttext="\boldsymbol{a}^{(j)}" class="ltx_Math" display="inline" id="S4.F3.21.10.m10.1"><semantics id="S4.F3.21.10.m10.1b"><msup id="S4.F3.21.10.m10.1.2" xref="S4.F3.21.10.m10.1.2.cmml"><mi id="S4.F3.21.10.m10.1.2.2" xref="S4.F3.21.10.m10.1.2.2.cmml">𝒂</mi><mrow id="S4.F3.21.10.m10.1.1.1.3" xref="S4.F3.21.10.m10.1.2.cmml"><mo id="S4.F3.21.10.m10.1.1.1.3.1" stretchy="false" xref="S4.F3.21.10.m10.1.2.cmml">(</mo><mi id="S4.F3.21.10.m10.1.1.1.1" xref="S4.F3.21.10.m10.1.1.1.1.cmml">j</mi><mo id="S4.F3.21.10.m10.1.1.1.3.2" stretchy="false" xref="S4.F3.21.10.m10.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.F3.21.10.m10.1c"><apply id="S4.F3.21.10.m10.1.2.cmml" xref="S4.F3.21.10.m10.1.2"><csymbol cd="ambiguous" id="S4.F3.21.10.m10.1.2.1.cmml" xref="S4.F3.21.10.m10.1.2">superscript</csymbol><ci id="S4.F3.21.10.m10.1.2.2.cmml" xref="S4.F3.21.10.m10.1.2.2">𝒂</ci><ci id="S4.F3.21.10.m10.1.1.1.1.cmml" xref="S4.F3.21.10.m10.1.1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.21.10.m10.1d">\boldsymbol{a}^{(j)}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.21.10.m10.1e">bold_italic_a start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT</annotation></semantics></math> relies on the Cholesky decomposition of the precision matrix <math alttext="\widehat{\mathbf{C}}^{-1}_{j}" class="ltx_Math" display="inline" id="S4.F3.22.11.m11.1"><semantics id="S4.F3.22.11.m11.1b"><msubsup id="S4.F3.22.11.m11.1.1" xref="S4.F3.22.11.m11.1.1.cmml"><mover accent="true" id="S4.F3.22.11.m11.1.1.2.2" xref="S4.F3.22.11.m11.1.1.2.2.cmml"><mi id="S4.F3.22.11.m11.1.1.2.2.2" xref="S4.F3.22.11.m11.1.1.2.2.2.cmml">𝐂</mi><mo id="S4.F3.22.11.m11.1.1.2.2.1" xref="S4.F3.22.11.m11.1.1.2.2.1.cmml">^</mo></mover><mi id="S4.F3.22.11.m11.1.1.3" xref="S4.F3.22.11.m11.1.1.3.cmml">j</mi><mrow id="S4.F3.22.11.m11.1.1.2.3" xref="S4.F3.22.11.m11.1.1.2.3.cmml"><mo id="S4.F3.22.11.m11.1.1.2.3b" xref="S4.F3.22.11.m11.1.1.2.3.cmml">−</mo><mn id="S4.F3.22.11.m11.1.1.2.3.2" xref="S4.F3.22.11.m11.1.1.2.3.2.cmml">1</mn></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S4.F3.22.11.m11.1c"><apply id="S4.F3.22.11.m11.1.1.cmml" xref="S4.F3.22.11.m11.1.1"><csymbol cd="ambiguous" id="S4.F3.22.11.m11.1.1.1.cmml" xref="S4.F3.22.11.m11.1.1">subscript</csymbol><apply id="S4.F3.22.11.m11.1.1.2.cmml" xref="S4.F3.22.11.m11.1.1"><csymbol cd="ambiguous" id="S4.F3.22.11.m11.1.1.2.1.cmml" xref="S4.F3.22.11.m11.1.1">superscript</csymbol><apply id="S4.F3.22.11.m11.1.1.2.2.cmml" xref="S4.F3.22.11.m11.1.1.2.2"><ci id="S4.F3.22.11.m11.1.1.2.2.1.cmml" xref="S4.F3.22.11.m11.1.1.2.2.1">^</ci><ci id="S4.F3.22.11.m11.1.1.2.2.2.cmml" xref="S4.F3.22.11.m11.1.1.2.2.2">𝐂</ci></apply><apply id="S4.F3.22.11.m11.1.1.2.3.cmml" xref="S4.F3.22.11.m11.1.1.2.3"><minus id="S4.F3.22.11.m11.1.1.2.3.1.cmml" xref="S4.F3.22.11.m11.1.1.2.3"></minus><cn id="S4.F3.22.11.m11.1.1.2.3.2.cmml" type="integer" xref="S4.F3.22.11.m11.1.1.2.3.2">1</cn></apply></apply><ci id="S4.F3.22.11.m11.1.1.3.cmml" xref="S4.F3.22.11.m11.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.22.11.m11.1d">\widehat{\mathbf{C}}^{-1}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.22.11.m11.1e">over^ start_ARG bold_C end_ARG start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. </figcaption> </figure> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Local Gaussian model of speckles.</h4> <div class="ltx_para ltx_noindent" id="S4.SS1.SSS0.Px1.p1"> Building on the PACO algorithm <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib26" title="">26</a>]</cite>, we propose to capture local spatial correlations between pixels of the nuisance term. 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id="S4.SS1.SSS0.Px1.p1.2.m2.1c">|\mathcal{G}_{p}|=HW</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.2.m2.1d">| caligraphic_G start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT | = italic_H italic_W</annotation></semantics></math> and <math alttext="\forall i" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.3.m3.1"><semantics id="S4.SS1.SSS0.Px1.p1.3.m3.1a"><mrow id="S4.SS1.SSS0.Px1.p1.3.m3.1.1" xref="S4.SS1.SSS0.Px1.p1.3.m3.1.1.cmml"><mo id="S4.SS1.SSS0.Px1.p1.3.m3.1.1.1" rspace="0.167em" xref="S4.SS1.SSS0.Px1.p1.3.m3.1.1.1.cmml">∀</mo><mi id="S4.SS1.SSS0.Px1.p1.3.m3.1.1.2" xref="S4.SS1.SSS0.Px1.p1.3.m3.1.1.2.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.3.m3.1b"><apply id="S4.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.3.m3.1.1"><csymbol cd="latexml" id="S4.SS1.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.3.m3.1.1.1">for-all</csymbol><ci id="S4.SS1.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.3.m3.1.1.2">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.3.m3.1c">\forall i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.3.m3.1d">∀ italic_i</annotation></semantics></math> in <math alttext="\llbracket 0,HW-1\rrbracket,\boldsymbol{x}_{i}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.4.m4.3"><semantics id="S4.SS1.SSS0.Px1.p1.4.m4.3a"><mrow id="S4.SS1.SSS0.Px1.p1.4.m4.3.3.2" xref="S4.SS1.SSS0.Px1.p1.4.m4.3.3.3.cmml"><mrow id="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1" xref="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.2.cmml"><mo id="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.2.cmml">⟦</mo><mn id="S4.SS1.SSS0.Px1.p1.4.m4.1.1" xref="S4.SS1.SSS0.Px1.p1.4.m4.1.1.cmml">0</mn><mo id="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1.3" xref="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.2.cmml">,</mo><mrow id="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1.1" 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id="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.2.2">𝐻</ci><ci id="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.2.3.cmml" xref="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.2.3">𝑊</ci></apply><cn id="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px1.p1.4.m4.2.2.1.1.1.1.3">1</cn></apply></list><apply id="S4.SS1.SSS0.Px1.p1.4.m4.3.3.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.4.m4.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.4.m4.3.3.2.2.1.cmml" xref="S4.SS1.SSS0.Px1.p1.4.m4.3.3.2.2">subscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.4.m4.3.3.2.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.4.m4.3.3.2.2.2">𝒙</ci><ci id="S4.SS1.SSS0.Px1.p1.4.m4.3.3.2.2.3.cmml" xref="S4.SS1.SSS0.Px1.p1.4.m4.3.3.2.2.3">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.4.m4.3c">\llbracket 0,HW-1\rrbracket,\boldsymbol{x}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.4.m4.3d">⟦ 0 , italic_H italic_W - 1 ⟧ , bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{2}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.5.m5.1"><semantics id="S4.SS1.SSS0.Px1.p1.5.m5.1a"><msup id="S4.SS1.SSS0.Px1.p1.5.m5.1.1" xref="S4.SS1.SSS0.Px1.p1.5.m5.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.5.m5.1.1.2" xref="S4.SS1.SSS0.Px1.p1.5.m5.1.1.2.cmml">ℝ</mi><mn id="S4.SS1.SSS0.Px1.p1.5.m5.1.1.3" xref="S4.SS1.SSS0.Px1.p1.5.m5.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.5.m5.1b"><apply id="S4.SS1.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.5.m5.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.5.m5.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.5.m5.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.5.m5.1.1.2">ℝ</ci><cn id="S4.SS1.SSS0.Px1.p1.5.m5.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px1.p1.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.5.m5.1c">\mathbb{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.5.m5.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Then, we model the statistical distribution of collections of patches positioned on a spatial grid <math alttext="\mathcal{G}_{d}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.6.m6.1"><semantics id="S4.SS1.SSS0.Px1.p1.6.m6.1a"><msub id="S4.SS1.SSS0.Px1.p1.6.m6.1.1" xref="S4.SS1.SSS0.Px1.p1.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px1.p1.6.m6.1.1.2" xref="S4.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml">𝒢</mi><mi id="S4.SS1.SSS0.Px1.p1.6.m6.1.1.3" xref="S4.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.6.m6.1b"><apply id="S4.SS1.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.6.m6.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.6.m6.1.1.2">𝒢</ci><ci id="S4.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.6.m6.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.6.m6.1c">\mathcal{G}_{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.6.m6.1d">caligraphic_G start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math>, such that <math alttext="\mathcal{G}_{d}\subset\mathcal{G}_{p}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.7.m7.1"><semantics id="S4.SS1.SSS0.Px1.p1.7.m7.1a"><mrow id="S4.SS1.SSS0.Px1.p1.7.m7.1.1" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.cmml"><msub id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.2" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.2.cmml">𝒢</mi><mi id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.3" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.3.cmml">d</mi></msub><mo id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.1" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.1.cmml">⊂</mo><msub id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.2" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.2.cmml">𝒢</mi><mi id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.3" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.7.m7.1b"><apply id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1"><subset id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.1"></subset><apply id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.2">𝒢</ci><ci id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.3.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.2.3">𝑑</ci></apply><apply id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.1.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3">subscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.2.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.2">𝒢</ci><ci id="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.3.cmml" xref="S4.SS1.SSS0.Px1.p1.7.m7.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.7.m7.1c">\mathcal{G}_{d}\subset\mathcal{G}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.7.m7.1d">caligraphic_G start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ⊂ caligraphic_G start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, with <math alttext="M\vcentcolon=|\mathcal{G}_{d}|" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.8.m8.1"><semantics id="S4.SS1.SSS0.Px1.p1.8.m8.1a"><mrow id="S4.SS1.SSS0.Px1.p1.8.m8.1.1" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.3" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml">M</mi><mo id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.2" lspace="0.278em" rspace="0.278em" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml">:=</mo><mrow id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.2.cmml"><mo id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.2.1.cmml">|</mo><msub id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.cmml">𝒢</mi><mi id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.cmml">d</mi></msub><mo id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.8.m8.1b"><apply id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1"><csymbol cd="latexml" id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.2">assign</csymbol><ci id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.3">𝑀</ci><apply id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1"><abs id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.2"></abs><apply id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2">𝒢</ci><ci id="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.8.m8.1c">M\vcentcolon=|\mathcal{G}_{d}|</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.8.m8.1d">italic_M := | caligraphic_G start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT |</annotation></semantics></math>. We denote by <math alttext="\boldsymbol{y}_{t}^{(j)}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.9.m9.1"><semantics id="S4.SS1.SSS0.Px1.p1.9.m9.1a"><msubsup id="S4.SS1.SSS0.Px1.p1.9.m9.1.2" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2.cmml"><mi id="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.2" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.2.cmml">𝒚</mi><mi id="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.3" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.3.cmml">t</mi><mrow id="S4.SS1.SSS0.Px1.p1.9.m9.1.1.1.3" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2.cmml"><mo id="S4.SS1.SSS0.Px1.p1.9.m9.1.1.1.3.1" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2.cmml">(</mo><mi id="S4.SS1.SSS0.Px1.p1.9.m9.1.1.1.1" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.1.1.1.cmml">j</mi><mo id="S4.SS1.SSS0.Px1.p1.9.m9.1.1.1.3.2" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.9.m9.1b"><apply id="S4.SS1.SSS0.Px1.p1.9.m9.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.9.m9.1.2.1.cmml" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2">superscript</csymbol><apply id="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.1.cmml" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2">subscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.2">𝒚</ci><ci id="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.3.cmml" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.2.2.3">𝑡</ci></apply><ci id="S4.SS1.SSS0.Px1.p1.9.m9.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.9.m9.1.1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.9.m9.1c">\boldsymbol{y}_{t}^{(j)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.9.m9.1d">bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{p},\ \forall(j,t)\in\llbracket 0,M-1\rrbracket\times\llbracket 0,T% -1\rrbracket" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.10.m10.8"><semantics id="S4.SS1.SSS0.Px1.p1.10.m10.8a"><mrow id="S4.SS1.SSS0.Px1.p1.10.m10.8.8" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.cmml"><mrow id="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2" xref="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.3.cmml"><msup id="S4.SS1.SSS0.Px1.p1.10.m10.5.5.1.1.1" xref="S4.SS1.SSS0.Px1.p1.10.m10.5.5.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.10.m10.5.5.1.1.1.2" xref="S4.SS1.SSS0.Px1.p1.10.m10.5.5.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS1.SSS0.Px1.p1.10.m10.5.5.1.1.1.3" xref="S4.SS1.SSS0.Px1.p1.10.m10.5.5.1.1.1.3.cmml">p</mi></msup><mo id="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.3" rspace="0.667em" xref="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.3.cmml">,</mo><mrow id="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2" xref="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.cmml"><mo id="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.1" xref="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.1.cmml">∀</mo><mrow id="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.2.2" xref="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.2.1.cmml"><mo id="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.2.2.1" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.2.1.cmml">(</mo><mi id="S4.SS1.SSS0.Px1.p1.10.m10.1.1" xref="S4.SS1.SSS0.Px1.p1.10.m10.1.1.cmml">j</mi><mo id="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.2.2.2" xref="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.2.1.cmml">,</mo><mi id="S4.SS1.SSS0.Px1.p1.10.m10.2.2" xref="S4.SS1.SSS0.Px1.p1.10.m10.2.2.cmml">t</mi><mo id="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.2.2.3" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.10.m10.6.6.2.2.2.2.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.5" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.5.cmml">∈</mo><mrow id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.cmml"><mrow id="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1" xref="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.2.cmml"><mo id="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.2.cmml">⟦</mo><mn id="S4.SS1.SSS0.Px1.p1.10.m10.3.3" xref="S4.SS1.SSS0.Px1.p1.10.m10.3.3.cmml">0</mn><mo id="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.3" xref="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.2.cmml">,</mo><mrow id="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.1" xref="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.1.2" xref="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.1.2.cmml">M</mi><mo id="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.1.1" xref="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.1.1.cmml">−</mo><mn id="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.1.3" xref="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.1.4" rspace="0.055em" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.10.m10.7.7.3.1.2.cmml">⟧</mo></mrow><mo id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.3" rspace="0.222em" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.3.cmml">×</mo><mrow id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.2.cmml"><mo id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.2.cmml">⟦</mo><mn id="S4.SS1.SSS0.Px1.p1.10.m10.4.4" xref="S4.SS1.SSS0.Px1.p1.10.m10.4.4.cmml">0</mn><mo id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.3" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.2.cmml">,</mo><mrow id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.1" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.1.2" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.1.2.cmml">T</mi><mo id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.1.1" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.1.1.cmml">−</mo><mn id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.1.3" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.1.3.cmml">1</mn></mrow><mo id="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.1.4" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.10.m10.8.8.4.2.2.cmml">⟧</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" 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0,M-1\rrbracket\times\llbracket 0,T% -1\rrbracket</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.10.m10.8d">blackboard_R start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT , ∀ ( italic_j , italic_t ) ∈ ⟦ 0 , italic_M - 1 ⟧ × ⟦ 0 , italic_T - 1 ⟧</annotation></semantics></math> the observed patch in spatial location <math alttext="j" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.11.m11.1"><semantics id="S4.SS1.SSS0.Px1.p1.11.m11.1a"><mi id="S4.SS1.SSS0.Px1.p1.11.m11.1.1" xref="S4.SS1.SSS0.Px1.p1.11.m11.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.11.m11.1b"><ci id="S4.SS1.SSS0.Px1.p1.11.m11.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.11.m11.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.11.m11.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.11.m11.1d">italic_j</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.12.m12.1"><semantics id="S4.SS1.SSS0.Px1.p1.12.m12.1a"><mi id="S4.SS1.SSS0.Px1.p1.12.m12.1.1" xref="S4.SS1.SSS0.Px1.p1.12.m12.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.12.m12.1b"><ci id="S4.SS1.SSS0.Px1.p1.12.m12.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.12.m12.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.12.m12.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.12.m12.1d">italic_t</annotation></semantics></math>. In absence of exoplanet, i.e., when only the nuisance component is present, each collection of patches <math alttext="\{\boldsymbol{y}_{t}^{(j)}\}_{t=0:T-1}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.13.m13.2"><semantics id="S4.SS1.SSS0.Px1.p1.13.m13.2a"><msub id="S4.SS1.SSS0.Px1.p1.13.m13.2.2" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.cmml"><mrow id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.1" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.2.cmml"><mo id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.2.cmml">{</mo><msubsup id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.1.1" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.1.1.2.2" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.1.1.2.2.cmml">𝒚</mi><mi id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.1.1.2.3" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.1.1.2.3.cmml">t</mi><mrow id="S4.SS1.SSS0.Px1.p1.13.m13.1.1.1.3" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.1.1.1.cmml"><mo 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id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.2.1.cmml" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.2.1"></eq><ci id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.2.2">𝑡</ci><cn id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.2.3.cmml" type="integer" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.2.3">0</cn></apply><apply id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.3.cmml" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.3"><minus id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.3.1.cmml" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.3.1"></minus><ci id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.3.2.cmml" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.3.2">𝑇</ci><cn id="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.3.3.cmml" type="integer" xref="S4.SS1.SSS0.Px1.p1.13.m13.2.2.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.13.m13.2c">\{\boldsymbol{y}_{t}^{(j)}\}_{t=0:T-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.13.m13.2d">{ bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_t = 0 : italic_T - 1 end_POSTSUBSCRIPT</annotation></semantics></math> is modeled by a multivariate Gaussian: <table class="ltx_equation ltx_eqn_table" id="S4.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall(j,t),\ \boldsymbol{y}_{t}^{(j)}\sim\mathcal{N}(\boldsymbol{m}_{j},% \mathbf{C}_{j})\,," class="ltx_Math" display="block" id="S4.E3.m1.4"><semantics id="S4.E3.m1.4a"><mrow id="S4.E3.m1.4.4.1" xref="S4.E3.m1.4.4.1.1.cmml"><mrow id="S4.E3.m1.4.4.1.1" xref="S4.E3.m1.4.4.1.1.cmml"><mrow id="S4.E3.m1.4.4.1.1.2.2" xref="S4.E3.m1.4.4.1.1.2.3.cmml"><mrow id="S4.E3.m1.4.4.1.1.1.1.1" xref="S4.E3.m1.4.4.1.1.1.1.1.cmml"><mo id="S4.E3.m1.4.4.1.1.1.1.1.1" xref="S4.E3.m1.4.4.1.1.1.1.1.1.cmml">∀</mo><mrow 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xref="S4.E3.m1.1.1.1.1.cmml">j</mi><mo id="S4.E3.m1.1.1.1.3.2" stretchy="false" xref="S4.E3.m1.4.4.1.1.2.2.2.cmml">)</mo></mrow></msubsup></mrow><mo id="S4.E3.m1.4.4.1.1.5" xref="S4.E3.m1.4.4.1.1.5.cmml">∼</mo><mrow id="S4.E3.m1.4.4.1.1.4" xref="S4.E3.m1.4.4.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E3.m1.4.4.1.1.4.4" xref="S4.E3.m1.4.4.1.1.4.4.cmml">𝒩</mi><mo id="S4.E3.m1.4.4.1.1.4.3" xref="S4.E3.m1.4.4.1.1.4.3.cmml">⁢</mo><mrow id="S4.E3.m1.4.4.1.1.4.2.2" xref="S4.E3.m1.4.4.1.1.4.2.3.cmml"><mo id="S4.E3.m1.4.4.1.1.4.2.2.3" stretchy="false" xref="S4.E3.m1.4.4.1.1.4.2.3.cmml">(</mo><msub id="S4.E3.m1.4.4.1.1.3.1.1.1" xref="S4.E3.m1.4.4.1.1.3.1.1.1.cmml"><mi id="S4.E3.m1.4.4.1.1.3.1.1.1.2" xref="S4.E3.m1.4.4.1.1.3.1.1.1.2.cmml">𝒎</mi><mi id="S4.E3.m1.4.4.1.1.3.1.1.1.3" xref="S4.E3.m1.4.4.1.1.3.1.1.1.3.cmml">j</mi></msub><mo id="S4.E3.m1.4.4.1.1.4.2.2.4" xref="S4.E3.m1.4.4.1.1.4.2.3.cmml">,</mo><msub id="S4.E3.m1.4.4.1.1.4.2.2.2" xref="S4.E3.m1.4.4.1.1.4.2.2.2.cmml"><mi id="S4.E3.m1.4.4.1.1.4.2.2.2.2" xref="S4.E3.m1.4.4.1.1.4.2.2.2.2.cmml">𝐂</mi><mi id="S4.E3.m1.4.4.1.1.4.2.2.2.3" xref="S4.E3.m1.4.4.1.1.4.2.2.2.3.cmml">j</mi></msub><mo id="S4.E3.m1.4.4.1.1.4.2.2.5" rspace="0.170em" stretchy="false" xref="S4.E3.m1.4.4.1.1.4.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E3.m1.4.4.1.2" xref="S4.E3.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E3.m1.4b"><apply id="S4.E3.m1.4.4.1.1.cmml" xref="S4.E3.m1.4.4.1"><csymbol cd="latexml" id="S4.E3.m1.4.4.1.1.5.cmml" xref="S4.E3.m1.4.4.1.1.5">similar-to</csymbol><list id="S4.E3.m1.4.4.1.1.2.3.cmml" xref="S4.E3.m1.4.4.1.1.2.2"><apply id="S4.E3.m1.4.4.1.1.1.1.1.cmml" xref="S4.E3.m1.4.4.1.1.1.1.1"><csymbol cd="latexml" id="S4.E3.m1.4.4.1.1.1.1.1.1.cmml" xref="S4.E3.m1.4.4.1.1.1.1.1.1">for-all</csymbol><interval closure="open" id="S4.E3.m1.4.4.1.1.1.1.1.2.1.cmml" xref="S4.E3.m1.4.4.1.1.1.1.1.2.2"><ci id="S4.E3.m1.2.2.cmml" xref="S4.E3.m1.2.2">𝑗</ci><ci id="S4.E3.m1.3.3.cmml" xref="S4.E3.m1.3.3">𝑡</ci></interval></apply><apply id="S4.E3.m1.4.4.1.1.2.2.2.cmml" xref="S4.E3.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.E3.m1.4.4.1.1.2.2.2.1.cmml" xref="S4.E3.m1.4.4.1.1.2.2.2">superscript</csymbol><apply id="S4.E3.m1.4.4.1.1.2.2.2.2.cmml" xref="S4.E3.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.E3.m1.4.4.1.1.2.2.2.2.1.cmml" xref="S4.E3.m1.4.4.1.1.2.2.2">subscript</csymbol><ci id="S4.E3.m1.4.4.1.1.2.2.2.2.2.cmml" xref="S4.E3.m1.4.4.1.1.2.2.2.2.2">𝒚</ci><ci id="S4.E3.m1.4.4.1.1.2.2.2.2.3.cmml" xref="S4.E3.m1.4.4.1.1.2.2.2.2.3">𝑡</ci></apply><ci id="S4.E3.m1.1.1.1.1.cmml" xref="S4.E3.m1.1.1.1.1">𝑗</ci></apply></list><apply id="S4.E3.m1.4.4.1.1.4.cmml" xref="S4.E3.m1.4.4.1.1.4"><times id="S4.E3.m1.4.4.1.1.4.3.cmml" xref="S4.E3.m1.4.4.1.1.4.3"></times><ci id="S4.E3.m1.4.4.1.1.4.4.cmml" xref="S4.E3.m1.4.4.1.1.4.4">𝒩</ci><interval closure="open" id="S4.E3.m1.4.4.1.1.4.2.3.cmml" xref="S4.E3.m1.4.4.1.1.4.2.2"><apply id="S4.E3.m1.4.4.1.1.3.1.1.1.cmml" xref="S4.E3.m1.4.4.1.1.3.1.1.1"><csymbol cd="ambiguous" id="S4.E3.m1.4.4.1.1.3.1.1.1.1.cmml" xref="S4.E3.m1.4.4.1.1.3.1.1.1">subscript</csymbol><ci id="S4.E3.m1.4.4.1.1.3.1.1.1.2.cmml" xref="S4.E3.m1.4.4.1.1.3.1.1.1.2">𝒎</ci><ci id="S4.E3.m1.4.4.1.1.3.1.1.1.3.cmml" xref="S4.E3.m1.4.4.1.1.3.1.1.1.3">𝑗</ci></apply><apply id="S4.E3.m1.4.4.1.1.4.2.2.2.cmml" xref="S4.E3.m1.4.4.1.1.4.2.2.2"><csymbol cd="ambiguous" id="S4.E3.m1.4.4.1.1.4.2.2.2.1.cmml" xref="S4.E3.m1.4.4.1.1.4.2.2.2">subscript</csymbol><ci id="S4.E3.m1.4.4.1.1.4.2.2.2.2.cmml" xref="S4.E3.m1.4.4.1.1.4.2.2.2.2">𝐂</ci><ci id="S4.E3.m1.4.4.1.1.4.2.2.2.3.cmml" xref="S4.E3.m1.4.4.1.1.4.2.2.2.3">𝑗</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E3.m1.4c">\forall(j,t),\ \boldsymbol{y}_{t}^{(j)}\sim\mathcal{N}(\boldsymbol{m}_{j},% \mathbf{C}_{j})\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m1.4d">∀ ( italic_j , italic_t ) , bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ∼ caligraphic_N ( bold_italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(3)</td> </tr></tbody> </table> with <math alttext="\boldsymbol{m}_{j}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.14.m1.1"><semantics id="S4.SS1.SSS0.Px1.p1.14.m1.1a"><msub id="S4.SS1.SSS0.Px1.p1.14.m1.1.1" xref="S4.SS1.SSS0.Px1.p1.14.m1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.14.m1.1.1.2" xref="S4.SS1.SSS0.Px1.p1.14.m1.1.1.2.cmml">𝒎</mi><mi id="S4.SS1.SSS0.Px1.p1.14.m1.1.1.3" xref="S4.SS1.SSS0.Px1.p1.14.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.14.m1.1b"><apply id="S4.SS1.SSS0.Px1.p1.14.m1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.14.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.14.m1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.14.m1.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.14.m1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.14.m1.1.1.2">𝒎</ci><ci id="S4.SS1.SSS0.Px1.p1.14.m1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.14.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.14.m1.1c">\boldsymbol{m}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.14.m1.1d">bold_italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{p}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.15.m2.1"><semantics id="S4.SS1.SSS0.Px1.p1.15.m2.1a"><msup id="S4.SS1.SSS0.Px1.p1.15.m2.1.1" xref="S4.SS1.SSS0.Px1.p1.15.m2.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.15.m2.1.1.2" xref="S4.SS1.SSS0.Px1.p1.15.m2.1.1.2.cmml">ℝ</mi><mi id="S4.SS1.SSS0.Px1.p1.15.m2.1.1.3" xref="S4.SS1.SSS0.Px1.p1.15.m2.1.1.3.cmml">p</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.15.m2.1b"><apply id="S4.SS1.SSS0.Px1.p1.15.m2.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.15.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.15.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.15.m2.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.15.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.15.m2.1.1.2">ℝ</ci><ci id="S4.SS1.SSS0.Px1.p1.15.m2.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.15.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.15.m2.1c">\mathbb{R}^{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.15.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathbf{C}_{j}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.16.m3.1"><semantics id="S4.SS1.SSS0.Px1.p1.16.m3.1a"><msub id="S4.SS1.SSS0.Px1.p1.16.m3.1.1" xref="S4.SS1.SSS0.Px1.p1.16.m3.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.16.m3.1.1.2" xref="S4.SS1.SSS0.Px1.p1.16.m3.1.1.2.cmml">𝐂</mi><mi id="S4.SS1.SSS0.Px1.p1.16.m3.1.1.3" xref="S4.SS1.SSS0.Px1.p1.16.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.16.m3.1b"><apply id="S4.SS1.SSS0.Px1.p1.16.m3.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.16.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.16.m3.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.16.m3.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.16.m3.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.16.m3.1.1.2">𝐂</ci><ci id="S4.SS1.SSS0.Px1.p1.16.m3.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.16.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.16.m3.1c">\mathbf{C}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.16.m3.1d">bold_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{p\times p}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.17.m4.1"><semantics id="S4.SS1.SSS0.Px1.p1.17.m4.1a"><msup id="S4.SS1.SSS0.Px1.p1.17.m4.1.1" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.2" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.2.cmml">ℝ</mi><mrow id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.cmml"><mi id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.2" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.2.cmml">p</mi><mo id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.1.cmml">×</mo><mi id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.3" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.3.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.17.m4.1b"><apply id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.2">ℝ</ci><apply id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3"><times id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.1.cmml" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.1"></times><ci id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.2.cmml" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.2">𝑝</ci><ci id="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.3.cmml" xref="S4.SS1.SSS0.Px1.p1.17.m4.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.17.m4.1c">\mathbb{R}^{p\times p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.17.m4.1d">blackboard_R start_POSTSUPERSCRIPT italic_p × italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> the mean and covariance of the Gaussian distribution. These parameters are estimated by maximum likelihood and shrinkage of the covariance matrix. They are denoted as <math alttext="\widehat{\boldsymbol{m}}_{j}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.18.m5.1"><semantics id="S4.SS1.SSS0.Px1.p1.18.m5.1a"><msub id="S4.SS1.SSS0.Px1.p1.18.m5.1.1" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1.cmml"><mover accent="true" id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.cmml"><mi id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.2" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.2.cmml">𝒎</mi><mo id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.1" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.1.cmml">^</mo></mover><mi id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.3" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.18.m5.1b"><apply id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1">subscript</csymbol><apply id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2"><ci id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.1">^</ci><ci id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1.2.2">𝒎</ci></apply><ci id="S4.SS1.SSS0.Px1.p1.18.m5.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.18.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.18.m5.1c">\widehat{\boldsymbol{m}}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.18.m5.1d">over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\widehat{\mathbf{C}}_{j}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.19.m6.1"><semantics id="S4.SS1.SSS0.Px1.p1.19.m6.1a"><msub id="S4.SS1.SSS0.Px1.p1.19.m6.1.1" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1.cmml"><mover accent="true" id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.cmml"><mi id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.2" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.2.cmml">𝐂</mi><mo id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.1" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.1.cmml">^</mo></mover><mi id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.3" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.19.m6.1b"><apply id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1">subscript</csymbol><apply id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2"><ci id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.1">^</ci><ci id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1.2.2">𝐂</ci></apply><ci id="S4.SS1.SSS0.Px1.p1.19.m6.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.19.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.19.m6.1c">\widehat{\mathbf{C}}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.19.m6.1d">over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in the following. Additional details are presented in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS1" title="A.1 Estimation of the statistical parameters ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.1</a>. </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Detection criterion at a given position.</h4> <div class="ltx_para ltx_noindent" id="S4.SS1.SSS0.Px2.p1"> This local statistical model estimates the time-invariant flux <math alttext="\widehat{\alpha}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.1.m1.1"><semantics id="S4.SS1.SSS0.Px2.p1.1.m1.1a"><mover accent="true" id="S4.SS1.SSS0.Px2.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.1.m1.1.1.2" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1.2.cmml">α</mi><mo id="S4.SS1.SSS0.Px2.p1.1.m1.1.1.1" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.1.m1.1b"><apply id="S4.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1"><ci id="S4.SS1.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1.1">^</ci><ci id="S4.SS1.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1.2">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.1.m1.1c">\widehat{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.1.m1.1d">over^ start_ARG italic_α end_ARG</annotation></semantics></math> of an exoplanet at position <math alttext="\boldsymbol{x}_{0}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.2.m2.1"><semantics id="S4.SS1.SSS0.Px2.p1.2.m2.1a"><msub id="S4.SS1.SSS0.Px2.p1.2.m2.1.1" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.2" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml">𝒙</mi><mn id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.2.m2.1b"><apply id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.2">𝒙</ci><cn id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.2.m2.1c">\boldsymbol{x}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.2.m2.1d">bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{2}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.3.m3.1"><semantics id="S4.SS1.SSS0.Px2.p1.3.m3.1a"><msup 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end_POSTSUPERSCRIPT</annotation></semantics></math> in the first frame by maximizing the global likelihood: <table class="ltx_equation ltx_eqn_table" id="S4.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widehat{\alpha}=\operatorname*{arg\,max}_{\alpha}\ell(\alpha,\boldsymbol{x}_{% 0})\,." class="ltx_Math" display="block" id="S4.E4.m1.2"><semantics id="S4.E4.m1.2a"><mrow id="S4.E4.m1.2.2.1" xref="S4.E4.m1.2.2.1.1.cmml"><mrow id="S4.E4.m1.2.2.1.1" xref="S4.E4.m1.2.2.1.1.cmml"><mover accent="true" id="S4.E4.m1.2.2.1.1.3" xref="S4.E4.m1.2.2.1.1.3.cmml"><mi id="S4.E4.m1.2.2.1.1.3.2" xref="S4.E4.m1.2.2.1.1.3.2.cmml">α</mi><mo id="S4.E4.m1.2.2.1.1.3.1" xref="S4.E4.m1.2.2.1.1.3.1.cmml">^</mo></mover><mo id="S4.E4.m1.2.2.1.1.2" xref="S4.E4.m1.2.2.1.1.2.cmml">=</mo><mrow id="S4.E4.m1.2.2.1.1.1" xref="S4.E4.m1.2.2.1.1.1.cmml"><mrow id="S4.E4.m1.2.2.1.1.1.3" xref="S4.E4.m1.2.2.1.1.1.3.cmml"><munder id="S4.E4.m1.2.2.1.1.1.3.1" xref="S4.E4.m1.2.2.1.1.1.3.1.cmml"><mrow id="S4.E4.m1.2.2.1.1.1.3.1.2" xref="S4.E4.m1.2.2.1.1.1.3.1.2.cmml"><mi id="S4.E4.m1.2.2.1.1.1.3.1.2.2" xref="S4.E4.m1.2.2.1.1.1.3.1.2.2.cmml">arg</mi><mo id="S4.E4.m1.2.2.1.1.1.3.1.2.1" lspace="0.170em" xref="S4.E4.m1.2.2.1.1.1.3.1.2.1.cmml">⁢</mo><mi id="S4.E4.m1.2.2.1.1.1.3.1.2.3" xref="S4.E4.m1.2.2.1.1.1.3.1.2.3.cmml">max</mi></mrow><mi id="S4.E4.m1.2.2.1.1.1.3.1.3" xref="S4.E4.m1.2.2.1.1.1.3.1.3.cmml">α</mi></munder><mo id="S4.E4.m1.2.2.1.1.1.3a" xref="S4.E4.m1.2.2.1.1.1.3.cmml">⁡</mo><mi id="S4.E4.m1.2.2.1.1.1.3.2" mathvariant="normal" xref="S4.E4.m1.2.2.1.1.1.3.2.cmml">ℓ</mi></mrow><mo id="S4.E4.m1.2.2.1.1.1.2" xref="S4.E4.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.E4.m1.2.2.1.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml"><mo id="S4.E4.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml">(</mo><mi id="S4.E4.m1.1.1" xref="S4.E4.m1.1.1.cmml">α</mi><mo id="S4.E4.m1.2.2.1.1.1.1.1.3" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml">,</mo><msub id="S4.E4.m1.2.2.1.1.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S4.E4.m1.2.2.1.1.1.1.1.1.2" xref="S4.E4.m1.2.2.1.1.1.1.1.1.2.cmml">𝒙</mi><mn id="S4.E4.m1.2.2.1.1.1.1.1.1.3" xref="S4.E4.m1.2.2.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.E4.m1.2.2.1.1.1.1.1.4" stretchy="false" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E4.m1.2.2.1.2" lspace="0.170em" xref="S4.E4.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E4.m1.2b"><apply id="S4.E4.m1.2.2.1.1.cmml" xref="S4.E4.m1.2.2.1"><eq id="S4.E4.m1.2.2.1.1.2.cmml" xref="S4.E4.m1.2.2.1.1.2"></eq><apply id="S4.E4.m1.2.2.1.1.3.cmml" xref="S4.E4.m1.2.2.1.1.3"><ci id="S4.E4.m1.2.2.1.1.3.1.cmml" xref="S4.E4.m1.2.2.1.1.3.1">^</ci><ci id="S4.E4.m1.2.2.1.1.3.2.cmml" xref="S4.E4.m1.2.2.1.1.3.2">𝛼</ci></apply><apply id="S4.E4.m1.2.2.1.1.1.cmml" xref="S4.E4.m1.2.2.1.1.1"><times id="S4.E4.m1.2.2.1.1.1.2.cmml" xref="S4.E4.m1.2.2.1.1.1.2"></times><apply id="S4.E4.m1.2.2.1.1.1.3.cmml" xref="S4.E4.m1.2.2.1.1.1.3"><apply id="S4.E4.m1.2.2.1.1.1.3.1.cmml" xref="S4.E4.m1.2.2.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.E4.m1.2.2.1.1.1.3.1.1.cmml" xref="S4.E4.m1.2.2.1.1.1.3.1">subscript</csymbol><apply id="S4.E4.m1.2.2.1.1.1.3.1.2.cmml" xref="S4.E4.m1.2.2.1.1.1.3.1.2"><times id="S4.E4.m1.2.2.1.1.1.3.1.2.1.cmml" xref="S4.E4.m1.2.2.1.1.1.3.1.2.1"></times><ci id="S4.E4.m1.2.2.1.1.1.3.1.2.2.cmml" xref="S4.E4.m1.2.2.1.1.1.3.1.2.2">arg</ci><ci id="S4.E4.m1.2.2.1.1.1.3.1.2.3.cmml" xref="S4.E4.m1.2.2.1.1.1.3.1.2.3">max</ci></apply><ci id="S4.E4.m1.2.2.1.1.1.3.1.3.cmml" xref="S4.E4.m1.2.2.1.1.1.3.1.3">𝛼</ci></apply><ci id="S4.E4.m1.2.2.1.1.1.3.2.cmml" xref="S4.E4.m1.2.2.1.1.1.3.2">ℓ</ci></apply><interval closure="open" id="S4.E4.m1.2.2.1.1.1.1.2.cmml" xref="S4.E4.m1.2.2.1.1.1.1.1"><ci id="S4.E4.m1.1.1.cmml" xref="S4.E4.m1.1.1">𝛼</ci><apply id="S4.E4.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.E4.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E4.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E4.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E4.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S4.E4.m1.2.2.1.1.1.1.1.1.2">𝒙</ci><cn id="S4.E4.m1.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.E4.m1.2.2.1.1.1.1.1.1.3">0</cn></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.2c">\widehat{\alpha}=\operatorname*{arg\,max}_{\alpha}\ell(\alpha,\boldsymbol{x}_{% 0})\,.</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.2d">over^ start_ARG italic_α end_ARG = start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT roman_ℓ ( italic_α , bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(4)</td> </tr></tbody> </table> Assuming that collections of patches on the trajectory of an exoplanet are independent, the likelihood becomes: <table class="ltx_equation ltx_eqn_table" id="S4.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ell(\alpha,\boldsymbol{x}_{0})=\prod\nolimits_{t}\mathbb{P}\left(\boldsymbol{% y}_{t}^{(i_{t})}-\alpha\boldsymbol{h}^{(i_{t})}(\boldsymbol{x}_{t})\,|\,% \widehat{\boldsymbol{m}}_{i_{t}},\widehat{\mathbf{C}}_{i_{t}}\right)\,," class="ltx_Math" display="block" id="S4.E5.m1.4"><semantics id="S4.E5.m1.4a"><mrow id="S4.E5.m1.4.4.1" xref="S4.E5.m1.4.4.1.1.cmml"><mrow id="S4.E5.m1.4.4.1.1" xref="S4.E5.m1.4.4.1.1.cmml"><mrow id="S4.E5.m1.4.4.1.1.1" xref="S4.E5.m1.4.4.1.1.1.cmml"><mi id="S4.E5.m1.4.4.1.1.1.3" mathvariant="normal" xref="S4.E5.m1.4.4.1.1.1.3.cmml">ℓ</mi><mo id="S4.E5.m1.4.4.1.1.1.2" xref="S4.E5.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S4.E5.m1.4.4.1.1.1.1.1" xref="S4.E5.m1.4.4.1.1.1.1.2.cmml"><mo id="S4.E5.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S4.E5.m1.4.4.1.1.1.1.2.cmml">(</mo><mi id="S4.E5.m1.3.3" xref="S4.E5.m1.3.3.cmml">α</mi><mo id="S4.E5.m1.4.4.1.1.1.1.1.3" xref="S4.E5.m1.4.4.1.1.1.1.2.cmml">,</mo><msub id="S4.E5.m1.4.4.1.1.1.1.1.1" xref="S4.E5.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S4.E5.m1.4.4.1.1.1.1.1.1.2" xref="S4.E5.m1.4.4.1.1.1.1.1.1.2.cmml">𝒙</mi><mn id="S4.E5.m1.4.4.1.1.1.1.1.1.3" xref="S4.E5.m1.4.4.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.E5.m1.4.4.1.1.1.1.1.4" stretchy="false" xref="S4.E5.m1.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.4.4.1.1.3" rspace="0.111em" xref="S4.E5.m1.4.4.1.1.3.cmml">=</mo><mrow id="S4.E5.m1.4.4.1.1.2" xref="S4.E5.m1.4.4.1.1.2.cmml"><msub id="S4.E5.m1.4.4.1.1.2.2" xref="S4.E5.m1.4.4.1.1.2.2.cmml"><mo id="S4.E5.m1.4.4.1.1.2.2.2" xref="S4.E5.m1.4.4.1.1.2.2.2.cmml">∏</mo><mi id="S4.E5.m1.4.4.1.1.2.2.3" xref="S4.E5.m1.4.4.1.1.2.2.3.cmml">t</mi></msub><mrow id="S4.E5.m1.4.4.1.1.2.1" xref="S4.E5.m1.4.4.1.1.2.1.cmml"><mi id="S4.E5.m1.4.4.1.1.2.1.3" xref="S4.E5.m1.4.4.1.1.2.1.3.cmml">ℙ</mi><mo id="S4.E5.m1.4.4.1.1.2.1.2" xref="S4.E5.m1.4.4.1.1.2.1.2.cmml">⁢</mo><mrow id="S4.E5.m1.4.4.1.1.2.1.1.1" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.cmml"><mo id="S4.E5.m1.4.4.1.1.2.1.1.1.2" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.cmml">(</mo><mrow id="S4.E5.m1.4.4.1.1.2.1.1.1.1" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.cmml"><msubsup id="S4.E5.m1.4.4.1.1.2.1.1.1.1.5" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.5.cmml"><mi id="S4.E5.m1.4.4.1.1.2.1.1.1.1.5.2.2" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.5.2.2.cmml">𝒚</mi><mi id="S4.E5.m1.4.4.1.1.2.1.1.1.1.5.2.3" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.5.2.3.cmml">t</mi><mrow id="S4.E5.m1.1.1.1.1" xref="S4.E5.m1.1.1.1.1.1.cmml"><mo id="S4.E5.m1.1.1.1.1.2" stretchy="false" xref="S4.E5.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E5.m1.1.1.1.1.1" xref="S4.E5.m1.1.1.1.1.1.cmml"><mi id="S4.E5.m1.1.1.1.1.1.2" xref="S4.E5.m1.1.1.1.1.1.2.cmml">i</mi><mi id="S4.E5.m1.1.1.1.1.1.3" xref="S4.E5.m1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.E5.m1.1.1.1.1.3" stretchy="false" xref="S4.E5.m1.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S4.E5.m1.4.4.1.1.2.1.1.1.1.4" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.4.cmml">−</mo><mrow id="S4.E5.m1.4.4.1.1.2.1.1.1.1.3" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.3.cmml"><mrow id="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1.cmml"><mi id="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1.3" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1.3.cmml">α</mi><mo id="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1.2" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1.2.cmml">⁢</mo><msup id="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1.4" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1.4.cmml"><mi id="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1.4.2" xref="S4.E5.m1.4.4.1.1.2.1.1.1.1.1.1.4.2.cmml">𝒉</mi><mrow id="S4.E5.m1.2.2.1.1" xref="S4.E5.m1.2.2.1.1.1.cmml"><mo id="S4.E5.m1.2.2.1.1.2" stretchy="false" xref="S4.E5.m1.2.2.1.1.1.cmml">(</mo><msub id="S4.E5.m1.2.2.1.1.1" xref="S4.E5.m1.2.2.1.1.1.cmml"><mi 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encoding="application/x-llamapun" id="S4.E5.m1.4d">roman_ℓ ( italic_α , bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = ∏ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT blackboard_P ( bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT - italic_α bold_italic_h start_POSTSUPERSCRIPT ( italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) | over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT , over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(5)</td> </tr></tbody> </table> where <math alttext="\boldsymbol{x}_{t}=r(\boldsymbol{x}_{0},\phi_{t})" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.4.m1.2"><semantics id="S4.SS1.SSS0.Px2.p1.4.m1.2a"><mrow id="S4.SS1.SSS0.Px2.p1.4.m1.2.2" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.cmml"><msub id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.cmml"><mi id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.2" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.2.cmml">𝒙</mi><mi id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.3" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.3.cmml">t</mi></msub><mo id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.3" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.3.cmml">=</mo><mrow id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.cmml"><mi id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.4" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.4.cmml">r</mi><mo id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.3" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.3.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.3.cmml"><mo id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.3" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.3.cmml">(</mo><msub id="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1" xref="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.2" xref="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.2.cmml">𝒙</mi><mn id="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.3" xref="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.4" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.3.cmml">,</mo><msub id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.cmml"><mi id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.2" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.2.cmml">ϕ</mi><mi id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.3" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.3.cmml">t</mi></msub><mo id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.5" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.4.m1.2b"><apply id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2"><eq id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.3.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.3"></eq><apply id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.1.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.2.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.2">𝒙</ci><ci id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.3.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.4.3">𝑡</ci></apply><apply id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2"><times id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.3.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.3"></times><ci id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.4.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.4">𝑟</ci><interval closure="open" id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.3.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2"><apply id="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.2">𝒙</ci><cn id="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.4.m1.1.1.1.1.1.1.3">0</cn></apply><apply id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.1.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.2">italic-ϕ</ci><ci id="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.3.cmml" xref="S4.SS1.SSS0.Px2.p1.4.m1.2.2.2.2.2.2.3">𝑡</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.4.m1.2c">\boldsymbol{x}_{t}=r(\boldsymbol{x}_{0},\phi_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.4.m1.2d">bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_r ( bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> is the predictive position of the exoplanet at time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.5.m2.1"><semantics id="S4.SS1.SSS0.Px2.p1.5.m2.1a"><mi id="S4.SS1.SSS0.Px2.p1.5.m2.1.1" xref="S4.SS1.SSS0.Px2.p1.5.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.5.m2.1b"><ci id="S4.SS1.SSS0.Px2.p1.5.m2.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.5.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.5.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.5.m2.1d">italic_t</annotation></semantics></math>, <math alttext="i_{t}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.6.m3.1"><semantics id="S4.SS1.SSS0.Px2.p1.6.m3.1a"><msub id="S4.SS1.SSS0.Px2.p1.6.m3.1.1" xref="S4.SS1.SSS0.Px2.p1.6.m3.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.6.m3.1.1.2" xref="S4.SS1.SSS0.Px2.p1.6.m3.1.1.2.cmml">i</mi><mi id="S4.SS1.SSS0.Px2.p1.6.m3.1.1.3" xref="S4.SS1.SSS0.Px2.p1.6.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.6.m3.1b"><apply id="S4.SS1.SSS0.Px2.p1.6.m3.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.6.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.6.m3.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.6.m3.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.6.m3.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.6.m3.1.1.2">𝑖</ci><ci id="S4.SS1.SSS0.Px2.p1.6.m3.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.6.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.6.m3.1c">i_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.6.m3.1d">italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> the index of the patch centered on position <math alttext="\lfloor\boldsymbol{x}_{t}\rceil" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.7.m4.1"><semantics id="S4.SS1.SSS0.Px2.p1.7.m4.1a"><mrow id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.2.cmml"><mo id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.2.1.cmml">⌊</mo><msub id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.2" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.3" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.3" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.2.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.7.m4.1b"><apply id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1"><csymbol cd="latexml" id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.2">delimited-⌊⌉</csymbol><apply id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.2">𝒙</ci><ci id="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.7.m4.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.7.m4.1c">\lfloor\boldsymbol{x}_{t}\rceil</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.7.m4.1d">⌊ bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ⌉</annotation></semantics></math> in <math alttext="\mathcal{G}_{p}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.8.m5.1"><semantics id="S4.SS1.SSS0.Px2.p1.8.m5.1a"><msub id="S4.SS1.SSS0.Px2.p1.8.m5.1.1" xref="S4.SS1.SSS0.Px2.p1.8.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px2.p1.8.m5.1.1.2" xref="S4.SS1.SSS0.Px2.p1.8.m5.1.1.2.cmml">𝒢</mi><mi id="S4.SS1.SSS0.Px2.p1.8.m5.1.1.3" xref="S4.SS1.SSS0.Px2.p1.8.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.8.m5.1b"><apply id="S4.SS1.SSS0.Px2.p1.8.m5.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.8.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.8.m5.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.8.m5.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.8.m5.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.8.m5.1.1.2">𝒢</ci><ci id="S4.SS1.SSS0.Px2.p1.8.m5.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.8.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.8.m5.1c">\mathcal{G}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.8.m5.1d">caligraphic_G start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\boldsymbol{h}^{(i_{t})}(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.9.m6.2"><semantics id="S4.SS1.SSS0.Px2.p1.9.m6.2a"><mrow id="S4.SS1.SSS0.Px2.p1.9.m6.2.2" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.cmml"><msup id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3.cmml"><mi id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3.2" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3.2.cmml">𝒉</mi><mrow id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.cmml"><mo id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.2" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.2.cmml">i</mi><mi id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.3" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.3" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.2" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.cmml"><mo id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.cmml">(</mo><msub id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.2" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.2.cmml">𝒙</mi><mi id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.3" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.3" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.9.m6.2b"><apply id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2"><times id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.2"></times><apply id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3.1.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3">superscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3.2.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.3.2">𝒉</ci><apply id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.2">𝑖</ci><ci id="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.1.1.1.1.1.3">𝑡</ci></apply></apply><apply id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.2">𝒙</ci><ci id="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.9.m6.2.2.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.9.m6.2c">\boldsymbol{h}^{(i_{t})}(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.9.m6.2d">bold_italic_h start_POSTSUPERSCRIPT ( italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> in <math alttext="\mathbb{R}^{p}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.10.m7.1"><semantics id="S4.SS1.SSS0.Px2.p1.10.m7.1a"><msup id="S4.SS1.SSS0.Px2.p1.10.m7.1.1" xref="S4.SS1.SSS0.Px2.p1.10.m7.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.10.m7.1.1.2" xref="S4.SS1.SSS0.Px2.p1.10.m7.1.1.2.cmml">ℝ</mi><mi id="S4.SS1.SSS0.Px2.p1.10.m7.1.1.3" xref="S4.SS1.SSS0.Px2.p1.10.m7.1.1.3.cmml">p</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.10.m7.1b"><apply id="S4.SS1.SSS0.Px2.p1.10.m7.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.10.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.10.m7.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.10.m7.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.10.m7.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.10.m7.1.1.2">ℝ</ci><ci id="S4.SS1.SSS0.Px2.p1.10.m7.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.10.m7.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.10.m7.1c">\mathbb{R}^{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.10.m7.1d">blackboard_R start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> the patch <math alttext="i_{t}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.11.m8.1"><semantics id="S4.SS1.SSS0.Px2.p1.11.m8.1a"><msub id="S4.SS1.SSS0.Px2.p1.11.m8.1.1" xref="S4.SS1.SSS0.Px2.p1.11.m8.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.11.m8.1.1.2" xref="S4.SS1.SSS0.Px2.p1.11.m8.1.1.2.cmml">i</mi><mi id="S4.SS1.SSS0.Px2.p1.11.m8.1.1.3" xref="S4.SS1.SSS0.Px2.p1.11.m8.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.11.m8.1b"><apply id="S4.SS1.SSS0.Px2.p1.11.m8.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.11.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.11.m8.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.11.m8.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.11.m8.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.11.m8.1.1.2">𝑖</ci><ci id="S4.SS1.SSS0.Px2.p1.11.m8.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.11.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.11.m8.1c">i_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.11.m8.1d">italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> of the off-axis PSF model centered on <math alttext="\boldsymbol{x}_{t}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.12.m9.1"><semantics id="S4.SS1.SSS0.Px2.p1.12.m9.1a"><msub id="S4.SS1.SSS0.Px2.p1.12.m9.1.1" xref="S4.SS1.SSS0.Px2.p1.12.m9.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.12.m9.1.1.2" xref="S4.SS1.SSS0.Px2.p1.12.m9.1.1.2.cmml">𝒙</mi><mi id="S4.SS1.SSS0.Px2.p1.12.m9.1.1.3" xref="S4.SS1.SSS0.Px2.p1.12.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.12.m9.1b"><apply id="S4.SS1.SSS0.Px2.p1.12.m9.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.12.m9.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.12.m9.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.12.m9.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.12.m9.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.12.m9.1.1.2">𝒙</ci><ci id="S4.SS1.SSS0.Px2.p1.12.m9.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.12.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.12.m9.1c">\boldsymbol{x}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.12.m9.1d">bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. We propose to consider the modified likelihood: <table class="ltx_equation ltx_eqn_table" id="S4.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ell(\alpha,\boldsymbol{x}_{0})=\prod_{t}\prod_{j\in S(\boldsymbol{x}_{t})}% \mathbb{P}\left(\boldsymbol{y}_{t}^{(j)}-\alpha\boldsymbol{h}^{(j)}(% \boldsymbol{x}_{t})\,|\,\widehat{\boldsymbol{m}}_{j},\widehat{\mathbf{C}}_{j}% \right)^{w_{j}}\,," class="ltx_Math" display="block" id="S4.E6.m1.5"><semantics id="S4.E6.m1.5a"><mrow id="S4.E6.m1.5.5.1" xref="S4.E6.m1.5.5.1.1.cmml"><mrow id="S4.E6.m1.5.5.1.1" xref="S4.E6.m1.5.5.1.1.cmml"><mrow id="S4.E6.m1.5.5.1.1.1" xref="S4.E6.m1.5.5.1.1.1.cmml"><mi id="S4.E6.m1.5.5.1.1.1.3" mathvariant="normal" xref="S4.E6.m1.5.5.1.1.1.3.cmml">ℓ</mi><mo id="S4.E6.m1.5.5.1.1.1.2" xref="S4.E6.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S4.E6.m1.5.5.1.1.1.1.1" 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id="S4.E6.m1.5d">roman_ℓ ( italic_α , bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = ∏ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∏ start_POSTSUBSCRIPT italic_j ∈ italic_S ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT blackboard_P ( bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT - italic_α bold_italic_h start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) | over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(6)</td> </tr></tbody> </table> where <math alttext="S(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.13.m1.1"><semantics id="S4.SS1.SSS0.Px2.p1.13.m1.1a"><mrow id="S4.SS1.SSS0.Px2.p1.13.m1.1.1" xref="S4.SS1.SSS0.Px2.p1.13.m1.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.13.m1.1.1.3" xref="S4.SS1.SSS0.Px2.p1.13.m1.1.1.3.cmml">S</mi><mo id="S4.SS1.SSS0.Px2.p1.13.m1.1.1.2" xref="S4.SS1.SSS0.Px2.p1.13.m1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1" xref="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.1.cmml"><mo id="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.1" xref="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.1.2" xref="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.1.3" xref="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS1.SSS0.Px2.p1.13.m1.1.1.1.1.3" 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id="S4.SS1.SSS0.Px2.p1.15.m3.1d">italic_j</annotation></semantics></math> weighted by <math alttext="w_{j}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.16.m4.1"><semantics id="S4.SS1.SSS0.Px2.p1.16.m4.1a"><msub id="S4.SS1.SSS0.Px2.p1.16.m4.1.1" xref="S4.SS1.SSS0.Px2.p1.16.m4.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.16.m4.1.1.2" xref="S4.SS1.SSS0.Px2.p1.16.m4.1.1.2.cmml">w</mi><mi id="S4.SS1.SSS0.Px2.p1.16.m4.1.1.3" xref="S4.SS1.SSS0.Px2.p1.16.m4.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.16.m4.1b"><apply id="S4.SS1.SSS0.Px2.p1.16.m4.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.16.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.16.m4.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.16.m4.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.16.m4.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.16.m4.1.1.2">𝑤</ci><ci id="S4.SS1.SSS0.Px2.p1.16.m4.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.16.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.16.m4.1c">w_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.16.m4.1d">italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.17.m5.1"><semantics id="S4.SS1.SSS0.Px2.p1.17.m5.1a"><msup id="S4.SS1.SSS0.Px2.p1.17.m5.1.1" xref="S4.SS1.SSS0.Px2.p1.17.m5.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.17.m5.1.1.2" xref="S4.SS1.SSS0.Px2.p1.17.m5.1.1.2.cmml">ℝ</mi><mo id="S4.SS1.SSS0.Px2.p1.17.m5.1.1.3" xref="S4.SS1.SSS0.Px2.p1.17.m5.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.17.m5.1b"><apply id="S4.SS1.SSS0.Px2.p1.17.m5.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.17.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.17.m5.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.17.m5.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.17.m5.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.17.m5.1.1.2">ℝ</ci><plus id="S4.SS1.SSS0.Px2.p1.17.m5.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.17.m5.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.17.m5.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.17.m5.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\sum w_{j}=1" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.18.m6.1"><semantics id="S4.SS1.SSS0.Px2.p1.18.m6.1a"><mrow id="S4.SS1.SSS0.Px2.p1.18.m6.1.1" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.cmml"><mrow id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.cmml"><mo id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.1" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.1.cmml">∑</mo><msub id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.cmml"><mi id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.2" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.2.cmml">w</mi><mi id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.3" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.3.cmml">j</mi></msub></mrow><mo id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.1" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.1.cmml">=</mo><mn id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.3" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.18.m6.1b"><apply id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1"><eq id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.1"></eq><apply id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2"><sum id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.1"></sum><apply id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.1.cmml" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.2">𝑤</ci><ci id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.3.cmml" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.2.2.3">𝑗</ci></apply></apply><cn id="S4.SS1.SSS0.Px2.p1.18.m6.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.18.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.18.m6.1c">\sum w_{j}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.18.m6.1d">∑ italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>. Compared to (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.E5" title="Equation 5 ‣ Detection criterion at a given position. ‣ 4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">5</a>), this approach is more robust, as it aggregates overlapping patch contributions at each time-step, modeling the nuisance component as a convolutional process influenced by multiple, overlapping noise sources. It is also more computationally efficient, as it does not require a dense grid <math alttext="\mathcal{G}_{d}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.19.m7.1"><semantics id="S4.SS1.SSS0.Px2.p1.19.m7.1a"><msub id="S4.SS1.SSS0.Px2.p1.19.m7.1.1" xref="S4.SS1.SSS0.Px2.p1.19.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px2.p1.19.m7.1.1.2" xref="S4.SS1.SSS0.Px2.p1.19.m7.1.1.2.cmml">𝒢</mi><mi id="S4.SS1.SSS0.Px2.p1.19.m7.1.1.3" xref="S4.SS1.SSS0.Px2.p1.19.m7.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.19.m7.1b"><apply id="S4.SS1.SSS0.Px2.p1.19.m7.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.19.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.19.m7.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.19.m7.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.19.m7.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.19.m7.1.1.2">𝒢</ci><ci id="S4.SS1.SSS0.Px2.p1.19.m7.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.19.m7.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.19.m7.1c">\mathcal{G}_{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.19.m7.1d">caligraphic_G start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> for patch collections, thereby reducing the parameter estimation load. This efficiency is essential for incorporating the model into an end-to-end learning framework (see Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS3" title="4.3 End-to-end trainable approach ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.3</a>). Finally, this flexible formulation can accommodate additional correlation types (see Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS2" title="4.2 Extensions of the statistical model: mixture of distributions ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.2</a>). Solving problem (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.E6" title="Equation 6 ‣ Detection criterion at a given position. ‣ 4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">6</a>) leads to the flux estimator <math alttext="\widehat{\alpha}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.20.m8.1"><semantics id="S4.SS1.SSS0.Px2.p1.20.m8.1a"><mover accent="true" id="S4.SS1.SSS0.Px2.p1.20.m8.1.1" xref="S4.SS1.SSS0.Px2.p1.20.m8.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.20.m8.1.1.2" xref="S4.SS1.SSS0.Px2.p1.20.m8.1.1.2.cmml">α</mi><mo id="S4.SS1.SSS0.Px2.p1.20.m8.1.1.1" xref="S4.SS1.SSS0.Px2.p1.20.m8.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.20.m8.1b"><apply id="S4.SS1.SSS0.Px2.p1.20.m8.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.20.m8.1.1"><ci id="S4.SS1.SSS0.Px2.p1.20.m8.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.20.m8.1.1.1">^</ci><ci id="S4.SS1.SSS0.Px2.p1.20.m8.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.20.m8.1.1.2">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.20.m8.1c">\widehat{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.20.m8.1d">over^ start_ARG italic_α end_ARG</annotation></semantics></math> and its standard deviation <math alttext="\widehat{\sigma}_{\alpha}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.21.m9.1"><semantics id="S4.SS1.SSS0.Px2.p1.21.m9.1a"><msub id="S4.SS1.SSS0.Px2.p1.21.m9.1.1" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1.cmml"><mover accent="true" id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.cmml"><mi id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.2" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.2.cmml">σ</mi><mo id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.1" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.1.cmml">^</mo></mover><mi id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.3" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.21.m9.1b"><apply id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1">subscript</csymbol><apply id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2"><ci id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.1">^</ci><ci id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1.2.2">𝜎</ci></apply><ci id="S4.SS1.SSS0.Px2.p1.21.m9.1.1.3.cmml" xref="S4.SS1.SSS0.Px2.p1.21.m9.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.21.m9.1c">\widehat{\sigma}_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.21.m9.1d">over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>: <table class="ltx_equation ltx_eqn_table" id="S4.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widehat{\alpha}=\frac{\sum_{t}b_{t}(\boldsymbol{x}_{t})}{\sum_{t}a_{t}(% \boldsymbol{x}_{t})}\,,\quad\widehat{\sigma}_{\alpha}=\frac{1}{\sqrt{\sum_{t}a% (\boldsymbol{x}_{t})}}\,," class="ltx_Math" display="block" id="S4.E7.m1.4"><semantics id="S4.E7.m1.4a"><mrow id="S4.E7.m1.4.4.1"><mrow id="S4.E7.m1.4.4.1.1.2" xref="S4.E7.m1.4.4.1.1.3.cmml"><mrow id="S4.E7.m1.4.4.1.1.1.1" xref="S4.E7.m1.4.4.1.1.1.1.cmml"><mover accent="true" id="S4.E7.m1.4.4.1.1.1.1.2" xref="S4.E7.m1.4.4.1.1.1.1.2.cmml"><mi id="S4.E7.m1.4.4.1.1.1.1.2.2" xref="S4.E7.m1.4.4.1.1.1.1.2.2.cmml">α</mi><mo id="S4.E7.m1.4.4.1.1.1.1.2.1" xref="S4.E7.m1.4.4.1.1.1.1.2.1.cmml">^</mo></mover><mo id="S4.E7.m1.4.4.1.1.1.1.1" 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xref="S4.E7.m1.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E7.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.3.3.1.1.1.1.1.1.1.2">𝒙</ci><ci id="S4.E7.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S4.E7.m1.3.3.1.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.4c">\widehat{\alpha}=\frac{\sum_{t}b_{t}(\boldsymbol{x}_{t})}{\sum_{t}a_{t}(% \boldsymbol{x}_{t})}\,,\quad\widehat{\sigma}_{\alpha}=\frac{1}{\sqrt{\sum_{t}a% (\boldsymbol{x}_{t})}}\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.4d">over^ start_ARG italic_α end_ARG = divide start_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_ARG , over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_a ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_ARG end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(7)</td> </tr></tbody> </table> where <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx1"> <tbody id="S4.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle b_{t}(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="S4.E8.m1.1"><semantics id="S4.E8.m1.1a"><mrow id="S4.E8.m1.1.1" xref="S4.E8.m1.1.1.cmml"><msub id="S4.E8.m1.1.1.3" xref="S4.E8.m1.1.1.3.cmml"><mi id="S4.E8.m1.1.1.3.2" xref="S4.E8.m1.1.1.3.2.cmml">b</mi><mi id="S4.E8.m1.1.1.3.3" xref="S4.E8.m1.1.1.3.3.cmml">t</mi></msub><mo id="S4.E8.m1.1.1.2" xref="S4.E8.m1.1.1.2.cmml">⁢</mo><mrow id="S4.E8.m1.1.1.1.1" xref="S4.E8.m1.1.1.1.1.1.cmml"><mo id="S4.E8.m1.1.1.1.1.2" stretchy="false" xref="S4.E8.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E8.m1.1.1.1.1.1" xref="S4.E8.m1.1.1.1.1.1.cmml"><mi id="S4.E8.m1.1.1.1.1.1.2" xref="S4.E8.m1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.E8.m1.1.1.1.1.1.3" xref="S4.E8.m1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.E8.m1.1.1.1.1.3" stretchy="false" xref="S4.E8.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E8.m1.1b"><apply id="S4.E8.m1.1.1.cmml" xref="S4.E8.m1.1.1"><times id="S4.E8.m1.1.1.2.cmml" xref="S4.E8.m1.1.1.2"></times><apply id="S4.E8.m1.1.1.3.cmml" xref="S4.E8.m1.1.1.3"><csymbol cd="ambiguous" id="S4.E8.m1.1.1.3.1.cmml" xref="S4.E8.m1.1.1.3">subscript</csymbol><ci id="S4.E8.m1.1.1.3.2.cmml" xref="S4.E8.m1.1.1.3.2">𝑏</ci><ci id="S4.E8.m1.1.1.3.3.cmml" xref="S4.E8.m1.1.1.3.3">𝑡</ci></apply><apply id="S4.E8.m1.1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E8.m1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1">subscript</csymbol><ci id="S4.E8.m1.1.1.1.1.1.2.cmml" xref="S4.E8.m1.1.1.1.1.1.2">𝒙</ci><ci id="S4.E8.m1.1.1.1.1.1.3.cmml" xref="S4.E8.m1.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.1c">\displaystyle b_{t}(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.1d">italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{j\in S(\boldsymbol{x}_{t})}w_{j}\,\boldsymbol{h}^{(j)}(% 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( italic_j ) end_POSTSUPERSCRIPT - over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(8)</td> </tr></tbody> <tbody id="S4.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle a(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="S4.E9.m1.1"><semantics id="S4.E9.m1.1a"><mrow id="S4.E9.m1.1.1" xref="S4.E9.m1.1.1.cmml"><mi id="S4.E9.m1.1.1.3" xref="S4.E9.m1.1.1.3.cmml">a</mi><mo id="S4.E9.m1.1.1.2" xref="S4.E9.m1.1.1.2.cmml">⁢</mo><mrow id="S4.E9.m1.1.1.1.1" xref="S4.E9.m1.1.1.1.1.1.cmml"><mo id="S4.E9.m1.1.1.1.1.2" stretchy="false" xref="S4.E9.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.E9.m1.1.1.1.1.1" xref="S4.E9.m1.1.1.1.1.1.cmml"><mi id="S4.E9.m1.1.1.1.1.1.2" xref="S4.E9.m1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.E9.m1.1.1.1.1.1.3" xref="S4.E9.m1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.E9.m1.1.1.1.1.3" stretchy="false" xref="S4.E9.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E9.m1.1b"><apply id="S4.E9.m1.1.1.cmml" xref="S4.E9.m1.1.1"><times id="S4.E9.m1.1.1.2.cmml" xref="S4.E9.m1.1.1.2"></times><ci id="S4.E9.m1.1.1.3.cmml" xref="S4.E9.m1.1.1.3">𝑎</ci><apply id="S4.E9.m1.1.1.1.1.1.cmml" xref="S4.E9.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E9.m1.1.1.1.1.1.1.cmml" xref="S4.E9.m1.1.1.1.1">subscript</csymbol><ci id="S4.E9.m1.1.1.1.1.1.2.cmml" xref="S4.E9.m1.1.1.1.1.1.2">𝒙</ci><ci id="S4.E9.m1.1.1.1.1.1.3.cmml" xref="S4.E9.m1.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E9.m1.1c">\displaystyle a(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.E9.m1.1d">italic_a ( bold_italic_x 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xref="S4.E9.m2.4.4.1.1.2.2.2.1.1.2">𝒙</ci><ci id="S4.E9.m2.4.4.1.1.2.2.2.1.1.3.cmml" xref="S4.E9.m2.4.4.1.1.2.2.2.1.1.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E9.m2.4c">\displaystyle=\sum_{j\in S(\boldsymbol{x}_{t})}w_{j}\,\boldsymbol{h}^{(j)}(% \boldsymbol{x}_{t})^{\top}\,\widehat{\mathbf{C}}_{j}^{-1}\,\boldsymbol{h}^{(j)% }(\boldsymbol{x}_{t})\,.</annotation><annotation encoding="application/x-llamapun" id="S4.E9.m2.4d">= ∑ start_POSTSUBSCRIPT italic_j ∈ italic_S ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_italic_h start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT bold_italic_h start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(9)</td> </tr></tbody> </table> To assess the probability of an exoplanet at <math alttext="\boldsymbol{x}_{0}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.22.m1.1"><semantics id="S4.SS1.SSS0.Px2.p1.22.m1.1a"><msub id="S4.SS1.SSS0.Px2.p1.22.m1.1.1" xref="S4.SS1.SSS0.Px2.p1.22.m1.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.22.m1.1.1.2" xref="S4.SS1.SSS0.Px2.p1.22.m1.1.1.2.cmml">𝒙</mi><mn id="S4.SS1.SSS0.Px2.p1.22.m1.1.1.3" xref="S4.SS1.SSS0.Px2.p1.22.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.22.m1.1b"><apply id="S4.SS1.SSS0.Px2.p1.22.m1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.22.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.22.m1.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.22.m1.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.22.m1.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.22.m1.1.1.2">𝒙</ci><cn id="S4.SS1.SSS0.Px2.p1.22.m1.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.22.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.22.m1.1c">\boldsymbol{x}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.22.m1.1d">bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, we use the generalized likelihood ratio test (GLRT) to statistically evaluate the parameter <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.23.m2.1"><semantics id="S4.SS1.SSS0.Px2.p1.23.m2.1a"><mi id="S4.SS1.SSS0.Px2.p1.23.m2.1.1" xref="S4.SS1.SSS0.Px2.p1.23.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.23.m2.1b"><ci id="S4.SS1.SSS0.Px2.p1.23.m2.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.23.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.23.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.23.m2.1d">italic_α</annotation></semantics></math>. Under the null hypothesis <math alttext="\mathcal{H}_{0}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.24.m3.1"><semantics id="S4.SS1.SSS0.Px2.p1.24.m3.1a"><msub id="S4.SS1.SSS0.Px2.p1.24.m3.1.1" xref="S4.SS1.SSS0.Px2.p1.24.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px2.p1.24.m3.1.1.2" xref="S4.SS1.SSS0.Px2.p1.24.m3.1.1.2.cmml">ℋ</mi><mn id="S4.SS1.SSS0.Px2.p1.24.m3.1.1.3" xref="S4.SS1.SSS0.Px2.p1.24.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.24.m3.1b"><apply id="S4.SS1.SSS0.Px2.p1.24.m3.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.24.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.24.m3.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.24.m3.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.24.m3.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.24.m3.1.1.2">ℋ</ci><cn id="S4.SS1.SSS0.Px2.p1.24.m3.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.24.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.24.m3.1c">\mathcal{H}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.24.m3.1d">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="\alpha=0" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.25.m4.1"><semantics id="S4.SS1.SSS0.Px2.p1.25.m4.1a"><mrow id="S4.SS1.SSS0.Px2.p1.25.m4.1.1" xref="S4.SS1.SSS0.Px2.p1.25.m4.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.25.m4.1.1.2" xref="S4.SS1.SSS0.Px2.p1.25.m4.1.1.2.cmml">α</mi><mo id="S4.SS1.SSS0.Px2.p1.25.m4.1.1.1" xref="S4.SS1.SSS0.Px2.p1.25.m4.1.1.1.cmml">=</mo><mn id="S4.SS1.SSS0.Px2.p1.25.m4.1.1.3" xref="S4.SS1.SSS0.Px2.p1.25.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.25.m4.1b"><apply id="S4.SS1.SSS0.Px2.p1.25.m4.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.25.m4.1.1"><eq id="S4.SS1.SSS0.Px2.p1.25.m4.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.25.m4.1.1.1"></eq><ci id="S4.SS1.SSS0.Px2.p1.25.m4.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.25.m4.1.1.2">𝛼</ci><cn id="S4.SS1.SSS0.Px2.p1.25.m4.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.25.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.25.m4.1c">\alpha=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.25.m4.1d">italic_α = 0</annotation></semantics></math> (indicating no exoplanet), the statistics <table class="ltx_equation ltx_eqn_table" id="S4.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widehat{\gamma}=\widehat{\alpha}\big{/}{\widehat{\sigma}_{\alpha}}=\Big{(}% \sum\nolimits_{t}b_{t}(\boldsymbol{x}_{t})\Big{)}\Big{/}\Big{(}\sqrt{\sum% \nolimits_{t}a(\boldsymbol{x}_{t})}\Big{)}\,," class="ltx_Math" display="block" id="S4.E10.m1.2"><semantics id="S4.E10.m1.2a"><mrow 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id="S4.E10.m1.2.2.1.1.5.3.2.2" xref="S4.E10.m1.2.2.1.1.5.3.2.2.cmml">σ</mi><mo id="S4.E10.m1.2.2.1.1.5.3.2.1" xref="S4.E10.m1.2.2.1.1.5.3.2.1.cmml">^</mo></mover><mi id="S4.E10.m1.2.2.1.1.5.3.3" xref="S4.E10.m1.2.2.1.1.5.3.3.cmml">α</mi></msub></mrow><mo id="S4.E10.m1.2.2.1.1.6" xref="S4.E10.m1.2.2.1.1.6.cmml">=</mo><mrow id="S4.E10.m1.2.2.1.1.1" xref="S4.E10.m1.2.2.1.1.1.cmml"><mrow id="S4.E10.m1.2.2.1.1.1.1.1" xref="S4.E10.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S4.E10.m1.2.2.1.1.1.1.1.2" maxsize="160%" minsize="160%" xref="S4.E10.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E10.m1.2.2.1.1.1.1.1.1" xref="S4.E10.m1.2.2.1.1.1.1.1.1.cmml"><msub id="S4.E10.m1.2.2.1.1.1.1.1.1.2" xref="S4.E10.m1.2.2.1.1.1.1.1.1.2.cmml"><mo id="S4.E10.m1.2.2.1.1.1.1.1.1.2.2" lspace="0em" xref="S4.E10.m1.2.2.1.1.1.1.1.1.2.2.cmml">∑</mo><mi id="S4.E10.m1.2.2.1.1.1.1.1.1.2.3" xref="S4.E10.m1.2.2.1.1.1.1.1.1.2.3.cmml">t</mi></msub><mrow id="S4.E10.m1.2.2.1.1.1.1.1.1.1" xref="S4.E10.m1.2.2.1.1.1.1.1.1.1.cmml"><msub 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xref="S4.E10.m1.1.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E10.m1.2c">\widehat{\gamma}=\widehat{\alpha}\big{/}{\widehat{\sigma}_{\alpha}}=\Big{(}% \sum\nolimits_{t}b_{t}(\boldsymbol{x}_{t})\Big{)}\Big{/}\Big{(}\sqrt{\sum% \nolimits_{t}a(\boldsymbol{x}_{t})}\Big{)}\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E10.m1.2d">over^ start_ARG italic_γ end_ARG = over^ start_ARG italic_α end_ARG / over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = ( ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ) / ( square-root start_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_a ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(10)</td> </tr></tbody> </table> is controlled and follows a Gaussian distribution <math alttext="\mathcal{N}(0,1)" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.26.m1.2"><semantics id="S4.SS1.SSS0.Px2.p1.26.m1.2a"><mrow id="S4.SS1.SSS0.Px2.p1.26.m1.2.3" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.2" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.2.cmml">𝒩</mi><mo id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.1" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.2" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.1.cmml"><mo id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.2.1" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.1.cmml">(</mo><mn id="S4.SS1.SSS0.Px2.p1.26.m1.1.1" xref="S4.SS1.SSS0.Px2.p1.26.m1.1.1.cmml">0</mn><mo id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.2.2" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.1.cmml">,</mo><mn id="S4.SS1.SSS0.Px2.p1.26.m1.2.2" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.2.cmml">1</mn><mo id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.2.3" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.26.m1.2b"><apply id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.cmml" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3"><times id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.1.cmml" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.1"></times><ci id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.2.cmml" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.2">𝒩</ci><interval closure="open" id="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.1.cmml" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.3.3.2"><cn id="S4.SS1.SSS0.Px2.p1.26.m1.1.1.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.26.m1.1.1">0</cn><cn id="S4.SS1.SSS0.Px2.p1.26.m1.2.2.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.26.m1.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.26.m1.2c">\mathcal{N}(0,1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.26.m1.2d">caligraphic_N ( 0 , 1 )</annotation></semantics></math>. When evaluating the probability of presence of an exoplanet, we test against the alternative hypothesis <math alttext="\mathcal{H}_{1}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.27.m2.1"><semantics id="S4.SS1.SSS0.Px2.p1.27.m2.1a"><msub id="S4.SS1.SSS0.Px2.p1.27.m2.1.1" xref="S4.SS1.SSS0.Px2.p1.27.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px2.p1.27.m2.1.1.2" xref="S4.SS1.SSS0.Px2.p1.27.m2.1.1.2.cmml">ℋ</mi><mn id="S4.SS1.SSS0.Px2.p1.27.m2.1.1.3" xref="S4.SS1.SSS0.Px2.p1.27.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.27.m2.1b"><apply id="S4.SS1.SSS0.Px2.p1.27.m2.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.27.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.27.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.27.m2.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.27.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.27.m2.1.1.2">ℋ</ci><cn id="S4.SS1.SSS0.Px2.p1.27.m2.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.27.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.27.m2.1c">\mathcal{H}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.27.m2.1d">caligraphic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>: <math alttext="\alpha>0" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.28.m3.1"><semantics id="S4.SS1.SSS0.Px2.p1.28.m3.1a"><mrow id="S4.SS1.SSS0.Px2.p1.28.m3.1.1" xref="S4.SS1.SSS0.Px2.p1.28.m3.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.28.m3.1.1.2" xref="S4.SS1.SSS0.Px2.p1.28.m3.1.1.2.cmml">α</mi><mo id="S4.SS1.SSS0.Px2.p1.28.m3.1.1.1" xref="S4.SS1.SSS0.Px2.p1.28.m3.1.1.1.cmml">></mo><mn id="S4.SS1.SSS0.Px2.p1.28.m3.1.1.3" xref="S4.SS1.SSS0.Px2.p1.28.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.28.m3.1b"><apply id="S4.SS1.SSS0.Px2.p1.28.m3.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.28.m3.1.1"><gt id="S4.SS1.SSS0.Px2.p1.28.m3.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.28.m3.1.1.1"></gt><ci id="S4.SS1.SSS0.Px2.p1.28.m3.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.28.m3.1.1.2">𝛼</ci><cn id="S4.SS1.SSS0.Px2.p1.28.m3.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.28.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.28.m3.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.28.m3.1d">italic_α > 0</annotation></semantics></math>. The statistics <math alttext="\widehat{\gamma}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.29.m4.1"><semantics id="S4.SS1.SSS0.Px2.p1.29.m4.1a"><mover accent="true" id="S4.SS1.SSS0.Px2.p1.29.m4.1.1" xref="S4.SS1.SSS0.Px2.p1.29.m4.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.29.m4.1.1.2" xref="S4.SS1.SSS0.Px2.p1.29.m4.1.1.2.cmml">γ</mi><mo id="S4.SS1.SSS0.Px2.p1.29.m4.1.1.1" xref="S4.SS1.SSS0.Px2.p1.29.m4.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.29.m4.1b"><apply id="S4.SS1.SSS0.Px2.p1.29.m4.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.29.m4.1.1"><ci id="S4.SS1.SSS0.Px2.p1.29.m4.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.29.m4.1.1.1">^</ci><ci id="S4.SS1.SSS0.Px2.p1.29.m4.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.29.m4.1.1.2">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.29.m4.1c">\widehat{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.29.m4.1d">over^ start_ARG italic_γ end_ARG</annotation></semantics></math> can be directly mapped to a probability of detection. It can also be interpreted as the output signal-to-noise ratio through the statistical model, and in practice, this is the detection score used by astronomers <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib63" title="">63</a>]</cite>. 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xref="S4.SS1.SSS0.Px2.p1.31.m6.1.1.3"><times id="S4.SS1.SSS0.Px2.p1.31.m6.1.1.3.1.cmml" xref="S4.SS1.SSS0.Px2.p1.31.m6.1.1.3.1"></times><ci id="S4.SS1.SSS0.Px2.p1.31.m6.1.1.3.2.cmml" xref="S4.SS1.SSS0.Px2.p1.31.m6.1.1.3.2">𝐻</ci><ci id="S4.SS1.SSS0.Px2.p1.31.m6.1.1.3.3.cmml" xref="S4.SS1.SSS0.Px2.p1.31.m6.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.31.m6.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.31.m6.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math>, we adopt a fast approximation similar to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib25" title="">25</a>]</cite> detailed in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS2" title="A.2 Dense detection map computation details ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.2</a>. </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Iterative procedure for characterization.</h4> <div class="ltx_para ltx_noindent" id="S4.SS1.SSS0.Px3.p1"> Astrometry is the task of precisely estimating the flux <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.1.m1.1"><semantics id="S4.SS1.SSS0.Px3.p1.1.m1.1a"><mi id="S4.SS1.SSS0.Px3.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px3.p1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.1.m1.1b"><ci id="S4.SS1.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.1.m1.1d">italic_α</annotation></semantics></math> and the sub-pixel position <math alttext="\boldsymbol{x}_{0}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.2.m2.1"><semantics id="S4.SS1.SSS0.Px3.p1.2.m2.1a"><msub id="S4.SS1.SSS0.Px3.p1.2.m2.1.1" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.cmml"><mi id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.2" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.2.cmml">𝒙</mi><mn id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.3" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.2.m2.1b"><apply id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.2">𝒙</ci><cn id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.2.m2.1c">\boldsymbol{x}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.2.m2.1d">bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> of an exoplanet. This can be achieved by jointly optimizing the likelihood defined in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.E6" title="Equation 6 ‣ Detection criterion at a given position. ‣ 4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">6</a>). Denoting by <math alttext="\boldsymbol{z}=[\alpha,\boldsymbol{x}_{0}]" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.3.m3.2"><semantics id="S4.SS1.SSS0.Px3.p1.3.m3.2a"><mrow id="S4.SS1.SSS0.Px3.p1.3.m3.2.2" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.cmml"><mi id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.3" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.3.cmml">𝒛</mi><mo id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.2" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.2.cmml">=</mo><mrow id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.2.cmml"><mo id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.2.cmml">[</mo><mi id="S4.SS1.SSS0.Px3.p1.3.m3.1.1" xref="S4.SS1.SSS0.Px3.p1.3.m3.1.1.cmml">α</mi><mo id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.3" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.2.cmml">,</mo><msub id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.1" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.1.2" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.1.2.cmml">𝒙</mi><mn 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id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.1.2">𝒙</ci><cn id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.1.1.3">0</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.3.m3.2c">\boldsymbol{z}=[\alpha,\boldsymbol{x}_{0}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.3.m3.2d">bold_italic_z = [ italic_α , bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math> in <math alttext="\mathbb{R}^{3}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.4.m4.1"><semantics id="S4.SS1.SSS0.Px3.p1.4.m4.1a"><msup id="S4.SS1.SSS0.Px3.p1.4.m4.1.1" xref="S4.SS1.SSS0.Px3.p1.4.m4.1.1.cmml"><mi id="S4.SS1.SSS0.Px3.p1.4.m4.1.1.2" xref="S4.SS1.SSS0.Px3.p1.4.m4.1.1.2.cmml">ℝ</mi><mn id="S4.SS1.SSS0.Px3.p1.4.m4.1.1.3" xref="S4.SS1.SSS0.Px3.p1.4.m4.1.1.3.cmml">3</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.4.m4.1b"><apply id="S4.SS1.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px3.p1.4.m4.1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.4.m4.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px3.p1.4.m4.1.1.2.cmml" xref="S4.SS1.SSS0.Px3.p1.4.m4.1.1.2">ℝ</ci><cn id="S4.SS1.SSS0.Px3.p1.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px3.p1.4.m4.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.4.m4.1c">\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.4.m4.1d">blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>, the gradient <math alttext="\boldsymbol{g}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.5.m5.1"><semantics id="S4.SS1.SSS0.Px3.p1.5.m5.1a"><mi id="S4.SS1.SSS0.Px3.p1.5.m5.1.1" xref="S4.SS1.SSS0.Px3.p1.5.m5.1.1.cmml">𝒈</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.5.m5.1b"><ci id="S4.SS1.SSS0.Px3.p1.5.m5.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.5.m5.1.1">𝒈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.5.m5.1c">\boldsymbol{g}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.5.m5.1d">bold_italic_g</annotation></semantics></math> and Hessian matrix <math alttext="\mathbf{H}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.6.m6.1"><semantics id="S4.SS1.SSS0.Px3.p1.6.m6.1a"><mi id="S4.SS1.SSS0.Px3.p1.6.m6.1.1" xref="S4.SS1.SSS0.Px3.p1.6.m6.1.1.cmml">𝐇</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.6.m6.1b"><ci id="S4.SS1.SSS0.Px3.p1.6.m6.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.6.m6.1.1">𝐇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.6.m6.1c">\mathbf{H}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.6.m6.1d">bold_H</annotation></semantics></math> of the neg-log-likelihood are decomposed as follows: <table class="ltx_equation ltx_eqn_table" id="S4.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\boldsymbol{g}(\boldsymbol{z})=\sum_{t}\!\sum_{j\in S(\boldsymbol{x}_{t})}\!w_% {j}\boldsymbol{g}_{j}(\boldsymbol{z}),\,\mathbf{H}(\boldsymbol{z})=\sum_{t}\!% \sum_{j\in S(\boldsymbol{x}_{t})}\!w_{j}\mathbf{H}_{j}(\boldsymbol{z})" class="ltx_Math" display="block" id="S4.Ex1.m1.8"><semantics id="S4.Ex1.m1.8a"><mrow id="S4.Ex1.m1.8.8.2" xref="S4.Ex1.m1.8.8.3.cmml"><mrow id="S4.Ex1.m1.7.7.1.1" xref="S4.Ex1.m1.7.7.1.1.cmml"><mrow id="S4.Ex1.m1.7.7.1.1.2" xref="S4.Ex1.m1.7.7.1.1.2.cmml"><mi id="S4.Ex1.m1.7.7.1.1.2.2" xref="S4.Ex1.m1.7.7.1.1.2.2.cmml">𝒈</mi><mo id="S4.Ex1.m1.7.7.1.1.2.1" xref="S4.Ex1.m1.7.7.1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex1.m1.7.7.1.1.2.3.2" xref="S4.Ex1.m1.7.7.1.1.2.cmml"><mo id="S4.Ex1.m1.7.7.1.1.2.3.2.1" stretchy="false" xref="S4.Ex1.m1.7.7.1.1.2.cmml">(</mo><mi id="S4.Ex1.m1.3.3" xref="S4.Ex1.m1.3.3.cmml">𝒛</mi><mo id="S4.Ex1.m1.7.7.1.1.2.3.2.2" stretchy="false" xref="S4.Ex1.m1.7.7.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex1.m1.7.7.1.1.1" rspace="0.111em" xref="S4.Ex1.m1.7.7.1.1.1.cmml">=</mo><mrow id="S4.Ex1.m1.7.7.1.1.3" xref="S4.Ex1.m1.7.7.1.1.3.cmml"><munder id="S4.Ex1.m1.7.7.1.1.3.1" xref="S4.Ex1.m1.7.7.1.1.3.1.cmml"><mo id="S4.Ex1.m1.7.7.1.1.3.1.2" movablelimits="false" rspace="0em" xref="S4.Ex1.m1.7.7.1.1.3.1.2.cmml">∑</mo><mi id="S4.Ex1.m1.7.7.1.1.3.1.3" xref="S4.Ex1.m1.7.7.1.1.3.1.3.cmml">t</mi></munder><mrow id="S4.Ex1.m1.7.7.1.1.3.2" xref="S4.Ex1.m1.7.7.1.1.3.2.cmml"><munder id="S4.Ex1.m1.7.7.1.1.3.2.1" xref="S4.Ex1.m1.7.7.1.1.3.2.1.cmml"><mo id="S4.Ex1.m1.7.7.1.1.3.2.1.2" movablelimits="false" xref="S4.Ex1.m1.7.7.1.1.3.2.1.2.cmml">∑</mo><mrow id="S4.Ex1.m1.1.1.1" xref="S4.Ex1.m1.1.1.1.cmml"><mi id="S4.Ex1.m1.1.1.1.3" xref="S4.Ex1.m1.1.1.1.3.cmml">j</mi><mo id="S4.Ex1.m1.1.1.1.2" xref="S4.Ex1.m1.1.1.1.2.cmml">∈</mo><mrow id="S4.Ex1.m1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.cmml"><mi id="S4.Ex1.m1.1.1.1.1.3" xref="S4.Ex1.m1.1.1.1.1.3.cmml">S</mi><mo id="S4.Ex1.m1.1.1.1.1.2" xref="S4.Ex1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex1.m1.1.1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex1.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex1.m1.1.1.1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.Ex1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S4.Ex1.m1.7.7.1.1.3.2.2" xref="S4.Ex1.m1.7.7.1.1.3.2.2.cmml"><msub id="S4.Ex1.m1.7.7.1.1.3.2.2.2" xref="S4.Ex1.m1.7.7.1.1.3.2.2.2.cmml"><mi id="S4.Ex1.m1.7.7.1.1.3.2.2.2.2" xref="S4.Ex1.m1.7.7.1.1.3.2.2.2.2.cmml">w</mi><mi id="S4.Ex1.m1.7.7.1.1.3.2.2.2.3" xref="S4.Ex1.m1.7.7.1.1.3.2.2.2.3.cmml">j</mi></msub><mo id="S4.Ex1.m1.7.7.1.1.3.2.2.1" xref="S4.Ex1.m1.7.7.1.1.3.2.2.1.cmml">⁢</mo><msub id="S4.Ex1.m1.7.7.1.1.3.2.2.3" xref="S4.Ex1.m1.7.7.1.1.3.2.2.3.cmml"><mi id="S4.Ex1.m1.7.7.1.1.3.2.2.3.2" xref="S4.Ex1.m1.7.7.1.1.3.2.2.3.2.cmml">𝒈</mi><mi id="S4.Ex1.m1.7.7.1.1.3.2.2.3.3" xref="S4.Ex1.m1.7.7.1.1.3.2.2.3.3.cmml">j</mi></msub><mo id="S4.Ex1.m1.7.7.1.1.3.2.2.1a" xref="S4.Ex1.m1.7.7.1.1.3.2.2.1.cmml">⁢</mo><mrow id="S4.Ex1.m1.7.7.1.1.3.2.2.4.2" xref="S4.Ex1.m1.7.7.1.1.3.2.2.cmml"><mo id="S4.Ex1.m1.7.7.1.1.3.2.2.4.2.1" stretchy="false" xref="S4.Ex1.m1.7.7.1.1.3.2.2.cmml">(</mo><mi id="S4.Ex1.m1.4.4" xref="S4.Ex1.m1.4.4.cmml">𝒛</mi><mo id="S4.Ex1.m1.7.7.1.1.3.2.2.4.2.2" stretchy="false" xref="S4.Ex1.m1.7.7.1.1.3.2.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><mo id="S4.Ex1.m1.8.8.2.3" rspace="0.337em" xref="S4.Ex1.m1.8.8.3a.cmml">,</mo><mrow id="S4.Ex1.m1.8.8.2.2" xref="S4.Ex1.m1.8.8.2.2.cmml"><mrow id="S4.Ex1.m1.8.8.2.2.2" xref="S4.Ex1.m1.8.8.2.2.2.cmml"><mi id="S4.Ex1.m1.8.8.2.2.2.2" xref="S4.Ex1.m1.8.8.2.2.2.2.cmml">𝐇</mi><mo id="S4.Ex1.m1.8.8.2.2.2.1" xref="S4.Ex1.m1.8.8.2.2.2.1.cmml">⁢</mo><mrow id="S4.Ex1.m1.8.8.2.2.2.3.2" xref="S4.Ex1.m1.8.8.2.2.2.cmml"><mo id="S4.Ex1.m1.8.8.2.2.2.3.2.1" stretchy="false" xref="S4.Ex1.m1.8.8.2.2.2.cmml">(</mo><mi id="S4.Ex1.m1.5.5" xref="S4.Ex1.m1.5.5.cmml">𝒛</mi><mo id="S4.Ex1.m1.8.8.2.2.2.3.2.2" stretchy="false" xref="S4.Ex1.m1.8.8.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex1.m1.8.8.2.2.1" rspace="0.111em" xref="S4.Ex1.m1.8.8.2.2.1.cmml">=</mo><mrow id="S4.Ex1.m1.8.8.2.2.3" xref="S4.Ex1.m1.8.8.2.2.3.cmml"><munder id="S4.Ex1.m1.8.8.2.2.3.1" xref="S4.Ex1.m1.8.8.2.2.3.1.cmml"><mo id="S4.Ex1.m1.8.8.2.2.3.1.2" movablelimits="false" rspace="0em" xref="S4.Ex1.m1.8.8.2.2.3.1.2.cmml">∑</mo><mi 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id="S4.SS1.SSS0.Px3.p1.9.m3.1a"><mi id="S4.SS1.SSS0.Px3.p1.9.m3.1.1" xref="S4.SS1.SSS0.Px3.p1.9.m3.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.9.m3.1b"><ci id="S4.SS1.SSS0.Px3.p1.9.m3.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.9.m3.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.9.m3.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.9.m3.1d">italic_j</annotation></semantics></math>, respectively. These operators are responsible for updating the statistical parameters <math alttext="\boldsymbol{m}_{j}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.10.m4.1"><semantics id="S4.SS1.SSS0.Px3.p1.10.m4.1a"><msub id="S4.SS1.SSS0.Px3.p1.10.m4.1.1" xref="S4.SS1.SSS0.Px3.p1.10.m4.1.1.cmml"><mi id="S4.SS1.SSS0.Px3.p1.10.m4.1.1.2" xref="S4.SS1.SSS0.Px3.p1.10.m4.1.1.2.cmml">𝒎</mi><mi id="S4.SS1.SSS0.Px3.p1.10.m4.1.1.3" xref="S4.SS1.SSS0.Px3.p1.10.m4.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.10.m4.1b"><apply id="S4.SS1.SSS0.Px3.p1.10.m4.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.10.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px3.p1.10.m4.1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.10.m4.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px3.p1.10.m4.1.1.2.cmml" xref="S4.SS1.SSS0.Px3.p1.10.m4.1.1.2">𝒎</ci><ci id="S4.SS1.SSS0.Px3.p1.10.m4.1.1.3.cmml" xref="S4.SS1.SSS0.Px3.p1.10.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.10.m4.1c">\boldsymbol{m}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.10.m4.1d">bold_italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathbf{C}_{j}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.11.m5.1"><semantics id="S4.SS1.SSS0.Px3.p1.11.m5.1a"><msub id="S4.SS1.SSS0.Px3.p1.11.m5.1.1" xref="S4.SS1.SSS0.Px3.p1.11.m5.1.1.cmml"><mi id="S4.SS1.SSS0.Px3.p1.11.m5.1.1.2" xref="S4.SS1.SSS0.Px3.p1.11.m5.1.1.2.cmml">𝐂</mi><mi id="S4.SS1.SSS0.Px3.p1.11.m5.1.1.3" xref="S4.SS1.SSS0.Px3.p1.11.m5.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.11.m5.1b"><apply id="S4.SS1.SSS0.Px3.p1.11.m5.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.11.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px3.p1.11.m5.1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.11.m5.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px3.p1.11.m5.1.1.2.cmml" xref="S4.SS1.SSS0.Px3.p1.11.m5.1.1.2">𝐂</ci><ci id="S4.SS1.SSS0.Px3.p1.11.m5.1.1.3.cmml" xref="S4.SS1.SSS0.Px3.p1.11.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.11.m5.1c">\mathbf{C}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.11.m5.1d">bold_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, which are initially biased due to the presence of the exoplanet. Further details are provided in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS4" title="A.4 Iterative optimization scheme for astrometry ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.4</a>. Each iteration indexed by <math alttext="l" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.12.m6.1"><semantics id="S4.SS1.SSS0.Px3.p1.12.m6.1a"><mi id="S4.SS1.SSS0.Px3.p1.12.m6.1.1" xref="S4.SS1.SSS0.Px3.p1.12.m6.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.12.m6.1b"><ci id="S4.SS1.SSS0.Px3.p1.12.m6.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.12.m6.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.12.m6.1c">l</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.12.m6.1d">italic_l</annotation></semantics></math> writes as: <table class="ltx_equation ltx_eqn_table" id="S4.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\boldsymbol{z}^{(l+1)}=\boldsymbol{z}^{(l)}-\mathbf{H}(\boldsymbol{z}^{(l)})^{% -1}\boldsymbol{g}(\boldsymbol{z}^{(l)})\,." 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id="S4.E11.m1.5.5.1.1.2.4" xref="S4.E11.m1.5.5.1.1.2.4.cmml"><mi id="S4.E11.m1.5.5.1.1.2.4.2" xref="S4.E11.m1.5.5.1.1.2.4.2.cmml">𝒛</mi><mrow id="S4.E11.m1.2.2.1.3" xref="S4.E11.m1.5.5.1.1.2.4.cmml"><mo id="S4.E11.m1.2.2.1.3.1" stretchy="false" xref="S4.E11.m1.5.5.1.1.2.4.cmml">(</mo><mi id="S4.E11.m1.2.2.1.1" xref="S4.E11.m1.2.2.1.1.cmml">l</mi><mo id="S4.E11.m1.2.2.1.3.2" stretchy="false" xref="S4.E11.m1.5.5.1.1.2.4.cmml">)</mo></mrow></msup><mo id="S4.E11.m1.5.5.1.1.2.3" xref="S4.E11.m1.5.5.1.1.2.3.cmml">−</mo><mrow id="S4.E11.m1.5.5.1.1.2.2" xref="S4.E11.m1.5.5.1.1.2.2.cmml"><mi id="S4.E11.m1.5.5.1.1.2.2.4" xref="S4.E11.m1.5.5.1.1.2.2.4.cmml">𝐇</mi><mo id="S4.E11.m1.5.5.1.1.2.2.3" xref="S4.E11.m1.5.5.1.1.2.2.3.cmml">⁢</mo><msup id="S4.E11.m1.5.5.1.1.1.1.1" xref="S4.E11.m1.5.5.1.1.1.1.1.cmml"><mrow id="S4.E11.m1.5.5.1.1.1.1.1.1.1" xref="S4.E11.m1.5.5.1.1.1.1.1.1.1.1.cmml"><mo id="S4.E11.m1.5.5.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E11.m1.5.5.1.1.1.1.1.1.1.1.cmml">(</mo><msup 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xref="S4.E11.m1.5.5.1.1.2.2.5.cmml">𝒈</mi><mo id="S4.E11.m1.5.5.1.1.2.2.3b" xref="S4.E11.m1.5.5.1.1.2.2.3.cmml">⁢</mo><mrow id="S4.E11.m1.5.5.1.1.2.2.2.1" xref="S4.E11.m1.5.5.1.1.2.2.2.1.1.cmml"><mo id="S4.E11.m1.5.5.1.1.2.2.2.1.2" stretchy="false" xref="S4.E11.m1.5.5.1.1.2.2.2.1.1.cmml">(</mo><msup id="S4.E11.m1.5.5.1.1.2.2.2.1.1" xref="S4.E11.m1.5.5.1.1.2.2.2.1.1.cmml"><mi id="S4.E11.m1.5.5.1.1.2.2.2.1.1.2" xref="S4.E11.m1.5.5.1.1.2.2.2.1.1.2.cmml">𝒛</mi><mrow id="S4.E11.m1.4.4.1.3" xref="S4.E11.m1.5.5.1.1.2.2.2.1.1.cmml"><mo id="S4.E11.m1.4.4.1.3.1" stretchy="false" xref="S4.E11.m1.5.5.1.1.2.2.2.1.1.cmml">(</mo><mi id="S4.E11.m1.4.4.1.1" xref="S4.E11.m1.4.4.1.1.cmml">l</mi><mo id="S4.E11.m1.4.4.1.3.2" stretchy="false" xref="S4.E11.m1.5.5.1.1.2.2.2.1.1.cmml">)</mo></mrow></msup><mo id="S4.E11.m1.5.5.1.1.2.2.2.1.3" stretchy="false" xref="S4.E11.m1.5.5.1.1.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.E11.m1.5.5.1.2" lspace="0.170em" 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xref="S4.E11.m1.3.3.1.1">𝑙</ci></apply><apply id="S4.E11.m1.5.5.1.1.1.1.1.3.cmml" xref="S4.E11.m1.5.5.1.1.1.1.1.3"><minus id="S4.E11.m1.5.5.1.1.1.1.1.3.1.cmml" xref="S4.E11.m1.5.5.1.1.1.1.1.3"></minus><cn id="S4.E11.m1.5.5.1.1.1.1.1.3.2.cmml" type="integer" xref="S4.E11.m1.5.5.1.1.1.1.1.3.2">1</cn></apply></apply><ci id="S4.E11.m1.5.5.1.1.2.2.5.cmml" xref="S4.E11.m1.5.5.1.1.2.2.5">𝒈</ci><apply id="S4.E11.m1.5.5.1.1.2.2.2.1.1.cmml" xref="S4.E11.m1.5.5.1.1.2.2.2.1"><csymbol cd="ambiguous" id="S4.E11.m1.5.5.1.1.2.2.2.1.1.1.cmml" xref="S4.E11.m1.5.5.1.1.2.2.2.1">superscript</csymbol><ci id="S4.E11.m1.5.5.1.1.2.2.2.1.1.2.cmml" xref="S4.E11.m1.5.5.1.1.2.2.2.1.1.2">𝒛</ci><ci id="S4.E11.m1.4.4.1.1.cmml" xref="S4.E11.m1.4.4.1.1">𝑙</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E11.m1.5c">\boldsymbol{z}^{(l+1)}=\boldsymbol{z}^{(l)}-\mathbf{H}(\boldsymbol{z}^{(l)})^{% -1}\boldsymbol{g}(\boldsymbol{z}^{(l)})\,.</annotation><annotation encoding="application/x-llamapun" id="S4.E11.m1.5d">bold_italic_z start_POSTSUPERSCRIPT ( italic_l + 1 ) end_POSTSUPERSCRIPT = bold_italic_z start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT - bold_H ( bold_italic_z start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT bold_italic_g ( bold_italic_z start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(11)</td> </tr></tbody> </table> </div> </section> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> 4.2 Extensions of the statistical model: mixture of distributions</h3> <div class="ltx_para" id="S4.SS2.p1"> We now build on the convolutional statistical model presented in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1" title="4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.1</a> to introduce a multi-scale statistical model of the nuisance component. </div> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Linear projection.</h4> <div class="ltx_para ltx_noindent" id="S4.SS2.SSS0.Px1.p1"> Instead of modeling correlations between the pixels of a patch, we propose to model correlations between projected linear features: <table class="ltx_equation ltx_eqn_table" id="S4.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall(j,t),\quad\mathbf{A}\,\boldsymbol{y}^{(j)}_{t}\sim\mathcal{N}(% \boldsymbol{m}_{j},\mathbf{C}_{j})\,," class="ltx_Math" display="block" id="S4.E12.m1.4"><semantics 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, bold_A bold_italic_y start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∼ caligraphic_N ( bold_italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(12)</td> </tr></tbody> </table> where <math alttext="\mathbf{A}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.1.m1.1"><semantics id="S4.SS2.SSS0.Px1.p1.1.m1.1a"><mi id="S4.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S4.SS2.SSS0.Px1.p1.1.m1.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.1.m1.1b"><ci id="S4.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.1.m1.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.1.m1.1c">\mathbf{A}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.1.m1.1d">bold_A</annotation></semantics></math> in <math alttext="\mathbb{R}^{m\times p}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.2.m2.1"><semantics id="S4.SS2.SSS0.Px1.p1.2.m2.1a"><msup id="S4.SS2.SSS0.Px1.p1.2.m2.1.1" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.2" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.2.cmml">ℝ</mi><mrow id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.cmml"><mi id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.2" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.2.cmml">m</mi><mo id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.1.cmml">×</mo><mi id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.3" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.3.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.2.m2.1b"><apply id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1">superscript</csymbol><ci id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.2">ℝ</ci><apply id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3"><times id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.1.cmml" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.1"></times><ci id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.2.cmml" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.2">𝑚</ci><ci id="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.3.cmml" xref="S4.SS2.SSS0.Px1.p1.2.m2.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.2.m2.1c">\mathbb{R}^{m\times p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_m × italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> is the projection matrix and <math alttext="m\leq p" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.3.m3.1"><semantics id="S4.SS2.SSS0.Px1.p1.3.m3.1a"><mrow id="S4.SS2.SSS0.Px1.p1.3.m3.1.1" xref="S4.SS2.SSS0.Px1.p1.3.m3.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p1.3.m3.1.1.2" xref="S4.SS2.SSS0.Px1.p1.3.m3.1.1.2.cmml">m</mi><mo id="S4.SS2.SSS0.Px1.p1.3.m3.1.1.1" xref="S4.SS2.SSS0.Px1.p1.3.m3.1.1.1.cmml">≤</mo><mi id="S4.SS2.SSS0.Px1.p1.3.m3.1.1.3" xref="S4.SS2.SSS0.Px1.p1.3.m3.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.3.m3.1b"><apply id="S4.SS2.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.3.m3.1.1"><leq id="S4.SS2.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.3.m3.1.1.1"></leq><ci id="S4.SS2.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p1.3.m3.1.1.2">𝑚</ci><ci id="S4.SS2.SSS0.Px1.p1.3.m3.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p1.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.3.m3.1c">m\leq p</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.3.m3.1d">italic_m ≤ italic_p</annotation></semantics></math>. This is a generalization of the model presented in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1" title="4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.1</a>, for which <math alttext="\mathbf{A}=\mathbf{I}_{p}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.4.m4.1"><semantics id="S4.SS2.SSS0.Px1.p1.4.m4.1a"><mrow id="S4.SS2.SSS0.Px1.p1.4.m4.1.1" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.2" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.2.cmml">𝐀</mi><mo id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.1" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.1.cmml">=</mo><msub id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.cmml"><mi id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.2" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.2.cmml">𝐈</mi><mi id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.3" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.4.m4.1b"><apply id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1"><eq id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.1"></eq><ci id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.2">𝐀</ci><apply id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.1.cmml" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.2.cmml" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.2">𝐈</ci><ci id="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.3.cmml" xref="S4.SS2.SSS0.Px1.p1.4.m4.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.4.m4.1c">\mathbf{A}=\mathbf{I}_{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.4.m4.1d">bold_A = bold_I start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. The terms <math alttext="\boldsymbol{b}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.5.m5.1"><semantics id="S4.SS2.SSS0.Px1.p1.5.m5.1a"><msub id="S4.SS2.SSS0.Px1.p1.5.m5.1.1" xref="S4.SS2.SSS0.Px1.p1.5.m5.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p1.5.m5.1.1.2" xref="S4.SS2.SSS0.Px1.p1.5.m5.1.1.2.cmml">𝒃</mi><mi id="S4.SS2.SSS0.Px1.p1.5.m5.1.1.3" xref="S4.SS2.SSS0.Px1.p1.5.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.5.m5.1b"><apply id="S4.SS2.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p1.5.m5.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.5.m5.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p1.5.m5.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p1.5.m5.1.1.2">𝒃</ci><ci id="S4.SS2.SSS0.Px1.p1.5.m5.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p1.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.5.m5.1c">\boldsymbol{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.5.m5.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{a}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.6.m6.1"><semantics id="S4.SS2.SSS0.Px1.p1.6.m6.1a"><mi id="S4.SS2.SSS0.Px1.p1.6.m6.1.1" xref="S4.SS2.SSS0.Px1.p1.6.m6.1.1.cmml">𝒂</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.6.m6.1b"><ci id="S4.SS2.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.6.m6.1.1">𝒂</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.6.m6.1c">\boldsymbol{a}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.6.m6.1d">bold_italic_a</annotation></semantics></math> introduced in Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E26" title="Equation 26 ‣ A.2 Dense detection map computation details ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">26</a>)-(<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E27" title="Equation 27 ‣ A.2 Dense detection map computation details ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">27</a>) become: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx2"> <tbody id="S4.E13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle b_{i,t}" class="ltx_Math" display="inline" id="S4.E13.m1.2"><semantics id="S4.E13.m1.2a"><msub id="S4.E13.m1.2.3" xref="S4.E13.m1.2.3.cmml"><mi id="S4.E13.m1.2.3.2" xref="S4.E13.m1.2.3.2.cmml">b</mi><mrow id="S4.E13.m1.2.2.2.4" xref="S4.E13.m1.2.2.2.3.cmml"><mi id="S4.E13.m1.1.1.1.1" xref="S4.E13.m1.1.1.1.1.cmml">i</mi><mo id="S4.E13.m1.2.2.2.4.1" xref="S4.E13.m1.2.2.2.3.cmml">,</mo><mi id="S4.E13.m1.2.2.2.2" xref="S4.E13.m1.2.2.2.2.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.E13.m1.2b"><apply id="S4.E13.m1.2.3.cmml" xref="S4.E13.m1.2.3"><csymbol cd="ambiguous" id="S4.E13.m1.2.3.1.cmml" xref="S4.E13.m1.2.3">subscript</csymbol><ci id="S4.E13.m1.2.3.2.cmml" xref="S4.E13.m1.2.3.2">𝑏</ci><list id="S4.E13.m1.2.2.2.3.cmml" xref="S4.E13.m1.2.2.2.4"><ci id="S4.E13.m1.1.1.1.1.cmml" xref="S4.E13.m1.1.1.1.1">𝑖</ci><ci id="S4.E13.m1.2.2.2.2.cmml" xref="S4.E13.m1.2.2.2.2">𝑡</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E13.m1.2c">\displaystyle b_{i,t}</annotation><annotation encoding="application/x-llamapun" id="S4.E13.m1.2d">italic_b start_POSTSUBSCRIPT italic_i , italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{j\in S(\boldsymbol{x}_{i})}w_{j}\,\boldsymbol{h}_{ji}^{% \top}\,\mathbf{A}_{j}^{\top}\,\widehat{\mathbf{C}}_{j}^{-1}\ \big{(}\mathbf{A}% _{j}\,\boldsymbol{y}_{t}^{(j)}-\widehat{\boldsymbol{m}}_{j}\big{)}\,," class="ltx_Math" display="inline" id="S4.E13.m2.3"><semantics id="S4.E13.m2.3a"><mrow id="S4.E13.m2.3.3.1" xref="S4.E13.m2.3.3.1.1.cmml"><mrow id="S4.E13.m2.3.3.1.1" xref="S4.E13.m2.3.3.1.1.cmml"><mi id="S4.E13.m2.3.3.1.1.3" xref="S4.E13.m2.3.3.1.1.3.cmml"></mi><mo id="S4.E13.m2.3.3.1.1.2" xref="S4.E13.m2.3.3.1.1.2.cmml">=</mo><mrow id="S4.E13.m2.3.3.1.1.1" xref="S4.E13.m2.3.3.1.1.1.cmml"><mstyle displaystyle="true" id="S4.E13.m2.3.3.1.1.1.2" xref="S4.E13.m2.3.3.1.1.1.2.cmml"><munder id="S4.E13.m2.3.3.1.1.1.2a" xref="S4.E13.m2.3.3.1.1.1.2.cmml"><mo id="S4.E13.m2.3.3.1.1.1.2.2" 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xref="S4.E13.m2.3.3.1.1.1.1.1.1.1.3.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E13.m2.3c">\displaystyle=\sum_{j\in S(\boldsymbol{x}_{i})}w_{j}\,\boldsymbol{h}_{ji}^{% \top}\,\mathbf{A}_{j}^{\top}\,\widehat{\mathbf{C}}_{j}^{-1}\ \big{(}\mathbf{A}% _{j}\,\boldsymbol{y}_{t}^{(j)}-\widehat{\boldsymbol{m}}_{j}\big{)}\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E13.m2.3d">= ∑ start_POSTSUBSCRIPT italic_j ∈ italic_S ( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_italic_h start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT - over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(13)</td> </tr></tbody> <tbody id="S4.E14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle a_{i}" class="ltx_Math" display="inline" id="S4.E14.m1.1"><semantics id="S4.E14.m1.1a"><msub id="S4.E14.m1.1.1" xref="S4.E14.m1.1.1.cmml"><mi id="S4.E14.m1.1.1.2" xref="S4.E14.m1.1.1.2.cmml">a</mi><mi id="S4.E14.m1.1.1.3" xref="S4.E14.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.E14.m1.1b"><apply 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start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(14)</td> </tr></tbody> </table> The learnable projection <math alttext="\mathbf{A}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.7.m1.1"><semantics id="S4.SS2.SSS0.Px1.p1.7.m1.1a"><mi id="S4.SS2.SSS0.Px1.p1.7.m1.1.1" xref="S4.SS2.SSS0.Px1.p1.7.m1.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.7.m1.1b"><ci id="S4.SS2.SSS0.Px1.p1.7.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.7.m1.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.7.m1.1c">\mathbf{A}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.7.m1.1d">bold_A</annotation></semantics></math> decorrelates the feature space, where the statistical distribution is defined, from the pixel space, enabling effective handling of long-range correlations. </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Multi-scale approach.</h4> <div class="ltx_para" id="S4.SS2.SSS0.Px2.p1"> So far, we have shown how our model can combine multiple neighboring local distributions through weighted averaging of the terms <math alttext="\boldsymbol{b}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.1.m1.1"><semantics id="S4.SS2.SSS0.Px2.p1.1.m1.1a"><msub id="S4.SS2.SSS0.Px2.p1.1.m1.1.1" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.2" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.2.cmml">𝒃</mi><mi id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.3" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.1.m1.1b"><apply id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.cmml" 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Extending this approach, we propose to integrate distributions across multiple spatial scales, by varying the patch sizes <math alttext="p" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.3.m3.1"><semantics id="S4.SS2.SSS0.Px2.p1.3.m3.1a"><mi id="S4.SS2.SSS0.Px2.p1.3.m3.1.1" xref="S4.SS2.SSS0.Px2.p1.3.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.3.m3.1b"><ci id="S4.SS2.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.3.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.3.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.3.m3.1d">italic_p</annotation></semantics></math>. We denote <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.4.m4.1"><semantics id="S4.SS2.SSS0.Px2.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px2.p1.4.m4.1.1" xref="S4.SS2.SSS0.Px2.p1.4.m4.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.4.m4.1b"><ci id="S4.SS2.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.4.m4.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.4.m4.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.4.m4.1d">caligraphic_P</annotation></semantics></math> the set of patch sizes chosen. We expand the set <math alttext="S(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.5.m5.1"><semantics id="S4.SS2.SSS0.Px2.p1.5.m5.1a"><mrow id="S4.SS2.SSS0.Px2.p1.5.m5.1.1" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.cmml"><mi id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.3" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.3.cmml">S</mi><mo id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.2" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.2" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.3" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.5.m5.1b"><apply id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1"><times id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.2.cmml" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.2"></times><ci id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.3.cmml" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.3">𝑆</ci><apply id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.2">𝒙</ci><ci id="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px2.p1.5.m5.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.5.m5.1c">S(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.5.m5.1d">italic_S ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> of modeled distributions to include different scales, such that: <table class="ltx_equation ltx_eqn_table" id="S4.E15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S(\boldsymbol{x}_{t})=\bigcup\nolimits_{p\in\mathcal{P}}S_{p}(\boldsymbol{x}_{% t})\,," class="ltx_Math" display="block" id="S4.E15.m1.1"><semantics id="S4.E15.m1.1a"><mrow id="S4.E15.m1.1.1.1" xref="S4.E15.m1.1.1.1.1.cmml"><mrow id="S4.E15.m1.1.1.1.1" xref="S4.E15.m1.1.1.1.1.cmml"><mrow id="S4.E15.m1.1.1.1.1.1" xref="S4.E15.m1.1.1.1.1.1.cmml"><mi id="S4.E15.m1.1.1.1.1.1.3" xref="S4.E15.m1.1.1.1.1.1.3.cmml">S</mi><mo id="S4.E15.m1.1.1.1.1.1.2" xref="S4.E15.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E15.m1.1.1.1.1.1.1.1" xref="S4.E15.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.E15.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E15.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.E15.m1.1.1.1.1.1.1.1.1" xref="S4.E15.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.E15.m1.1.1.1.1.1.1.1.1.2" xref="S4.E15.m1.1.1.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.E15.m1.1.1.1.1.1.1.1.1.3" xref="S4.E15.m1.1.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.E15.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E15.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E15.m1.1.1.1.1.3" rspace="0.111em" xref="S4.E15.m1.1.1.1.1.3.cmml">=</mo><mrow id="S4.E15.m1.1.1.1.1.2" xref="S4.E15.m1.1.1.1.1.2.cmml"><msub id="S4.E15.m1.1.1.1.1.2.2" xref="S4.E15.m1.1.1.1.1.2.2.cmml"><mo id="S4.E15.m1.1.1.1.1.2.2.2" xref="S4.E15.m1.1.1.1.1.2.2.2.cmml">⋃</mo><mrow id="S4.E15.m1.1.1.1.1.2.2.3" xref="S4.E15.m1.1.1.1.1.2.2.3.cmml"><mi id="S4.E15.m1.1.1.1.1.2.2.3.2" xref="S4.E15.m1.1.1.1.1.2.2.3.2.cmml">p</mi><mo id="S4.E15.m1.1.1.1.1.2.2.3.1" xref="S4.E15.m1.1.1.1.1.2.2.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.E15.m1.1.1.1.1.2.2.3.3" xref="S4.E15.m1.1.1.1.1.2.2.3.3.cmml">𝒫</mi></mrow></msub><mrow id="S4.E15.m1.1.1.1.1.2.1" xref="S4.E15.m1.1.1.1.1.2.1.cmml"><msub id="S4.E15.m1.1.1.1.1.2.1.3" xref="S4.E15.m1.1.1.1.1.2.1.3.cmml"><mi id="S4.E15.m1.1.1.1.1.2.1.3.2" xref="S4.E15.m1.1.1.1.1.2.1.3.2.cmml">S</mi><mi id="S4.E15.m1.1.1.1.1.2.1.3.3" xref="S4.E15.m1.1.1.1.1.2.1.3.3.cmml">p</mi></msub><mo id="S4.E15.m1.1.1.1.1.2.1.2" xref="S4.E15.m1.1.1.1.1.2.1.2.cmml">⁢</mo><mrow id="S4.E15.m1.1.1.1.1.2.1.1.1" xref="S4.E15.m1.1.1.1.1.2.1.1.1.1.cmml"><mo id="S4.E15.m1.1.1.1.1.2.1.1.1.2" stretchy="false" xref="S4.E15.m1.1.1.1.1.2.1.1.1.1.cmml">(</mo><msub id="S4.E15.m1.1.1.1.1.2.1.1.1.1" xref="S4.E15.m1.1.1.1.1.2.1.1.1.1.cmml"><mi id="S4.E15.m1.1.1.1.1.2.1.1.1.1.2" xref="S4.E15.m1.1.1.1.1.2.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.E15.m1.1.1.1.1.2.1.1.1.1.3" xref="S4.E15.m1.1.1.1.1.2.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.E15.m1.1.1.1.1.2.1.1.1.3" rspace="0.170em" stretchy="false" xref="S4.E15.m1.1.1.1.1.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.E15.m1.1.1.1.2" xref="S4.E15.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E15.m1.1b"><apply id="S4.E15.m1.1.1.1.1.cmml" xref="S4.E15.m1.1.1.1"><eq id="S4.E15.m1.1.1.1.1.3.cmml" xref="S4.E15.m1.1.1.1.1.3"></eq><apply id="S4.E15.m1.1.1.1.1.1.cmml" xref="S4.E15.m1.1.1.1.1.1"><times id="S4.E15.m1.1.1.1.1.1.2.cmml" xref="S4.E15.m1.1.1.1.1.1.2"></times><ci id="S4.E15.m1.1.1.1.1.1.3.cmml" xref="S4.E15.m1.1.1.1.1.1.3">𝑆</ci><apply id="S4.E15.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E15.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E15.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E15.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E15.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E15.m1.1.1.1.1.1.1.1.1.2">𝒙</ci><ci id="S4.E15.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E15.m1.1.1.1.1.1.1.1.1.3">𝑡</ci></apply></apply><apply id="S4.E15.m1.1.1.1.1.2.cmml" xref="S4.E15.m1.1.1.1.1.2"><apply id="S4.E15.m1.1.1.1.1.2.2.cmml" xref="S4.E15.m1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.E15.m1.1.1.1.1.2.2.1.cmml" xref="S4.E15.m1.1.1.1.1.2.2">subscript</csymbol><union id="S4.E15.m1.1.1.1.1.2.2.2.cmml" xref="S4.E15.m1.1.1.1.1.2.2.2"></union><apply id="S4.E15.m1.1.1.1.1.2.2.3.cmml" xref="S4.E15.m1.1.1.1.1.2.2.3"><in id="S4.E15.m1.1.1.1.1.2.2.3.1.cmml" xref="S4.E15.m1.1.1.1.1.2.2.3.1"></in><ci id="S4.E15.m1.1.1.1.1.2.2.3.2.cmml" xref="S4.E15.m1.1.1.1.1.2.2.3.2">𝑝</ci><ci id="S4.E15.m1.1.1.1.1.2.2.3.3.cmml" xref="S4.E15.m1.1.1.1.1.2.2.3.3">𝒫</ci></apply></apply><apply id="S4.E15.m1.1.1.1.1.2.1.cmml" xref="S4.E15.m1.1.1.1.1.2.1"><times id="S4.E15.m1.1.1.1.1.2.1.2.cmml" xref="S4.E15.m1.1.1.1.1.2.1.2"></times><apply id="S4.E15.m1.1.1.1.1.2.1.3.cmml" xref="S4.E15.m1.1.1.1.1.2.1.3"><csymbol cd="ambiguous" id="S4.E15.m1.1.1.1.1.2.1.3.1.cmml" xref="S4.E15.m1.1.1.1.1.2.1.3">subscript</csymbol><ci id="S4.E15.m1.1.1.1.1.2.1.3.2.cmml" xref="S4.E15.m1.1.1.1.1.2.1.3.2">𝑆</ci><ci id="S4.E15.m1.1.1.1.1.2.1.3.3.cmml" xref="S4.E15.m1.1.1.1.1.2.1.3.3">𝑝</ci></apply><apply id="S4.E15.m1.1.1.1.1.2.1.1.1.1.cmml" xref="S4.E15.m1.1.1.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S4.E15.m1.1.1.1.1.2.1.1.1.1.1.cmml" xref="S4.E15.m1.1.1.1.1.2.1.1.1">subscript</csymbol><ci id="S4.E15.m1.1.1.1.1.2.1.1.1.1.2.cmml" xref="S4.E15.m1.1.1.1.1.2.1.1.1.1.2">𝒙</ci><ci id="S4.E15.m1.1.1.1.1.2.1.1.1.1.3.cmml" xref="S4.E15.m1.1.1.1.1.2.1.1.1.1.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E15.m1.1c">S(\boldsymbol{x}_{t})=\bigcup\nolimits_{p\in\mathcal{P}}S_{p}(\boldsymbol{x}_{% t})\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E15.m1.1d">italic_S ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) = ⋃ start_POSTSUBSCRIPT italic_p ∈ caligraphic_P end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(15)</td> </tr></tbody> </table> where <math alttext="S_{p}(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.6.m1.1"><semantics id="S4.SS2.SSS0.Px2.p1.6.m1.1a"><mrow id="S4.SS2.SSS0.Px2.p1.6.m1.1.1" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.cmml"><msub id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.cmml"><mi id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.2" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.2.cmml">S</mi><mi id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.3" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.3.cmml">p</mi></msub><mo id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.2" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.2" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.3" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.6.m1.1b"><apply id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1"><times id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.2.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.2"></times><apply id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.1.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.2.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.2">𝑆</ci><ci id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.3.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.3.3">𝑝</ci></apply><apply id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.2">𝒙</ci><ci id="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px2.p1.6.m1.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.6.m1.1c">S_{p}(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.6.m1.1d">italic_S start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> is the subset of patches with patch size <math alttext="p" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.7.m2.1"><semantics id="S4.SS2.SSS0.Px2.p1.7.m2.1a"><mi id="S4.SS2.SSS0.Px2.p1.7.m2.1.1" xref="S4.SS2.SSS0.Px2.p1.7.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.7.m2.1b"><ci id="S4.SS2.SSS0.Px2.p1.7.m2.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.7.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.7.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.7.m2.1d">italic_p</annotation></semantics></math> containing <math alttext="\boldsymbol{x}_{t}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.8.m3.1"><semantics id="S4.SS2.SSS0.Px2.p1.8.m3.1a"><msub id="S4.SS2.SSS0.Px2.p1.8.m3.1.1" xref="S4.SS2.SSS0.Px2.p1.8.m3.1.1.cmml"><mi id="S4.SS2.SSS0.Px2.p1.8.m3.1.1.2" xref="S4.SS2.SSS0.Px2.p1.8.m3.1.1.2.cmml">𝒙</mi><mi id="S4.SS2.SSS0.Px2.p1.8.m3.1.1.3" xref="S4.SS2.SSS0.Px2.p1.8.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.8.m3.1b"><apply id="S4.SS2.SSS0.Px2.p1.8.m3.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.8.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px2.p1.8.m3.1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.8.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px2.p1.8.m3.1.1.2.cmml" xref="S4.SS2.SSS0.Px2.p1.8.m3.1.1.2">𝒙</ci><ci id="S4.SS2.SSS0.Px2.p1.8.m3.1.1.3.cmml" xref="S4.SS2.SSS0.Px2.p1.8.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.8.m3.1c">\boldsymbol{x}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.8.m3.1d">bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. The linear projection is particularly useful in controlling the dimensionality of the feature space for larger patches by setting <math alttext="m<p" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.9.m4.1"><semantics id="S4.SS2.SSS0.Px2.p1.9.m4.1a"><mrow id="S4.SS2.SSS0.Px2.p1.9.m4.1.1" xref="S4.SS2.SSS0.Px2.p1.9.m4.1.1.cmml"><mi id="S4.SS2.SSS0.Px2.p1.9.m4.1.1.2" xref="S4.SS2.SSS0.Px2.p1.9.m4.1.1.2.cmml">m</mi><mo id="S4.SS2.SSS0.Px2.p1.9.m4.1.1.1" xref="S4.SS2.SSS0.Px2.p1.9.m4.1.1.1.cmml"><</mo><mi id="S4.SS2.SSS0.Px2.p1.9.m4.1.1.3" xref="S4.SS2.SSS0.Px2.p1.9.m4.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.9.m4.1b"><apply id="S4.SS2.SSS0.Px2.p1.9.m4.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.9.m4.1.1"><lt id="S4.SS2.SSS0.Px2.p1.9.m4.1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.9.m4.1.1.1"></lt><ci id="S4.SS2.SSS0.Px2.p1.9.m4.1.1.2.cmml" xref="S4.SS2.SSS0.Px2.p1.9.m4.1.1.2">𝑚</ci><ci id="S4.SS2.SSS0.Px2.p1.9.m4.1.1.3.cmml" xref="S4.SS2.SSS0.Px2.p1.9.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.9.m4.1c">m<p</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.9.m4.1d">italic_m < italic_p</annotation></semantics></math>, thereby maintaining computational efficiency. </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Leveraging rotational symmetries.</h4> <div class="ltx_para ltx_noindent" id="S4.SS2.SSS0.Px3.p1"> In direct imaging, the speckle pattern exhibits approximate central and rotational symmetries, see Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S1.F1" title="Figure A ‣ 1 Introduction ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A</a>. These symmetries arise from near-isotropic distortions of the wavefront and from the optical system’s near-circular symmetry. Symmetries of the speckles can be theoretically understood through a series expansion of the diffraction pattern, using increasing powers of the Fourier transform of the residual phase error <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib48" title="">48</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib59" title="">59</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib53" title="">53</a>]</cite>. Leveraging these symmetries is essential for constructing a more robust model of the speckles, particularly to mitigate the effects of self-subtraction. This effect is a well-known challenge in direct imaging that arises when limited parallactic rotation results in the contamination of speckle parameters by the exoplanet signal, ultimately reducing detection sensitivity <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib46" title="">46</a>]</cite>. To incorporate these symmetries, we propose modeling the joint distribution of patches extracted from the same location after rotating the observation by <math alttext="2\pi/N" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px3.p1.1.m1.1"><semantics id="S4.SS2.SSS0.Px3.p1.1.m1.1a"><mrow id="S4.SS2.SSS0.Px3.p1.1.m1.1.1" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.cmml"><mrow id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.cmml"><mn id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.2" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.2.cmml">2</mn><mo id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.1" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.3" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.3.cmml">π</mi></mrow><mo id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.1" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.1.cmml">/</mo><mi id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.3" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px3.p1.1.m1.1b"><apply id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1"><divide id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.1"></divide><apply id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2"><times id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.1.cmml" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.1"></times><cn id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.2.cmml" type="integer" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.2">2</cn><ci id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.3.cmml" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.2.3">𝜋</ci></apply><ci id="S4.SS2.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="S4.SS2.SSS0.Px3.p1.1.m1.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px3.p1.1.m1.1c">2\pi/N</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px3.p1.1.m1.1d">2 italic_π / italic_N</annotation></semantics></math>, also known as <math alttext="N" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px3.p1.2.m2.1"><semantics id="S4.SS2.SSS0.Px3.p1.2.m2.1a"><mi id="S4.SS2.SSS0.Px3.p1.2.m2.1.1" xref="S4.SS2.SSS0.Px3.p1.2.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px3.p1.2.m2.1b"><ci id="S4.SS2.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS0.Px3.p1.2.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px3.p1.2.m2.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px3.p1.2.m2.1d">italic_N</annotation></semantics></math>-fold rotational symmetry: <table class="ltx_equation ltx_eqn_table" id="S4.E16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall(j,t),\quad\mathbf{A}\big{[}{\mathbf{R}}_{2\pi n/N}(\boldsymbol{y}_{t})% ^{(j)}\big{]}_{n=0:N-1}\sim\mathcal{N}(\boldsymbol{m}_{j},\mathbf{C}_{j})\,," class="ltx_Math" display="block" id="S4.E16.m1.4"><semantics id="S4.E16.m1.4a"><mrow 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xref="S4.E16.m1.4.4.1.1.4.3"></times><ci id="S4.E16.m1.4.4.1.1.4.4.cmml" xref="S4.E16.m1.4.4.1.1.4.4">𝒩</ci><interval closure="open" id="S4.E16.m1.4.4.1.1.4.2.3.cmml" xref="S4.E16.m1.4.4.1.1.4.2.2"><apply id="S4.E16.m1.4.4.1.1.3.1.1.1.cmml" xref="S4.E16.m1.4.4.1.1.3.1.1.1"><csymbol cd="ambiguous" id="S4.E16.m1.4.4.1.1.3.1.1.1.1.cmml" xref="S4.E16.m1.4.4.1.1.3.1.1.1">subscript</csymbol><ci id="S4.E16.m1.4.4.1.1.3.1.1.1.2.cmml" xref="S4.E16.m1.4.4.1.1.3.1.1.1.2">𝒎</ci><ci id="S4.E16.m1.4.4.1.1.3.1.1.1.3.cmml" xref="S4.E16.m1.4.4.1.1.3.1.1.1.3">𝑗</ci></apply><apply id="S4.E16.m1.4.4.1.1.4.2.2.2.cmml" xref="S4.E16.m1.4.4.1.1.4.2.2.2"><csymbol cd="ambiguous" id="S4.E16.m1.4.4.1.1.4.2.2.2.1.cmml" xref="S4.E16.m1.4.4.1.1.4.2.2.2">subscript</csymbol><ci id="S4.E16.m1.4.4.1.1.4.2.2.2.2.cmml" xref="S4.E16.m1.4.4.1.1.4.2.2.2.2">𝐂</ci><ci id="S4.E16.m1.4.4.1.1.4.2.2.2.3.cmml" xref="S4.E16.m1.4.4.1.1.4.2.2.2.3">𝑗</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E16.m1.4c">\forall(j,t),\quad\mathbf{A}\big{[}{\mathbf{R}}_{2\pi n/N}(\boldsymbol{y}_{t})% ^{(j)}\big{]}_{n=0:N-1}\sim\mathcal{N}(\boldsymbol{m}_{j},\mathbf{C}_{j})\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E16.m1.4d">∀ ( italic_j , italic_t ) , bold_A [ bold_R start_POSTSUBSCRIPT 2 italic_π italic_n / italic_N end_POSTSUBSCRIPT ( bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_n = 0 : italic_N - 1 end_POSTSUBSCRIPT ∼ caligraphic_N ( bold_italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(16)</td> </tr></tbody> </table> where <math alttext="\mathbf{A}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px3.p1.3.m1.1"><semantics id="S4.SS2.SSS0.Px3.p1.3.m1.1a"><mi id="S4.SS2.SSS0.Px3.p1.3.m1.1.1" xref="S4.SS2.SSS0.Px3.p1.3.m1.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px3.p1.3.m1.1b"><ci id="S4.SS2.SSS0.Px3.p1.3.m1.1.1.cmml" xref="S4.SS2.SSS0.Px3.p1.3.m1.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px3.p1.3.m1.1c">\mathbf{A}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px3.p1.3.m1.1d">bold_A</annotation></semantics></math> is in <math alttext="\mathbb{R}^{m\times Np}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px3.p1.4.m2.1"><semantics id="S4.SS2.SSS0.Px3.p1.4.m2.1a"><msup id="S4.SS2.SSS0.Px3.p1.4.m2.1.1" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.cmml"><mi id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.2" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.2.cmml">ℝ</mi><mrow id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.cmml"><mrow id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.cmml"><mi id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.2" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.2.cmml">m</mi><mo id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.1.cmml">×</mo><mi id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.3" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.3.cmml">N</mi></mrow><mo id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.1" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.3" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.3.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px3.p1.4.m2.1b"><apply id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.1.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1">superscript</csymbol><ci id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.2.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.2">ℝ</ci><apply id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3"><times id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.1.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.1"></times><apply id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2"><times id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.1.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.1"></times><ci id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.2.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.2">𝑚</ci><ci id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.3.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.2.3">𝑁</ci></apply><ci id="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.3.cmml" xref="S4.SS2.SSS0.Px3.p1.4.m2.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px3.p1.4.m2.1c">\mathbb{R}^{m\times Np}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px3.p1.4.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_m × italic_N italic_p end_POSTSUPERSCRIPT</annotation></semantics></math>. The parameters of this joint distribution are less subject to self-subtraction as it is very unlikely to observe exoplanets simultaneously in all jointly modeled patches. We use a mixture model with <math alttext="N=1,2,4" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px3.p1.5.m3.3"><semantics id="S4.SS2.SSS0.Px3.p1.5.m3.3a"><mrow id="S4.SS2.SSS0.Px3.p1.5.m3.3.4" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4.cmml"><mi id="S4.SS2.SSS0.Px3.p1.5.m3.3.4.2" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4.2.cmml">N</mi><mo id="S4.SS2.SSS0.Px3.p1.5.m3.3.4.1" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4.1.cmml">=</mo><mrow id="S4.SS2.SSS0.Px3.p1.5.m3.3.4.3.2" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4.3.1.cmml"><mn id="S4.SS2.SSS0.Px3.p1.5.m3.1.1" xref="S4.SS2.SSS0.Px3.p1.5.m3.1.1.cmml">1</mn><mo id="S4.SS2.SSS0.Px3.p1.5.m3.3.4.3.2.1" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4.3.1.cmml">,</mo><mn id="S4.SS2.SSS0.Px3.p1.5.m3.2.2" xref="S4.SS2.SSS0.Px3.p1.5.m3.2.2.cmml">2</mn><mo id="S4.SS2.SSS0.Px3.p1.5.m3.3.4.3.2.2" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4.3.1.cmml">,</mo><mn id="S4.SS2.SSS0.Px3.p1.5.m3.3.3" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.3.cmml">4</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px3.p1.5.m3.3b"><apply id="S4.SS2.SSS0.Px3.p1.5.m3.3.4.cmml" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4"><eq id="S4.SS2.SSS0.Px3.p1.5.m3.3.4.1.cmml" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4.1"></eq><ci id="S4.SS2.SSS0.Px3.p1.5.m3.3.4.2.cmml" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4.2">𝑁</ci><list id="S4.SS2.SSS0.Px3.p1.5.m3.3.4.3.1.cmml" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.4.3.2"><cn id="S4.SS2.SSS0.Px3.p1.5.m3.1.1.cmml" type="integer" xref="S4.SS2.SSS0.Px3.p1.5.m3.1.1">1</cn><cn id="S4.SS2.SSS0.Px3.p1.5.m3.2.2.cmml" type="integer" xref="S4.SS2.SSS0.Px3.p1.5.m3.2.2">2</cn><cn id="S4.SS2.SSS0.Px3.p1.5.m3.3.3.cmml" type="integer" xref="S4.SS2.SSS0.Px3.p1.5.m3.3.3">4</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px3.p1.5.m3.3c">N=1,2,4</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px3.p1.5.m3.3d">italic_N = 1 , 2 , 4</annotation></semantics></math>. </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Joint multi-spectral modeling.</h4> <div class="ltx_para" id="S4.SS2.SSS0.Px4.p1"> In ASDI, the spectral channels are related by a homothety, as given in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S3.E2" title="Equation 2 ‣ 3.2 Angular Spectral Differential Imaging (ASDI) ‣ 3 Image formation models ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">2</a>). We propose to leverage this relationship, and model: <table class="ltx_equation ltx_eqn_table" id="S4.E17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall(j,c,t),\quad\mathbf{A}\,{\beta_{c,j}^{-1}}\,{\mathbf{D}}_{\lambda_{0}/% \lambda_{c}}(\boldsymbol{y}_{c,t})^{(j)}\sim\mathcal{N}(\boldsymbol{m}_{j},% \mathbf{C}_{j})\,." class="ltx_Math" display="block" id="S4.E17.m1.9"><semantics id="S4.E17.m1.9a"><mrow id="S4.E17.m1.9.9.1" xref="S4.E17.m1.9.9.1.1.cmml"><mrow id="S4.E17.m1.9.9.1.1" xref="S4.E17.m1.9.9.1.1.cmml"><mrow id="S4.E17.m1.9.9.1.1.2.2" xref="S4.E17.m1.9.9.1.1.2.3.cmml"><mrow id="S4.E17.m1.9.9.1.1.1.1.1" xref="S4.E17.m1.9.9.1.1.1.1.1.cmml"><mo id="S4.E17.m1.9.9.1.1.1.1.1.1" xref="S4.E17.m1.9.9.1.1.1.1.1.1.cmml">∀</mo><mrow id="S4.E17.m1.9.9.1.1.1.1.1.2.2" xref="S4.E17.m1.9.9.1.1.1.1.1.2.1.cmml"><mo id="S4.E17.m1.9.9.1.1.1.1.1.2.2.1" stretchy="false" xref="S4.E17.m1.9.9.1.1.1.1.1.2.1.cmml">(</mo><mi id="S4.E17.m1.6.6" xref="S4.E17.m1.6.6.cmml">j</mi><mo id="S4.E17.m1.9.9.1.1.1.1.1.2.2.2" xref="S4.E17.m1.9.9.1.1.1.1.1.2.1.cmml">,</mo><mi id="S4.E17.m1.7.7" xref="S4.E17.m1.7.7.cmml">c</mi><mo id="S4.E17.m1.9.9.1.1.1.1.1.2.2.3" xref="S4.E17.m1.9.9.1.1.1.1.1.2.1.cmml">,</mo><mi id="S4.E17.m1.8.8" xref="S4.E17.m1.8.8.cmml">t</mi><mo id="S4.E17.m1.9.9.1.1.1.1.1.2.2.4" stretchy="false" xref="S4.E17.m1.9.9.1.1.1.1.1.2.1.cmml">)</mo></mrow></mrow><mo id="S4.E17.m1.9.9.1.1.2.2.3" rspace="1.167em" xref="S4.E17.m1.9.9.1.1.2.3.cmml">,</mo><mrow id="S4.E17.m1.9.9.1.1.2.2.2" xref="S4.E17.m1.9.9.1.1.2.2.2.cmml"><mi id="S4.E17.m1.9.9.1.1.2.2.2.3" xref="S4.E17.m1.9.9.1.1.2.2.2.3.cmml">𝐀</mi><mo id="S4.E17.m1.9.9.1.1.2.2.2.2" lspace="0.170em" xref="S4.E17.m1.9.9.1.1.2.2.2.2.cmml">⁢</mo><msubsup id="S4.E17.m1.9.9.1.1.2.2.2.4" xref="S4.E17.m1.9.9.1.1.2.2.2.4.cmml"><mi id="S4.E17.m1.9.9.1.1.2.2.2.4.2.2" xref="S4.E17.m1.9.9.1.1.2.2.2.4.2.2.cmml">β</mi><mrow id="S4.E17.m1.2.2.2.4" xref="S4.E17.m1.2.2.2.3.cmml"><mi id="S4.E17.m1.1.1.1.1" xref="S4.E17.m1.1.1.1.1.cmml">c</mi><mo id="S4.E17.m1.2.2.2.4.1" xref="S4.E17.m1.2.2.2.3.cmml">,</mo><mi id="S4.E17.m1.2.2.2.2" xref="S4.E17.m1.2.2.2.2.cmml">j</mi></mrow><mrow id="S4.E17.m1.9.9.1.1.2.2.2.4.3" xref="S4.E17.m1.9.9.1.1.2.2.2.4.3.cmml"><mo id="S4.E17.m1.9.9.1.1.2.2.2.4.3a" xref="S4.E17.m1.9.9.1.1.2.2.2.4.3.cmml">−</mo><mn id="S4.E17.m1.9.9.1.1.2.2.2.4.3.2" xref="S4.E17.m1.9.9.1.1.2.2.2.4.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.E17.m1.9.9.1.1.2.2.2.2a" xref="S4.E17.m1.9.9.1.1.2.2.2.2.cmml">⁢</mo><msub id="S4.E17.m1.9.9.1.1.2.2.2.5" xref="S4.E17.m1.9.9.1.1.2.2.2.5.cmml"><mi id="S4.E17.m1.9.9.1.1.2.2.2.5.2" xref="S4.E17.m1.9.9.1.1.2.2.2.5.2.cmml">𝐃</mi><mrow id="S4.E17.m1.9.9.1.1.2.2.2.5.3" xref="S4.E17.m1.9.9.1.1.2.2.2.5.3.cmml"><msub id="S4.E17.m1.9.9.1.1.2.2.2.5.3.2" xref="S4.E17.m1.9.9.1.1.2.2.2.5.3.2.cmml"><mi 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id="S4.E17.m1.9.9.1.1.4.4" xref="S4.E17.m1.9.9.1.1.4.4.cmml">𝒩</mi><mo id="S4.E17.m1.9.9.1.1.4.3" xref="S4.E17.m1.9.9.1.1.4.3.cmml">⁢</mo><mrow id="S4.E17.m1.9.9.1.1.4.2.2" xref="S4.E17.m1.9.9.1.1.4.2.3.cmml"><mo id="S4.E17.m1.9.9.1.1.4.2.2.3" stretchy="false" xref="S4.E17.m1.9.9.1.1.4.2.3.cmml">(</mo><msub id="S4.E17.m1.9.9.1.1.3.1.1.1" xref="S4.E17.m1.9.9.1.1.3.1.1.1.cmml"><mi id="S4.E17.m1.9.9.1.1.3.1.1.1.2" xref="S4.E17.m1.9.9.1.1.3.1.1.1.2.cmml">𝒎</mi><mi id="S4.E17.m1.9.9.1.1.3.1.1.1.3" xref="S4.E17.m1.9.9.1.1.3.1.1.1.3.cmml">j</mi></msub><mo id="S4.E17.m1.9.9.1.1.4.2.2.4" xref="S4.E17.m1.9.9.1.1.4.2.3.cmml">,</mo><msub id="S4.E17.m1.9.9.1.1.4.2.2.2" xref="S4.E17.m1.9.9.1.1.4.2.2.2.cmml"><mi id="S4.E17.m1.9.9.1.1.4.2.2.2.2" xref="S4.E17.m1.9.9.1.1.4.2.2.2.2.cmml">𝐂</mi><mi id="S4.E17.m1.9.9.1.1.4.2.2.2.3" xref="S4.E17.m1.9.9.1.1.4.2.2.2.3.cmml">j</mi></msub><mo id="S4.E17.m1.9.9.1.1.4.2.2.5" stretchy="false" xref="S4.E17.m1.9.9.1.1.4.2.3.cmml">)</mo></mrow></mrow></mrow><mo 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encoding="application/x-tex" id="S4.SS2.SSS0.Px4.p1.4.m4.4c">\{{\mathbf{D}}_{\lambda_{0}/\lambda_{c}}(\boldsymbol{y}_{c,t})^{(j)}\}_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px4.p1.4.m4.4d">{ bold_D start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_y start_POSTSUBSCRIPT italic_c , italic_t end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. </div> </section> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> 4.3 End-to-end trainable approach</h3> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Problem statement.</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px1.p1"> We propose to combine our statistical model on the nuisance component with a learnable prior on the exoplanet signals. 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}))+\psi_{\nu}(\boldsymbol{\alpha})\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E18.m1.4d">over^ start_ARG bold_italic_α end_ARG = start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT bold_italic_α ∈ blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_φ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( bold_italic_α , italic_χ ( bold_italic_y ) ) + italic_ψ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ( bold_italic_α ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(18)</td> </tr></tbody> </table> where <math alttext="\varphi" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px1.p1.1.m1.1a"><mi id="S4.SS3.SSS0.Px1.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p1.1.m1.1b"><ci id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.1.m1.1d">italic_φ</annotation></semantics></math> is the data-fitting term, corresponding to the negative log-likelihood of our statistical model of the nuisance component described in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1" title="4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.1</a>, and <math alttext="\psi" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p1.2.m2.1"><semantics id="S4.SS3.SSS0.Px1.p1.2.m2.1a"><mi id="S4.SS3.SSS0.Px1.p1.2.m2.1.1" xref="S4.SS3.SSS0.Px1.p1.2.m2.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p1.2.m2.1b"><ci id="S4.SS3.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.2.m2.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.2.m2.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.2.m2.1d">italic_ψ</annotation></semantics></math> is a prior on exoplanet signals. We denote by <math alttext="\theta,\nu" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p1.3.m3.2"><semantics id="S4.SS3.SSS0.Px1.p1.3.m3.2a"><mrow id="S4.SS3.SSS0.Px1.p1.3.m3.2.3.2" xref="S4.SS3.SSS0.Px1.p1.3.m3.2.3.1.cmml"><mi id="S4.SS3.SSS0.Px1.p1.3.m3.1.1" xref="S4.SS3.SSS0.Px1.p1.3.m3.1.1.cmml">θ</mi><mo id="S4.SS3.SSS0.Px1.p1.3.m3.2.3.2.1" xref="S4.SS3.SSS0.Px1.p1.3.m3.2.3.1.cmml">,</mo><mi id="S4.SS3.SSS0.Px1.p1.3.m3.2.2" xref="S4.SS3.SSS0.Px1.p1.3.m3.2.2.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p1.3.m3.2b"><list id="S4.SS3.SSS0.Px1.p1.3.m3.2.3.1.cmml" xref="S4.SS3.SSS0.Px1.p1.3.m3.2.3.2"><ci id="S4.SS3.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.3.m3.1.1">𝜃</ci><ci id="S4.SS3.SSS0.Px1.p1.3.m3.2.2.cmml" xref="S4.SS3.SSS0.Px1.p1.3.m3.2.2">𝜈</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.3.m3.2c">\theta,\nu</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.3.m3.2d">italic_θ , italic_ν</annotation></semantics></math> the learnable parameters associated with these terms, and by <math alttext="\chi(\boldsymbol{y})" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p1.4.m4.1"><semantics id="S4.SS3.SSS0.Px1.p1.4.m4.1a"><mrow id="S4.SS3.SSS0.Px1.p1.4.m4.1.2" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.2.cmml"><mi id="S4.SS3.SSS0.Px1.p1.4.m4.1.2.2" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.2.2.cmml">χ</mi><mo id="S4.SS3.SSS0.Px1.p1.4.m4.1.2.1" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.SS3.SSS0.Px1.p1.4.m4.1.2.3.2" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.2.cmml"><mo id="S4.SS3.SSS0.Px1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.2.cmml">(</mo><mi id="S4.SS3.SSS0.Px1.p1.4.m4.1.1" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.1.cmml">𝒚</mi><mo id="S4.SS3.SSS0.Px1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p1.4.m4.1b"><apply id="S4.SS3.SSS0.Px1.p1.4.m4.1.2.cmml" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.2"><times id="S4.SS3.SSS0.Px1.p1.4.m4.1.2.1.cmml" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.2.1"></times><ci id="S4.SS3.SSS0.Px1.p1.4.m4.1.2.2.cmml" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.2.2">𝜒</ci><ci id="S4.SS3.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.4.m4.1.1">𝒚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.4.m4.1c">\chi(\boldsymbol{y})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.4.m4.1d">italic_χ ( bold_italic_y )</annotation></semantics></math> the parameters estimated for each observation. 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xref="S4.SS3.SSS0.Px1.p1.5.m5.6.6.3.3.3.2.2">𝛽</ci></apply><list id="S4.SS3.SSS0.Px1.p1.5.m5.2.2.2.3.cmml" xref="S4.SS3.SSS0.Px1.p1.5.m5.2.2.2.4"><ci id="S4.SS3.SSS0.Px1.p1.5.m5.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.5.m5.1.1.1.1">𝑐</ci><ci id="S4.SS3.SSS0.Px1.p1.5.m5.2.2.2.2.cmml" xref="S4.SS3.SSS0.Px1.p1.5.m5.2.2.2.2">𝑗</ci></list></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.5.m5.6c">\chi(\boldsymbol{y})=\{\widehat{\boldsymbol{m}}_{j},\widehat{\mathbf{C}}_{j},% \widehat{\beta}_{c,j}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.5.m5.6d">italic_χ ( bold_italic_y ) = { over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , over^ start_ARG italic_β end_ARG start_POSTSUBSCRIPT italic_c , italic_j end_POSTSUBSCRIPT }</annotation></semantics></math>, <math 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xref="S4.SS3.SSS0.Px1.p1.6.m6.2.2.4">𝜃</ci><apply id="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.3.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2">subscript</csymbol><set id="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.3.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.2"><apply id="S4.SS3.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.2">𝐀</ci><ci id="S4.SS3.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.3">𝑗</ci></apply><apply id="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.2.2.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.2.2.1.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.2.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.2.2.2.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.2.2.2">𝑤</ci><ci id="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.2.2.3.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.2.2.2.3">𝑗</ci></apply></set><ci id="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.4.cmml" xref="S4.SS3.SSS0.Px1.p1.6.m6.2.2.2.4">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.6.m6.2c">\theta=\{\mathbf{A}_{j},w_{j}\}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.6.m6.2d">italic_θ = { bold_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\nu" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p1.7.m7.1"><semantics id="S4.SS3.SSS0.Px1.p1.7.m7.1a"><mi id="S4.SS3.SSS0.Px1.p1.7.m7.1.1" xref="S4.SS3.SSS0.Px1.p1.7.m7.1.1.cmml">ν</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p1.7.m7.1b"><ci id="S4.SS3.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.7.m7.1.1">𝜈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.7.m7.1c">\nu</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.7.m7.1d">italic_ν</annotation></semantics></math> corresponds to the parameters of the neural network implementing <math alttext="\psi" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p1.8.m8.1"><semantics id="S4.SS3.SSS0.Px1.p1.8.m8.1a"><mi id="S4.SS3.SSS0.Px1.p1.8.m8.1.1" xref="S4.SS3.SSS0.Px1.p1.8.m8.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p1.8.m8.1b"><ci id="S4.SS3.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.8.m8.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.8.m8.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.8.m8.1d">italic_ψ</annotation></semantics></math>. </div> </section> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Detection by denoising.</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px2.p1"> We propose a two-step approach to obtain the detection score corresponding to Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.E18" title="Equation 18 ‣ Problem statement. ‣ 4.3 End-to-end trainable approach ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">18</a>). First, we compute the detection score corresponding to the statistical model <math alttext="\varphi" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px2.p1.1.m1.1a"><mi id="S4.SS3.SSS0.Px2.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.1.m1.1b"><ci id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.1.m1.1d">italic_φ</annotation></semantics></math>, which admits a fast approximation denoted by <math alttext="\boldsymbol{\gamma}\in\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.2.m2.1"><semantics id="S4.SS3.SSS0.Px2.p1.2.m2.1a"><mrow id="S4.SS3.SSS0.Px2.p1.2.m2.1.1" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.2" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.2.cmml">𝜸</mi><mo id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.1" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.1.cmml">∈</mo><msup id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.cmml"><mi id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.2" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.cmml"><mi id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.2" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.2.cmml">H</mi><mo id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.1.cmml">×</mo><mi id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.3" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.3.cmml">W</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.2.m2.1b"><apply id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1"><in id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.1"></in><ci id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.2">𝜸</ci><apply id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3">superscript</csymbol><ci id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.2">ℝ</ci><apply id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3"><times id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.1.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.1"></times><ci id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.2.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.2">𝐻</ci><ci id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.3.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.3.3.3">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.2.m2.1c">\boldsymbol{\gamma}\in\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.2.m2.1d">bold_italic_γ ∈ blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> in the following, and given by Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E28" title="Equation 28 ‣ A.2 Dense detection map computation details ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">28</a>). We recall that under the null hypothesis, i.e., when no exoplanet is present, the elements of <math alttext="\boldsymbol{\gamma}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.3.m3.1"><semantics id="S4.SS3.SSS0.Px2.p1.3.m3.1a"><mi id="S4.SS3.SSS0.Px2.p1.3.m3.1.1" xref="S4.SS3.SSS0.Px2.p1.3.m3.1.1.cmml">𝜸</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.3.m3.1b"><ci id="S4.SS3.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.3.m3.1.1">𝜸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.3.m3.1c">\boldsymbol{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.3.m3.1d">bold_italic_γ</annotation></semantics></math> follow a Gaussian distribution <math alttext="\mathcal{N}(0,1)" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.4.m4.2"><semantics id="S4.SS3.SSS0.Px2.p1.4.m4.2a"><mrow id="S4.SS3.SSS0.Px2.p1.4.m4.2.3" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.2" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.2.cmml">𝒩</mi><mo id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.1" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.1.cmml">⁢</mo><mrow id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.2" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.1.cmml"><mo id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.1.cmml">(</mo><mn id="S4.SS3.SSS0.Px2.p1.4.m4.1.1" xref="S4.SS3.SSS0.Px2.p1.4.m4.1.1.cmml">0</mn><mo id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.2.2" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.1.cmml">,</mo><mn id="S4.SS3.SSS0.Px2.p1.4.m4.2.2" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.4.m4.2b"><apply id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.cmml" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3"><times id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.1.cmml" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.1"></times><ci id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.2.cmml" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.2">𝒩</ci><interval closure="open" id="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.1.cmml" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.3.3.2"><cn id="S4.SS3.SSS0.Px2.p1.4.m4.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p1.4.m4.1.1">0</cn><cn id="S4.SS3.SSS0.Px2.p1.4.m4.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p1.4.m4.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.4.m4.2c">\mathcal{N}(0,1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.4.m4.2d">caligraphic_N ( 0 , 1 )</annotation></semantics></math>. Therefore, extracting the signals of exoplanets corresponds to denoising <math alttext="\boldsymbol{\gamma}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.5.m5.1"><semantics id="S4.SS3.SSS0.Px2.p1.5.m5.1a"><mi id="S4.SS3.SSS0.Px2.p1.5.m5.1.1" xref="S4.SS3.SSS0.Px2.p1.5.m5.1.1.cmml">𝜸</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.5.m5.1b"><ci id="S4.SS3.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.5.m5.1.1">𝜸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.5.m5.1c">\boldsymbol{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.5.m5.1d">bold_italic_γ</annotation></semantics></math> to remove this background noise. We propose to achieve this step using a neural network, such that: <table class="ltx_equation ltx_eqn_table" id="S4.E19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\boldsymbol{\gamma}}=f_{\nu}\left(\widehat{\boldsymbol{\gamma}}% \right)\,," class="ltx_Math" display="block" id="S4.E19.m1.2"><semantics id="S4.E19.m1.2a"><mrow id="S4.E19.m1.2.2.1" xref="S4.E19.m1.2.2.1.1.cmml"><mrow id="S4.E19.m1.2.2.1.1" xref="S4.E19.m1.2.2.1.1.cmml"><mover accent="true" id="S4.E19.m1.2.2.1.1.2" xref="S4.E19.m1.2.2.1.1.2.cmml"><mi id="S4.E19.m1.2.2.1.1.2.2" xref="S4.E19.m1.2.2.1.1.2.2.cmml">𝜸</mi><mo id="S4.E19.m1.2.2.1.1.2.1" xref="S4.E19.m1.2.2.1.1.2.1.cmml">~</mo></mover><mo id="S4.E19.m1.2.2.1.1.1" xref="S4.E19.m1.2.2.1.1.1.cmml">=</mo><mrow id="S4.E19.m1.2.2.1.1.3" xref="S4.E19.m1.2.2.1.1.3.cmml"><msub id="S4.E19.m1.2.2.1.1.3.2" xref="S4.E19.m1.2.2.1.1.3.2.cmml"><mi id="S4.E19.m1.2.2.1.1.3.2.2" xref="S4.E19.m1.2.2.1.1.3.2.2.cmml">f</mi><mi id="S4.E19.m1.2.2.1.1.3.2.3" xref="S4.E19.m1.2.2.1.1.3.2.3.cmml">ν</mi></msub><mo id="S4.E19.m1.2.2.1.1.3.1" xref="S4.E19.m1.2.2.1.1.3.1.cmml">⁢</mo><mrow id="S4.E19.m1.2.2.1.1.3.3.2" xref="S4.E19.m1.1.1.cmml"><mo id="S4.E19.m1.2.2.1.1.3.3.2.1" xref="S4.E19.m1.1.1.cmml">(</mo><mover accent="true" id="S4.E19.m1.1.1" xref="S4.E19.m1.1.1.cmml"><mi id="S4.E19.m1.1.1.2" xref="S4.E19.m1.1.1.2.cmml">𝜸</mi><mo id="S4.E19.m1.1.1.1" xref="S4.E19.m1.1.1.1.cmml">^</mo></mover><mo id="S4.E19.m1.2.2.1.1.3.3.2.2" rspace="0.170em" xref="S4.E19.m1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E19.m1.2.2.1.2" xref="S4.E19.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E19.m1.2b"><apply id="S4.E19.m1.2.2.1.1.cmml" xref="S4.E19.m1.2.2.1"><eq id="S4.E19.m1.2.2.1.1.1.cmml" xref="S4.E19.m1.2.2.1.1.1"></eq><apply id="S4.E19.m1.2.2.1.1.2.cmml" 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id="S4.E19.m1.2c">\widetilde{\boldsymbol{\gamma}}=f_{\nu}\left(\widehat{\boldsymbol{\gamma}}% \right)\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E19.m1.2d">over~ start_ARG bold_italic_γ end_ARG = italic_f start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ( over^ start_ARG bold_italic_γ end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(19)</td> </tr></tbody> </table> where <math alttext="f_{\nu}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.6.m1.1"><semantics id="S4.SS3.SSS0.Px2.p1.6.m1.1a"><msub id="S4.SS3.SSS0.Px2.p1.6.m1.1.1" xref="S4.SS3.SSS0.Px2.p1.6.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p1.6.m1.1.1.2" xref="S4.SS3.SSS0.Px2.p1.6.m1.1.1.2.cmml">f</mi><mi id="S4.SS3.SSS0.Px2.p1.6.m1.1.1.3" xref="S4.SS3.SSS0.Px2.p1.6.m1.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.6.m1.1b"><apply id="S4.SS3.SSS0.Px2.p1.6.m1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.6.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p1.6.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.6.m1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p1.6.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p1.6.m1.1.1.2">𝑓</ci><ci id="S4.SS3.SSS0.Px2.p1.6.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p1.6.m1.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.6.m1.1c">f_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.6.m1.1d">italic_f start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> is the denoiser implemented by the neural network, and <math alttext="\widetilde{\boldsymbol{\gamma}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.7.m2.1"><semantics id="S4.SS3.SSS0.Px2.p1.7.m2.1a"><mover accent="true" id="S4.SS3.SSS0.Px2.p1.7.m2.1.1" xref="S4.SS3.SSS0.Px2.p1.7.m2.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p1.7.m2.1.1.2" xref="S4.SS3.SSS0.Px2.p1.7.m2.1.1.2.cmml">𝜸</mi><mo id="S4.SS3.SSS0.Px2.p1.7.m2.1.1.1" xref="S4.SS3.SSS0.Px2.p1.7.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.7.m2.1b"><apply id="S4.SS3.SSS0.Px2.p1.7.m2.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.7.m2.1.1"><ci id="S4.SS3.SSS0.Px2.p1.7.m2.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.7.m2.1.1.1">~</ci><ci id="S4.SS3.SSS0.Px2.p1.7.m2.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p1.7.m2.1.1.2">𝜸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.7.m2.1c">\widetilde{\boldsymbol{\gamma}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.7.m2.1d">over~ start_ARG bold_italic_γ end_ARG</annotation></semantics></math> the final detection score. </div> </section> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Training objective.</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px3.p1"> We suppose that the denoised detection score can be decomposed similarly to the GLRT form provided in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.E10" title="Equation 10 ‣ Detection criterion at a given position. ‣ 4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">10</a>) for the statistical model. This leads to <math alttext="\widetilde{\boldsymbol{\gamma}}=\widetilde{\boldsymbol{\alpha}}/\widetilde{% \boldsymbol{\sigma}}_{e}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px3.p1.1.m1.1a"><mrow id="S4.SS3.SSS0.Px3.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.cmml"><mover accent="true" id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2.cmml"><mi id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2.2" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2.2.cmml">𝜸</mi><mo id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2.1" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2.1.cmml">~</mo></mover><mo id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.1" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.cmml"><mover accent="true" id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.cmml"><mi id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.2" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.2.cmml">𝜶</mi><mo 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xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.2"><ci id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.2.1">~</ci><ci id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.2.2">𝝈</ci></apply><ci id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.3.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.3">𝑒</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.1.m1.1c">\widetilde{\boldsymbol{\gamma}}=\widetilde{\boldsymbol{\alpha}}/\widetilde{% \boldsymbol{\sigma}}_{e}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.1.m1.1d">over~ start_ARG bold_italic_γ end_ARG = over~ start_ARG bold_italic_α end_ARG / over~ start_ARG bold_italic_σ end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\widetilde{\boldsymbol{\alpha}},\widetilde{\boldsymbol{\sigma}}" class="ltx_Math" display="inline" 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Additionally, we assume that the uncertainty <math alttext="\widetilde{\boldsymbol{\sigma}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.4.m4.1"><semantics id="S4.SS3.SSS0.Px3.p1.4.m4.1a"><mover accent="true" id="S4.SS3.SSS0.Px3.p1.4.m4.1.1" xref="S4.SS3.SSS0.Px3.p1.4.m4.1.1.cmml"><mi id="S4.SS3.SSS0.Px3.p1.4.m4.1.1.2" xref="S4.SS3.SSS0.Px3.p1.4.m4.1.1.2.cmml">𝝈</mi><mo id="S4.SS3.SSS0.Px3.p1.4.m4.1.1.1" xref="S4.SS3.SSS0.Px3.p1.4.m4.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.4.m4.1b"><apply id="S4.SS3.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.4.m4.1.1"><ci id="S4.SS3.SSS0.Px3.p1.4.m4.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.4.m4.1.1.1">~</ci><ci id="S4.SS3.SSS0.Px3.p1.4.m4.1.1.2.cmml" xref="S4.SS3.SSS0.Px3.p1.4.m4.1.1.2">𝝈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.4.m4.1c">\widetilde{\boldsymbol{\sigma}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.4.m4.1d">over~ start_ARG bold_italic_σ end_ARG</annotation></semantics></math> remains unaffected by the neural network, as it is primarily driven by the high variability of the speckles already captured by the statistical model. Consequently, both components can be expressed as: <table class="ltx_equation ltx_eqn_table" id="S4.E20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\boldsymbol{\alpha}}=f_{\nu}(\widehat{\boldsymbol{\alpha}}/\widehat% {\boldsymbol{\sigma}})\times\widehat{\boldsymbol{\sigma}},\,\quad\widetilde{% \boldsymbol{\sigma}}=\widehat{\boldsymbol{\sigma}}\,." class="ltx_Math" display="block" id="S4.E20.m1.1"><semantics id="S4.E20.m1.1a"><mrow id="S4.E20.m1.1.1.1"><mrow id="S4.E20.m1.1.1.1.1.2" xref="S4.E20.m1.1.1.1.1.3.cmml"><mrow id="S4.E20.m1.1.1.1.1.1.1" xref="S4.E20.m1.1.1.1.1.1.1.cmml"><mover accent="true" id="S4.E20.m1.1.1.1.1.1.1.3" xref="S4.E20.m1.1.1.1.1.1.1.3.cmml"><mi id="S4.E20.m1.1.1.1.1.1.1.3.2" xref="S4.E20.m1.1.1.1.1.1.1.3.2.cmml">𝜶</mi><mo id="S4.E20.m1.1.1.1.1.1.1.3.1" xref="S4.E20.m1.1.1.1.1.1.1.3.1.cmml">~</mo></mover><mo 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xref="S4.E21.m1.3.3.1.1.2.3.2.1">~</ci><ci id="S4.E21.m1.3.3.1.1.2.3.2.2.cmml" xref="S4.E21.m1.3.3.1.1.2.3.2.2">𝝈</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E21.m1.3c">\mathcal{L}(\widetilde{\boldsymbol{\alpha}},\widetilde{\boldsymbol{\sigma}},% \boldsymbol{\alpha}_{\text{gt}})=0.5\,(\widetilde{\boldsymbol{\alpha}}-% \boldsymbol{\alpha}_{\text{gt}})^{2}/\widetilde{\boldsymbol{\sigma}}^{2}+\log% \widetilde{\boldsymbol{\sigma}}\,.</annotation><annotation encoding="application/x-llamapun" id="S4.E21.m1.3d">caligraphic_L ( over~ start_ARG bold_italic_α end_ARG , over~ start_ARG bold_italic_σ end_ARG , bold_italic_α start_POSTSUBSCRIPT gt end_POSTSUBSCRIPT ) = 0.5 ( over~ start_ARG bold_italic_α end_ARG - bold_italic_α start_POSTSUBSCRIPT gt end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / over~ start_ARG bold_italic_σ end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + roman_log over~ start_ARG bold_italic_σ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(21)</td> </tr></tbody> </table> </div> </section> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Model ensembling.</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px4.p1"> To improve robustness, we combine outputs from multiple trained models. Specifically, given outputs <math alttext="\widetilde{\boldsymbol{\alpha}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px4.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px4.p1.1.m1.1a"><mover accent="true" id="S4.SS3.SSS0.Px4.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px4.p1.1.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px4.p1.1.m1.1.1.2" xref="S4.SS3.SSS0.Px4.p1.1.m1.1.1.2.cmml">𝜶</mi><mo id="S4.SS3.SSS0.Px4.p1.1.m1.1.1.1" xref="S4.SS3.SSS0.Px4.p1.1.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px4.p1.1.m1.1b"><apply id="S4.SS3.SSS0.Px4.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.1.m1.1.1"><ci id="S4.SS3.SSS0.Px4.p1.1.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.1.m1.1.1.1">~</ci><ci id="S4.SS3.SSS0.Px4.p1.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px4.p1.1.m1.1.1.2">𝜶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px4.p1.1.m1.1c">\widetilde{\boldsymbol{\alpha}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px4.p1.1.m1.1d">over~ start_ARG bold_italic_α end_ARG</annotation></semantics></math> and <math alttext="\widetilde{\boldsymbol{\sigma}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px4.p1.2.m2.1"><semantics id="S4.SS3.SSS0.Px4.p1.2.m2.1a"><mover accent="true" id="S4.SS3.SSS0.Px4.p1.2.m2.1.1" xref="S4.SS3.SSS0.Px4.p1.2.m2.1.1.cmml"><mi id="S4.SS3.SSS0.Px4.p1.2.m2.1.1.2" xref="S4.SS3.SSS0.Px4.p1.2.m2.1.1.2.cmml">𝝈</mi><mo id="S4.SS3.SSS0.Px4.p1.2.m2.1.1.1" xref="S4.SS3.SSS0.Px4.p1.2.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px4.p1.2.m2.1b"><apply id="S4.SS3.SSS0.Px4.p1.2.m2.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.2.m2.1.1"><ci id="S4.SS3.SSS0.Px4.p1.2.m2.1.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.2.m2.1.1.1">~</ci><ci id="S4.SS3.SSS0.Px4.p1.2.m2.1.1.2.cmml" xref="S4.SS3.SSS0.Px4.p1.2.m2.1.1.2">𝝈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px4.p1.2.m2.1c">\widetilde{\boldsymbol{\sigma}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px4.p1.2.m2.1d">over~ start_ARG bold_italic_σ end_ARG</annotation></semantics></math>, we can define <math alttext="\widetilde{\boldsymbol{a}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px4.p1.3.m3.1"><semantics id="S4.SS3.SSS0.Px4.p1.3.m3.1a"><mover accent="true" id="S4.SS3.SSS0.Px4.p1.3.m3.1.1" xref="S4.SS3.SSS0.Px4.p1.3.m3.1.1.cmml"><mi id="S4.SS3.SSS0.Px4.p1.3.m3.1.1.2" xref="S4.SS3.SSS0.Px4.p1.3.m3.1.1.2.cmml">𝒂</mi><mo id="S4.SS3.SSS0.Px4.p1.3.m3.1.1.1" xref="S4.SS3.SSS0.Px4.p1.3.m3.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px4.p1.3.m3.1b"><apply id="S4.SS3.SSS0.Px4.p1.3.m3.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.3.m3.1.1"><ci id="S4.SS3.SSS0.Px4.p1.3.m3.1.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.3.m3.1.1.1">~</ci><ci id="S4.SS3.SSS0.Px4.p1.3.m3.1.1.2.cmml" 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xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.1"></eq><apply id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.2.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.2"><ci id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.2.1">~</ci><ci id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.2.2">𝝈</ci></apply><apply id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3"><divide id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.1"></divide><cn id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.2.cmml" type="integer" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.2">1</cn><apply id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3"><root id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3a.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3"></root><apply id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3.2.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3.2"><ci id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3.2.1.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3.2.1">~</ci><ci id="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3.2.2.cmml" xref="S4.SS3.SSS0.Px4.p1.5.m5.1.1.3.3.2.2">𝒂</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px4.p1.5.m5.1c">\widetilde{\boldsymbol{\sigma}}=1/\sqrt{\widetilde{\boldsymbol{a}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px4.p1.5.m5.1d">over~ start_ARG bold_italic_σ end_ARG = 1 / square-root start_ARG over~ start_ARG bold_italic_a end_ARG end_ARG</annotation></semantics></math> and <math alttext="\widetilde{\boldsymbol{\alpha}}=\widetilde{\boldsymbol{b}}/\widetilde{% \boldsymbol{a}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px4.p1.6.m6.1"><semantics id="S4.SS3.SSS0.Px4.p1.6.m6.1a"><mrow id="S4.SS3.SSS0.Px4.p1.6.m6.1.1" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.cmml"><mover accent="true" id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.cmml"><mi id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.2" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.2.cmml">𝜶</mi><mo id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.1" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.1.cmml">~</mo></mover><mo id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.1" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.cmml"><mover accent="true" id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.cmml"><mi id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.2" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.2.cmml">𝒃</mi><mo id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.1" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.1.cmml">~</mo></mover><mo id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.1" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.1.cmml">/</mo><mover accent="true" id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.cmml"><mi id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.2" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.2.cmml">𝒂</mi><mo id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.1" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px4.p1.6.m6.1b"><apply id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1"><eq id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.1"></eq><apply id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2"><ci id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.1">~</ci><ci id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.2.2">𝜶</ci></apply><apply id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3"><divide id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.1"></divide><apply id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2"><ci id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.1.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.1">~</ci><ci id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.2.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.2.2">𝒃</ci></apply><apply id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3"><ci id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.1.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.1">~</ci><ci id="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.2.cmml" xref="S4.SS3.SSS0.Px4.p1.6.m6.1.1.3.3.2">𝒂</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px4.p1.6.m6.1c">\widetilde{\boldsymbol{\alpha}}=\widetilde{\boldsymbol{b}}/\widetilde{% \boldsymbol{a}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px4.p1.6.m6.1d">over~ start_ARG bold_italic_α end_ARG = over~ start_ARG bold_italic_b end_ARG / over~ start_ARG bold_italic_a end_ARG</annotation></semantics></math>. For <math alttext="Q" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px4.p1.7.m7.1"><semantics id="S4.SS3.SSS0.Px4.p1.7.m7.1a"><mi id="S4.SS3.SSS0.Px4.p1.7.m7.1.1" xref="S4.SS3.SSS0.Px4.p1.7.m7.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px4.p1.7.m7.1b"><ci id="S4.SS3.SSS0.Px4.p1.7.m7.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.7.m7.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px4.p1.7.m7.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px4.p1.7.m7.1d">italic_Q</annotation></semantics></math> models indexed by <math alttext="q" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px4.p1.8.m8.1"><semantics id="S4.SS3.SSS0.Px4.p1.8.m8.1a"><mi id="S4.SS3.SSS0.Px4.p1.8.m8.1.1" xref="S4.SS3.SSS0.Px4.p1.8.m8.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px4.p1.8.m8.1b"><ci id="S4.SS3.SSS0.Px4.p1.8.m8.1.1.cmml" xref="S4.SS3.SSS0.Px4.p1.8.m8.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px4.p1.8.m8.1c">q</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px4.p1.8.m8.1d">italic_q</annotation></semantics></math>, the outputs are aggregated as follows: <table class="ltx_equation ltx_eqn_table" id="S4.E22"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\boldsymbol{a}}_{Q}=\sum\nolimits_{q}\widetilde{\boldsymbol{a}}_{q}% /Q,\quad\widetilde{\boldsymbol{b}}_{Q}=\sum\nolimits_{q}\widetilde{\boldsymbol% {b}}_{q}/Q\,." class="ltx_Math" display="block" id="S4.E22.m1.1"><semantics id="S4.E22.m1.1a"><mrow id="S4.E22.m1.1.1.1"><mrow id="S4.E22.m1.1.1.1.1.2" xref="S4.E22.m1.1.1.1.1.3.cmml"><mrow id="S4.E22.m1.1.1.1.1.1.1" xref="S4.E22.m1.1.1.1.1.1.1.cmml"><msub id="S4.E22.m1.1.1.1.1.1.1.2" xref="S4.E22.m1.1.1.1.1.1.1.2.cmml"><mover accent="true" id="S4.E22.m1.1.1.1.1.1.1.2.2" xref="S4.E22.m1.1.1.1.1.1.1.2.2.cmml"><mi id="S4.E22.m1.1.1.1.1.1.1.2.2.2" xref="S4.E22.m1.1.1.1.1.1.1.2.2.2.cmml">𝒂</mi><mo id="S4.E22.m1.1.1.1.1.1.1.2.2.1" xref="S4.E22.m1.1.1.1.1.1.1.2.2.1.cmml">~</mo></mover><mi id="S4.E22.m1.1.1.1.1.1.1.2.3" xref="S4.E22.m1.1.1.1.1.1.1.2.3.cmml">Q</mi></msub><mo id="S4.E22.m1.1.1.1.1.1.1.1" rspace="0.111em" xref="S4.E22.m1.1.1.1.1.1.1.1.cmml">=</mo><mrow id="S4.E22.m1.1.1.1.1.1.1.3" xref="S4.E22.m1.1.1.1.1.1.1.3.cmml"><msub id="S4.E22.m1.1.1.1.1.1.1.3.1" xref="S4.E22.m1.1.1.1.1.1.1.3.1.cmml"><mo id="S4.E22.m1.1.1.1.1.1.1.3.1.2" xref="S4.E22.m1.1.1.1.1.1.1.3.1.2.cmml">∑</mo><mi id="S4.E22.m1.1.1.1.1.1.1.3.1.3" xref="S4.E22.m1.1.1.1.1.1.1.3.1.3.cmml">q</mi></msub><mrow id="S4.E22.m1.1.1.1.1.1.1.3.2" xref="S4.E22.m1.1.1.1.1.1.1.3.2.cmml"><msub id="S4.E22.m1.1.1.1.1.1.1.3.2.2" xref="S4.E22.m1.1.1.1.1.1.1.3.2.2.cmml"><mover accent="true" id="S4.E22.m1.1.1.1.1.1.1.3.2.2.2" xref="S4.E22.m1.1.1.1.1.1.1.3.2.2.2.cmml"><mi id="S4.E22.m1.1.1.1.1.1.1.3.2.2.2.2" xref="S4.E22.m1.1.1.1.1.1.1.3.2.2.2.2.cmml">𝒂</mi><mo id="S4.E22.m1.1.1.1.1.1.1.3.2.2.2.1" xref="S4.E22.m1.1.1.1.1.1.1.3.2.2.2.1.cmml">~</mo></mover><mi id="S4.E22.m1.1.1.1.1.1.1.3.2.2.3" xref="S4.E22.m1.1.1.1.1.1.1.3.2.2.3.cmml">q</mi></msub><mo id="S4.E22.m1.1.1.1.1.1.1.3.2.1" xref="S4.E22.m1.1.1.1.1.1.1.3.2.1.cmml">/</mo><mi id="S4.E22.m1.1.1.1.1.1.1.3.2.3" xref="S4.E22.m1.1.1.1.1.1.1.3.2.3.cmml">Q</mi></mrow></mrow></mrow><mo id="S4.E22.m1.1.1.1.1.2.3" rspace="1.167em" xref="S4.E22.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S4.E22.m1.1.1.1.1.2.2" xref="S4.E22.m1.1.1.1.1.2.2.cmml"><msub id="S4.E22.m1.1.1.1.1.2.2.2" xref="S4.E22.m1.1.1.1.1.2.2.2.cmml"><mover accent="true" id="S4.E22.m1.1.1.1.1.2.2.2.2" xref="S4.E22.m1.1.1.1.1.2.2.2.2.cmml"><mi id="S4.E22.m1.1.1.1.1.2.2.2.2.2" xref="S4.E22.m1.1.1.1.1.2.2.2.2.2.cmml">𝒃</mi><mo id="S4.E22.m1.1.1.1.1.2.2.2.2.1" xref="S4.E22.m1.1.1.1.1.2.2.2.2.1.cmml">~</mo></mover><mi id="S4.E22.m1.1.1.1.1.2.2.2.3" xref="S4.E22.m1.1.1.1.1.2.2.2.3.cmml">Q</mi></msub><mo id="S4.E22.m1.1.1.1.1.2.2.1" rspace="0.111em" xref="S4.E22.m1.1.1.1.1.2.2.1.cmml">=</mo><mrow id="S4.E22.m1.1.1.1.1.2.2.3" xref="S4.E22.m1.1.1.1.1.2.2.3.cmml"><msub id="S4.E22.m1.1.1.1.1.2.2.3.1" xref="S4.E22.m1.1.1.1.1.2.2.3.1.cmml"><mo id="S4.E22.m1.1.1.1.1.2.2.3.1.2" xref="S4.E22.m1.1.1.1.1.2.2.3.1.2.cmml">∑</mo><mi id="S4.E22.m1.1.1.1.1.2.2.3.1.3" xref="S4.E22.m1.1.1.1.1.2.2.3.1.3.cmml">q</mi></msub><mrow id="S4.E22.m1.1.1.1.1.2.2.3.2" xref="S4.E22.m1.1.1.1.1.2.2.3.2.cmml"><msub id="S4.E22.m1.1.1.1.1.2.2.3.2.2" xref="S4.E22.m1.1.1.1.1.2.2.3.2.2.cmml"><mover accent="true" id="S4.E22.m1.1.1.1.1.2.2.3.2.2.2" xref="S4.E22.m1.1.1.1.1.2.2.3.2.2.2.cmml"><mi id="S4.E22.m1.1.1.1.1.2.2.3.2.2.2.2" xref="S4.E22.m1.1.1.1.1.2.2.3.2.2.2.2.cmml">𝒃</mi><mo id="S4.E22.m1.1.1.1.1.2.2.3.2.2.2.1" xref="S4.E22.m1.1.1.1.1.2.2.3.2.2.2.1.cmml">~</mo></mover><mi 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start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT over~ start_ARG bold_italic_a end_ARG start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT / italic_Q , over~ start_ARG bold_italic_b end_ARG start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT over~ start_ARG bold_italic_b end_ARG start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT / italic_Q .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(22)</td> </tr></tbody> </table> The combined detection score, <math alttext="\widetilde{\boldsymbol{\gamma}}_{Q}=\widetilde{\boldsymbol{b}}_{Q}/\sqrt{% \widetilde{\boldsymbol{a}}_{Q}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px4.p1.9.m1.1"><semantics id="S4.SS3.SSS0.Px4.p1.9.m1.1a"><mrow id="S4.SS3.SSS0.Px4.p1.9.m1.1.1" xref="S4.SS3.SSS0.Px4.p1.9.m1.1.1.cmml"><msub id="S4.SS3.SSS0.Px4.p1.9.m1.1.1.2" 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id="S4.SS3.SSS0.Px4.p1.9.m1.1.1.3.3.2.2.cmml" xref="S4.SS3.SSS0.Px4.p1.9.m1.1.1.3.3.2.2"><ci id="S4.SS3.SSS0.Px4.p1.9.m1.1.1.3.3.2.2.1.cmml" xref="S4.SS3.SSS0.Px4.p1.9.m1.1.1.3.3.2.2.1">~</ci><ci id="S4.SS3.SSS0.Px4.p1.9.m1.1.1.3.3.2.2.2.cmml" xref="S4.SS3.SSS0.Px4.p1.9.m1.1.1.3.3.2.2.2">𝒂</ci></apply><ci id="S4.SS3.SSS0.Px4.p1.9.m1.1.1.3.3.2.3.cmml" xref="S4.SS3.SSS0.Px4.p1.9.m1.1.1.3.3.2.3">𝑄</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px4.p1.9.m1.1c">\widetilde{\boldsymbol{\gamma}}_{Q}=\widetilde{\boldsymbol{b}}_{Q}/\sqrt{% \widetilde{\boldsymbol{a}}_{Q}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px4.p1.9.m1.1d">over~ start_ARG bold_italic_γ end_ARG start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = over~ start_ARG bold_italic_b end_ARG start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT / square-root start_ARG over~ start_ARG bold_italic_a end_ARG start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>, represents a weighted average of the individual detection scores. </div> </section> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px5"> <h4 class="ltx_title ltx_title_paragraph">Calibration.</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px5.p1"> The trained model needs to be calibrated in order to relate the detection score to a probability of false alarm. We follow the procedure outlined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib5" title="">5</a>]</cite>, and rely on a separate calibration dataset to estimate the cumulative distribution function of <math alttext="\widetilde{\boldsymbol{\gamma}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px5.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px5.p1.1.m1.1a"><mover accent="true" id="S4.SS3.SSS0.Px5.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px5.p1.1.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px5.p1.1.m1.1.1.2" xref="S4.SS3.SSS0.Px5.p1.1.m1.1.1.2.cmml">𝜸</mi><mo id="S4.SS3.SSS0.Px5.p1.1.m1.1.1.1" xref="S4.SS3.SSS0.Px5.p1.1.m1.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px5.p1.1.m1.1b"><apply id="S4.SS3.SSS0.Px5.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px5.p1.1.m1.1.1"><ci id="S4.SS3.SSS0.Px5.p1.1.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px5.p1.1.m1.1.1.1">~</ci><ci id="S4.SS3.SSS0.Px5.p1.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px5.p1.1.m1.1.1.2">𝜸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px5.p1.1.m1.1c">\widetilde{\boldsymbol{\gamma}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px5.p1.1.m1.1d">over~ start_ARG bold_italic_γ end_ARG</annotation></semantics></math> under the null hypothesis. Additional details are provided in the Supplementary Material. </div> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> 5 Experiments</h2> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> 5.1 Data, algorithms and evaluation protocol</h3> <figure class="ltx_table" id="S5.T1"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S5.T1.32.32" style="width:216.8pt;height:112.6pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T1.32.32.32"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T1.32.32.32.33.1"> <th class="ltx_td ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" id="S5.T1.32.32.32.33.1.1"></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" id="S5.T1.32.32.32.33.1.2">Spatial scales</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T1.32.32.32.33.1.3">N=1</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T1.32.32.32.33.1.4">N=1,2</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T1.32.32.32.33.1.5">N=1, 2, 4</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T1.4.4.4.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T1.4.4.4.4.5" rowspan="4"> ADI </th> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T1.1.1.1.1.1"><math alttext="8\times 8" class="ltx_Math" display="inline" id="S5.T1.1.1.1.1.1.m1.1"><semantics id="S5.T1.1.1.1.1.1.m1.1a"><mrow id="S5.T1.1.1.1.1.1.m1.1.1" xref="S5.T1.1.1.1.1.1.m1.1.1.cmml"><mn id="S5.T1.1.1.1.1.1.m1.1.1.2" xref="S5.T1.1.1.1.1.1.m1.1.1.2.cmml">8</mn><mo id="S5.T1.1.1.1.1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.T1.1.1.1.1.1.m1.1.1.1.cmml">×</mo><mn id="S5.T1.1.1.1.1.1.m1.1.1.3" xref="S5.T1.1.1.1.1.1.m1.1.1.3.cmml">8</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.1.1.1.1.1.m1.1b"><apply id="S5.T1.1.1.1.1.1.m1.1.1.cmml" xref="S5.T1.1.1.1.1.1.m1.1.1"><times id="S5.T1.1.1.1.1.1.m1.1.1.1.cmml" xref="S5.T1.1.1.1.1.1.m1.1.1.1"></times><cn id="S5.T1.1.1.1.1.1.m1.1.1.2.cmml" type="integer" xref="S5.T1.1.1.1.1.1.m1.1.1.2">8</cn><cn id="S5.T1.1.1.1.1.1.m1.1.1.3.cmml" type="integer" xref="S5.T1.1.1.1.1.1.m1.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.1.1.1.1.1.m1.1c">8\times 8</annotation><annotation encoding="application/x-llamapun" id="S5.T1.1.1.1.1.1.m1.1d">8 × 8</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.2.2.2.2.2"><math alttext="0.554\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.2.2.2.2.2.m1.1"><semantics id="S5.T1.2.2.2.2.2.m1.1a"><mrow id="S5.T1.2.2.2.2.2.m1.1.1" xref="S5.T1.2.2.2.2.2.m1.1.1.cmml"><mn id="S5.T1.2.2.2.2.2.m1.1.1.2" xref="S5.T1.2.2.2.2.2.m1.1.1.2.cmml">0.554</mn><mo id="S5.T1.2.2.2.2.2.m1.1.1.1" xref="S5.T1.2.2.2.2.2.m1.1.1.1.cmml">±</mo><mn id="S5.T1.2.2.2.2.2.m1.1.1.3" xref="S5.T1.2.2.2.2.2.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.2.2.2.2.2.m1.1b"><apply id="S5.T1.2.2.2.2.2.m1.1.1.cmml" xref="S5.T1.2.2.2.2.2.m1.1.1"><csymbol cd="latexml" id="S5.T1.2.2.2.2.2.m1.1.1.1.cmml" xref="S5.T1.2.2.2.2.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.2.2.2.2.2.m1.1.1.2.cmml" type="float" xref="S5.T1.2.2.2.2.2.m1.1.1.2">0.554</cn><cn id="S5.T1.2.2.2.2.2.m1.1.1.3.cmml" type="float" xref="S5.T1.2.2.2.2.2.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.2.2.2.2.2.m1.1c">0.554\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.2.2.2.2.2.m1.1d">0.554 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.3.3.3.3.3"><math alttext="0.571\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.3.3.3.3.3.m1.1"><semantics id="S5.T1.3.3.3.3.3.m1.1a"><mrow id="S5.T1.3.3.3.3.3.m1.1.1" xref="S5.T1.3.3.3.3.3.m1.1.1.cmml"><mn id="S5.T1.3.3.3.3.3.m1.1.1.2" xref="S5.T1.3.3.3.3.3.m1.1.1.2.cmml">0.571</mn><mo id="S5.T1.3.3.3.3.3.m1.1.1.1" xref="S5.T1.3.3.3.3.3.m1.1.1.1.cmml">±</mo><mn id="S5.T1.3.3.3.3.3.m1.1.1.3" xref="S5.T1.3.3.3.3.3.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.3.3.3.3.3.m1.1b"><apply id="S5.T1.3.3.3.3.3.m1.1.1.cmml" xref="S5.T1.3.3.3.3.3.m1.1.1"><csymbol cd="latexml" id="S5.T1.3.3.3.3.3.m1.1.1.1.cmml" xref="S5.T1.3.3.3.3.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.3.3.3.3.3.m1.1.1.2.cmml" type="float" xref="S5.T1.3.3.3.3.3.m1.1.1.2">0.571</cn><cn id="S5.T1.3.3.3.3.3.m1.1.1.3.cmml" type="float" xref="S5.T1.3.3.3.3.3.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.3.3.3.3.3.m1.1c">0.571\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.3.3.3.3.3.m1.1d">0.571 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.4.4.4.4.4"><math alttext="0.575\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.4.4.4.4.4.m1.1"><semantics id="S5.T1.4.4.4.4.4.m1.1a"><mrow id="S5.T1.4.4.4.4.4.m1.1.1" xref="S5.T1.4.4.4.4.4.m1.1.1.cmml"><mn id="S5.T1.4.4.4.4.4.m1.1.1.2" xref="S5.T1.4.4.4.4.4.m1.1.1.2.cmml">0.575</mn><mo id="S5.T1.4.4.4.4.4.m1.1.1.1" xref="S5.T1.4.4.4.4.4.m1.1.1.1.cmml">±</mo><mn id="S5.T1.4.4.4.4.4.m1.1.1.3" xref="S5.T1.4.4.4.4.4.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.4.4.4.4.4.m1.1b"><apply id="S5.T1.4.4.4.4.4.m1.1.1.cmml" xref="S5.T1.4.4.4.4.4.m1.1.1"><csymbol cd="latexml" id="S5.T1.4.4.4.4.4.m1.1.1.1.cmml" xref="S5.T1.4.4.4.4.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.4.4.4.4.4.m1.1.1.2.cmml" type="float" xref="S5.T1.4.4.4.4.4.m1.1.1.2">0.575</cn><cn id="S5.T1.4.4.4.4.4.m1.1.1.3.cmml" type="float" xref="S5.T1.4.4.4.4.4.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.4.4.4.4.4.m1.1c">0.575\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.4.4.4.4.4.m1.1d">0.575 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.8.8.8.8"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.5.5.5.5.1">+<math alttext="16\times 16" class="ltx_Math" display="inline" id="S5.T1.5.5.5.5.1.m1.1"><semantics id="S5.T1.5.5.5.5.1.m1.1a"><mrow id="S5.T1.5.5.5.5.1.m1.1.1" xref="S5.T1.5.5.5.5.1.m1.1.1.cmml"><mn id="S5.T1.5.5.5.5.1.m1.1.1.2" xref="S5.T1.5.5.5.5.1.m1.1.1.2.cmml">16</mn><mo id="S5.T1.5.5.5.5.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.T1.5.5.5.5.1.m1.1.1.1.cmml">×</mo><mn id="S5.T1.5.5.5.5.1.m1.1.1.3" xref="S5.T1.5.5.5.5.1.m1.1.1.3.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.5.5.5.5.1.m1.1b"><apply id="S5.T1.5.5.5.5.1.m1.1.1.cmml" xref="S5.T1.5.5.5.5.1.m1.1.1"><times id="S5.T1.5.5.5.5.1.m1.1.1.1.cmml" xref="S5.T1.5.5.5.5.1.m1.1.1.1"></times><cn id="S5.T1.5.5.5.5.1.m1.1.1.2.cmml" type="integer" xref="S5.T1.5.5.5.5.1.m1.1.1.2">16</cn><cn id="S5.T1.5.5.5.5.1.m1.1.1.3.cmml" type="integer" xref="S5.T1.5.5.5.5.1.m1.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.5.5.5.5.1.m1.1c">16\times 16</annotation><annotation encoding="application/x-llamapun" id="S5.T1.5.5.5.5.1.m1.1d">16 × 16</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center" id="S5.T1.6.6.6.6.2"><math alttext="0.561\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.6.6.6.6.2.m1.1"><semantics id="S5.T1.6.6.6.6.2.m1.1a"><mrow id="S5.T1.6.6.6.6.2.m1.1.1" xref="S5.T1.6.6.6.6.2.m1.1.1.cmml"><mn id="S5.T1.6.6.6.6.2.m1.1.1.2" xref="S5.T1.6.6.6.6.2.m1.1.1.2.cmml">0.561</mn><mo id="S5.T1.6.6.6.6.2.m1.1.1.1" xref="S5.T1.6.6.6.6.2.m1.1.1.1.cmml">±</mo><mn id="S5.T1.6.6.6.6.2.m1.1.1.3" xref="S5.T1.6.6.6.6.2.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.6.6.6.6.2.m1.1b"><apply id="S5.T1.6.6.6.6.2.m1.1.1.cmml" xref="S5.T1.6.6.6.6.2.m1.1.1"><csymbol cd="latexml" id="S5.T1.6.6.6.6.2.m1.1.1.1.cmml" xref="S5.T1.6.6.6.6.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.6.6.6.6.2.m1.1.1.2.cmml" type="float" xref="S5.T1.6.6.6.6.2.m1.1.1.2">0.561</cn><cn id="S5.T1.6.6.6.6.2.m1.1.1.3.cmml" type="float" xref="S5.T1.6.6.6.6.2.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.6.6.6.6.2.m1.1c">0.561\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.6.6.6.6.2.m1.1d">0.561 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.7.7.7.7.3"><math alttext="0.573\pm 0.004" class="ltx_Math" display="inline" id="S5.T1.7.7.7.7.3.m1.1"><semantics id="S5.T1.7.7.7.7.3.m1.1a"><mrow id="S5.T1.7.7.7.7.3.m1.1.1" xref="S5.T1.7.7.7.7.3.m1.1.1.cmml"><mn id="S5.T1.7.7.7.7.3.m1.1.1.2" xref="S5.T1.7.7.7.7.3.m1.1.1.2.cmml">0.573</mn><mo id="S5.T1.7.7.7.7.3.m1.1.1.1" xref="S5.T1.7.7.7.7.3.m1.1.1.1.cmml">±</mo><mn id="S5.T1.7.7.7.7.3.m1.1.1.3" xref="S5.T1.7.7.7.7.3.m1.1.1.3.cmml">0.004</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.7.7.7.7.3.m1.1b"><apply id="S5.T1.7.7.7.7.3.m1.1.1.cmml" xref="S5.T1.7.7.7.7.3.m1.1.1"><csymbol cd="latexml" id="S5.T1.7.7.7.7.3.m1.1.1.1.cmml" xref="S5.T1.7.7.7.7.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.7.7.7.7.3.m1.1.1.2.cmml" type="float" xref="S5.T1.7.7.7.7.3.m1.1.1.2">0.573</cn><cn id="S5.T1.7.7.7.7.3.m1.1.1.3.cmml" type="float" xref="S5.T1.7.7.7.7.3.m1.1.1.3">0.004</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.7.7.7.7.3.m1.1c">0.573\pm 0.004</annotation><annotation encoding="application/x-llamapun" id="S5.T1.7.7.7.7.3.m1.1d">0.573 ± 0.004</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.8.8.8.8.4"><math alttext="0.579\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.8.8.8.8.4.m1.1"><semantics id="S5.T1.8.8.8.8.4.m1.1a"><mrow id="S5.T1.8.8.8.8.4.m1.1.1" xref="S5.T1.8.8.8.8.4.m1.1.1.cmml"><mn id="S5.T1.8.8.8.8.4.m1.1.1.2" xref="S5.T1.8.8.8.8.4.m1.1.1.2.cmml">0.579</mn><mo id="S5.T1.8.8.8.8.4.m1.1.1.1" xref="S5.T1.8.8.8.8.4.m1.1.1.1.cmml">±</mo><mn id="S5.T1.8.8.8.8.4.m1.1.1.3" xref="S5.T1.8.8.8.8.4.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.8.8.8.8.4.m1.1b"><apply id="S5.T1.8.8.8.8.4.m1.1.1.cmml" xref="S5.T1.8.8.8.8.4.m1.1.1"><csymbol cd="latexml" id="S5.T1.8.8.8.8.4.m1.1.1.1.cmml" xref="S5.T1.8.8.8.8.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.8.8.8.8.4.m1.1.1.2.cmml" type="float" xref="S5.T1.8.8.8.8.4.m1.1.1.2">0.579</cn><cn id="S5.T1.8.8.8.8.4.m1.1.1.3.cmml" type="float" xref="S5.T1.8.8.8.8.4.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.8.8.8.8.4.m1.1c">0.579\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.8.8.8.8.4.m1.1d">0.579 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.12.12.12.12"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.9.9.9.9.1">+<math alttext="32\times 32" class="ltx_Math" display="inline" id="S5.T1.9.9.9.9.1.m1.1"><semantics id="S5.T1.9.9.9.9.1.m1.1a"><mrow id="S5.T1.9.9.9.9.1.m1.1.1" xref="S5.T1.9.9.9.9.1.m1.1.1.cmml"><mn id="S5.T1.9.9.9.9.1.m1.1.1.2" xref="S5.T1.9.9.9.9.1.m1.1.1.2.cmml">32</mn><mo id="S5.T1.9.9.9.9.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.T1.9.9.9.9.1.m1.1.1.1.cmml">×</mo><mn id="S5.T1.9.9.9.9.1.m1.1.1.3" xref="S5.T1.9.9.9.9.1.m1.1.1.3.cmml">32</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.9.9.9.9.1.m1.1b"><apply id="S5.T1.9.9.9.9.1.m1.1.1.cmml" xref="S5.T1.9.9.9.9.1.m1.1.1"><times id="S5.T1.9.9.9.9.1.m1.1.1.1.cmml" xref="S5.T1.9.9.9.9.1.m1.1.1.1"></times><cn id="S5.T1.9.9.9.9.1.m1.1.1.2.cmml" type="integer" xref="S5.T1.9.9.9.9.1.m1.1.1.2">32</cn><cn id="S5.T1.9.9.9.9.1.m1.1.1.3.cmml" type="integer" xref="S5.T1.9.9.9.9.1.m1.1.1.3">32</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.9.9.9.9.1.m1.1c">32\times 32</annotation><annotation encoding="application/x-llamapun" id="S5.T1.9.9.9.9.1.m1.1d">32 × 32</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center" id="S5.T1.10.10.10.10.2"><math alttext="0.567\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.10.10.10.10.2.m1.1"><semantics id="S5.T1.10.10.10.10.2.m1.1a"><mrow id="S5.T1.10.10.10.10.2.m1.1.1" xref="S5.T1.10.10.10.10.2.m1.1.1.cmml"><mn id="S5.T1.10.10.10.10.2.m1.1.1.2" xref="S5.T1.10.10.10.10.2.m1.1.1.2.cmml">0.567</mn><mo id="S5.T1.10.10.10.10.2.m1.1.1.1" xref="S5.T1.10.10.10.10.2.m1.1.1.1.cmml">±</mo><mn id="S5.T1.10.10.10.10.2.m1.1.1.3" xref="S5.T1.10.10.10.10.2.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.10.10.10.10.2.m1.1b"><apply id="S5.T1.10.10.10.10.2.m1.1.1.cmml" xref="S5.T1.10.10.10.10.2.m1.1.1"><csymbol cd="latexml" id="S5.T1.10.10.10.10.2.m1.1.1.1.cmml" xref="S5.T1.10.10.10.10.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.10.10.10.10.2.m1.1.1.2.cmml" type="float" xref="S5.T1.10.10.10.10.2.m1.1.1.2">0.567</cn><cn id="S5.T1.10.10.10.10.2.m1.1.1.3.cmml" type="float" xref="S5.T1.10.10.10.10.2.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.10.10.10.10.2.m1.1c">0.567\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.10.10.10.10.2.m1.1d">0.567 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.11.11.11.11.3"><math alttext="0.575\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.11.11.11.11.3.m1.1"><semantics id="S5.T1.11.11.11.11.3.m1.1a"><mrow id="S5.T1.11.11.11.11.3.m1.1.1" xref="S5.T1.11.11.11.11.3.m1.1.1.cmml"><mn id="S5.T1.11.11.11.11.3.m1.1.1.2" xref="S5.T1.11.11.11.11.3.m1.1.1.2.cmml">0.575</mn><mo id="S5.T1.11.11.11.11.3.m1.1.1.1" xref="S5.T1.11.11.11.11.3.m1.1.1.1.cmml">±</mo><mn id="S5.T1.11.11.11.11.3.m1.1.1.3" xref="S5.T1.11.11.11.11.3.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.11.11.11.11.3.m1.1b"><apply id="S5.T1.11.11.11.11.3.m1.1.1.cmml" xref="S5.T1.11.11.11.11.3.m1.1.1"><csymbol cd="latexml" id="S5.T1.11.11.11.11.3.m1.1.1.1.cmml" xref="S5.T1.11.11.11.11.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.11.11.11.11.3.m1.1.1.2.cmml" type="float" xref="S5.T1.11.11.11.11.3.m1.1.1.2">0.575</cn><cn id="S5.T1.11.11.11.11.3.m1.1.1.3.cmml" type="float" xref="S5.T1.11.11.11.11.3.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.11.11.11.11.3.m1.1c">0.575\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.11.11.11.11.3.m1.1d">0.575 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.12.12.12.12.4"><math alttext="0.580\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.12.12.12.12.4.m1.1"><semantics id="S5.T1.12.12.12.12.4.m1.1a"><mrow id="S5.T1.12.12.12.12.4.m1.1.1" xref="S5.T1.12.12.12.12.4.m1.1.1.cmml"><mn id="S5.T1.12.12.12.12.4.m1.1.1.2" xref="S5.T1.12.12.12.12.4.m1.1.1.2.cmml">0.580</mn><mo id="S5.T1.12.12.12.12.4.m1.1.1.1" xref="S5.T1.12.12.12.12.4.m1.1.1.1.cmml">±</mo><mn id="S5.T1.12.12.12.12.4.m1.1.1.3" xref="S5.T1.12.12.12.12.4.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.12.12.12.12.4.m1.1b"><apply id="S5.T1.12.12.12.12.4.m1.1.1.cmml" xref="S5.T1.12.12.12.12.4.m1.1.1"><csymbol cd="latexml" id="S5.T1.12.12.12.12.4.m1.1.1.1.cmml" xref="S5.T1.12.12.12.12.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.12.12.12.12.4.m1.1.1.2.cmml" type="float" xref="S5.T1.12.12.12.12.4.m1.1.1.2">0.580</cn><cn id="S5.T1.12.12.12.12.4.m1.1.1.3.cmml" type="float" xref="S5.T1.12.12.12.12.4.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.12.12.12.12.4.m1.1c">0.580\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.12.12.12.12.4.m1.1d">0.580 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.16.16.16.16"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.13.13.13.13.1">+<math alttext="64\times 64" class="ltx_Math" display="inline" id="S5.T1.13.13.13.13.1.m1.1"><semantics id="S5.T1.13.13.13.13.1.m1.1a"><mrow id="S5.T1.13.13.13.13.1.m1.1.1" xref="S5.T1.13.13.13.13.1.m1.1.1.cmml"><mn id="S5.T1.13.13.13.13.1.m1.1.1.2" xref="S5.T1.13.13.13.13.1.m1.1.1.2.cmml">64</mn><mo id="S5.T1.13.13.13.13.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.T1.13.13.13.13.1.m1.1.1.1.cmml">×</mo><mn id="S5.T1.13.13.13.13.1.m1.1.1.3" xref="S5.T1.13.13.13.13.1.m1.1.1.3.cmml">64</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.13.13.13.13.1.m1.1b"><apply id="S5.T1.13.13.13.13.1.m1.1.1.cmml" xref="S5.T1.13.13.13.13.1.m1.1.1"><times id="S5.T1.13.13.13.13.1.m1.1.1.1.cmml" xref="S5.T1.13.13.13.13.1.m1.1.1.1"></times><cn id="S5.T1.13.13.13.13.1.m1.1.1.2.cmml" type="integer" xref="S5.T1.13.13.13.13.1.m1.1.1.2">64</cn><cn id="S5.T1.13.13.13.13.1.m1.1.1.3.cmml" type="integer" xref="S5.T1.13.13.13.13.1.m1.1.1.3">64</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.13.13.13.13.1.m1.1c">64\times 64</annotation><annotation encoding="application/x-llamapun" id="S5.T1.13.13.13.13.1.m1.1d">64 × 64</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center" id="S5.T1.14.14.14.14.2"><math alttext="0.569\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.14.14.14.14.2.m1.1"><semantics id="S5.T1.14.14.14.14.2.m1.1a"><mrow id="S5.T1.14.14.14.14.2.m1.1.1" xref="S5.T1.14.14.14.14.2.m1.1.1.cmml"><mn id="S5.T1.14.14.14.14.2.m1.1.1.2" xref="S5.T1.14.14.14.14.2.m1.1.1.2.cmml">0.569</mn><mo id="S5.T1.14.14.14.14.2.m1.1.1.1" xref="S5.T1.14.14.14.14.2.m1.1.1.1.cmml">±</mo><mn id="S5.T1.14.14.14.14.2.m1.1.1.3" xref="S5.T1.14.14.14.14.2.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.14.14.14.14.2.m1.1b"><apply id="S5.T1.14.14.14.14.2.m1.1.1.cmml" xref="S5.T1.14.14.14.14.2.m1.1.1"><csymbol cd="latexml" id="S5.T1.14.14.14.14.2.m1.1.1.1.cmml" xref="S5.T1.14.14.14.14.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.14.14.14.14.2.m1.1.1.2.cmml" type="float" xref="S5.T1.14.14.14.14.2.m1.1.1.2">0.569</cn><cn id="S5.T1.14.14.14.14.2.m1.1.1.3.cmml" type="float" xref="S5.T1.14.14.14.14.2.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.14.14.14.14.2.m1.1c">0.569\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.14.14.14.14.2.m1.1d">0.569 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.15.15.15.15.3"><math alttext="0.577\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.15.15.15.15.3.m1.1"><semantics id="S5.T1.15.15.15.15.3.m1.1a"><mrow id="S5.T1.15.15.15.15.3.m1.1.1" xref="S5.T1.15.15.15.15.3.m1.1.1.cmml"><mn id="S5.T1.15.15.15.15.3.m1.1.1.2" xref="S5.T1.15.15.15.15.3.m1.1.1.2.cmml">0.577</mn><mo id="S5.T1.15.15.15.15.3.m1.1.1.1" xref="S5.T1.15.15.15.15.3.m1.1.1.1.cmml">±</mo><mn id="S5.T1.15.15.15.15.3.m1.1.1.3" xref="S5.T1.15.15.15.15.3.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.15.15.15.15.3.m1.1b"><apply id="S5.T1.15.15.15.15.3.m1.1.1.cmml" xref="S5.T1.15.15.15.15.3.m1.1.1"><csymbol cd="latexml" id="S5.T1.15.15.15.15.3.m1.1.1.1.cmml" xref="S5.T1.15.15.15.15.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.15.15.15.15.3.m1.1.1.2.cmml" type="float" xref="S5.T1.15.15.15.15.3.m1.1.1.2">0.577</cn><cn id="S5.T1.15.15.15.15.3.m1.1.1.3.cmml" type="float" xref="S5.T1.15.15.15.15.3.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.15.15.15.15.3.m1.1c">0.577\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.15.15.15.15.3.m1.1d">0.577 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.16.16.16.16.4"><math alttext="\mathbf{0.581}\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.16.16.16.16.4.m1.1"><semantics id="S5.T1.16.16.16.16.4.m1.1a"><mrow id="S5.T1.16.16.16.16.4.m1.1.1" xref="S5.T1.16.16.16.16.4.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S5.T1.16.16.16.16.4.m1.1.1.2" mathvariant="bold" xref="S5.T1.16.16.16.16.4.m1.1.1.2.cmml">0.581</mn><mo id="S5.T1.16.16.16.16.4.m1.1.1.1" xref="S5.T1.16.16.16.16.4.m1.1.1.1.cmml">±</mo><mn id="S5.T1.16.16.16.16.4.m1.1.1.3" xref="S5.T1.16.16.16.16.4.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.16.16.16.16.4.m1.1b"><apply id="S5.T1.16.16.16.16.4.m1.1.1.cmml" xref="S5.T1.16.16.16.16.4.m1.1.1"><csymbol cd="latexml" id="S5.T1.16.16.16.16.4.m1.1.1.1.cmml" xref="S5.T1.16.16.16.16.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.16.16.16.16.4.m1.1.1.2.cmml" type="float" xref="S5.T1.16.16.16.16.4.m1.1.1.2">0.581</cn><cn id="S5.T1.16.16.16.16.4.m1.1.1.3.cmml" type="float" xref="S5.T1.16.16.16.16.4.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.16.16.16.16.4.m1.1c">\mathbf{0.581}\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.16.16.16.16.4.m1.1d">bold_0.581 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.20.20.20.20"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_r ltx_border_t" id="S5.T1.20.20.20.20.5" rowspan="4"> ASDI </th> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T1.17.17.17.17.1"><math alttext="8\times 8" class="ltx_Math" display="inline" id="S5.T1.17.17.17.17.1.m1.1"><semantics id="S5.T1.17.17.17.17.1.m1.1a"><mrow id="S5.T1.17.17.17.17.1.m1.1.1" xref="S5.T1.17.17.17.17.1.m1.1.1.cmml"><mn id="S5.T1.17.17.17.17.1.m1.1.1.2" xref="S5.T1.17.17.17.17.1.m1.1.1.2.cmml">8</mn><mo id="S5.T1.17.17.17.17.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.T1.17.17.17.17.1.m1.1.1.1.cmml">×</mo><mn id="S5.T1.17.17.17.17.1.m1.1.1.3" xref="S5.T1.17.17.17.17.1.m1.1.1.3.cmml">8</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.17.17.17.17.1.m1.1b"><apply id="S5.T1.17.17.17.17.1.m1.1.1.cmml" xref="S5.T1.17.17.17.17.1.m1.1.1"><times id="S5.T1.17.17.17.17.1.m1.1.1.1.cmml" xref="S5.T1.17.17.17.17.1.m1.1.1.1"></times><cn id="S5.T1.17.17.17.17.1.m1.1.1.2.cmml" type="integer" xref="S5.T1.17.17.17.17.1.m1.1.1.2">8</cn><cn id="S5.T1.17.17.17.17.1.m1.1.1.3.cmml" type="integer" xref="S5.T1.17.17.17.17.1.m1.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.17.17.17.17.1.m1.1c">8\times 8</annotation><annotation encoding="application/x-llamapun" id="S5.T1.17.17.17.17.1.m1.1d">8 × 8</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.18.18.18.18.2"><math alttext="0.713\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.18.18.18.18.2.m1.1"><semantics id="S5.T1.18.18.18.18.2.m1.1a"><mrow id="S5.T1.18.18.18.18.2.m1.1.1" xref="S5.T1.18.18.18.18.2.m1.1.1.cmml"><mn id="S5.T1.18.18.18.18.2.m1.1.1.2" xref="S5.T1.18.18.18.18.2.m1.1.1.2.cmml">0.713</mn><mo id="S5.T1.18.18.18.18.2.m1.1.1.1" xref="S5.T1.18.18.18.18.2.m1.1.1.1.cmml">±</mo><mn id="S5.T1.18.18.18.18.2.m1.1.1.3" xref="S5.T1.18.18.18.18.2.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.18.18.18.18.2.m1.1b"><apply id="S5.T1.18.18.18.18.2.m1.1.1.cmml" xref="S5.T1.18.18.18.18.2.m1.1.1"><csymbol cd="latexml" id="S5.T1.18.18.18.18.2.m1.1.1.1.cmml" xref="S5.T1.18.18.18.18.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.18.18.18.18.2.m1.1.1.2.cmml" type="float" xref="S5.T1.18.18.18.18.2.m1.1.1.2">0.713</cn><cn id="S5.T1.18.18.18.18.2.m1.1.1.3.cmml" type="float" xref="S5.T1.18.18.18.18.2.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.18.18.18.18.2.m1.1c">0.713\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.18.18.18.18.2.m1.1d">0.713 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.19.19.19.19.3"><math alttext="0.719\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.19.19.19.19.3.m1.1"><semantics id="S5.T1.19.19.19.19.3.m1.1a"><mrow id="S5.T1.19.19.19.19.3.m1.1.1" xref="S5.T1.19.19.19.19.3.m1.1.1.cmml"><mn id="S5.T1.19.19.19.19.3.m1.1.1.2" xref="S5.T1.19.19.19.19.3.m1.1.1.2.cmml">0.719</mn><mo id="S5.T1.19.19.19.19.3.m1.1.1.1" xref="S5.T1.19.19.19.19.3.m1.1.1.1.cmml">±</mo><mn id="S5.T1.19.19.19.19.3.m1.1.1.3" xref="S5.T1.19.19.19.19.3.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.19.19.19.19.3.m1.1b"><apply id="S5.T1.19.19.19.19.3.m1.1.1.cmml" xref="S5.T1.19.19.19.19.3.m1.1.1"><csymbol cd="latexml" id="S5.T1.19.19.19.19.3.m1.1.1.1.cmml" xref="S5.T1.19.19.19.19.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.19.19.19.19.3.m1.1.1.2.cmml" type="float" xref="S5.T1.19.19.19.19.3.m1.1.1.2">0.719</cn><cn id="S5.T1.19.19.19.19.3.m1.1.1.3.cmml" type="float" xref="S5.T1.19.19.19.19.3.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.19.19.19.19.3.m1.1c">0.719\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.19.19.19.19.3.m1.1d">0.719 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.20.20.20.20.4"><math alttext="0.723\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.20.20.20.20.4.m1.1"><semantics id="S5.T1.20.20.20.20.4.m1.1a"><mrow id="S5.T1.20.20.20.20.4.m1.1.1" xref="S5.T1.20.20.20.20.4.m1.1.1.cmml"><mn id="S5.T1.20.20.20.20.4.m1.1.1.2" xref="S5.T1.20.20.20.20.4.m1.1.1.2.cmml">0.723</mn><mo id="S5.T1.20.20.20.20.4.m1.1.1.1" xref="S5.T1.20.20.20.20.4.m1.1.1.1.cmml">±</mo><mn id="S5.T1.20.20.20.20.4.m1.1.1.3" xref="S5.T1.20.20.20.20.4.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.20.20.20.20.4.m1.1b"><apply id="S5.T1.20.20.20.20.4.m1.1.1.cmml" xref="S5.T1.20.20.20.20.4.m1.1.1"><csymbol cd="latexml" id="S5.T1.20.20.20.20.4.m1.1.1.1.cmml" xref="S5.T1.20.20.20.20.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.20.20.20.20.4.m1.1.1.2.cmml" type="float" xref="S5.T1.20.20.20.20.4.m1.1.1.2">0.723</cn><cn id="S5.T1.20.20.20.20.4.m1.1.1.3.cmml" type="float" xref="S5.T1.20.20.20.20.4.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.20.20.20.20.4.m1.1c">0.723\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.20.20.20.20.4.m1.1d">0.723 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.24.24.24.24"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.21.21.21.21.1">+<math alttext="16\times 16" class="ltx_Math" display="inline" id="S5.T1.21.21.21.21.1.m1.1"><semantics id="S5.T1.21.21.21.21.1.m1.1a"><mrow id="S5.T1.21.21.21.21.1.m1.1.1" xref="S5.T1.21.21.21.21.1.m1.1.1.cmml"><mn id="S5.T1.21.21.21.21.1.m1.1.1.2" xref="S5.T1.21.21.21.21.1.m1.1.1.2.cmml">16</mn><mo id="S5.T1.21.21.21.21.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.T1.21.21.21.21.1.m1.1.1.1.cmml">×</mo><mn id="S5.T1.21.21.21.21.1.m1.1.1.3" xref="S5.T1.21.21.21.21.1.m1.1.1.3.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.21.21.21.21.1.m1.1b"><apply id="S5.T1.21.21.21.21.1.m1.1.1.cmml" xref="S5.T1.21.21.21.21.1.m1.1.1"><times id="S5.T1.21.21.21.21.1.m1.1.1.1.cmml" xref="S5.T1.21.21.21.21.1.m1.1.1.1"></times><cn id="S5.T1.21.21.21.21.1.m1.1.1.2.cmml" type="integer" xref="S5.T1.21.21.21.21.1.m1.1.1.2">16</cn><cn id="S5.T1.21.21.21.21.1.m1.1.1.3.cmml" type="integer" xref="S5.T1.21.21.21.21.1.m1.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.21.21.21.21.1.m1.1c">16\times 16</annotation><annotation encoding="application/x-llamapun" id="S5.T1.21.21.21.21.1.m1.1d">16 × 16</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center" id="S5.T1.22.22.22.22.2"><math alttext="0.720\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.22.22.22.22.2.m1.1"><semantics id="S5.T1.22.22.22.22.2.m1.1a"><mrow id="S5.T1.22.22.22.22.2.m1.1.1" xref="S5.T1.22.22.22.22.2.m1.1.1.cmml"><mn id="S5.T1.22.22.22.22.2.m1.1.1.2" xref="S5.T1.22.22.22.22.2.m1.1.1.2.cmml">0.720</mn><mo id="S5.T1.22.22.22.22.2.m1.1.1.1" xref="S5.T1.22.22.22.22.2.m1.1.1.1.cmml">±</mo><mn id="S5.T1.22.22.22.22.2.m1.1.1.3" xref="S5.T1.22.22.22.22.2.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.22.22.22.22.2.m1.1b"><apply id="S5.T1.22.22.22.22.2.m1.1.1.cmml" xref="S5.T1.22.22.22.22.2.m1.1.1"><csymbol cd="latexml" id="S5.T1.22.22.22.22.2.m1.1.1.1.cmml" xref="S5.T1.22.22.22.22.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.22.22.22.22.2.m1.1.1.2.cmml" type="float" xref="S5.T1.22.22.22.22.2.m1.1.1.2">0.720</cn><cn id="S5.T1.22.22.22.22.2.m1.1.1.3.cmml" type="float" xref="S5.T1.22.22.22.22.2.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.22.22.22.22.2.m1.1c">0.720\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.22.22.22.22.2.m1.1d">0.720 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.23.23.23.23.3"><math alttext="0.723\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.23.23.23.23.3.m1.1"><semantics id="S5.T1.23.23.23.23.3.m1.1a"><mrow id="S5.T1.23.23.23.23.3.m1.1.1" xref="S5.T1.23.23.23.23.3.m1.1.1.cmml"><mn id="S5.T1.23.23.23.23.3.m1.1.1.2" xref="S5.T1.23.23.23.23.3.m1.1.1.2.cmml">0.723</mn><mo id="S5.T1.23.23.23.23.3.m1.1.1.1" xref="S5.T1.23.23.23.23.3.m1.1.1.1.cmml">±</mo><mn id="S5.T1.23.23.23.23.3.m1.1.1.3" xref="S5.T1.23.23.23.23.3.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.23.23.23.23.3.m1.1b"><apply id="S5.T1.23.23.23.23.3.m1.1.1.cmml" xref="S5.T1.23.23.23.23.3.m1.1.1"><csymbol cd="latexml" id="S5.T1.23.23.23.23.3.m1.1.1.1.cmml" xref="S5.T1.23.23.23.23.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.23.23.23.23.3.m1.1.1.2.cmml" type="float" xref="S5.T1.23.23.23.23.3.m1.1.1.2">0.723</cn><cn id="S5.T1.23.23.23.23.3.m1.1.1.3.cmml" type="float" xref="S5.T1.23.23.23.23.3.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.23.23.23.23.3.m1.1c">0.723\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.23.23.23.23.3.m1.1d">0.723 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.24.24.24.24.4"><math alttext="0.725\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.24.24.24.24.4.m1.1"><semantics id="S5.T1.24.24.24.24.4.m1.1a"><mrow id="S5.T1.24.24.24.24.4.m1.1.1" xref="S5.T1.24.24.24.24.4.m1.1.1.cmml"><mn id="S5.T1.24.24.24.24.4.m1.1.1.2" xref="S5.T1.24.24.24.24.4.m1.1.1.2.cmml">0.725</mn><mo id="S5.T1.24.24.24.24.4.m1.1.1.1" xref="S5.T1.24.24.24.24.4.m1.1.1.1.cmml">±</mo><mn id="S5.T1.24.24.24.24.4.m1.1.1.3" xref="S5.T1.24.24.24.24.4.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.24.24.24.24.4.m1.1b"><apply id="S5.T1.24.24.24.24.4.m1.1.1.cmml" xref="S5.T1.24.24.24.24.4.m1.1.1"><csymbol cd="latexml" id="S5.T1.24.24.24.24.4.m1.1.1.1.cmml" xref="S5.T1.24.24.24.24.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.24.24.24.24.4.m1.1.1.2.cmml" type="float" xref="S5.T1.24.24.24.24.4.m1.1.1.2">0.725</cn><cn id="S5.T1.24.24.24.24.4.m1.1.1.3.cmml" type="float" xref="S5.T1.24.24.24.24.4.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.24.24.24.24.4.m1.1c">0.725\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.24.24.24.24.4.m1.1d">0.725 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.28.28.28.28"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.25.25.25.25.1">+<math alttext="32\times 32" class="ltx_Math" display="inline" id="S5.T1.25.25.25.25.1.m1.1"><semantics id="S5.T1.25.25.25.25.1.m1.1a"><mrow id="S5.T1.25.25.25.25.1.m1.1.1" xref="S5.T1.25.25.25.25.1.m1.1.1.cmml"><mn id="S5.T1.25.25.25.25.1.m1.1.1.2" xref="S5.T1.25.25.25.25.1.m1.1.1.2.cmml">32</mn><mo id="S5.T1.25.25.25.25.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.T1.25.25.25.25.1.m1.1.1.1.cmml">×</mo><mn id="S5.T1.25.25.25.25.1.m1.1.1.3" xref="S5.T1.25.25.25.25.1.m1.1.1.3.cmml">32</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.25.25.25.25.1.m1.1b"><apply id="S5.T1.25.25.25.25.1.m1.1.1.cmml" xref="S5.T1.25.25.25.25.1.m1.1.1"><times id="S5.T1.25.25.25.25.1.m1.1.1.1.cmml" xref="S5.T1.25.25.25.25.1.m1.1.1.1"></times><cn id="S5.T1.25.25.25.25.1.m1.1.1.2.cmml" type="integer" xref="S5.T1.25.25.25.25.1.m1.1.1.2">32</cn><cn id="S5.T1.25.25.25.25.1.m1.1.1.3.cmml" type="integer" xref="S5.T1.25.25.25.25.1.m1.1.1.3">32</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.25.25.25.25.1.m1.1c">32\times 32</annotation><annotation encoding="application/x-llamapun" id="S5.T1.25.25.25.25.1.m1.1d">32 × 32</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center" id="S5.T1.26.26.26.26.2"><math alttext="0.720\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.26.26.26.26.2.m1.1"><semantics id="S5.T1.26.26.26.26.2.m1.1a"><mrow id="S5.T1.26.26.26.26.2.m1.1.1" xref="S5.T1.26.26.26.26.2.m1.1.1.cmml"><mn id="S5.T1.26.26.26.26.2.m1.1.1.2" xref="S5.T1.26.26.26.26.2.m1.1.1.2.cmml">0.720</mn><mo id="S5.T1.26.26.26.26.2.m1.1.1.1" xref="S5.T1.26.26.26.26.2.m1.1.1.1.cmml">±</mo><mn id="S5.T1.26.26.26.26.2.m1.1.1.3" xref="S5.T1.26.26.26.26.2.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.26.26.26.26.2.m1.1b"><apply id="S5.T1.26.26.26.26.2.m1.1.1.cmml" xref="S5.T1.26.26.26.26.2.m1.1.1"><csymbol cd="latexml" id="S5.T1.26.26.26.26.2.m1.1.1.1.cmml" xref="S5.T1.26.26.26.26.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.26.26.26.26.2.m1.1.1.2.cmml" type="float" xref="S5.T1.26.26.26.26.2.m1.1.1.2">0.720</cn><cn id="S5.T1.26.26.26.26.2.m1.1.1.3.cmml" type="float" xref="S5.T1.26.26.26.26.2.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.26.26.26.26.2.m1.1c">0.720\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.26.26.26.26.2.m1.1d">0.720 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.27.27.27.27.3"><math alttext="0.725\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.27.27.27.27.3.m1.1"><semantics id="S5.T1.27.27.27.27.3.m1.1a"><mrow id="S5.T1.27.27.27.27.3.m1.1.1" xref="S5.T1.27.27.27.27.3.m1.1.1.cmml"><mn id="S5.T1.27.27.27.27.3.m1.1.1.2" xref="S5.T1.27.27.27.27.3.m1.1.1.2.cmml">0.725</mn><mo id="S5.T1.27.27.27.27.3.m1.1.1.1" xref="S5.T1.27.27.27.27.3.m1.1.1.1.cmml">±</mo><mn id="S5.T1.27.27.27.27.3.m1.1.1.3" xref="S5.T1.27.27.27.27.3.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.27.27.27.27.3.m1.1b"><apply id="S5.T1.27.27.27.27.3.m1.1.1.cmml" xref="S5.T1.27.27.27.27.3.m1.1.1"><csymbol cd="latexml" id="S5.T1.27.27.27.27.3.m1.1.1.1.cmml" xref="S5.T1.27.27.27.27.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.27.27.27.27.3.m1.1.1.2.cmml" type="float" xref="S5.T1.27.27.27.27.3.m1.1.1.2">0.725</cn><cn id="S5.T1.27.27.27.27.3.m1.1.1.3.cmml" type="float" xref="S5.T1.27.27.27.27.3.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.27.27.27.27.3.m1.1c">0.725\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.27.27.27.27.3.m1.1d">0.725 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.28.28.28.28.4"><math alttext="\mathbf{0.726}\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.28.28.28.28.4.m1.1"><semantics id="S5.T1.28.28.28.28.4.m1.1a"><mrow id="S5.T1.28.28.28.28.4.m1.1.1" xref="S5.T1.28.28.28.28.4.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S5.T1.28.28.28.28.4.m1.1.1.2" mathvariant="bold" xref="S5.T1.28.28.28.28.4.m1.1.1.2.cmml">0.726</mn><mo id="S5.T1.28.28.28.28.4.m1.1.1.1" xref="S5.T1.28.28.28.28.4.m1.1.1.1.cmml">±</mo><mn id="S5.T1.28.28.28.28.4.m1.1.1.3" xref="S5.T1.28.28.28.28.4.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.28.28.28.28.4.m1.1b"><apply id="S5.T1.28.28.28.28.4.m1.1.1.cmml" xref="S5.T1.28.28.28.28.4.m1.1.1"><csymbol cd="latexml" id="S5.T1.28.28.28.28.4.m1.1.1.1.cmml" xref="S5.T1.28.28.28.28.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.28.28.28.28.4.m1.1.1.2.cmml" type="float" xref="S5.T1.28.28.28.28.4.m1.1.1.2">0.726</cn><cn id="S5.T1.28.28.28.28.4.m1.1.1.3.cmml" type="float" xref="S5.T1.28.28.28.28.4.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.28.28.28.28.4.m1.1c">\mathbf{0.726}\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.28.28.28.28.4.m1.1d">bold_0.726 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.32.32.32.32"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_r" id="S5.T1.29.29.29.29.1">+<math alttext="64\times 64" class="ltx_Math" display="inline" id="S5.T1.29.29.29.29.1.m1.1"><semantics id="S5.T1.29.29.29.29.1.m1.1a"><mrow id="S5.T1.29.29.29.29.1.m1.1.1" xref="S5.T1.29.29.29.29.1.m1.1.1.cmml"><mn id="S5.T1.29.29.29.29.1.m1.1.1.2" xref="S5.T1.29.29.29.29.1.m1.1.1.2.cmml">64</mn><mo id="S5.T1.29.29.29.29.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.T1.29.29.29.29.1.m1.1.1.1.cmml">×</mo><mn id="S5.T1.29.29.29.29.1.m1.1.1.3" xref="S5.T1.29.29.29.29.1.m1.1.1.3.cmml">64</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.29.29.29.29.1.m1.1b"><apply id="S5.T1.29.29.29.29.1.m1.1.1.cmml" xref="S5.T1.29.29.29.29.1.m1.1.1"><times id="S5.T1.29.29.29.29.1.m1.1.1.1.cmml" xref="S5.T1.29.29.29.29.1.m1.1.1.1"></times><cn id="S5.T1.29.29.29.29.1.m1.1.1.2.cmml" type="integer" xref="S5.T1.29.29.29.29.1.m1.1.1.2">64</cn><cn id="S5.T1.29.29.29.29.1.m1.1.1.3.cmml" type="integer" xref="S5.T1.29.29.29.29.1.m1.1.1.3">64</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.29.29.29.29.1.m1.1c">64\times 64</annotation><annotation encoding="application/x-llamapun" id="S5.T1.29.29.29.29.1.m1.1d">64 × 64</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.30.30.30.30.2"><math alttext="0.720\pm 0.005" class="ltx_Math" display="inline" id="S5.T1.30.30.30.30.2.m1.1"><semantics id="S5.T1.30.30.30.30.2.m1.1a"><mrow id="S5.T1.30.30.30.30.2.m1.1.1" xref="S5.T1.30.30.30.30.2.m1.1.1.cmml"><mn id="S5.T1.30.30.30.30.2.m1.1.1.2" xref="S5.T1.30.30.30.30.2.m1.1.1.2.cmml">0.720</mn><mo id="S5.T1.30.30.30.30.2.m1.1.1.1" xref="S5.T1.30.30.30.30.2.m1.1.1.1.cmml">±</mo><mn id="S5.T1.30.30.30.30.2.m1.1.1.3" xref="S5.T1.30.30.30.30.2.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.30.30.30.30.2.m1.1b"><apply id="S5.T1.30.30.30.30.2.m1.1.1.cmml" xref="S5.T1.30.30.30.30.2.m1.1.1"><csymbol cd="latexml" id="S5.T1.30.30.30.30.2.m1.1.1.1.cmml" xref="S5.T1.30.30.30.30.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.30.30.30.30.2.m1.1.1.2.cmml" type="float" xref="S5.T1.30.30.30.30.2.m1.1.1.2">0.720</cn><cn id="S5.T1.30.30.30.30.2.m1.1.1.3.cmml" type="float" xref="S5.T1.30.30.30.30.2.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.30.30.30.30.2.m1.1c">0.720\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T1.30.30.30.30.2.m1.1d">0.720 ± 0.005</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.31.31.31.31.3"><math alttext="0.725\pm 0.004" class="ltx_Math" display="inline" id="S5.T1.31.31.31.31.3.m1.1"><semantics id="S5.T1.31.31.31.31.3.m1.1a"><mrow id="S5.T1.31.31.31.31.3.m1.1.1" xref="S5.T1.31.31.31.31.3.m1.1.1.cmml"><mn id="S5.T1.31.31.31.31.3.m1.1.1.2" xref="S5.T1.31.31.31.31.3.m1.1.1.2.cmml">0.725</mn><mo id="S5.T1.31.31.31.31.3.m1.1.1.1" xref="S5.T1.31.31.31.31.3.m1.1.1.1.cmml">±</mo><mn id="S5.T1.31.31.31.31.3.m1.1.1.3" xref="S5.T1.31.31.31.31.3.m1.1.1.3.cmml">0.004</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.31.31.31.31.3.m1.1b"><apply id="S5.T1.31.31.31.31.3.m1.1.1.cmml" xref="S5.T1.31.31.31.31.3.m1.1.1"><csymbol cd="latexml" id="S5.T1.31.31.31.31.3.m1.1.1.1.cmml" xref="S5.T1.31.31.31.31.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.31.31.31.31.3.m1.1.1.2.cmml" type="float" xref="S5.T1.31.31.31.31.3.m1.1.1.2">0.725</cn><cn id="S5.T1.31.31.31.31.3.m1.1.1.3.cmml" type="float" xref="S5.T1.31.31.31.31.3.m1.1.1.3">0.004</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.31.31.31.31.3.m1.1c">0.725\pm 0.004</annotation><annotation encoding="application/x-llamapun" id="S5.T1.31.31.31.31.3.m1.1d">0.725 ± 0.004</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T1.32.32.32.32.4"><math alttext="\mathbf{0.726}\pm 0.004" class="ltx_Math" display="inline" id="S5.T1.32.32.32.32.4.m1.1"><semantics id="S5.T1.32.32.32.32.4.m1.1a"><mrow id="S5.T1.32.32.32.32.4.m1.1.1" xref="S5.T1.32.32.32.32.4.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S5.T1.32.32.32.32.4.m1.1.1.2" mathvariant="bold" xref="S5.T1.32.32.32.32.4.m1.1.1.2.cmml">0.726</mn><mo id="S5.T1.32.32.32.32.4.m1.1.1.1" xref="S5.T1.32.32.32.32.4.m1.1.1.1.cmml">±</mo><mn id="S5.T1.32.32.32.32.4.m1.1.1.3" xref="S5.T1.32.32.32.32.4.m1.1.1.3.cmml">0.004</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T1.32.32.32.32.4.m1.1b"><apply id="S5.T1.32.32.32.32.4.m1.1.1.cmml" xref="S5.T1.32.32.32.32.4.m1.1.1"><csymbol cd="latexml" id="S5.T1.32.32.32.32.4.m1.1.1.1.cmml" xref="S5.T1.32.32.32.32.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T1.32.32.32.32.4.m1.1.1.2.cmml" type="float" xref="S5.T1.32.32.32.32.4.m1.1.1.2">0.726</cn><cn id="S5.T1.32.32.32.32.4.m1.1.1.3.cmml" type="float" xref="S5.T1.32.32.32.32.4.m1.1.1.3">0.004</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.32.32.32.32.4.m1.1c">\mathbf{0.726}\pm 0.004</annotation><annotation encoding="application/x-llamapun" id="S5.T1.32.32.32.32.4.m1.1d">bold_0.726 ± 0.004</annotation></semantics></math></td> </tr> </tbody> </table> </div> <figcaption class="ltx_caption ltx_centering">Table A: Impact of multi-scale and <math alttext="N" class="ltx_Math" display="inline" id="S5.T1.34.34.1.m1.1"><semantics id="S5.T1.34.34.1.m1.1b"><mi id="S5.T1.34.34.1.m1.1.1" xref="S5.T1.34.34.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S5.T1.34.34.1.m1.1c"><ci id="S5.T1.34.34.1.m1.1.1.cmml" xref="S5.T1.34.34.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.34.34.1.m1.1d">N</annotation><annotation encoding="application/x-llamapun" id="S5.T1.34.34.1.m1.1e">italic_N</annotation></semantics></math>-fold rotational symmetries on detection performance (AUC) of our statistical model. </figcaption> </figure> <section class="ltx_paragraph" id="S5.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Datasets.</h4> <div class="ltx_para" id="S5.SS1.SSS0.Px1.p1"> We use SPHERE datasets, a cutting-edge exoplanet finder instrument at the VLT <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib4" title="">4</a>]</cite>. Raw observations were sourced from the public data archive of the European Southern Observatory and calibrated with public tools <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib47" title="">47</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib21" title="">21</a>]</cite> from the High-Contrast Data Center. This results in science-ready 4-D datasets <math alttext="\boldsymbol{y}" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px1.p1.1.m1.1"><semantics id="S5.SS1.SSS0.Px1.p1.1.m1.1a"><mi id="S5.SS1.SSS0.Px1.p1.1.m1.1.1" xref="S5.SS1.SSS0.Px1.p1.1.m1.1.1.cmml">𝒚</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS0.Px1.p1.1.m1.1b"><ci id="S5.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.1.m1.1.1">𝒚</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px1.p1.1.m1.1c">\boldsymbol{y}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px1.p1.1.m1.1d">bold_italic_y</annotation></semantics></math> (with <math alttext="L=2" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px1.p1.2.m2.1"><semantics id="S5.SS1.SSS0.Px1.p1.2.m2.1a"><mrow id="S5.SS1.SSS0.Px1.p1.2.m2.1.1" xref="S5.SS1.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S5.SS1.SSS0.Px1.p1.2.m2.1.1.2" xref="S5.SS1.SSS0.Px1.p1.2.m2.1.1.2.cmml">L</mi><mo 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xref="S5.SS1.SSS0.Px1.p1.3.m3.2.3"><in id="S5.SS1.SSS0.Px1.p1.3.m3.2.3.1.cmml" xref="S5.SS1.SSS0.Px1.p1.3.m3.2.3.1"></in><ci id="S5.SS1.SSS0.Px1.p1.3.m3.2.3.2.cmml" xref="S5.SS1.SSS0.Px1.p1.3.m3.2.3.2">𝑇</ci><list id="S5.SS1.SSS0.Px1.p1.3.m3.2.3.3.1.cmml" xref="S5.SS1.SSS0.Px1.p1.3.m3.2.3.3.2"><cn id="S5.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" type="integer" xref="S5.SS1.SSS0.Px1.p1.3.m3.1.1">15</cn><cn id="S5.SS1.SSS0.Px1.p1.3.m3.2.2.cmml" type="integer" xref="S5.SS1.SSS0.Px1.p1.3.m3.2.2">300</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px1.p1.3.m3.2c">T\in\llbracket 15,300\rrbracket</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px1.p1.3.m3.2d">italic_T ∈ ⟦ 15 , 300 ⟧</annotation></semantics></math> temporal frames, and <math alttext="H\times W=1024^{2}" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px1.p1.4.m4.1"><semantics id="S5.SS1.SSS0.Px1.p1.4.m4.1a"><mrow id="S5.SS1.SSS0.Px1.p1.4.m4.1.1" 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id="S5.SS1.SSS0.Px1.p1.6.m6.1a"><msub id="S5.SS1.SSS0.Px1.p1.6.m6.1.1" xref="S5.SS1.SSS0.Px1.p1.6.m6.1.1.cmml"><mi id="S5.SS1.SSS0.Px1.p1.6.m6.1.1.2" xref="S5.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml">λ</mi><mi id="S5.SS1.SSS0.Px1.p1.6.m6.1.1.3" xref="S5.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS0.Px1.p1.6.m6.1b"><apply id="S5.SS1.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.6.m6.1.1">subscript</csymbol><ci id="S5.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="S5.SS1.SSS0.Px1.p1.6.m6.1.1.2">𝜆</ci><ci id="S5.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="S5.SS1.SSS0.Px1.p1.6.m6.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px1.p1.6.m6.1c">\lambda_{c}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px1.p1.6.m6.1d">italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math>, and parallactic angles <math alttext="\phi_{t}" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px1.p1.7.m7.1"><semantics id="S5.SS1.SSS0.Px1.p1.7.m7.1a"><msub id="S5.SS1.SSS0.Px1.p1.7.m7.1.1" xref="S5.SS1.SSS0.Px1.p1.7.m7.1.1.cmml"><mi id="S5.SS1.SSS0.Px1.p1.7.m7.1.1.2" xref="S5.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml">ϕ</mi><mi id="S5.SS1.SSS0.Px1.p1.7.m7.1.1.3" xref="S5.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS0.Px1.p1.7.m7.1b"><apply id="S5.SS1.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS0.Px1.p1.7.m7.1.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.7.m7.1.1">subscript</csymbol><ci id="S5.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S5.SS1.SSS0.Px1.p1.7.m7.1.1.2">italic-ϕ</ci><ci id="S5.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="S5.SS1.SSS0.Px1.p1.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px1.p1.7.m7.1c">\phi_{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px1.p1.7.m7.1d">italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> for algorithm input. Many algorithms have achieved optimal detection sensitivity far from the star, limited by photon noise <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib5" title="">5</a>]</cite>. Thus, our analysis focuses on a smaller, star-centered region of <math alttext="H=W=256" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px1.p1.8.m8.1"><semantics id="S5.SS1.SSS0.Px1.p1.8.m8.1a"><mrow id="S5.SS1.SSS0.Px1.p1.8.m8.1.1" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.cmml"><mi id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.2" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml">H</mi><mo id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.3" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml">=</mo><mi id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.4" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.4.cmml">W</mi><mo id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.5" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.5.cmml">=</mo><mn id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.6" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.6.cmml">256</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS0.Px1.p1.8.m8.1b"><apply id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1"><and id="S5.SS1.SSS0.Px1.p1.8.m8.1.1a.cmml" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1"></and><apply id="S5.SS1.SSS0.Px1.p1.8.m8.1.1b.cmml" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1"><eq id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.3"></eq><ci id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.2">𝐻</ci><ci id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.4.cmml" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.4">𝑊</ci></apply><apply id="S5.SS1.SSS0.Px1.p1.8.m8.1.1c.cmml" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1"><eq id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.5.cmml" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.5"></eq><share href="https://arxiv.org/html/2503.17117v1#S5.SS1.SSS0.Px1.p1.8.m8.1.1.4.cmml" id="S5.SS1.SSS0.Px1.p1.8.m8.1.1d.cmml" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1"></share><cn id="S5.SS1.SSS0.Px1.p1.8.m8.1.1.6.cmml" type="integer" xref="S5.SS1.SSS0.Px1.p1.8.m8.1.1.6">256</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px1.p1.8.m8.1c">H=W=256</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px1.p1.8.m8.1d">italic_H = italic_W = 256</annotation></semantics></math> pixels, where detection sensitivity can still be significantly improved <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib5" title="">5</a>]</cite>. </div> </section> <section class="ltx_paragraph" id="S5.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Training procedure.</h4> <div class="ltx_para" id="S5.SS1.SSS0.Px2.p1"> For training, we use 220 observations from the SHINE-F150 large survey of SPHERE <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib39" title="">39</a>]</cite>. 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Five test datasets (on stars HD 159911, HD 216803, HD 206860, HD 188228, HD 102647) are used for benchmarking against state of the art methods and conducting a model ablation analysis. To evaluate performance, we simulate synthetic faint point-like sources mimicking exoplanet signatures and inject them into real data. This simulation procedure is common practice in direct imaging to ground the actual performance of algorithms because it is very realistic <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib28" title="">28</a>]</cite>. Any real source indeed takes the form of the off-axis PSF, which is measured immediately before and after the main observation sequence by offsetting the star from the coronagraph. This simulation procedure is essential due to the lack of ground truth and the limited number of exoplanets (only a few dozen) detected by direct imaging to date. In addition, we rely on simulated sources for training our deep model, which is also a standard practice for learning-based techniques. Three datasets of star HR 8799, hosting three known exoplanets in the field of view <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib42" title="">42</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib43" title="">43</a>]</cite>, are also used as full real data. </div> </section> <section class="ltx_paragraph" id="S5.SS1.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Baselines.</h4> <div class="ltx_para" id="S5.SS1.SSS0.Px3.p1"> For detection, we benchmark the proposed algorithms against methods from different classes described in Sect. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S2" title="2 Related work ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">2</a>. Selection criteria are (i) code availability (often limited in direct imaging), (ii) relevance, and (iii) widespread use. In this context, we include the cADI <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib41" title="">41</a>]</cite>, KLIP/PCA <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib60" title="">60</a>]</cite> subtraction-based methods, as these are implemented in most data processing pipelines <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib62" title="">62</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib13" title="">13</a>]</cite>, have been responsible for detecting nearly all imaged exoplanets –including the most recent discoveries <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib67" title="">67</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib45" title="">45</a>]</cite>– and remain heavily utilized by astronomers. Concerning statistical methods, we focus on PACO <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib26" title="">26</a>]</cite> that uniquely accounts for data correlations. PACO has consistently outperformed KLIP/PCA in large observational surveys <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib22" title="">22</a>]</cite> and achieved state of the art performance on SPHERE data in a community benchmark, surpassing various subtraction-based, statistical, and learning-based approaches <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib7" title="">7</a>]</cite>. We also evaluate the proposed approach against MODEL&CO <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib5" title="">5</a>]</cite>, a recent hybrid method that shown to outperform cADI, KLIP/PCA, and (deep) PACO <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib5" title="">5</a>]</cite>. For cADI and KLIP/PCA, we use the VIP Python package <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib13" title="">13</a>]</cite>, fine-tuning parameters (e.g., PCA modes) to optimize detection scores guided by the ground truth. PACO and MODEL&CO were processed by their authors using data-driven settings. For flux estimation, we compare only with PACO due to computational constraints. We conduct all analyses in both ADI and ASDI modes (where supported) to assess the gains from joint spectral processing. </div> <figure class="ltx_table" id="S5.T2"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S5.T2.4.4" style="width:173.4pt;height:37pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T2.4.4.4"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T2.4.4.4.5.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="S5.T2.4.4.4.5.1.1">Method</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T2.4.4.4.5.1.2">flux error (ARE)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T2.4.4.4.5.1.3">position error (RMSE)</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T2.2.2.2.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S5.T2.2.2.2.2.3">PACO</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.1.1.1.1.1"><math alttext="0.56" class="ltx_Math" display="inline" id="S5.T2.1.1.1.1.1.m1.1"><semantics id="S5.T2.1.1.1.1.1.m1.1a"><mn id="S5.T2.1.1.1.1.1.m1.1.1" xref="S5.T2.1.1.1.1.1.m1.1.1.cmml">0.56</mn><annotation-xml encoding="MathML-Content" id="S5.T2.1.1.1.1.1.m1.1b"><cn id="S5.T2.1.1.1.1.1.m1.1.1.cmml" type="float" xref="S5.T2.1.1.1.1.1.m1.1.1">0.56</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.1.1.1.1.1.m1.1c">0.56</annotation><annotation encoding="application/x-llamapun" id="S5.T2.1.1.1.1.1.m1.1d">0.56</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.2.2.2.2.2"><math alttext="0.21" class="ltx_Math" display="inline" id="S5.T2.2.2.2.2.2.m1.1"><semantics id="S5.T2.2.2.2.2.2.m1.1a"><mn id="S5.T2.2.2.2.2.2.m1.1.1" xref="S5.T2.2.2.2.2.2.m1.1.1.cmml">0.21</mn><annotation-xml encoding="MathML-Content" id="S5.T2.2.2.2.2.2.m1.1b"><cn id="S5.T2.2.2.2.2.2.m1.1.1.cmml" type="float" xref="S5.T2.2.2.2.2.2.m1.1.1">0.21</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.2.2.2.2.2.m1.1c">0.21</annotation><annotation encoding="application/x-llamapun" id="S5.T2.2.2.2.2.2.m1.1d">0.21</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T2.4.4.4.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_bb" id="S5.T2.4.4.4.4.3">Proposed</th> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T2.3.3.3.3.1"><math alttext="\mathbf{0.51}" class="ltx_Math" display="inline" id="S5.T2.3.3.3.3.1.m1.1"><semantics id="S5.T2.3.3.3.3.1.m1.1a"><mn class="ltx_mathvariant_bold" id="S5.T2.3.3.3.3.1.m1.1.1" mathvariant="bold" xref="S5.T2.3.3.3.3.1.m1.1.1.cmml">0.51</mn><annotation-xml encoding="MathML-Content" id="S5.T2.3.3.3.3.1.m1.1b"><cn id="S5.T2.3.3.3.3.1.m1.1.1.cmml" type="float" xref="S5.T2.3.3.3.3.1.m1.1.1">0.51</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.3.3.3.3.1.m1.1c">\mathbf{0.51}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.3.3.3.3.1.m1.1d">bold_0.51</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T2.4.4.4.4.2"><math alttext="\mathbf{0.11}" class="ltx_Math" display="inline" id="S5.T2.4.4.4.4.2.m1.1"><semantics id="S5.T2.4.4.4.4.2.m1.1a"><mn class="ltx_mathvariant_bold" id="S5.T2.4.4.4.4.2.m1.1.1" mathvariant="bold" xref="S5.T2.4.4.4.4.2.m1.1.1.cmml">0.11</mn><annotation-xml encoding="MathML-Content" id="S5.T2.4.4.4.4.2.m1.1b"><cn id="S5.T2.4.4.4.4.2.m1.1.1.cmml" type="float" xref="S5.T2.4.4.4.4.2.m1.1.1">0.11</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.4.4.4.4.2.m1.1c">\mathbf{0.11}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.4.4.4.4.2.m1.1d">bold_0.11</annotation></semantics></math></td> </tr> </tbody> </table> </div> <figcaption class="ltx_caption ltx_centering">Table B: Comparison of statistical models for flux estimation.</figcaption> </figure> </section> <section class="ltx_paragraph" id="S5.SS1.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Metrics.</h4> <div class="ltx_para" id="S5.SS1.SSS0.Px4.p1"> The detection metric used is the area under the receiver operating characteristic curve (AUC), representing the true positive rate against the false discovery rate obtained by varying the detection threshold. Higher AUC values indicate better performance. This standard metric in direct imaging <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib7" title="">7</a>]</cite> captures the precision-recall trade-off and allows fair algorithm comparisons, as a common threshold does not ensure consistent false alarm rates due to the lack of statistical grounding in some detection maps, see Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S1" title="1 Introduction ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">1</a>. The primary metric is the absolute relative error (ARE) between the ground truth and estimated flux, with lower values indicating better performance. We also report the root mean square error (RMSE) for sub-pixel localization of exoplanets. </div> <figure class="ltx_table" id="S5.T3"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S5.T3.54.54" style="width:422.8pt;height:124.3pt;vertical-align:-0.7pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T3.54.54.54"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T3.54.54.54.55.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="S5.T3.54.54.54.55.1.1">Modality</th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" id="S5.T3.54.54.54.55.1.2">Method</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T3.54.54.54.55.1.3">HD159911 (54°)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T3.54.54.54.55.1.4">HD216803 (23°)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T3.54.54.54.55.1.5">HD206860 (11°)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T3.54.54.54.55.1.6">HD188228 (6°)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T3.54.54.54.55.1.7">HD102647 (2°)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S5.T3.54.54.54.55.1.8">average AUC</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T3.6.6.6.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S5.T3.6.6.6.6.7" rowspan="5">ADI</th> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.6.6.6.6.8">cADI</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.1.1.1.1.1"><math alttext="0.288\pm 0.014" class="ltx_Math" display="inline" id="S5.T3.1.1.1.1.1.m1.1"><semantics id="S5.T3.1.1.1.1.1.m1.1a"><mrow id="S5.T3.1.1.1.1.1.m1.1.1" xref="S5.T3.1.1.1.1.1.m1.1.1.cmml"><mn id="S5.T3.1.1.1.1.1.m1.1.1.2" xref="S5.T3.1.1.1.1.1.m1.1.1.2.cmml">0.288</mn><mo id="S5.T3.1.1.1.1.1.m1.1.1.1" xref="S5.T3.1.1.1.1.1.m1.1.1.1.cmml">±</mo><mn id="S5.T3.1.1.1.1.1.m1.1.1.3" xref="S5.T3.1.1.1.1.1.m1.1.1.3.cmml">0.014</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.1.1.1.1.1.m1.1b"><apply id="S5.T3.1.1.1.1.1.m1.1.1.cmml" xref="S5.T3.1.1.1.1.1.m1.1.1"><csymbol cd="latexml" id="S5.T3.1.1.1.1.1.m1.1.1.1.cmml" xref="S5.T3.1.1.1.1.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.1.1.1.1.1.m1.1.1.2.cmml" type="float" xref="S5.T3.1.1.1.1.1.m1.1.1.2">0.288</cn><cn id="S5.T3.1.1.1.1.1.m1.1.1.3.cmml" type="float" xref="S5.T3.1.1.1.1.1.m1.1.1.3">0.014</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.1.1.1.1.1.m1.1c">0.288\pm 0.014</annotation><annotation encoding="application/x-llamapun" id="S5.T3.1.1.1.1.1.m1.1d">0.288 ± 0.014</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.2.2.2.2.2"><math alttext="0.422\pm 0.007" class="ltx_Math" display="inline" id="S5.T3.2.2.2.2.2.m1.1"><semantics id="S5.T3.2.2.2.2.2.m1.1a"><mrow id="S5.T3.2.2.2.2.2.m1.1.1" xref="S5.T3.2.2.2.2.2.m1.1.1.cmml"><mn id="S5.T3.2.2.2.2.2.m1.1.1.2" xref="S5.T3.2.2.2.2.2.m1.1.1.2.cmml">0.422</mn><mo id="S5.T3.2.2.2.2.2.m1.1.1.1" xref="S5.T3.2.2.2.2.2.m1.1.1.1.cmml">±</mo><mn id="S5.T3.2.2.2.2.2.m1.1.1.3" xref="S5.T3.2.2.2.2.2.m1.1.1.3.cmml">0.007</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.2.2.2.2.2.m1.1b"><apply id="S5.T3.2.2.2.2.2.m1.1.1.cmml" xref="S5.T3.2.2.2.2.2.m1.1.1"><csymbol cd="latexml" id="S5.T3.2.2.2.2.2.m1.1.1.1.cmml" xref="S5.T3.2.2.2.2.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.2.2.2.2.2.m1.1.1.2.cmml" type="float" xref="S5.T3.2.2.2.2.2.m1.1.1.2">0.422</cn><cn id="S5.T3.2.2.2.2.2.m1.1.1.3.cmml" type="float" xref="S5.T3.2.2.2.2.2.m1.1.1.3">0.007</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.2.2.2.2.2.m1.1c">0.422\pm 0.007</annotation><annotation encoding="application/x-llamapun" id="S5.T3.2.2.2.2.2.m1.1d">0.422 ± 0.007</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.3.3.3.3.3"><math alttext="0.489\pm 0.010" class="ltx_Math" display="inline" id="S5.T3.3.3.3.3.3.m1.1"><semantics id="S5.T3.3.3.3.3.3.m1.1a"><mrow id="S5.T3.3.3.3.3.3.m1.1.1" xref="S5.T3.3.3.3.3.3.m1.1.1.cmml"><mn id="S5.T3.3.3.3.3.3.m1.1.1.2" xref="S5.T3.3.3.3.3.3.m1.1.1.2.cmml">0.489</mn><mo id="S5.T3.3.3.3.3.3.m1.1.1.1" xref="S5.T3.3.3.3.3.3.m1.1.1.1.cmml">±</mo><mn id="S5.T3.3.3.3.3.3.m1.1.1.3" xref="S5.T3.3.3.3.3.3.m1.1.1.3.cmml">0.010</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.3.3.3.3.3.m1.1b"><apply id="S5.T3.3.3.3.3.3.m1.1.1.cmml" xref="S5.T3.3.3.3.3.3.m1.1.1"><csymbol cd="latexml" id="S5.T3.3.3.3.3.3.m1.1.1.1.cmml" xref="S5.T3.3.3.3.3.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.3.3.3.3.3.m1.1.1.2.cmml" type="float" xref="S5.T3.3.3.3.3.3.m1.1.1.2">0.489</cn><cn id="S5.T3.3.3.3.3.3.m1.1.1.3.cmml" type="float" xref="S5.T3.3.3.3.3.3.m1.1.1.3">0.010</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.3.3.3.3.3.m1.1c">0.489\pm 0.010</annotation><annotation encoding="application/x-llamapun" id="S5.T3.3.3.3.3.3.m1.1d">0.489 ± 0.010</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.4.4.4.4.4"><math alttext="0.303\pm 0.014" class="ltx_Math" display="inline" id="S5.T3.4.4.4.4.4.m1.1"><semantics id="S5.T3.4.4.4.4.4.m1.1a"><mrow id="S5.T3.4.4.4.4.4.m1.1.1" xref="S5.T3.4.4.4.4.4.m1.1.1.cmml"><mn id="S5.T3.4.4.4.4.4.m1.1.1.2" xref="S5.T3.4.4.4.4.4.m1.1.1.2.cmml">0.303</mn><mo id="S5.T3.4.4.4.4.4.m1.1.1.1" xref="S5.T3.4.4.4.4.4.m1.1.1.1.cmml">±</mo><mn id="S5.T3.4.4.4.4.4.m1.1.1.3" xref="S5.T3.4.4.4.4.4.m1.1.1.3.cmml">0.014</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.4.4.4.4.4.m1.1b"><apply id="S5.T3.4.4.4.4.4.m1.1.1.cmml" xref="S5.T3.4.4.4.4.4.m1.1.1"><csymbol cd="latexml" id="S5.T3.4.4.4.4.4.m1.1.1.1.cmml" xref="S5.T3.4.4.4.4.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.4.4.4.4.4.m1.1.1.2.cmml" type="float" xref="S5.T3.4.4.4.4.4.m1.1.1.2">0.303</cn><cn id="S5.T3.4.4.4.4.4.m1.1.1.3.cmml" type="float" xref="S5.T3.4.4.4.4.4.m1.1.1.3">0.014</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.4.4.4.4.4.m1.1c">0.303\pm 0.014</annotation><annotation encoding="application/x-llamapun" id="S5.T3.4.4.4.4.4.m1.1d">0.303 ± 0.014</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.5.5.5.5.5"><math alttext="0.343\pm 0.009" class="ltx_Math" display="inline" id="S5.T3.5.5.5.5.5.m1.1"><semantics id="S5.T3.5.5.5.5.5.m1.1a"><mrow id="S5.T3.5.5.5.5.5.m1.1.1" xref="S5.T3.5.5.5.5.5.m1.1.1.cmml"><mn id="S5.T3.5.5.5.5.5.m1.1.1.2" xref="S5.T3.5.5.5.5.5.m1.1.1.2.cmml">0.343</mn><mo id="S5.T3.5.5.5.5.5.m1.1.1.1" xref="S5.T3.5.5.5.5.5.m1.1.1.1.cmml">±</mo><mn id="S5.T3.5.5.5.5.5.m1.1.1.3" xref="S5.T3.5.5.5.5.5.m1.1.1.3.cmml">0.009</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.5.5.5.5.5.m1.1b"><apply id="S5.T3.5.5.5.5.5.m1.1.1.cmml" xref="S5.T3.5.5.5.5.5.m1.1.1"><csymbol cd="latexml" id="S5.T3.5.5.5.5.5.m1.1.1.1.cmml" xref="S5.T3.5.5.5.5.5.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.5.5.5.5.5.m1.1.1.2.cmml" type="float" xref="S5.T3.5.5.5.5.5.m1.1.1.2">0.343</cn><cn id="S5.T3.5.5.5.5.5.m1.1.1.3.cmml" type="float" xref="S5.T3.5.5.5.5.5.m1.1.1.3">0.009</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.5.5.5.5.5.m1.1c">0.343\pm 0.009</annotation><annotation encoding="application/x-llamapun" id="S5.T3.5.5.5.5.5.m1.1d">0.343 ± 0.009</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.6.6.6.6.6"><math alttext="0.369\pm 0.005" class="ltx_Math" display="inline" id="S5.T3.6.6.6.6.6.m1.1"><semantics id="S5.T3.6.6.6.6.6.m1.1a"><mrow id="S5.T3.6.6.6.6.6.m1.1.1" xref="S5.T3.6.6.6.6.6.m1.1.1.cmml"><mn id="S5.T3.6.6.6.6.6.m1.1.1.2" xref="S5.T3.6.6.6.6.6.m1.1.1.2.cmml">0.369</mn><mo id="S5.T3.6.6.6.6.6.m1.1.1.1" xref="S5.T3.6.6.6.6.6.m1.1.1.1.cmml">±</mo><mn id="S5.T3.6.6.6.6.6.m1.1.1.3" xref="S5.T3.6.6.6.6.6.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.6.6.6.6.6.m1.1b"><apply id="S5.T3.6.6.6.6.6.m1.1.1.cmml" xref="S5.T3.6.6.6.6.6.m1.1.1"><csymbol cd="latexml" id="S5.T3.6.6.6.6.6.m1.1.1.1.cmml" xref="S5.T3.6.6.6.6.6.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.6.6.6.6.6.m1.1.1.2.cmml" type="float" xref="S5.T3.6.6.6.6.6.m1.1.1.2">0.369</cn><cn id="S5.T3.6.6.6.6.6.m1.1.1.3.cmml" type="float" xref="S5.T3.6.6.6.6.6.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.6.6.6.6.6.m1.1c">0.369\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T3.6.6.6.6.6.m1.1d">0.369 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T3.12.12.12.12"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.12.12.12.12.7">PCA</th> <td class="ltx_td ltx_align_center" id="S5.T3.7.7.7.7.1"><math alttext="0.634\pm 0.010" class="ltx_Math" display="inline" id="S5.T3.7.7.7.7.1.m1.1"><semantics id="S5.T3.7.7.7.7.1.m1.1a"><mrow id="S5.T3.7.7.7.7.1.m1.1.1" xref="S5.T3.7.7.7.7.1.m1.1.1.cmml"><mn id="S5.T3.7.7.7.7.1.m1.1.1.2" xref="S5.T3.7.7.7.7.1.m1.1.1.2.cmml">0.634</mn><mo id="S5.T3.7.7.7.7.1.m1.1.1.1" xref="S5.T3.7.7.7.7.1.m1.1.1.1.cmml">±</mo><mn id="S5.T3.7.7.7.7.1.m1.1.1.3" xref="S5.T3.7.7.7.7.1.m1.1.1.3.cmml">0.010</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.7.7.7.7.1.m1.1b"><apply id="S5.T3.7.7.7.7.1.m1.1.1.cmml" xref="S5.T3.7.7.7.7.1.m1.1.1"><csymbol cd="latexml" id="S5.T3.7.7.7.7.1.m1.1.1.1.cmml" xref="S5.T3.7.7.7.7.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.7.7.7.7.1.m1.1.1.2.cmml" type="float" xref="S5.T3.7.7.7.7.1.m1.1.1.2">0.634</cn><cn id="S5.T3.7.7.7.7.1.m1.1.1.3.cmml" type="float" xref="S5.T3.7.7.7.7.1.m1.1.1.3">0.010</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.7.7.7.7.1.m1.1c">0.634\pm 0.010</annotation><annotation encoding="application/x-llamapun" id="S5.T3.7.7.7.7.1.m1.1d">0.634 ± 0.010</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.8.8.8.8.2"><math alttext="0.643\pm 0.011" class="ltx_Math" display="inline" id="S5.T3.8.8.8.8.2.m1.1"><semantics id="S5.T3.8.8.8.8.2.m1.1a"><mrow id="S5.T3.8.8.8.8.2.m1.1.1" xref="S5.T3.8.8.8.8.2.m1.1.1.cmml"><mn id="S5.T3.8.8.8.8.2.m1.1.1.2" xref="S5.T3.8.8.8.8.2.m1.1.1.2.cmml">0.643</mn><mo id="S5.T3.8.8.8.8.2.m1.1.1.1" xref="S5.T3.8.8.8.8.2.m1.1.1.1.cmml">±</mo><mn id="S5.T3.8.8.8.8.2.m1.1.1.3" xref="S5.T3.8.8.8.8.2.m1.1.1.3.cmml">0.011</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.8.8.8.8.2.m1.1b"><apply id="S5.T3.8.8.8.8.2.m1.1.1.cmml" xref="S5.T3.8.8.8.8.2.m1.1.1"><csymbol cd="latexml" id="S5.T3.8.8.8.8.2.m1.1.1.1.cmml" xref="S5.T3.8.8.8.8.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.8.8.8.8.2.m1.1.1.2.cmml" type="float" xref="S5.T3.8.8.8.8.2.m1.1.1.2">0.643</cn><cn id="S5.T3.8.8.8.8.2.m1.1.1.3.cmml" type="float" xref="S5.T3.8.8.8.8.2.m1.1.1.3">0.011</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.8.8.8.8.2.m1.1c">0.643\pm 0.011</annotation><annotation encoding="application/x-llamapun" id="S5.T3.8.8.8.8.2.m1.1d">0.643 ± 0.011</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.9.9.9.9.3"><math alttext="0.505\pm 0.009" class="ltx_Math" display="inline" id="S5.T3.9.9.9.9.3.m1.1"><semantics id="S5.T3.9.9.9.9.3.m1.1a"><mrow id="S5.T3.9.9.9.9.3.m1.1.1" xref="S5.T3.9.9.9.9.3.m1.1.1.cmml"><mn id="S5.T3.9.9.9.9.3.m1.1.1.2" xref="S5.T3.9.9.9.9.3.m1.1.1.2.cmml">0.505</mn><mo id="S5.T3.9.9.9.9.3.m1.1.1.1" xref="S5.T3.9.9.9.9.3.m1.1.1.1.cmml">±</mo><mn id="S5.T3.9.9.9.9.3.m1.1.1.3" xref="S5.T3.9.9.9.9.3.m1.1.1.3.cmml">0.009</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.9.9.9.9.3.m1.1b"><apply id="S5.T3.9.9.9.9.3.m1.1.1.cmml" xref="S5.T3.9.9.9.9.3.m1.1.1"><csymbol cd="latexml" id="S5.T3.9.9.9.9.3.m1.1.1.1.cmml" xref="S5.T3.9.9.9.9.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.9.9.9.9.3.m1.1.1.2.cmml" type="float" xref="S5.T3.9.9.9.9.3.m1.1.1.2">0.505</cn><cn id="S5.T3.9.9.9.9.3.m1.1.1.3.cmml" type="float" xref="S5.T3.9.9.9.9.3.m1.1.1.3">0.009</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.9.9.9.9.3.m1.1c">0.505\pm 0.009</annotation><annotation encoding="application/x-llamapun" id="S5.T3.9.9.9.9.3.m1.1d">0.505 ± 0.009</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.10.10.10.10.4"><math alttext="0.392\pm 0.011" class="ltx_Math" display="inline" id="S5.T3.10.10.10.10.4.m1.1"><semantics id="S5.T3.10.10.10.10.4.m1.1a"><mrow id="S5.T3.10.10.10.10.4.m1.1.1" xref="S5.T3.10.10.10.10.4.m1.1.1.cmml"><mn id="S5.T3.10.10.10.10.4.m1.1.1.2" xref="S5.T3.10.10.10.10.4.m1.1.1.2.cmml">0.392</mn><mo id="S5.T3.10.10.10.10.4.m1.1.1.1" xref="S5.T3.10.10.10.10.4.m1.1.1.1.cmml">±</mo><mn id="S5.T3.10.10.10.10.4.m1.1.1.3" xref="S5.T3.10.10.10.10.4.m1.1.1.3.cmml">0.011</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.10.10.10.10.4.m1.1b"><apply id="S5.T3.10.10.10.10.4.m1.1.1.cmml" xref="S5.T3.10.10.10.10.4.m1.1.1"><csymbol cd="latexml" id="S5.T3.10.10.10.10.4.m1.1.1.1.cmml" xref="S5.T3.10.10.10.10.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.10.10.10.10.4.m1.1.1.2.cmml" type="float" xref="S5.T3.10.10.10.10.4.m1.1.1.2">0.392</cn><cn id="S5.T3.10.10.10.10.4.m1.1.1.3.cmml" type="float" xref="S5.T3.10.10.10.10.4.m1.1.1.3">0.011</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.10.10.10.10.4.m1.1c">0.392\pm 0.011</annotation><annotation encoding="application/x-llamapun" id="S5.T3.10.10.10.10.4.m1.1d">0.392 ± 0.011</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.11.11.11.11.5"><math alttext="0.218\pm 0.011" class="ltx_Math" display="inline" id="S5.T3.11.11.11.11.5.m1.1"><semantics id="S5.T3.11.11.11.11.5.m1.1a"><mrow id="S5.T3.11.11.11.11.5.m1.1.1" xref="S5.T3.11.11.11.11.5.m1.1.1.cmml"><mn id="S5.T3.11.11.11.11.5.m1.1.1.2" xref="S5.T3.11.11.11.11.5.m1.1.1.2.cmml">0.218</mn><mo id="S5.T3.11.11.11.11.5.m1.1.1.1" xref="S5.T3.11.11.11.11.5.m1.1.1.1.cmml">±</mo><mn id="S5.T3.11.11.11.11.5.m1.1.1.3" xref="S5.T3.11.11.11.11.5.m1.1.1.3.cmml">0.011</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.11.11.11.11.5.m1.1b"><apply id="S5.T3.11.11.11.11.5.m1.1.1.cmml" xref="S5.T3.11.11.11.11.5.m1.1.1"><csymbol cd="latexml" id="S5.T3.11.11.11.11.5.m1.1.1.1.cmml" xref="S5.T3.11.11.11.11.5.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.11.11.11.11.5.m1.1.1.2.cmml" type="float" xref="S5.T3.11.11.11.11.5.m1.1.1.2">0.218</cn><cn id="S5.T3.11.11.11.11.5.m1.1.1.3.cmml" type="float" xref="S5.T3.11.11.11.11.5.m1.1.1.3">0.011</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.11.11.11.11.5.m1.1c">0.218\pm 0.011</annotation><annotation encoding="application/x-llamapun" id="S5.T3.11.11.11.11.5.m1.1d">0.218 ± 0.011</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.12.12.12.12.6"><math alttext="0.478\pm 0.005" class="ltx_Math" display="inline" id="S5.T3.12.12.12.12.6.m1.1"><semantics id="S5.T3.12.12.12.12.6.m1.1a"><mrow id="S5.T3.12.12.12.12.6.m1.1.1" xref="S5.T3.12.12.12.12.6.m1.1.1.cmml"><mn id="S5.T3.12.12.12.12.6.m1.1.1.2" xref="S5.T3.12.12.12.12.6.m1.1.1.2.cmml">0.478</mn><mo id="S5.T3.12.12.12.12.6.m1.1.1.1" xref="S5.T3.12.12.12.12.6.m1.1.1.1.cmml">±</mo><mn id="S5.T3.12.12.12.12.6.m1.1.1.3" xref="S5.T3.12.12.12.12.6.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.12.12.12.12.6.m1.1b"><apply id="S5.T3.12.12.12.12.6.m1.1.1.cmml" xref="S5.T3.12.12.12.12.6.m1.1.1"><csymbol cd="latexml" id="S5.T3.12.12.12.12.6.m1.1.1.1.cmml" xref="S5.T3.12.12.12.12.6.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.12.12.12.12.6.m1.1.1.2.cmml" type="float" xref="S5.T3.12.12.12.12.6.m1.1.1.2">0.478</cn><cn id="S5.T3.12.12.12.12.6.m1.1.1.3.cmml" type="float" xref="S5.T3.12.12.12.12.6.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.12.12.12.12.6.m1.1c">0.478\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T3.12.12.12.12.6.m1.1d">0.478 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T3.18.18.18.18"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.18.18.18.18.7">PACO</th> <td class="ltx_td ltx_align_center" id="S5.T3.13.13.13.13.1"><math alttext="0.629\pm 0.006" class="ltx_Math" display="inline" id="S5.T3.13.13.13.13.1.m1.1"><semantics id="S5.T3.13.13.13.13.1.m1.1a"><mrow id="S5.T3.13.13.13.13.1.m1.1.1" xref="S5.T3.13.13.13.13.1.m1.1.1.cmml"><mn id="S5.T3.13.13.13.13.1.m1.1.1.2" xref="S5.T3.13.13.13.13.1.m1.1.1.2.cmml">0.629</mn><mo id="S5.T3.13.13.13.13.1.m1.1.1.1" xref="S5.T3.13.13.13.13.1.m1.1.1.1.cmml">±</mo><mn id="S5.T3.13.13.13.13.1.m1.1.1.3" xref="S5.T3.13.13.13.13.1.m1.1.1.3.cmml">0.006</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.13.13.13.13.1.m1.1b"><apply id="S5.T3.13.13.13.13.1.m1.1.1.cmml" xref="S5.T3.13.13.13.13.1.m1.1.1"><csymbol cd="latexml" id="S5.T3.13.13.13.13.1.m1.1.1.1.cmml" xref="S5.T3.13.13.13.13.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.13.13.13.13.1.m1.1.1.2.cmml" type="float" xref="S5.T3.13.13.13.13.1.m1.1.1.2">0.629</cn><cn id="S5.T3.13.13.13.13.1.m1.1.1.3.cmml" type="float" xref="S5.T3.13.13.13.13.1.m1.1.1.3">0.006</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.13.13.13.13.1.m1.1c">0.629\pm 0.006</annotation><annotation encoding="application/x-llamapun" id="S5.T3.13.13.13.13.1.m1.1d">0.629 ± 0.006</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.14.14.14.14.2"><math alttext="0.669\pm 0.012" class="ltx_Math" display="inline" id="S5.T3.14.14.14.14.2.m1.1"><semantics id="S5.T3.14.14.14.14.2.m1.1a"><mrow id="S5.T3.14.14.14.14.2.m1.1.1" xref="S5.T3.14.14.14.14.2.m1.1.1.cmml"><mn id="S5.T3.14.14.14.14.2.m1.1.1.2" xref="S5.T3.14.14.14.14.2.m1.1.1.2.cmml">0.669</mn><mo id="S5.T3.14.14.14.14.2.m1.1.1.1" xref="S5.T3.14.14.14.14.2.m1.1.1.1.cmml">±</mo><mn id="S5.T3.14.14.14.14.2.m1.1.1.3" xref="S5.T3.14.14.14.14.2.m1.1.1.3.cmml">0.012</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.14.14.14.14.2.m1.1b"><apply id="S5.T3.14.14.14.14.2.m1.1.1.cmml" xref="S5.T3.14.14.14.14.2.m1.1.1"><csymbol cd="latexml" id="S5.T3.14.14.14.14.2.m1.1.1.1.cmml" xref="S5.T3.14.14.14.14.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.14.14.14.14.2.m1.1.1.2.cmml" type="float" xref="S5.T3.14.14.14.14.2.m1.1.1.2">0.669</cn><cn id="S5.T3.14.14.14.14.2.m1.1.1.3.cmml" type="float" xref="S5.T3.14.14.14.14.2.m1.1.1.3">0.012</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.14.14.14.14.2.m1.1c">0.669\pm 0.012</annotation><annotation encoding="application/x-llamapun" id="S5.T3.14.14.14.14.2.m1.1d">0.669 ± 0.012</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.15.15.15.15.3"><math alttext="0.579\pm 0.015" class="ltx_Math" display="inline" id="S5.T3.15.15.15.15.3.m1.1"><semantics id="S5.T3.15.15.15.15.3.m1.1a"><mrow id="S5.T3.15.15.15.15.3.m1.1.1" xref="S5.T3.15.15.15.15.3.m1.1.1.cmml"><mn id="S5.T3.15.15.15.15.3.m1.1.1.2" xref="S5.T3.15.15.15.15.3.m1.1.1.2.cmml">0.579</mn><mo id="S5.T3.15.15.15.15.3.m1.1.1.1" xref="S5.T3.15.15.15.15.3.m1.1.1.1.cmml">±</mo><mn id="S5.T3.15.15.15.15.3.m1.1.1.3" xref="S5.T3.15.15.15.15.3.m1.1.1.3.cmml">0.015</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.15.15.15.15.3.m1.1b"><apply id="S5.T3.15.15.15.15.3.m1.1.1.cmml" xref="S5.T3.15.15.15.15.3.m1.1.1"><csymbol cd="latexml" id="S5.T3.15.15.15.15.3.m1.1.1.1.cmml" xref="S5.T3.15.15.15.15.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.15.15.15.15.3.m1.1.1.2.cmml" type="float" xref="S5.T3.15.15.15.15.3.m1.1.1.2">0.579</cn><cn id="S5.T3.15.15.15.15.3.m1.1.1.3.cmml" type="float" xref="S5.T3.15.15.15.15.3.m1.1.1.3">0.015</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.15.15.15.15.3.m1.1c">0.579\pm 0.015</annotation><annotation encoding="application/x-llamapun" id="S5.T3.15.15.15.15.3.m1.1d">0.579 ± 0.015</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.16.16.16.16.4"><math alttext="0.517\pm 0.015" class="ltx_Math" display="inline" id="S5.T3.16.16.16.16.4.m1.1"><semantics id="S5.T3.16.16.16.16.4.m1.1a"><mrow id="S5.T3.16.16.16.16.4.m1.1.1" xref="S5.T3.16.16.16.16.4.m1.1.1.cmml"><mn id="S5.T3.16.16.16.16.4.m1.1.1.2" xref="S5.T3.16.16.16.16.4.m1.1.1.2.cmml">0.517</mn><mo id="S5.T3.16.16.16.16.4.m1.1.1.1" xref="S5.T3.16.16.16.16.4.m1.1.1.1.cmml">±</mo><mn id="S5.T3.16.16.16.16.4.m1.1.1.3" xref="S5.T3.16.16.16.16.4.m1.1.1.3.cmml">0.015</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.16.16.16.16.4.m1.1b"><apply id="S5.T3.16.16.16.16.4.m1.1.1.cmml" xref="S5.T3.16.16.16.16.4.m1.1.1"><csymbol cd="latexml" id="S5.T3.16.16.16.16.4.m1.1.1.1.cmml" xref="S5.T3.16.16.16.16.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.16.16.16.16.4.m1.1.1.2.cmml" type="float" xref="S5.T3.16.16.16.16.4.m1.1.1.2">0.517</cn><cn id="S5.T3.16.16.16.16.4.m1.1.1.3.cmml" type="float" xref="S5.T3.16.16.16.16.4.m1.1.1.3">0.015</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.16.16.16.16.4.m1.1c">0.517\pm 0.015</annotation><annotation encoding="application/x-llamapun" id="S5.T3.16.16.16.16.4.m1.1d">0.517 ± 0.015</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.17.17.17.17.5"><math alttext="0.207\pm 0.012" class="ltx_Math" display="inline" id="S5.T3.17.17.17.17.5.m1.1"><semantics id="S5.T3.17.17.17.17.5.m1.1a"><mrow id="S5.T3.17.17.17.17.5.m1.1.1" xref="S5.T3.17.17.17.17.5.m1.1.1.cmml"><mn id="S5.T3.17.17.17.17.5.m1.1.1.2" xref="S5.T3.17.17.17.17.5.m1.1.1.2.cmml">0.207</mn><mo id="S5.T3.17.17.17.17.5.m1.1.1.1" xref="S5.T3.17.17.17.17.5.m1.1.1.1.cmml">±</mo><mn id="S5.T3.17.17.17.17.5.m1.1.1.3" xref="S5.T3.17.17.17.17.5.m1.1.1.3.cmml">0.012</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.17.17.17.17.5.m1.1b"><apply id="S5.T3.17.17.17.17.5.m1.1.1.cmml" xref="S5.T3.17.17.17.17.5.m1.1.1"><csymbol cd="latexml" id="S5.T3.17.17.17.17.5.m1.1.1.1.cmml" xref="S5.T3.17.17.17.17.5.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.17.17.17.17.5.m1.1.1.2.cmml" type="float" xref="S5.T3.17.17.17.17.5.m1.1.1.2">0.207</cn><cn id="S5.T3.17.17.17.17.5.m1.1.1.3.cmml" type="float" xref="S5.T3.17.17.17.17.5.m1.1.1.3">0.012</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.17.17.17.17.5.m1.1c">0.207\pm 0.012</annotation><annotation encoding="application/x-llamapun" id="S5.T3.17.17.17.17.5.m1.1d">0.207 ± 0.012</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.18.18.18.18.6"><math alttext="0.520\pm 0.006" class="ltx_Math" display="inline" id="S5.T3.18.18.18.18.6.m1.1"><semantics id="S5.T3.18.18.18.18.6.m1.1a"><mrow id="S5.T3.18.18.18.18.6.m1.1.1" xref="S5.T3.18.18.18.18.6.m1.1.1.cmml"><mn id="S5.T3.18.18.18.18.6.m1.1.1.2" xref="S5.T3.18.18.18.18.6.m1.1.1.2.cmml">0.520</mn><mo id="S5.T3.18.18.18.18.6.m1.1.1.1" xref="S5.T3.18.18.18.18.6.m1.1.1.1.cmml">±</mo><mn id="S5.T3.18.18.18.18.6.m1.1.1.3" xref="S5.T3.18.18.18.18.6.m1.1.1.3.cmml">0.006</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.18.18.18.18.6.m1.1b"><apply id="S5.T3.18.18.18.18.6.m1.1.1.cmml" xref="S5.T3.18.18.18.18.6.m1.1.1"><csymbol cd="latexml" id="S5.T3.18.18.18.18.6.m1.1.1.1.cmml" xref="S5.T3.18.18.18.18.6.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.18.18.18.18.6.m1.1.1.2.cmml" type="float" xref="S5.T3.18.18.18.18.6.m1.1.1.2">0.520</cn><cn id="S5.T3.18.18.18.18.6.m1.1.1.3.cmml" type="float" xref="S5.T3.18.18.18.18.6.m1.1.1.3">0.006</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.18.18.18.18.6.m1.1c">0.520\pm 0.006</annotation><annotation encoding="application/x-llamapun" id="S5.T3.18.18.18.18.6.m1.1d">0.520 ± 0.006</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T3.24.24.24.24"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.24.24.24.24.7">MODEL&CO</th> <td class="ltx_td ltx_align_center" id="S5.T3.19.19.19.19.1"><math alttext="0.653\pm 0.010" class="ltx_Math" display="inline" id="S5.T3.19.19.19.19.1.m1.1"><semantics id="S5.T3.19.19.19.19.1.m1.1a"><mrow id="S5.T3.19.19.19.19.1.m1.1.1" xref="S5.T3.19.19.19.19.1.m1.1.1.cmml"><mn id="S5.T3.19.19.19.19.1.m1.1.1.2" xref="S5.T3.19.19.19.19.1.m1.1.1.2.cmml">0.653</mn><mo id="S5.T3.19.19.19.19.1.m1.1.1.1" xref="S5.T3.19.19.19.19.1.m1.1.1.1.cmml">±</mo><mn id="S5.T3.19.19.19.19.1.m1.1.1.3" xref="S5.T3.19.19.19.19.1.m1.1.1.3.cmml">0.010</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.19.19.19.19.1.m1.1b"><apply id="S5.T3.19.19.19.19.1.m1.1.1.cmml" xref="S5.T3.19.19.19.19.1.m1.1.1"><csymbol cd="latexml" id="S5.T3.19.19.19.19.1.m1.1.1.1.cmml" xref="S5.T3.19.19.19.19.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.19.19.19.19.1.m1.1.1.2.cmml" type="float" xref="S5.T3.19.19.19.19.1.m1.1.1.2">0.653</cn><cn id="S5.T3.19.19.19.19.1.m1.1.1.3.cmml" type="float" xref="S5.T3.19.19.19.19.1.m1.1.1.3">0.010</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.19.19.19.19.1.m1.1c">0.653\pm 0.010</annotation><annotation encoding="application/x-llamapun" id="S5.T3.19.19.19.19.1.m1.1d">0.653 ± 0.010</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.20.20.20.20.2"><math alttext="0.731\pm 0.011" class="ltx_Math" display="inline" id="S5.T3.20.20.20.20.2.m1.1"><semantics id="S5.T3.20.20.20.20.2.m1.1a"><mrow id="S5.T3.20.20.20.20.2.m1.1.1" xref="S5.T3.20.20.20.20.2.m1.1.1.cmml"><mn id="S5.T3.20.20.20.20.2.m1.1.1.2" xref="S5.T3.20.20.20.20.2.m1.1.1.2.cmml">0.731</mn><mo id="S5.T3.20.20.20.20.2.m1.1.1.1" xref="S5.T3.20.20.20.20.2.m1.1.1.1.cmml">±</mo><mn id="S5.T3.20.20.20.20.2.m1.1.1.3" xref="S5.T3.20.20.20.20.2.m1.1.1.3.cmml">0.011</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.20.20.20.20.2.m1.1b"><apply id="S5.T3.20.20.20.20.2.m1.1.1.cmml" xref="S5.T3.20.20.20.20.2.m1.1.1"><csymbol cd="latexml" id="S5.T3.20.20.20.20.2.m1.1.1.1.cmml" xref="S5.T3.20.20.20.20.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.20.20.20.20.2.m1.1.1.2.cmml" type="float" xref="S5.T3.20.20.20.20.2.m1.1.1.2">0.731</cn><cn id="S5.T3.20.20.20.20.2.m1.1.1.3.cmml" type="float" xref="S5.T3.20.20.20.20.2.m1.1.1.3">0.011</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.20.20.20.20.2.m1.1c">0.731\pm 0.011</annotation><annotation encoding="application/x-llamapun" id="S5.T3.20.20.20.20.2.m1.1d">0.731 ± 0.011</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.21.21.21.21.3"><math alttext="0.646\pm 0.012" class="ltx_Math" display="inline" id="S5.T3.21.21.21.21.3.m1.1"><semantics id="S5.T3.21.21.21.21.3.m1.1a"><mrow id="S5.T3.21.21.21.21.3.m1.1.1" xref="S5.T3.21.21.21.21.3.m1.1.1.cmml"><mn id="S5.T3.21.21.21.21.3.m1.1.1.2" xref="S5.T3.21.21.21.21.3.m1.1.1.2.cmml">0.646</mn><mo id="S5.T3.21.21.21.21.3.m1.1.1.1" xref="S5.T3.21.21.21.21.3.m1.1.1.1.cmml">±</mo><mn id="S5.T3.21.21.21.21.3.m1.1.1.3" xref="S5.T3.21.21.21.21.3.m1.1.1.3.cmml">0.012</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.21.21.21.21.3.m1.1b"><apply id="S5.T3.21.21.21.21.3.m1.1.1.cmml" xref="S5.T3.21.21.21.21.3.m1.1.1"><csymbol cd="latexml" id="S5.T3.21.21.21.21.3.m1.1.1.1.cmml" xref="S5.T3.21.21.21.21.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.21.21.21.21.3.m1.1.1.2.cmml" type="float" xref="S5.T3.21.21.21.21.3.m1.1.1.2">0.646</cn><cn id="S5.T3.21.21.21.21.3.m1.1.1.3.cmml" type="float" xref="S5.T3.21.21.21.21.3.m1.1.1.3">0.012</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.21.21.21.21.3.m1.1c">0.646\pm 0.012</annotation><annotation encoding="application/x-llamapun" id="S5.T3.21.21.21.21.3.m1.1d">0.646 ± 0.012</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.22.22.22.22.4"><math alttext="\textbf{0.638}\pm 0.014" class="ltx_Math" display="inline" id="S5.T3.22.22.22.22.4.m1.1"><semantics id="S5.T3.22.22.22.22.4.m1.1a"><mrow id="S5.T3.22.22.22.22.4.m1.1.1" xref="S5.T3.22.22.22.22.4.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S5.T3.22.22.22.22.4.m1.1.1.2" xref="S5.T3.22.22.22.22.4.m1.1.1.2a.cmml">0.638</mtext><mo id="S5.T3.22.22.22.22.4.m1.1.1.1" xref="S5.T3.22.22.22.22.4.m1.1.1.1.cmml">±</mo><mn id="S5.T3.22.22.22.22.4.m1.1.1.3" xref="S5.T3.22.22.22.22.4.m1.1.1.3.cmml">0.014</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.22.22.22.22.4.m1.1b"><apply id="S5.T3.22.22.22.22.4.m1.1.1.cmml" xref="S5.T3.22.22.22.22.4.m1.1.1"><csymbol cd="latexml" id="S5.T3.22.22.22.22.4.m1.1.1.1.cmml" xref="S5.T3.22.22.22.22.4.m1.1.1.1">plus-or-minus</csymbol><ci id="S5.T3.22.22.22.22.4.m1.1.1.2a.cmml" xref="S5.T3.22.22.22.22.4.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S5.T3.22.22.22.22.4.m1.1.1.2.cmml" xref="S5.T3.22.22.22.22.4.m1.1.1.2">0.638</mtext></ci><cn id="S5.T3.22.22.22.22.4.m1.1.1.3.cmml" type="float" xref="S5.T3.22.22.22.22.4.m1.1.1.3">0.014</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.22.22.22.22.4.m1.1c">\textbf{0.638}\pm 0.014</annotation><annotation encoding="application/x-llamapun" id="S5.T3.22.22.22.22.4.m1.1d">0.638 ± 0.014</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.23.23.23.23.5"><math alttext="\textbf{0.554}\pm 0.009" class="ltx_Math" display="inline" id="S5.T3.23.23.23.23.5.m1.1"><semantics id="S5.T3.23.23.23.23.5.m1.1a"><mrow id="S5.T3.23.23.23.23.5.m1.1.1" xref="S5.T3.23.23.23.23.5.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S5.T3.23.23.23.23.5.m1.1.1.2" xref="S5.T3.23.23.23.23.5.m1.1.1.2a.cmml">0.554</mtext><mo id="S5.T3.23.23.23.23.5.m1.1.1.1" xref="S5.T3.23.23.23.23.5.m1.1.1.1.cmml">±</mo><mn id="S5.T3.23.23.23.23.5.m1.1.1.3" xref="S5.T3.23.23.23.23.5.m1.1.1.3.cmml">0.009</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.23.23.23.23.5.m1.1b"><apply id="S5.T3.23.23.23.23.5.m1.1.1.cmml" xref="S5.T3.23.23.23.23.5.m1.1.1"><csymbol cd="latexml" id="S5.T3.23.23.23.23.5.m1.1.1.1.cmml" xref="S5.T3.23.23.23.23.5.m1.1.1.1">plus-or-minus</csymbol><ci id="S5.T3.23.23.23.23.5.m1.1.1.2a.cmml" xref="S5.T3.23.23.23.23.5.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S5.T3.23.23.23.23.5.m1.1.1.2.cmml" xref="S5.T3.23.23.23.23.5.m1.1.1.2">0.554</mtext></ci><cn id="S5.T3.23.23.23.23.5.m1.1.1.3.cmml" type="float" xref="S5.T3.23.23.23.23.5.m1.1.1.3">0.009</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.23.23.23.23.5.m1.1c">\textbf{0.554}\pm 0.009</annotation><annotation encoding="application/x-llamapun" id="S5.T3.23.23.23.23.5.m1.1d">0.554 ± 0.009</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.24.24.24.24.6"><math alttext="\mathbf{0.645}\pm 0.005" class="ltx_Math" display="inline" id="S5.T3.24.24.24.24.6.m1.1"><semantics id="S5.T3.24.24.24.24.6.m1.1a"><mrow id="S5.T3.24.24.24.24.6.m1.1.1" xref="S5.T3.24.24.24.24.6.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S5.T3.24.24.24.24.6.m1.1.1.2" mathvariant="bold" xref="S5.T3.24.24.24.24.6.m1.1.1.2.cmml">0.645</mn><mo id="S5.T3.24.24.24.24.6.m1.1.1.1" xref="S5.T3.24.24.24.24.6.m1.1.1.1.cmml">±</mo><mn id="S5.T3.24.24.24.24.6.m1.1.1.3" xref="S5.T3.24.24.24.24.6.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.24.24.24.24.6.m1.1b"><apply id="S5.T3.24.24.24.24.6.m1.1.1.cmml" xref="S5.T3.24.24.24.24.6.m1.1.1"><csymbol cd="latexml" id="S5.T3.24.24.24.24.6.m1.1.1.1.cmml" xref="S5.T3.24.24.24.24.6.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.24.24.24.24.6.m1.1.1.2.cmml" type="float" xref="S5.T3.24.24.24.24.6.m1.1.1.2">0.645</cn><cn id="S5.T3.24.24.24.24.6.m1.1.1.3.cmml" type="float" xref="S5.T3.24.24.24.24.6.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.24.24.24.24.6.m1.1c">\mathbf{0.645}\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T3.24.24.24.24.6.m1.1d">bold_0.645 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T3.30.30.30.30"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.30.30.30.30.7">Proposed</th> <td class="ltx_td ltx_align_center" id="S5.T3.25.25.25.25.1"><math alttext="\mathbf{0.673}\pm 0.009" class="ltx_Math" display="inline" id="S5.T3.25.25.25.25.1.m1.1"><semantics id="S5.T3.25.25.25.25.1.m1.1a"><mrow id="S5.T3.25.25.25.25.1.m1.1.1" xref="S5.T3.25.25.25.25.1.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S5.T3.25.25.25.25.1.m1.1.1.2" mathvariant="bold" xref="S5.T3.25.25.25.25.1.m1.1.1.2.cmml">0.673</mn><mo id="S5.T3.25.25.25.25.1.m1.1.1.1" xref="S5.T3.25.25.25.25.1.m1.1.1.1.cmml">±</mo><mn id="S5.T3.25.25.25.25.1.m1.1.1.3" xref="S5.T3.25.25.25.25.1.m1.1.1.3.cmml">0.009</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.25.25.25.25.1.m1.1b"><apply id="S5.T3.25.25.25.25.1.m1.1.1.cmml" xref="S5.T3.25.25.25.25.1.m1.1.1"><csymbol cd="latexml" id="S5.T3.25.25.25.25.1.m1.1.1.1.cmml" xref="S5.T3.25.25.25.25.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.25.25.25.25.1.m1.1.1.2.cmml" type="float" xref="S5.T3.25.25.25.25.1.m1.1.1.2">0.673</cn><cn id="S5.T3.25.25.25.25.1.m1.1.1.3.cmml" type="float" xref="S5.T3.25.25.25.25.1.m1.1.1.3">0.009</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.25.25.25.25.1.m1.1c">\mathbf{0.673}\pm 0.009</annotation><annotation encoding="application/x-llamapun" id="S5.T3.25.25.25.25.1.m1.1d">bold_0.673 ± 0.009</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.26.26.26.26.2"><math alttext="\mathbf{0.740}\pm 0.010" class="ltx_Math" display="inline" id="S5.T3.26.26.26.26.2.m1.1"><semantics id="S5.T3.26.26.26.26.2.m1.1a"><mrow id="S5.T3.26.26.26.26.2.m1.1.1" xref="S5.T3.26.26.26.26.2.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S5.T3.26.26.26.26.2.m1.1.1.2" mathvariant="bold" xref="S5.T3.26.26.26.26.2.m1.1.1.2.cmml">0.740</mn><mo id="S5.T3.26.26.26.26.2.m1.1.1.1" xref="S5.T3.26.26.26.26.2.m1.1.1.1.cmml">±</mo><mn id="S5.T3.26.26.26.26.2.m1.1.1.3" xref="S5.T3.26.26.26.26.2.m1.1.1.3.cmml">0.010</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.26.26.26.26.2.m1.1b"><apply id="S5.T3.26.26.26.26.2.m1.1.1.cmml" xref="S5.T3.26.26.26.26.2.m1.1.1"><csymbol cd="latexml" id="S5.T3.26.26.26.26.2.m1.1.1.1.cmml" xref="S5.T3.26.26.26.26.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.26.26.26.26.2.m1.1.1.2.cmml" type="float" xref="S5.T3.26.26.26.26.2.m1.1.1.2">0.740</cn><cn id="S5.T3.26.26.26.26.2.m1.1.1.3.cmml" type="float" xref="S5.T3.26.26.26.26.2.m1.1.1.3">0.010</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.26.26.26.26.2.m1.1c">\mathbf{0.740}\pm 0.010</annotation><annotation encoding="application/x-llamapun" id="S5.T3.26.26.26.26.2.m1.1d">bold_0.740 ± 0.010</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.27.27.27.27.3"><math alttext="\textbf{0.661}\pm 0.012" class="ltx_Math" display="inline" id="S5.T3.27.27.27.27.3.m1.1"><semantics id="S5.T3.27.27.27.27.3.m1.1a"><mrow id="S5.T3.27.27.27.27.3.m1.1.1" xref="S5.T3.27.27.27.27.3.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S5.T3.27.27.27.27.3.m1.1.1.2" xref="S5.T3.27.27.27.27.3.m1.1.1.2a.cmml">0.661</mtext><mo id="S5.T3.27.27.27.27.3.m1.1.1.1" xref="S5.T3.27.27.27.27.3.m1.1.1.1.cmml">±</mo><mn id="S5.T3.27.27.27.27.3.m1.1.1.3" xref="S5.T3.27.27.27.27.3.m1.1.1.3.cmml">0.012</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.27.27.27.27.3.m1.1b"><apply id="S5.T3.27.27.27.27.3.m1.1.1.cmml" xref="S5.T3.27.27.27.27.3.m1.1.1"><csymbol cd="latexml" id="S5.T3.27.27.27.27.3.m1.1.1.1.cmml" xref="S5.T3.27.27.27.27.3.m1.1.1.1">plus-or-minus</csymbol><ci id="S5.T3.27.27.27.27.3.m1.1.1.2a.cmml" xref="S5.T3.27.27.27.27.3.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S5.T3.27.27.27.27.3.m1.1.1.2.cmml" xref="S5.T3.27.27.27.27.3.m1.1.1.2">0.661</mtext></ci><cn id="S5.T3.27.27.27.27.3.m1.1.1.3.cmml" type="float" xref="S5.T3.27.27.27.27.3.m1.1.1.3">0.012</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.27.27.27.27.3.m1.1c">\textbf{0.661}\pm 0.012</annotation><annotation encoding="application/x-llamapun" id="S5.T3.27.27.27.27.3.m1.1d">0.661 ± 0.012</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.28.28.28.28.4"><math alttext="0.631\pm 0.013" class="ltx_Math" display="inline" id="S5.T3.28.28.28.28.4.m1.1"><semantics id="S5.T3.28.28.28.28.4.m1.1a"><mrow id="S5.T3.28.28.28.28.4.m1.1.1" xref="S5.T3.28.28.28.28.4.m1.1.1.cmml"><mn id="S5.T3.28.28.28.28.4.m1.1.1.2" xref="S5.T3.28.28.28.28.4.m1.1.1.2.cmml">0.631</mn><mo id="S5.T3.28.28.28.28.4.m1.1.1.1" xref="S5.T3.28.28.28.28.4.m1.1.1.1.cmml">±</mo><mn id="S5.T3.28.28.28.28.4.m1.1.1.3" xref="S5.T3.28.28.28.28.4.m1.1.1.3.cmml">0.013</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.28.28.28.28.4.m1.1b"><apply id="S5.T3.28.28.28.28.4.m1.1.1.cmml" xref="S5.T3.28.28.28.28.4.m1.1.1"><csymbol cd="latexml" id="S5.T3.28.28.28.28.4.m1.1.1.1.cmml" xref="S5.T3.28.28.28.28.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.28.28.28.28.4.m1.1.1.2.cmml" type="float" xref="S5.T3.28.28.28.28.4.m1.1.1.2">0.631</cn><cn id="S5.T3.28.28.28.28.4.m1.1.1.3.cmml" type="float" xref="S5.T3.28.28.28.28.4.m1.1.1.3">0.013</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.28.28.28.28.4.m1.1c">0.631\pm 0.013</annotation><annotation encoding="application/x-llamapun" id="S5.T3.28.28.28.28.4.m1.1d">0.631 ± 0.013</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.29.29.29.29.5"><math alttext="0.518\pm 0.012" class="ltx_Math" display="inline" id="S5.T3.29.29.29.29.5.m1.1"><semantics id="S5.T3.29.29.29.29.5.m1.1a"><mrow id="S5.T3.29.29.29.29.5.m1.1.1" xref="S5.T3.29.29.29.29.5.m1.1.1.cmml"><mn id="S5.T3.29.29.29.29.5.m1.1.1.2" xref="S5.T3.29.29.29.29.5.m1.1.1.2.cmml">0.518</mn><mo id="S5.T3.29.29.29.29.5.m1.1.1.1" xref="S5.T3.29.29.29.29.5.m1.1.1.1.cmml">±</mo><mn id="S5.T3.29.29.29.29.5.m1.1.1.3" xref="S5.T3.29.29.29.29.5.m1.1.1.3.cmml">0.012</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.29.29.29.29.5.m1.1b"><apply id="S5.T3.29.29.29.29.5.m1.1.1.cmml" xref="S5.T3.29.29.29.29.5.m1.1.1"><csymbol cd="latexml" id="S5.T3.29.29.29.29.5.m1.1.1.1.cmml" xref="S5.T3.29.29.29.29.5.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.29.29.29.29.5.m1.1.1.2.cmml" type="float" xref="S5.T3.29.29.29.29.5.m1.1.1.2">0.518</cn><cn id="S5.T3.29.29.29.29.5.m1.1.1.3.cmml" type="float" xref="S5.T3.29.29.29.29.5.m1.1.1.3">0.012</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.29.29.29.29.5.m1.1c">0.518\pm 0.012</annotation><annotation encoding="application/x-llamapun" id="S5.T3.29.29.29.29.5.m1.1d">0.518 ± 0.012</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.30.30.30.30.6"><math alttext="\mathbf{0.645}\pm 0.005" class="ltx_Math" display="inline" id="S5.T3.30.30.30.30.6.m1.1"><semantics id="S5.T3.30.30.30.30.6.m1.1a"><mrow id="S5.T3.30.30.30.30.6.m1.1.1" xref="S5.T3.30.30.30.30.6.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S5.T3.30.30.30.30.6.m1.1.1.2" mathvariant="bold" xref="S5.T3.30.30.30.30.6.m1.1.1.2.cmml">0.645</mn><mo id="S5.T3.30.30.30.30.6.m1.1.1.1" xref="S5.T3.30.30.30.30.6.m1.1.1.1.cmml">±</mo><mn id="S5.T3.30.30.30.30.6.m1.1.1.3" xref="S5.T3.30.30.30.30.6.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.30.30.30.30.6.m1.1b"><apply id="S5.T3.30.30.30.30.6.m1.1.1.cmml" xref="S5.T3.30.30.30.30.6.m1.1.1"><csymbol cd="latexml" id="S5.T3.30.30.30.30.6.m1.1.1.1.cmml" xref="S5.T3.30.30.30.30.6.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.30.30.30.30.6.m1.1.1.2.cmml" type="float" xref="S5.T3.30.30.30.30.6.m1.1.1.2">0.645</cn><cn id="S5.T3.30.30.30.30.6.m1.1.1.3.cmml" type="float" xref="S5.T3.30.30.30.30.6.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.30.30.30.30.6.m1.1c">\mathbf{0.645}\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T3.30.30.30.30.6.m1.1d">bold_0.645 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T3.36.36.36.36"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_bb ltx_border_t" id="S5.T3.36.36.36.36.7" rowspan="4">ASDI</th> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.36.36.36.36.8">cASDI</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.31.31.31.31.1"><math alttext="0.448\pm 0.009" class="ltx_Math" display="inline" id="S5.T3.31.31.31.31.1.m1.1"><semantics id="S5.T3.31.31.31.31.1.m1.1a"><mrow id="S5.T3.31.31.31.31.1.m1.1.1" xref="S5.T3.31.31.31.31.1.m1.1.1.cmml"><mn id="S5.T3.31.31.31.31.1.m1.1.1.2" xref="S5.T3.31.31.31.31.1.m1.1.1.2.cmml">0.448</mn><mo id="S5.T3.31.31.31.31.1.m1.1.1.1" xref="S5.T3.31.31.31.31.1.m1.1.1.1.cmml">±</mo><mn id="S5.T3.31.31.31.31.1.m1.1.1.3" xref="S5.T3.31.31.31.31.1.m1.1.1.3.cmml">0.009</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.31.31.31.31.1.m1.1b"><apply id="S5.T3.31.31.31.31.1.m1.1.1.cmml" xref="S5.T3.31.31.31.31.1.m1.1.1"><csymbol cd="latexml" id="S5.T3.31.31.31.31.1.m1.1.1.1.cmml" xref="S5.T3.31.31.31.31.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.31.31.31.31.1.m1.1.1.2.cmml" type="float" xref="S5.T3.31.31.31.31.1.m1.1.1.2">0.448</cn><cn id="S5.T3.31.31.31.31.1.m1.1.1.3.cmml" type="float" xref="S5.T3.31.31.31.31.1.m1.1.1.3">0.009</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.31.31.31.31.1.m1.1c">0.448\pm 0.009</annotation><annotation encoding="application/x-llamapun" id="S5.T3.31.31.31.31.1.m1.1d">0.448 ± 0.009</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.32.32.32.32.2"><math alttext="0.537\pm 0.013" class="ltx_Math" display="inline" id="S5.T3.32.32.32.32.2.m1.1"><semantics id="S5.T3.32.32.32.32.2.m1.1a"><mrow id="S5.T3.32.32.32.32.2.m1.1.1" xref="S5.T3.32.32.32.32.2.m1.1.1.cmml"><mn id="S5.T3.32.32.32.32.2.m1.1.1.2" xref="S5.T3.32.32.32.32.2.m1.1.1.2.cmml">0.537</mn><mo id="S5.T3.32.32.32.32.2.m1.1.1.1" xref="S5.T3.32.32.32.32.2.m1.1.1.1.cmml">±</mo><mn id="S5.T3.32.32.32.32.2.m1.1.1.3" xref="S5.T3.32.32.32.32.2.m1.1.1.3.cmml">0.013</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.32.32.32.32.2.m1.1b"><apply id="S5.T3.32.32.32.32.2.m1.1.1.cmml" xref="S5.T3.32.32.32.32.2.m1.1.1"><csymbol cd="latexml" id="S5.T3.32.32.32.32.2.m1.1.1.1.cmml" xref="S5.T3.32.32.32.32.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.32.32.32.32.2.m1.1.1.2.cmml" type="float" xref="S5.T3.32.32.32.32.2.m1.1.1.2">0.537</cn><cn id="S5.T3.32.32.32.32.2.m1.1.1.3.cmml" type="float" xref="S5.T3.32.32.32.32.2.m1.1.1.3">0.013</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.32.32.32.32.2.m1.1c">0.537\pm 0.013</annotation><annotation encoding="application/x-llamapun" id="S5.T3.32.32.32.32.2.m1.1d">0.537 ± 0.013</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.33.33.33.33.3"><math alttext="0.451\pm 0.007" class="ltx_Math" display="inline" id="S5.T3.33.33.33.33.3.m1.1"><semantics id="S5.T3.33.33.33.33.3.m1.1a"><mrow id="S5.T3.33.33.33.33.3.m1.1.1" xref="S5.T3.33.33.33.33.3.m1.1.1.cmml"><mn id="S5.T3.33.33.33.33.3.m1.1.1.2" xref="S5.T3.33.33.33.33.3.m1.1.1.2.cmml">0.451</mn><mo id="S5.T3.33.33.33.33.3.m1.1.1.1" xref="S5.T3.33.33.33.33.3.m1.1.1.1.cmml">±</mo><mn id="S5.T3.33.33.33.33.3.m1.1.1.3" xref="S5.T3.33.33.33.33.3.m1.1.1.3.cmml">0.007</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.33.33.33.33.3.m1.1b"><apply id="S5.T3.33.33.33.33.3.m1.1.1.cmml" xref="S5.T3.33.33.33.33.3.m1.1.1"><csymbol cd="latexml" id="S5.T3.33.33.33.33.3.m1.1.1.1.cmml" xref="S5.T3.33.33.33.33.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.33.33.33.33.3.m1.1.1.2.cmml" type="float" xref="S5.T3.33.33.33.33.3.m1.1.1.2">0.451</cn><cn id="S5.T3.33.33.33.33.3.m1.1.1.3.cmml" type="float" xref="S5.T3.33.33.33.33.3.m1.1.1.3">0.007</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.33.33.33.33.3.m1.1c">0.451\pm 0.007</annotation><annotation encoding="application/x-llamapun" id="S5.T3.33.33.33.33.3.m1.1d">0.451 ± 0.007</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.34.34.34.34.4"><math alttext="0.352\pm 0.017" class="ltx_Math" display="inline" id="S5.T3.34.34.34.34.4.m1.1"><semantics id="S5.T3.34.34.34.34.4.m1.1a"><mrow id="S5.T3.34.34.34.34.4.m1.1.1" xref="S5.T3.34.34.34.34.4.m1.1.1.cmml"><mn id="S5.T3.34.34.34.34.4.m1.1.1.2" xref="S5.T3.34.34.34.34.4.m1.1.1.2.cmml">0.352</mn><mo id="S5.T3.34.34.34.34.4.m1.1.1.1" xref="S5.T3.34.34.34.34.4.m1.1.1.1.cmml">±</mo><mn id="S5.T3.34.34.34.34.4.m1.1.1.3" xref="S5.T3.34.34.34.34.4.m1.1.1.3.cmml">0.017</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.34.34.34.34.4.m1.1b"><apply id="S5.T3.34.34.34.34.4.m1.1.1.cmml" xref="S5.T3.34.34.34.34.4.m1.1.1"><csymbol cd="latexml" id="S5.T3.34.34.34.34.4.m1.1.1.1.cmml" xref="S5.T3.34.34.34.34.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.34.34.34.34.4.m1.1.1.2.cmml" type="float" xref="S5.T3.34.34.34.34.4.m1.1.1.2">0.352</cn><cn id="S5.T3.34.34.34.34.4.m1.1.1.3.cmml" type="float" xref="S5.T3.34.34.34.34.4.m1.1.1.3">0.017</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.34.34.34.34.4.m1.1c">0.352\pm 0.017</annotation><annotation encoding="application/x-llamapun" id="S5.T3.34.34.34.34.4.m1.1d">0.352 ± 0.017</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.35.35.35.35.5"><math alttext="0.294\pm 0.012" class="ltx_Math" display="inline" id="S5.T3.35.35.35.35.5.m1.1"><semantics id="S5.T3.35.35.35.35.5.m1.1a"><mrow id="S5.T3.35.35.35.35.5.m1.1.1" xref="S5.T3.35.35.35.35.5.m1.1.1.cmml"><mn id="S5.T3.35.35.35.35.5.m1.1.1.2" xref="S5.T3.35.35.35.35.5.m1.1.1.2.cmml">0.294</mn><mo id="S5.T3.35.35.35.35.5.m1.1.1.1" xref="S5.T3.35.35.35.35.5.m1.1.1.1.cmml">±</mo><mn id="S5.T3.35.35.35.35.5.m1.1.1.3" xref="S5.T3.35.35.35.35.5.m1.1.1.3.cmml">0.012</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.35.35.35.35.5.m1.1b"><apply id="S5.T3.35.35.35.35.5.m1.1.1.cmml" xref="S5.T3.35.35.35.35.5.m1.1.1"><csymbol cd="latexml" id="S5.T3.35.35.35.35.5.m1.1.1.1.cmml" xref="S5.T3.35.35.35.35.5.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.35.35.35.35.5.m1.1.1.2.cmml" type="float" xref="S5.T3.35.35.35.35.5.m1.1.1.2">0.294</cn><cn id="S5.T3.35.35.35.35.5.m1.1.1.3.cmml" type="float" xref="S5.T3.35.35.35.35.5.m1.1.1.3">0.012</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.35.35.35.35.5.m1.1c">0.294\pm 0.012</annotation><annotation encoding="application/x-llamapun" id="S5.T3.35.35.35.35.5.m1.1d">0.294 ± 0.012</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T3.36.36.36.36.6"><math alttext="0.417\pm 0.006" class="ltx_Math" display="inline" id="S5.T3.36.36.36.36.6.m1.1"><semantics id="S5.T3.36.36.36.36.6.m1.1a"><mrow id="S5.T3.36.36.36.36.6.m1.1.1" xref="S5.T3.36.36.36.36.6.m1.1.1.cmml"><mn id="S5.T3.36.36.36.36.6.m1.1.1.2" xref="S5.T3.36.36.36.36.6.m1.1.1.2.cmml">0.417</mn><mo id="S5.T3.36.36.36.36.6.m1.1.1.1" xref="S5.T3.36.36.36.36.6.m1.1.1.1.cmml">±</mo><mn id="S5.T3.36.36.36.36.6.m1.1.1.3" xref="S5.T3.36.36.36.36.6.m1.1.1.3.cmml">0.006</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.36.36.36.36.6.m1.1b"><apply id="S5.T3.36.36.36.36.6.m1.1.1.cmml" xref="S5.T3.36.36.36.36.6.m1.1.1"><csymbol cd="latexml" id="S5.T3.36.36.36.36.6.m1.1.1.1.cmml" xref="S5.T3.36.36.36.36.6.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.36.36.36.36.6.m1.1.1.2.cmml" type="float" xref="S5.T3.36.36.36.36.6.m1.1.1.2">0.417</cn><cn id="S5.T3.36.36.36.36.6.m1.1.1.3.cmml" type="float" xref="S5.T3.36.36.36.36.6.m1.1.1.3">0.006</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.36.36.36.36.6.m1.1c">0.417\pm 0.006</annotation><annotation encoding="application/x-llamapun" id="S5.T3.36.36.36.36.6.m1.1d">0.417 ± 0.006</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T3.42.42.42.42"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.42.42.42.42.7">PCA</th> <td class="ltx_td ltx_align_center" id="S5.T3.37.37.37.37.1"><math alttext="0.694\pm 0.013" class="ltx_Math" display="inline" id="S5.T3.37.37.37.37.1.m1.1"><semantics id="S5.T3.37.37.37.37.1.m1.1a"><mrow id="S5.T3.37.37.37.37.1.m1.1.1" xref="S5.T3.37.37.37.37.1.m1.1.1.cmml"><mn id="S5.T3.37.37.37.37.1.m1.1.1.2" xref="S5.T3.37.37.37.37.1.m1.1.1.2.cmml">0.694</mn><mo id="S5.T3.37.37.37.37.1.m1.1.1.1" xref="S5.T3.37.37.37.37.1.m1.1.1.1.cmml">±</mo><mn id="S5.T3.37.37.37.37.1.m1.1.1.3" xref="S5.T3.37.37.37.37.1.m1.1.1.3.cmml">0.013</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.37.37.37.37.1.m1.1b"><apply id="S5.T3.37.37.37.37.1.m1.1.1.cmml" xref="S5.T3.37.37.37.37.1.m1.1.1"><csymbol cd="latexml" id="S5.T3.37.37.37.37.1.m1.1.1.1.cmml" xref="S5.T3.37.37.37.37.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.37.37.37.37.1.m1.1.1.2.cmml" type="float" xref="S5.T3.37.37.37.37.1.m1.1.1.2">0.694</cn><cn id="S5.T3.37.37.37.37.1.m1.1.1.3.cmml" type="float" xref="S5.T3.37.37.37.37.1.m1.1.1.3">0.013</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.37.37.37.37.1.m1.1c">0.694\pm 0.013</annotation><annotation encoding="application/x-llamapun" id="S5.T3.37.37.37.37.1.m1.1d">0.694 ± 0.013</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.38.38.38.38.2"><math alttext="0.696\pm 0.007" class="ltx_Math" display="inline" id="S5.T3.38.38.38.38.2.m1.1"><semantics id="S5.T3.38.38.38.38.2.m1.1a"><mrow id="S5.T3.38.38.38.38.2.m1.1.1" xref="S5.T3.38.38.38.38.2.m1.1.1.cmml"><mn id="S5.T3.38.38.38.38.2.m1.1.1.2" xref="S5.T3.38.38.38.38.2.m1.1.1.2.cmml">0.696</mn><mo id="S5.T3.38.38.38.38.2.m1.1.1.1" xref="S5.T3.38.38.38.38.2.m1.1.1.1.cmml">±</mo><mn id="S5.T3.38.38.38.38.2.m1.1.1.3" xref="S5.T3.38.38.38.38.2.m1.1.1.3.cmml">0.007</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.38.38.38.38.2.m1.1b"><apply id="S5.T3.38.38.38.38.2.m1.1.1.cmml" xref="S5.T3.38.38.38.38.2.m1.1.1"><csymbol cd="latexml" id="S5.T3.38.38.38.38.2.m1.1.1.1.cmml" xref="S5.T3.38.38.38.38.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.38.38.38.38.2.m1.1.1.2.cmml" type="float" xref="S5.T3.38.38.38.38.2.m1.1.1.2">0.696</cn><cn id="S5.T3.38.38.38.38.2.m1.1.1.3.cmml" type="float" xref="S5.T3.38.38.38.38.2.m1.1.1.3">0.007</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.38.38.38.38.2.m1.1c">0.696\pm 0.007</annotation><annotation encoding="application/x-llamapun" id="S5.T3.38.38.38.38.2.m1.1d">0.696 ± 0.007</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.39.39.39.39.3"><math alttext="0.552\pm 0.011" class="ltx_Math" display="inline" id="S5.T3.39.39.39.39.3.m1.1"><semantics id="S5.T3.39.39.39.39.3.m1.1a"><mrow id="S5.T3.39.39.39.39.3.m1.1.1" xref="S5.T3.39.39.39.39.3.m1.1.1.cmml"><mn id="S5.T3.39.39.39.39.3.m1.1.1.2" xref="S5.T3.39.39.39.39.3.m1.1.1.2.cmml">0.552</mn><mo id="S5.T3.39.39.39.39.3.m1.1.1.1" xref="S5.T3.39.39.39.39.3.m1.1.1.1.cmml">±</mo><mn id="S5.T3.39.39.39.39.3.m1.1.1.3" xref="S5.T3.39.39.39.39.3.m1.1.1.3.cmml">0.011</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.39.39.39.39.3.m1.1b"><apply id="S5.T3.39.39.39.39.3.m1.1.1.cmml" xref="S5.T3.39.39.39.39.3.m1.1.1"><csymbol cd="latexml" id="S5.T3.39.39.39.39.3.m1.1.1.1.cmml" xref="S5.T3.39.39.39.39.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.39.39.39.39.3.m1.1.1.2.cmml" type="float" xref="S5.T3.39.39.39.39.3.m1.1.1.2">0.552</cn><cn id="S5.T3.39.39.39.39.3.m1.1.1.3.cmml" type="float" xref="S5.T3.39.39.39.39.3.m1.1.1.3">0.011</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.39.39.39.39.3.m1.1c">0.552\pm 0.011</annotation><annotation encoding="application/x-llamapun" id="S5.T3.39.39.39.39.3.m1.1d">0.552 ± 0.011</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.40.40.40.40.4"><math alttext="0.398\pm 0.011" class="ltx_Math" display="inline" id="S5.T3.40.40.40.40.4.m1.1"><semantics id="S5.T3.40.40.40.40.4.m1.1a"><mrow id="S5.T3.40.40.40.40.4.m1.1.1" xref="S5.T3.40.40.40.40.4.m1.1.1.cmml"><mn id="S5.T3.40.40.40.40.4.m1.1.1.2" xref="S5.T3.40.40.40.40.4.m1.1.1.2.cmml">0.398</mn><mo id="S5.T3.40.40.40.40.4.m1.1.1.1" xref="S5.T3.40.40.40.40.4.m1.1.1.1.cmml">±</mo><mn id="S5.T3.40.40.40.40.4.m1.1.1.3" xref="S5.T3.40.40.40.40.4.m1.1.1.3.cmml">0.011</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.40.40.40.40.4.m1.1b"><apply id="S5.T3.40.40.40.40.4.m1.1.1.cmml" xref="S5.T3.40.40.40.40.4.m1.1.1"><csymbol cd="latexml" id="S5.T3.40.40.40.40.4.m1.1.1.1.cmml" xref="S5.T3.40.40.40.40.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.40.40.40.40.4.m1.1.1.2.cmml" type="float" xref="S5.T3.40.40.40.40.4.m1.1.1.2">0.398</cn><cn id="S5.T3.40.40.40.40.4.m1.1.1.3.cmml" type="float" xref="S5.T3.40.40.40.40.4.m1.1.1.3">0.011</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.40.40.40.40.4.m1.1c">0.398\pm 0.011</annotation><annotation encoding="application/x-llamapun" id="S5.T3.40.40.40.40.4.m1.1d">0.398 ± 0.011</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.41.41.41.41.5"><math alttext="0.236\pm 0.009" class="ltx_Math" display="inline" id="S5.T3.41.41.41.41.5.m1.1"><semantics id="S5.T3.41.41.41.41.5.m1.1a"><mrow id="S5.T3.41.41.41.41.5.m1.1.1" xref="S5.T3.41.41.41.41.5.m1.1.1.cmml"><mn id="S5.T3.41.41.41.41.5.m1.1.1.2" xref="S5.T3.41.41.41.41.5.m1.1.1.2.cmml">0.236</mn><mo id="S5.T3.41.41.41.41.5.m1.1.1.1" xref="S5.T3.41.41.41.41.5.m1.1.1.1.cmml">±</mo><mn id="S5.T3.41.41.41.41.5.m1.1.1.3" xref="S5.T3.41.41.41.41.5.m1.1.1.3.cmml">0.009</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.41.41.41.41.5.m1.1b"><apply id="S5.T3.41.41.41.41.5.m1.1.1.cmml" xref="S5.T3.41.41.41.41.5.m1.1.1"><csymbol cd="latexml" id="S5.T3.41.41.41.41.5.m1.1.1.1.cmml" xref="S5.T3.41.41.41.41.5.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.41.41.41.41.5.m1.1.1.2.cmml" type="float" xref="S5.T3.41.41.41.41.5.m1.1.1.2">0.236</cn><cn id="S5.T3.41.41.41.41.5.m1.1.1.3.cmml" type="float" xref="S5.T3.41.41.41.41.5.m1.1.1.3">0.009</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.41.41.41.41.5.m1.1c">0.236\pm 0.009</annotation><annotation encoding="application/x-llamapun" id="S5.T3.41.41.41.41.5.m1.1d">0.236 ± 0.009</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.42.42.42.42.6"><math alttext="0.515\pm 0.005" class="ltx_Math" display="inline" id="S5.T3.42.42.42.42.6.m1.1"><semantics id="S5.T3.42.42.42.42.6.m1.1a"><mrow id="S5.T3.42.42.42.42.6.m1.1.1" xref="S5.T3.42.42.42.42.6.m1.1.1.cmml"><mn id="S5.T3.42.42.42.42.6.m1.1.1.2" xref="S5.T3.42.42.42.42.6.m1.1.1.2.cmml">0.515</mn><mo id="S5.T3.42.42.42.42.6.m1.1.1.1" xref="S5.T3.42.42.42.42.6.m1.1.1.1.cmml">±</mo><mn id="S5.T3.42.42.42.42.6.m1.1.1.3" xref="S5.T3.42.42.42.42.6.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.42.42.42.42.6.m1.1b"><apply id="S5.T3.42.42.42.42.6.m1.1.1.cmml" xref="S5.T3.42.42.42.42.6.m1.1.1"><csymbol cd="latexml" id="S5.T3.42.42.42.42.6.m1.1.1.1.cmml" xref="S5.T3.42.42.42.42.6.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.42.42.42.42.6.m1.1.1.2.cmml" type="float" xref="S5.T3.42.42.42.42.6.m1.1.1.2">0.515</cn><cn id="S5.T3.42.42.42.42.6.m1.1.1.3.cmml" type="float" xref="S5.T3.42.42.42.42.6.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.42.42.42.42.6.m1.1c">0.515\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T3.42.42.42.42.6.m1.1d">0.515 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T3.48.48.48.48"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.48.48.48.48.7">PACO</th> <td class="ltx_td ltx_align_center" id="S5.T3.43.43.43.43.1"><math alttext="0.698\pm 0.015" class="ltx_Math" display="inline" id="S5.T3.43.43.43.43.1.m1.1"><semantics id="S5.T3.43.43.43.43.1.m1.1a"><mrow id="S5.T3.43.43.43.43.1.m1.1.1" xref="S5.T3.43.43.43.43.1.m1.1.1.cmml"><mn id="S5.T3.43.43.43.43.1.m1.1.1.2" xref="S5.T3.43.43.43.43.1.m1.1.1.2.cmml">0.698</mn><mo id="S5.T3.43.43.43.43.1.m1.1.1.1" xref="S5.T3.43.43.43.43.1.m1.1.1.1.cmml">±</mo><mn id="S5.T3.43.43.43.43.1.m1.1.1.3" xref="S5.T3.43.43.43.43.1.m1.1.1.3.cmml">0.015</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.43.43.43.43.1.m1.1b"><apply id="S5.T3.43.43.43.43.1.m1.1.1.cmml" xref="S5.T3.43.43.43.43.1.m1.1.1"><csymbol cd="latexml" id="S5.T3.43.43.43.43.1.m1.1.1.1.cmml" xref="S5.T3.43.43.43.43.1.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.43.43.43.43.1.m1.1.1.2.cmml" type="float" xref="S5.T3.43.43.43.43.1.m1.1.1.2">0.698</cn><cn id="S5.T3.43.43.43.43.1.m1.1.1.3.cmml" type="float" xref="S5.T3.43.43.43.43.1.m1.1.1.3">0.015</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.43.43.43.43.1.m1.1c">0.698\pm 0.015</annotation><annotation encoding="application/x-llamapun" id="S5.T3.43.43.43.43.1.m1.1d">0.698 ± 0.015</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.44.44.44.44.2"><math alttext="0.768\pm 0.008" class="ltx_Math" display="inline" id="S5.T3.44.44.44.44.2.m1.1"><semantics id="S5.T3.44.44.44.44.2.m1.1a"><mrow id="S5.T3.44.44.44.44.2.m1.1.1" xref="S5.T3.44.44.44.44.2.m1.1.1.cmml"><mn id="S5.T3.44.44.44.44.2.m1.1.1.2" xref="S5.T3.44.44.44.44.2.m1.1.1.2.cmml">0.768</mn><mo id="S5.T3.44.44.44.44.2.m1.1.1.1" xref="S5.T3.44.44.44.44.2.m1.1.1.1.cmml">±</mo><mn id="S5.T3.44.44.44.44.2.m1.1.1.3" xref="S5.T3.44.44.44.44.2.m1.1.1.3.cmml">0.008</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.44.44.44.44.2.m1.1b"><apply id="S5.T3.44.44.44.44.2.m1.1.1.cmml" xref="S5.T3.44.44.44.44.2.m1.1.1"><csymbol cd="latexml" id="S5.T3.44.44.44.44.2.m1.1.1.1.cmml" xref="S5.T3.44.44.44.44.2.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.44.44.44.44.2.m1.1.1.2.cmml" type="float" xref="S5.T3.44.44.44.44.2.m1.1.1.2">0.768</cn><cn id="S5.T3.44.44.44.44.2.m1.1.1.3.cmml" type="float" xref="S5.T3.44.44.44.44.2.m1.1.1.3">0.008</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.44.44.44.44.2.m1.1c">0.768\pm 0.008</annotation><annotation encoding="application/x-llamapun" id="S5.T3.44.44.44.44.2.m1.1d">0.768 ± 0.008</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.45.45.45.45.3"><math alttext="0.710\pm 0.014" class="ltx_Math" display="inline" id="S5.T3.45.45.45.45.3.m1.1"><semantics id="S5.T3.45.45.45.45.3.m1.1a"><mrow id="S5.T3.45.45.45.45.3.m1.1.1" xref="S5.T3.45.45.45.45.3.m1.1.1.cmml"><mn id="S5.T3.45.45.45.45.3.m1.1.1.2" xref="S5.T3.45.45.45.45.3.m1.1.1.2.cmml">0.710</mn><mo id="S5.T3.45.45.45.45.3.m1.1.1.1" xref="S5.T3.45.45.45.45.3.m1.1.1.1.cmml">±</mo><mn id="S5.T3.45.45.45.45.3.m1.1.1.3" xref="S5.T3.45.45.45.45.3.m1.1.1.3.cmml">0.014</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.45.45.45.45.3.m1.1b"><apply id="S5.T3.45.45.45.45.3.m1.1.1.cmml" xref="S5.T3.45.45.45.45.3.m1.1.1"><csymbol cd="latexml" id="S5.T3.45.45.45.45.3.m1.1.1.1.cmml" xref="S5.T3.45.45.45.45.3.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.45.45.45.45.3.m1.1.1.2.cmml" type="float" xref="S5.T3.45.45.45.45.3.m1.1.1.2">0.710</cn><cn id="S5.T3.45.45.45.45.3.m1.1.1.3.cmml" type="float" xref="S5.T3.45.45.45.45.3.m1.1.1.3">0.014</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.45.45.45.45.3.m1.1c">0.710\pm 0.014</annotation><annotation encoding="application/x-llamapun" id="S5.T3.45.45.45.45.3.m1.1d">0.710 ± 0.014</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.46.46.46.46.4"><math alttext="0.700\pm 0.009" class="ltx_Math" display="inline" id="S5.T3.46.46.46.46.4.m1.1"><semantics id="S5.T3.46.46.46.46.4.m1.1a"><mrow id="S5.T3.46.46.46.46.4.m1.1.1" xref="S5.T3.46.46.46.46.4.m1.1.1.cmml"><mn id="S5.T3.46.46.46.46.4.m1.1.1.2" xref="S5.T3.46.46.46.46.4.m1.1.1.2.cmml">0.700</mn><mo id="S5.T3.46.46.46.46.4.m1.1.1.1" xref="S5.T3.46.46.46.46.4.m1.1.1.1.cmml">±</mo><mn id="S5.T3.46.46.46.46.4.m1.1.1.3" xref="S5.T3.46.46.46.46.4.m1.1.1.3.cmml">0.009</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.46.46.46.46.4.m1.1b"><apply id="S5.T3.46.46.46.46.4.m1.1.1.cmml" xref="S5.T3.46.46.46.46.4.m1.1.1"><csymbol cd="latexml" id="S5.T3.46.46.46.46.4.m1.1.1.1.cmml" xref="S5.T3.46.46.46.46.4.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.46.46.46.46.4.m1.1.1.2.cmml" type="float" xref="S5.T3.46.46.46.46.4.m1.1.1.2">0.700</cn><cn id="S5.T3.46.46.46.46.4.m1.1.1.3.cmml" type="float" xref="S5.T3.46.46.46.46.4.m1.1.1.3">0.009</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.46.46.46.46.4.m1.1c">0.700\pm 0.009</annotation><annotation encoding="application/x-llamapun" id="S5.T3.46.46.46.46.4.m1.1d">0.700 ± 0.009</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.47.47.47.47.5"><math alttext="0.589\pm 0.014" class="ltx_Math" display="inline" id="S5.T3.47.47.47.47.5.m1.1"><semantics id="S5.T3.47.47.47.47.5.m1.1a"><mrow id="S5.T3.47.47.47.47.5.m1.1.1" xref="S5.T3.47.47.47.47.5.m1.1.1.cmml"><mn id="S5.T3.47.47.47.47.5.m1.1.1.2" xref="S5.T3.47.47.47.47.5.m1.1.1.2.cmml">0.589</mn><mo id="S5.T3.47.47.47.47.5.m1.1.1.1" xref="S5.T3.47.47.47.47.5.m1.1.1.1.cmml">±</mo><mn id="S5.T3.47.47.47.47.5.m1.1.1.3" xref="S5.T3.47.47.47.47.5.m1.1.1.3.cmml">0.014</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.47.47.47.47.5.m1.1b"><apply id="S5.T3.47.47.47.47.5.m1.1.1.cmml" xref="S5.T3.47.47.47.47.5.m1.1.1"><csymbol cd="latexml" id="S5.T3.47.47.47.47.5.m1.1.1.1.cmml" xref="S5.T3.47.47.47.47.5.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.47.47.47.47.5.m1.1.1.2.cmml" type="float" xref="S5.T3.47.47.47.47.5.m1.1.1.2">0.589</cn><cn id="S5.T3.47.47.47.47.5.m1.1.1.3.cmml" type="float" xref="S5.T3.47.47.47.47.5.m1.1.1.3">0.014</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.47.47.47.47.5.m1.1c">0.589\pm 0.014</annotation><annotation encoding="application/x-llamapun" id="S5.T3.47.47.47.47.5.m1.1d">0.589 ± 0.014</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T3.48.48.48.48.6"><math alttext="0.693\pm 0.005" class="ltx_Math" display="inline" id="S5.T3.48.48.48.48.6.m1.1"><semantics id="S5.T3.48.48.48.48.6.m1.1a"><mrow id="S5.T3.48.48.48.48.6.m1.1.1" xref="S5.T3.48.48.48.48.6.m1.1.1.cmml"><mn id="S5.T3.48.48.48.48.6.m1.1.1.2" xref="S5.T3.48.48.48.48.6.m1.1.1.2.cmml">0.693</mn><mo id="S5.T3.48.48.48.48.6.m1.1.1.1" xref="S5.T3.48.48.48.48.6.m1.1.1.1.cmml">±</mo><mn id="S5.T3.48.48.48.48.6.m1.1.1.3" xref="S5.T3.48.48.48.48.6.m1.1.1.3.cmml">0.005</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.48.48.48.48.6.m1.1b"><apply id="S5.T3.48.48.48.48.6.m1.1.1.cmml" xref="S5.T3.48.48.48.48.6.m1.1.1"><csymbol cd="latexml" id="S5.T3.48.48.48.48.6.m1.1.1.1.cmml" xref="S5.T3.48.48.48.48.6.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.48.48.48.48.6.m1.1.1.2.cmml" type="float" xref="S5.T3.48.48.48.48.6.m1.1.1.2">0.693</cn><cn id="S5.T3.48.48.48.48.6.m1.1.1.3.cmml" type="float" xref="S5.T3.48.48.48.48.6.m1.1.1.3">0.005</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.48.48.48.48.6.m1.1c">0.693\pm 0.005</annotation><annotation encoding="application/x-llamapun" id="S5.T3.48.48.48.48.6.m1.1d">0.693 ± 0.005</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T3.54.54.54.54"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_r" id="S5.T3.54.54.54.54.7">Proposed</th> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.49.49.49.49.1"><math alttext="\textbf{0.731}\pm 0.009" class="ltx_Math" display="inline" id="S5.T3.49.49.49.49.1.m1.1"><semantics id="S5.T3.49.49.49.49.1.m1.1a"><mrow id="S5.T3.49.49.49.49.1.m1.1.1" xref="S5.T3.49.49.49.49.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S5.T3.49.49.49.49.1.m1.1.1.2" xref="S5.T3.49.49.49.49.1.m1.1.1.2a.cmml">0.731</mtext><mo id="S5.T3.49.49.49.49.1.m1.1.1.1" xref="S5.T3.49.49.49.49.1.m1.1.1.1.cmml">±</mo><mn id="S5.T3.49.49.49.49.1.m1.1.1.3" xref="S5.T3.49.49.49.49.1.m1.1.1.3.cmml">0.009</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.49.49.49.49.1.m1.1b"><apply id="S5.T3.49.49.49.49.1.m1.1.1.cmml" xref="S5.T3.49.49.49.49.1.m1.1.1"><csymbol cd="latexml" id="S5.T3.49.49.49.49.1.m1.1.1.1.cmml" xref="S5.T3.49.49.49.49.1.m1.1.1.1">plus-or-minus</csymbol><ci id="S5.T3.49.49.49.49.1.m1.1.1.2a.cmml" xref="S5.T3.49.49.49.49.1.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S5.T3.49.49.49.49.1.m1.1.1.2.cmml" xref="S5.T3.49.49.49.49.1.m1.1.1.2">0.731</mtext></ci><cn id="S5.T3.49.49.49.49.1.m1.1.1.3.cmml" type="float" xref="S5.T3.49.49.49.49.1.m1.1.1.3">0.009</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.49.49.49.49.1.m1.1c">\textbf{0.731}\pm 0.009</annotation><annotation encoding="application/x-llamapun" id="S5.T3.49.49.49.49.1.m1.1d">0.731 ± 0.009</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.50.50.50.50.2"><math alttext="\textbf{0.804}\pm 0.013" class="ltx_Math" display="inline" id="S5.T3.50.50.50.50.2.m1.1"><semantics id="S5.T3.50.50.50.50.2.m1.1a"><mrow id="S5.T3.50.50.50.50.2.m1.1.1" xref="S5.T3.50.50.50.50.2.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S5.T3.50.50.50.50.2.m1.1.1.2" xref="S5.T3.50.50.50.50.2.m1.1.1.2a.cmml">0.804</mtext><mo id="S5.T3.50.50.50.50.2.m1.1.1.1" xref="S5.T3.50.50.50.50.2.m1.1.1.1.cmml">±</mo><mn id="S5.T3.50.50.50.50.2.m1.1.1.3" xref="S5.T3.50.50.50.50.2.m1.1.1.3.cmml">0.013</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.50.50.50.50.2.m1.1b"><apply id="S5.T3.50.50.50.50.2.m1.1.1.cmml" xref="S5.T3.50.50.50.50.2.m1.1.1"><csymbol cd="latexml" id="S5.T3.50.50.50.50.2.m1.1.1.1.cmml" xref="S5.T3.50.50.50.50.2.m1.1.1.1">plus-or-minus</csymbol><ci id="S5.T3.50.50.50.50.2.m1.1.1.2a.cmml" xref="S5.T3.50.50.50.50.2.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S5.T3.50.50.50.50.2.m1.1.1.2.cmml" xref="S5.T3.50.50.50.50.2.m1.1.1.2">0.804</mtext></ci><cn id="S5.T3.50.50.50.50.2.m1.1.1.3.cmml" type="float" xref="S5.T3.50.50.50.50.2.m1.1.1.3">0.013</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.50.50.50.50.2.m1.1c">\textbf{0.804}\pm 0.013</annotation><annotation encoding="application/x-llamapun" id="S5.T3.50.50.50.50.2.m1.1d">0.804 ± 0.013</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.51.51.51.51.3"><math alttext="\textbf{0.747}\pm 0.010" class="ltx_Math" display="inline" id="S5.T3.51.51.51.51.3.m1.1"><semantics id="S5.T3.51.51.51.51.3.m1.1a"><mrow id="S5.T3.51.51.51.51.3.m1.1.1" xref="S5.T3.51.51.51.51.3.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S5.T3.51.51.51.51.3.m1.1.1.2" xref="S5.T3.51.51.51.51.3.m1.1.1.2a.cmml">0.747</mtext><mo id="S5.T3.51.51.51.51.3.m1.1.1.1" xref="S5.T3.51.51.51.51.3.m1.1.1.1.cmml">±</mo><mn id="S5.T3.51.51.51.51.3.m1.1.1.3" xref="S5.T3.51.51.51.51.3.m1.1.1.3.cmml">0.010</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.51.51.51.51.3.m1.1b"><apply id="S5.T3.51.51.51.51.3.m1.1.1.cmml" xref="S5.T3.51.51.51.51.3.m1.1.1"><csymbol cd="latexml" id="S5.T3.51.51.51.51.3.m1.1.1.1.cmml" xref="S5.T3.51.51.51.51.3.m1.1.1.1">plus-or-minus</csymbol><ci id="S5.T3.51.51.51.51.3.m1.1.1.2a.cmml" xref="S5.T3.51.51.51.51.3.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S5.T3.51.51.51.51.3.m1.1.1.2.cmml" xref="S5.T3.51.51.51.51.3.m1.1.1.2">0.747</mtext></ci><cn id="S5.T3.51.51.51.51.3.m1.1.1.3.cmml" type="float" xref="S5.T3.51.51.51.51.3.m1.1.1.3">0.010</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.51.51.51.51.3.m1.1c">\textbf{0.747}\pm 0.010</annotation><annotation encoding="application/x-llamapun" id="S5.T3.51.51.51.51.3.m1.1d">0.747 ± 0.010</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.52.52.52.52.4"><math alttext="\textbf{0.782}\pm 0.008" class="ltx_Math" display="inline" id="S5.T3.52.52.52.52.4.m1.1"><semantics id="S5.T3.52.52.52.52.4.m1.1a"><mrow id="S5.T3.52.52.52.52.4.m1.1.1" xref="S5.T3.52.52.52.52.4.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S5.T3.52.52.52.52.4.m1.1.1.2" xref="S5.T3.52.52.52.52.4.m1.1.1.2a.cmml">0.782</mtext><mo id="S5.T3.52.52.52.52.4.m1.1.1.1" xref="S5.T3.52.52.52.52.4.m1.1.1.1.cmml">±</mo><mn id="S5.T3.52.52.52.52.4.m1.1.1.3" xref="S5.T3.52.52.52.52.4.m1.1.1.3.cmml">0.008</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.52.52.52.52.4.m1.1b"><apply id="S5.T3.52.52.52.52.4.m1.1.1.cmml" xref="S5.T3.52.52.52.52.4.m1.1.1"><csymbol cd="latexml" id="S5.T3.52.52.52.52.4.m1.1.1.1.cmml" xref="S5.T3.52.52.52.52.4.m1.1.1.1">plus-or-minus</csymbol><ci id="S5.T3.52.52.52.52.4.m1.1.1.2a.cmml" xref="S5.T3.52.52.52.52.4.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S5.T3.52.52.52.52.4.m1.1.1.2.cmml" xref="S5.T3.52.52.52.52.4.m1.1.1.2">0.782</mtext></ci><cn id="S5.T3.52.52.52.52.4.m1.1.1.3.cmml" type="float" xref="S5.T3.52.52.52.52.4.m1.1.1.3">0.008</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.52.52.52.52.4.m1.1c">\textbf{0.782}\pm 0.008</annotation><annotation encoding="application/x-llamapun" id="S5.T3.52.52.52.52.4.m1.1d">0.782 ± 0.008</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.53.53.53.53.5"><math alttext="\mathbf{0.744}\pm 0.006" class="ltx_Math" display="inline" id="S5.T3.53.53.53.53.5.m1.1"><semantics id="S5.T3.53.53.53.53.5.m1.1a"><mrow id="S5.T3.53.53.53.53.5.m1.1.1" xref="S5.T3.53.53.53.53.5.m1.1.1.cmml"><mn class="ltx_mathvariant_bold" id="S5.T3.53.53.53.53.5.m1.1.1.2" mathvariant="bold" xref="S5.T3.53.53.53.53.5.m1.1.1.2.cmml">0.744</mn><mo id="S5.T3.53.53.53.53.5.m1.1.1.1" xref="S5.T3.53.53.53.53.5.m1.1.1.1.cmml">±</mo><mn id="S5.T3.53.53.53.53.5.m1.1.1.3" xref="S5.T3.53.53.53.53.5.m1.1.1.3.cmml">0.006</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.53.53.53.53.5.m1.1b"><apply id="S5.T3.53.53.53.53.5.m1.1.1.cmml" xref="S5.T3.53.53.53.53.5.m1.1.1"><csymbol cd="latexml" id="S5.T3.53.53.53.53.5.m1.1.1.1.cmml" xref="S5.T3.53.53.53.53.5.m1.1.1.1">plus-or-minus</csymbol><cn id="S5.T3.53.53.53.53.5.m1.1.1.2.cmml" type="float" xref="S5.T3.53.53.53.53.5.m1.1.1.2">0.744</cn><cn id="S5.T3.53.53.53.53.5.m1.1.1.3.cmml" type="float" xref="S5.T3.53.53.53.53.5.m1.1.1.3">0.006</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.53.53.53.53.5.m1.1c">\mathbf{0.744}\pm 0.006</annotation><annotation encoding="application/x-llamapun" id="S5.T3.53.53.53.53.5.m1.1d">bold_0.744 ± 0.006</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T3.54.54.54.54.6"><math alttext="\textbf{0.761}\pm 0.004" class="ltx_Math" display="inline" id="S5.T3.54.54.54.54.6.m1.1"><semantics id="S5.T3.54.54.54.54.6.m1.1a"><mrow id="S5.T3.54.54.54.54.6.m1.1.1" xref="S5.T3.54.54.54.54.6.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S5.T3.54.54.54.54.6.m1.1.1.2" xref="S5.T3.54.54.54.54.6.m1.1.1.2a.cmml">0.761</mtext><mo id="S5.T3.54.54.54.54.6.m1.1.1.1" xref="S5.T3.54.54.54.54.6.m1.1.1.1.cmml">±</mo><mn id="S5.T3.54.54.54.54.6.m1.1.1.3" xref="S5.T3.54.54.54.54.6.m1.1.1.3.cmml">0.004</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.54.54.54.54.6.m1.1b"><apply id="S5.T3.54.54.54.54.6.m1.1.1.cmml" xref="S5.T3.54.54.54.54.6.m1.1.1"><csymbol cd="latexml" id="S5.T3.54.54.54.54.6.m1.1.1.1.cmml" xref="S5.T3.54.54.54.54.6.m1.1.1.1">plus-or-minus</csymbol><ci id="S5.T3.54.54.54.54.6.m1.1.1.2a.cmml" xref="S5.T3.54.54.54.54.6.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S5.T3.54.54.54.54.6.m1.1.1.2.cmml" xref="S5.T3.54.54.54.54.6.m1.1.1.2">0.761</mtext></ci><cn id="S5.T3.54.54.54.54.6.m1.1.1.3.cmml" type="float" xref="S5.T3.54.54.54.54.6.m1.1.1.3">0.004</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.54.54.54.54.6.m1.1c">\textbf{0.761}\pm 0.004</annotation><annotation encoding="application/x-llamapun" id="S5.T3.54.54.54.54.6.m1.1d">0.761 ± 0.004</annotation></semantics></math></td> </tr> </tbody> </table> </div> <figcaption class="ltx_caption ltx_centering">Table C: Comparative detection scores (AUC) in A(S)DI modes. Dataset names (stars) and parallactic amplitude <math alttext="\Delta_{\phi}" class="ltx_Math" display="inline" id="S5.T3.56.56.1.m1.1"><semantics id="S5.T3.56.56.1.m1.1b"><msub id="S5.T3.56.56.1.m1.1.1" xref="S5.T3.56.56.1.m1.1.1.cmml"><mi id="S5.T3.56.56.1.m1.1.1.2" mathvariant="normal" xref="S5.T3.56.56.1.m1.1.1.2.cmml">Δ</mi><mi id="S5.T3.56.56.1.m1.1.1.3" xref="S5.T3.56.56.1.m1.1.1.3.cmml">ϕ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.T3.56.56.1.m1.1c"><apply id="S5.T3.56.56.1.m1.1.1.cmml" xref="S5.T3.56.56.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T3.56.56.1.m1.1.1.1.cmml" xref="S5.T3.56.56.1.m1.1.1">subscript</csymbol><ci id="S5.T3.56.56.1.m1.1.1.2.cmml" xref="S5.T3.56.56.1.m1.1.1.2">Δ</ci><ci id="S5.T3.56.56.1.m1.1.1.3.cmml" xref="S5.T3.56.56.1.m1.1.1.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.56.56.1.m1.1d">\Delta_{\phi}</annotation><annotation encoding="application/x-llamapun" id="S5.T3.56.56.1.m1.1e">roman_Δ start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT</annotation></semantics></math> are reported on top. ASDI mode (which is better than ADI) is not supported by MODEL&CO.</figcaption> </figure> </section> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> 5.2 Quantitative and qualitative evaluations</h3> <section class="ltx_paragraph" id="S5.SS2.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Statistical model.</h4> <div class="ltx_para" id="S5.SS2.SSS0.Px1.p1"> We evaluate the impact of leveraging multiple spatial scales and symmetries in our statistical model by testing its detection performance across configurations listed in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.T1" title="Table A ‣ 5.1 Data, algorithms and evaluation protocol ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A</a>. Results show that using both symmetries and scales is crucial in ADI, where statistical parameters suffer from self-subtraction without the robustness of added spectral diversity. </div> <figure class="ltx_figure" id="S5.F4"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="464" id="S5.F4.g1" src="x4.png" width="447"/> <figcaption class="ltx_caption">Figure D: Detection maps on observations of HD 159911 star with synthetic exoplanets. The (calibrated) detection threshold is equivalent for all methods. The proposed approach here detect 1 additional source compared to the second best method (PACO ASDI).</figcaption> </figure> <div class="ltx_para" id="S5.SS2.SSS0.Px1.p2"> We then evaluate the statistical model’s performance in estimating flux and sub-pixel position (regression tasks) using the optimization process from Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1" title="4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.1</a>. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.T2" title="Table B ‣ Baselines. ‣ 5.1 Data, algorithms and evaluation protocol ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">B</a> shows average errors across 1,351 synthetic exoplanets detectable by both considered methods. Our approach outperforming others in almost all cases. </div> </section> <section class="ltx_paragraph" id="S5.SS2.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">End-to-end learnable model.</h4> <div class="ltx_para" id="S5.SS2.SSS0.Px2.p1"> We evaluate detection performance on 5 observations using synthetic exoplanet injections through direct models (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S3.E1" title="Equation 1 ‣ 3.1 Angular Differential Imaging (ADI) ‣ 3 Image formation models ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">1</a>)-(<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S3.E2" title="Equation 2 ‣ 3.2 Angular Spectral Differential Imaging (ASDI) ‣ 3 Image formation models ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">2</a>). For each observation, 100 cubes are generated, totaling 1,000 injected exoplanets. This process is repeated 5 times with different random seeds, and AUC scores are reported in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.T3" title="Table C ‣ Metrics. ‣ 5.1 Data, algorithms and evaluation protocol ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">C</a>. Examples of representative detection maps are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A4.F11" title="Figure K ‣ Appendix D Additional results ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">K</a>. The proposed approach matches or outperforms state of the art algorithms in ADI mode. In ASDI mode, it shows a significant boost in detection sensitivity, consistently surpassing comparative methods. </div> </section> <section class="ltx_paragraph" id="S5.SS2.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Validation on real data.</h4> <div class="ltx_para" id="S5.SS2.SSS0.Px3.p1"> We compare detection methods using 3 observations, spanning several years, of the HR 8799 star. The stacked detection maps, highlighting exoplanets’ orbital motion, are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S5.F5" title="Figure E ‣ Validation on real data. ‣ 5.2 Quantitative and qualitative evaluations ‣ 5 Experiments ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">E</a>. All detection maps use a consistent unit (signal-to-noise score) and dynamic range. The proposed method maximizes detection confidence with no false alarms. </div> <figure class="ltx_figure" id="S5.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="297" id="S5.F5.g1" src="x5.png" width="447"/> <figcaption class="ltx_caption ltx_centering">Figure E: Detection maps on 3 observations (stacked in false RGB colors) of HR 8799 star. The elliptical arcs depict the estimated (projected) orbits of three known exoplanets, with the detection results shown as red, green and blue dots for the corresponding 2016, 2018 and 2021 observations. Squares are for false alarms identified in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib5" title="">5</a>]</cite>, Fig. 17.</figcaption> </figure> </section> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> 6 Conclusion</h2> <div class="ltx_para" id="S6.p1"> We propose a novel hybrid approach for exoplanet imaging that combines a multi-scale statistical model with deep learning, capturing spatial correlations in the nuisance component for simultaneous detection and flux estimation. This approach provides statistically grounded detection scores, unbiased estimates, and native uncertainty quantification. Tested on VLT/SPHERE data, it outperforms state-of-the-art techniques, demonstrating efficiency and robustness across varied data qualities. Its versatility and reduced computational complexity make it ideal for large-scale surveys. The approach will be extended to handle higher spectral resolution data. Additionally, the nuisance model can be adapted for reconstructing spatially extended objects like circumstellar disks, namely birthplaces of exoplanets. </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgments</h2> <div class="ltx_para ltx_noindent" id="Sx1.p1"> This work was supported by the French government under management of Agence Nationale de la Recherche as part of the “France 2030” program, PR[AI]RIE-PSAI projet (reference ANR-23-IACL-0008) and MIAI 3IA Institute (reference ANR-19-P3IA-0003), and by the European Research Council (ERC) under grant agreement 101087696 (APHELEIA project). This work was also supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (COBREX; grant agreement 885593), the ANR under the France 2030 program (PEPR Origins, reference ANR-22-EXOR-0016), the French National Programs (PNP and PNPS), and the Action Spécifique Haute Résolution Angulaire (ASHRA) of CNRS/INSU co-funded by CNES. This work was granted access to the HPC resources of IDRIS under the allocation 2022-AD011013643 made by GENCI. JP was supported in part by the Louis Vuitton/ENS chair in artificial intelligence and a Global Distinguished Professorship at the Courant Institute of Mathematical Sciences and the Center for Data Science at New York University. </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography" style="font-size:90%;">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> Aharon et al. [2006] Michal Aharon, Michael Elad, and Alfred Bruckstein. 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IEEE, 2011. </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> <div class="ltx_pagination ltx_role_newpage"></div> <div class="ltx_para ltx_noindent" id="p1"> \thetitle Supplementary Material </div> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix" style="font-size:144%;"> Appendix A Technical details</h2> <section class="ltx_subsection" id="A1.SS1"> <h3 class="ltx_title ltx_title_subsection" style="font-size:144%;"> A.1 Estimation of the statistical parameters</h3> <div class="ltx_para ltx_noindent" id="A1.SS1.p1"> In this section, we provide additional information regarding the estimation of the statistical parameters of the Gaussian distribution (see Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1" style="font-size:144%;" title="4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.1</a>). These parameters are estimated by maximum likelihood: <table class="ltx_equation ltx_eqn_table" id="A1.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widehat{\boldsymbol{m}}_{j}=\frac{1}{T}\sum_{t}\boldsymbol{y}_{t}^{(j)},\,% \widehat{\mathbf{S}}_{j}=\frac{1}{T}\sum_{t}\big{(}\boldsymbol{y}_{t}^{(j)}-% \widehat{\boldsymbol{m}}_{j}\big{)}\big{(}\boldsymbol{y}_{t}^{(j)}-\widehat{% \boldsymbol{m}}_{j}\big{)}^{\top}," class="ltx_Math" display="block" id="A1.Ex2.m1.4"><semantics id="A1.Ex2.m1.4a"><mrow id="A1.Ex2.m1.4.4.1"><mrow id="A1.Ex2.m1.4.4.1.1.2" xref="A1.Ex2.m1.4.4.1.1.3.cmml"><mrow id="A1.Ex2.m1.4.4.1.1.1.1" xref="A1.Ex2.m1.4.4.1.1.1.1.cmml"><msub id="A1.Ex2.m1.4.4.1.1.1.1.2" xref="A1.Ex2.m1.4.4.1.1.1.1.2.cmml"><mover accent="true" id="A1.Ex2.m1.4.4.1.1.1.1.2.2" xref="A1.Ex2.m1.4.4.1.1.1.1.2.2.cmml"><mi id="A1.Ex2.m1.4.4.1.1.1.1.2.2.2" 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end_POSTSUPERSCRIPT , over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_T end_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT - over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ( bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT - over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> where <math alttext="\widehat{\mathbf{S}}_{j}" class="ltx_Math" display="inline" id="A1.SS1.p1.1.m1.1"><semantics id="A1.SS1.p1.1.m1.1a"><msub id="A1.SS1.p1.1.m1.1.1" xref="A1.SS1.p1.1.m1.1.1.cmml"><mover accent="true" id="A1.SS1.p1.1.m1.1.1.2" xref="A1.SS1.p1.1.m1.1.1.2.cmml"><mi id="A1.SS1.p1.1.m1.1.1.2.2" mathsize="144%" xref="A1.SS1.p1.1.m1.1.1.2.2.cmml">𝐒</mi><mo id="A1.SS1.p1.1.m1.1.1.2.1" mathsize="144%" xref="A1.SS1.p1.1.m1.1.1.2.1.cmml">^</mo></mover><mi id="A1.SS1.p1.1.m1.1.1.3" mathsize="144%" xref="A1.SS1.p1.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.1.m1.1b"><apply id="A1.SS1.p1.1.m1.1.1.cmml" xref="A1.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.1.m1.1.1.1.cmml" xref="A1.SS1.p1.1.m1.1.1">subscript</csymbol><apply id="A1.SS1.p1.1.m1.1.1.2.cmml" xref="A1.SS1.p1.1.m1.1.1.2"><ci id="A1.SS1.p1.1.m1.1.1.2.1.cmml" xref="A1.SS1.p1.1.m1.1.1.2.1">^</ci><ci id="A1.SS1.p1.1.m1.1.1.2.2.cmml" xref="A1.SS1.p1.1.m1.1.1.2.2">𝐒</ci></apply><ci id="A1.SS1.p1.1.m1.1.1.3.cmml" xref="A1.SS1.p1.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.1.m1.1c">\widehat{\mathbf{S}}_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.1.m1.1d">over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{p\times p}" class="ltx_Math" display="inline" id="A1.SS1.p1.2.m2.1"><semantics id="A1.SS1.p1.2.m2.1a"><msup id="A1.SS1.p1.2.m2.1.1" xref="A1.SS1.p1.2.m2.1.1.cmml"><mi id="A1.SS1.p1.2.m2.1.1.2" mathsize="144%" xref="A1.SS1.p1.2.m2.1.1.2.cmml">ℝ</mi><mrow id="A1.SS1.p1.2.m2.1.1.3" xref="A1.SS1.p1.2.m2.1.1.3.cmml"><mi id="A1.SS1.p1.2.m2.1.1.3.2" mathsize="144%" xref="A1.SS1.p1.2.m2.1.1.3.2.cmml">p</mi><mo id="A1.SS1.p1.2.m2.1.1.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS1.p1.2.m2.1.1.3.1.cmml">×</mo><mi id="A1.SS1.p1.2.m2.1.1.3.3" mathsize="144%" xref="A1.SS1.p1.2.m2.1.1.3.3.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.2.m2.1b"><apply id="A1.SS1.p1.2.m2.1.1.cmml" xref="A1.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.2.m2.1.1.1.cmml" xref="A1.SS1.p1.2.m2.1.1">superscript</csymbol><ci id="A1.SS1.p1.2.m2.1.1.2.cmml" xref="A1.SS1.p1.2.m2.1.1.2">ℝ</ci><apply id="A1.SS1.p1.2.m2.1.1.3.cmml" xref="A1.SS1.p1.2.m2.1.1.3"><times id="A1.SS1.p1.2.m2.1.1.3.1.cmml" xref="A1.SS1.p1.2.m2.1.1.3.1"></times><ci id="A1.SS1.p1.2.m2.1.1.3.2.cmml" xref="A1.SS1.p1.2.m2.1.1.3.2">𝑝</ci><ci id="A1.SS1.p1.2.m2.1.1.3.3.cmml" xref="A1.SS1.p1.2.m2.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.2.m2.1c">\mathbb{R}^{p\times p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_p × italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> is the sample covariance matrix. Given that <math alttext="\widehat{\mathbf{S}}_{j}" class="ltx_Math" display="inline" id="A1.SS1.p1.3.m3.1"><semantics id="A1.SS1.p1.3.m3.1a"><msub id="A1.SS1.p1.3.m3.1.1" xref="A1.SS1.p1.3.m3.1.1.cmml"><mover accent="true" id="A1.SS1.p1.3.m3.1.1.2" xref="A1.SS1.p1.3.m3.1.1.2.cmml"><mi id="A1.SS1.p1.3.m3.1.1.2.2" mathsize="144%" xref="A1.SS1.p1.3.m3.1.1.2.2.cmml">𝐒</mi><mo id="A1.SS1.p1.3.m3.1.1.2.1" mathsize="144%" xref="A1.SS1.p1.3.m3.1.1.2.1.cmml">^</mo></mover><mi id="A1.SS1.p1.3.m3.1.1.3" mathsize="144%" xref="A1.SS1.p1.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.3.m3.1b"><apply id="A1.SS1.p1.3.m3.1.1.cmml" xref="A1.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.3.m3.1.1.1.cmml" xref="A1.SS1.p1.3.m3.1.1">subscript</csymbol><apply id="A1.SS1.p1.3.m3.1.1.2.cmml" xref="A1.SS1.p1.3.m3.1.1.2"><ci id="A1.SS1.p1.3.m3.1.1.2.1.cmml" xref="A1.SS1.p1.3.m3.1.1.2.1">^</ci><ci id="A1.SS1.p1.3.m3.1.1.2.2.cmml" xref="A1.SS1.p1.3.m3.1.1.2.2">𝐒</ci></apply><ci id="A1.SS1.p1.3.m3.1.1.3.cmml" xref="A1.SS1.p1.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.3.m3.1c">\widehat{\mathbf{S}}_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.3.m3.1d">over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> usually suffers from a large variance due to small sample statistics (i.e., <math alttext="p\geq T\simeq 70" class="ltx_Math" display="inline" id="A1.SS1.p1.4.m4.1"><semantics id="A1.SS1.p1.4.m4.1a"><mrow id="A1.SS1.p1.4.m4.1.1" xref="A1.SS1.p1.4.m4.1.1.cmml"><mi id="A1.SS1.p1.4.m4.1.1.2" mathsize="144%" xref="A1.SS1.p1.4.m4.1.1.2.cmml">p</mi><mo id="A1.SS1.p1.4.m4.1.1.3" mathsize="144%" xref="A1.SS1.p1.4.m4.1.1.3.cmml">≥</mo><mi id="A1.SS1.p1.4.m4.1.1.4" mathsize="144%" xref="A1.SS1.p1.4.m4.1.1.4.cmml">T</mi><mo id="A1.SS1.p1.4.m4.1.1.5" mathsize="144%" xref="A1.SS1.p1.4.m4.1.1.5.cmml">≃</mo><mn id="A1.SS1.p1.4.m4.1.1.6" mathsize="144%" xref="A1.SS1.p1.4.m4.1.1.6.cmml">70</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.4.m4.1b"><apply id="A1.SS1.p1.4.m4.1.1.cmml" xref="A1.SS1.p1.4.m4.1.1"><and id="A1.SS1.p1.4.m4.1.1a.cmml" xref="A1.SS1.p1.4.m4.1.1"></and><apply id="A1.SS1.p1.4.m4.1.1b.cmml" xref="A1.SS1.p1.4.m4.1.1"><geq id="A1.SS1.p1.4.m4.1.1.3.cmml" xref="A1.SS1.p1.4.m4.1.1.3"></geq><ci id="A1.SS1.p1.4.m4.1.1.2.cmml" xref="A1.SS1.p1.4.m4.1.1.2">𝑝</ci><ci id="A1.SS1.p1.4.m4.1.1.4.cmml" xref="A1.SS1.p1.4.m4.1.1.4">𝑇</ci></apply><apply id="A1.SS1.p1.4.m4.1.1c.cmml" xref="A1.SS1.p1.4.m4.1.1"><csymbol cd="latexml" id="A1.SS1.p1.4.m4.1.1.5.cmml" xref="A1.SS1.p1.4.m4.1.1.5">similar-to-or-equals</csymbol><share href="https://arxiv.org/html/2503.17117v1#A1.SS1.p1.4.m4.1.1.4.cmml" id="A1.SS1.p1.4.m4.1.1d.cmml" xref="A1.SS1.p1.4.m4.1.1"></share><cn id="A1.SS1.p1.4.m4.1.1.6.cmml" type="integer" xref="A1.SS1.p1.4.m4.1.1.6">70</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.4.m4.1c">p\geq T\simeq 70</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.4.m4.1d">italic_p ≥ italic_T ≃ 70</annotation></semantics></math>), we use a shrinkage estimator leading to the final covariance estimator <math alttext="\widehat{\mathbf{C}}_{j}" class="ltx_Math" display="inline" id="A1.SS1.p1.5.m5.1"><semantics id="A1.SS1.p1.5.m5.1a"><msub id="A1.SS1.p1.5.m5.1.1" xref="A1.SS1.p1.5.m5.1.1.cmml"><mover accent="true" id="A1.SS1.p1.5.m5.1.1.2" xref="A1.SS1.p1.5.m5.1.1.2.cmml"><mi id="A1.SS1.p1.5.m5.1.1.2.2" mathsize="144%" xref="A1.SS1.p1.5.m5.1.1.2.2.cmml">𝐂</mi><mo id="A1.SS1.p1.5.m5.1.1.2.1" mathsize="144%" xref="A1.SS1.p1.5.m5.1.1.2.1.cmml">^</mo></mover><mi id="A1.SS1.p1.5.m5.1.1.3" mathsize="144%" xref="A1.SS1.p1.5.m5.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.5.m5.1b"><apply 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id="A1.SS1.p1.6.m1.1.1.3" mathsize="144%" xref="A1.SS1.p1.6.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.6.m1.1b"><apply id="A1.SS1.p1.6.m1.1.1.cmml" xref="A1.SS1.p1.6.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.6.m1.1.1.1.cmml" xref="A1.SS1.p1.6.m1.1.1">subscript</csymbol><apply id="A1.SS1.p1.6.m1.1.1.2.cmml" xref="A1.SS1.p1.6.m1.1.1.2"><ci id="A1.SS1.p1.6.m1.1.1.2.1.cmml" xref="A1.SS1.p1.6.m1.1.1.2.1">^</ci><ci id="A1.SS1.p1.6.m1.1.1.2.2.cmml" xref="A1.SS1.p1.6.m1.1.1.2.2">𝐃</ci></apply><ci id="A1.SS1.p1.6.m1.1.1.3.cmml" xref="A1.SS1.p1.6.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.6.m1.1c">\widehat{\mathbf{D}}_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.6.m1.1d">over^ start_ARG bold_D end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{p\times p}" class="ltx_Math" display="inline" id="A1.SS1.p1.7.m2.1"><semantics id="A1.SS1.p1.7.m2.1a"><msup id="A1.SS1.p1.7.m2.1.1" xref="A1.SS1.p1.7.m2.1.1.cmml"><mi id="A1.SS1.p1.7.m2.1.1.2" mathsize="144%" xref="A1.SS1.p1.7.m2.1.1.2.cmml">ℝ</mi><mrow id="A1.SS1.p1.7.m2.1.1.3" xref="A1.SS1.p1.7.m2.1.1.3.cmml"><mi id="A1.SS1.p1.7.m2.1.1.3.2" mathsize="144%" xref="A1.SS1.p1.7.m2.1.1.3.2.cmml">p</mi><mo id="A1.SS1.p1.7.m2.1.1.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS1.p1.7.m2.1.1.3.1.cmml">×</mo><mi id="A1.SS1.p1.7.m2.1.1.3.3" mathsize="144%" xref="A1.SS1.p1.7.m2.1.1.3.3.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.7.m2.1b"><apply id="A1.SS1.p1.7.m2.1.1.cmml" xref="A1.SS1.p1.7.m2.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.7.m2.1.1.1.cmml" xref="A1.SS1.p1.7.m2.1.1">superscript</csymbol><ci id="A1.SS1.p1.7.m2.1.1.2.cmml" xref="A1.SS1.p1.7.m2.1.1.2">ℝ</ci><apply id="A1.SS1.p1.7.m2.1.1.3.cmml" xref="A1.SS1.p1.7.m2.1.1.3"><times id="A1.SS1.p1.7.m2.1.1.3.1.cmml" xref="A1.SS1.p1.7.m2.1.1.3.1"></times><ci id="A1.SS1.p1.7.m2.1.1.3.2.cmml" xref="A1.SS1.p1.7.m2.1.1.3.2">𝑝</ci><ci id="A1.SS1.p1.7.m2.1.1.3.3.cmml" xref="A1.SS1.p1.7.m2.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.7.m2.1c">\mathbb{R}^{p\times p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.7.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_p × italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> is the low-variance/high-bias diagonal matrix formed with diagonal elements of <math alttext="\widehat{\mathbf{S}}_{j}" class="ltx_Math" display="inline" id="A1.SS1.p1.8.m3.1"><semantics id="A1.SS1.p1.8.m3.1a"><msub id="A1.SS1.p1.8.m3.1.1" xref="A1.SS1.p1.8.m3.1.1.cmml"><mover accent="true" id="A1.SS1.p1.8.m3.1.1.2" xref="A1.SS1.p1.8.m3.1.1.2.cmml"><mi id="A1.SS1.p1.8.m3.1.1.2.2" mathsize="144%" xref="A1.SS1.p1.8.m3.1.1.2.2.cmml">𝐒</mi><mo id="A1.SS1.p1.8.m3.1.1.2.1" mathsize="144%" xref="A1.SS1.p1.8.m3.1.1.2.1.cmml">^</mo></mover><mi id="A1.SS1.p1.8.m3.1.1.3" mathsize="144%" xref="A1.SS1.p1.8.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.8.m3.1b"><apply id="A1.SS1.p1.8.m3.1.1.cmml" xref="A1.SS1.p1.8.m3.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.8.m3.1.1.1.cmml" xref="A1.SS1.p1.8.m3.1.1">subscript</csymbol><apply id="A1.SS1.p1.8.m3.1.1.2.cmml" xref="A1.SS1.p1.8.m3.1.1.2"><ci id="A1.SS1.p1.8.m3.1.1.2.1.cmml" xref="A1.SS1.p1.8.m3.1.1.2.1">^</ci><ci id="A1.SS1.p1.8.m3.1.1.2.2.cmml" xref="A1.SS1.p1.8.m3.1.1.2.2">𝐒</ci></apply><ci id="A1.SS1.p1.8.m3.1.1.3.cmml" xref="A1.SS1.p1.8.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.8.m3.1c">\widehat{\mathbf{S}}_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.8.m3.1d">over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\widehat{\rho}_{j}" class="ltx_Math" display="inline" id="A1.SS1.p1.9.m4.1"><semantics id="A1.SS1.p1.9.m4.1a"><msub id="A1.SS1.p1.9.m4.1.1" xref="A1.SS1.p1.9.m4.1.1.cmml"><mover accent="true" id="A1.SS1.p1.9.m4.1.1.2" xref="A1.SS1.p1.9.m4.1.1.2.cmml"><mi id="A1.SS1.p1.9.m4.1.1.2.2" mathsize="144%" xref="A1.SS1.p1.9.m4.1.1.2.2.cmml">ρ</mi><mo id="A1.SS1.p1.9.m4.1.1.2.1" mathsize="144%" xref="A1.SS1.p1.9.m4.1.1.2.1.cmml">^</mo></mover><mi id="A1.SS1.p1.9.m4.1.1.3" mathsize="144%" xref="A1.SS1.p1.9.m4.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.9.m4.1b"><apply id="A1.SS1.p1.9.m4.1.1.cmml" xref="A1.SS1.p1.9.m4.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.9.m4.1.1.1.cmml" xref="A1.SS1.p1.9.m4.1.1">subscript</csymbol><apply id="A1.SS1.p1.9.m4.1.1.2.cmml" xref="A1.SS1.p1.9.m4.1.1.2"><ci id="A1.SS1.p1.9.m4.1.1.2.1.cmml" xref="A1.SS1.p1.9.m4.1.1.2.1">^</ci><ci id="A1.SS1.p1.9.m4.1.1.2.2.cmml" xref="A1.SS1.p1.9.m4.1.1.2.2">𝜌</ci></apply><ci id="A1.SS1.p1.9.m4.1.1.3.cmml" xref="A1.SS1.p1.9.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.9.m4.1c">\widehat{\rho}_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.9.m4.1d">over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="[0,\,1]" class="ltx_Math" display="inline" id="A1.SS1.p1.10.m5.2"><semantics id="A1.SS1.p1.10.m5.2a"><mrow id="A1.SS1.p1.10.m5.2.3.2" xref="A1.SS1.p1.10.m5.2.3.1.cmml"><mo id="A1.SS1.p1.10.m5.2.3.2.1" maxsize="144%" minsize="144%" xref="A1.SS1.p1.10.m5.2.3.1.cmml">[</mo><mn id="A1.SS1.p1.10.m5.1.1" mathsize="144%" xref="A1.SS1.p1.10.m5.1.1.cmml">0</mn><mo id="A1.SS1.p1.10.m5.2.3.2.2" mathsize="144%" xref="A1.SS1.p1.10.m5.2.3.1.cmml">,</mo><mn id="A1.SS1.p1.10.m5.2.2" mathsize="144%" xref="A1.SS1.p1.10.m5.2.2.cmml"> 1</mn><mo id="A1.SS1.p1.10.m5.2.3.2.3" maxsize="144%" minsize="144%" xref="A1.SS1.p1.10.m5.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.10.m5.2b"><interval closure="closed" id="A1.SS1.p1.10.m5.2.3.1.cmml" xref="A1.SS1.p1.10.m5.2.3.2"><cn id="A1.SS1.p1.10.m5.1.1.cmml" type="integer" xref="A1.SS1.p1.10.m5.1.1">0</cn><cn id="A1.SS1.p1.10.m5.2.2.cmml" type="integer" xref="A1.SS1.p1.10.m5.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.10.m5.2c">[0,\,1]</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.10.m5.2d">[ 0 , 1 ]</annotation></semantics></math>. Hyper-parameter <math alttext="\widehat{\rho}_{j}" class="ltx_Math" display="inline" id="A1.SS1.p1.11.m6.1"><semantics id="A1.SS1.p1.11.m6.1a"><msub id="A1.SS1.p1.11.m6.1.1" xref="A1.SS1.p1.11.m6.1.1.cmml"><mover accent="true" id="A1.SS1.p1.11.m6.1.1.2" xref="A1.SS1.p1.11.m6.1.1.2.cmml"><mi id="A1.SS1.p1.11.m6.1.1.2.2" mathsize="144%" xref="A1.SS1.p1.11.m6.1.1.2.2.cmml">ρ</mi><mo id="A1.SS1.p1.11.m6.1.1.2.1" mathsize="144%" xref="A1.SS1.p1.11.m6.1.1.2.1.cmml">^</mo></mover><mi id="A1.SS1.p1.11.m6.1.1.3" mathsize="144%" xref="A1.SS1.p1.11.m6.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.11.m6.1b"><apply id="A1.SS1.p1.11.m6.1.1.cmml" xref="A1.SS1.p1.11.m6.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.11.m6.1.1.1.cmml" xref="A1.SS1.p1.11.m6.1.1">subscript</csymbol><apply id="A1.SS1.p1.11.m6.1.1.2.cmml" xref="A1.SS1.p1.11.m6.1.1.2"><ci id="A1.SS1.p1.11.m6.1.1.2.1.cmml" xref="A1.SS1.p1.11.m6.1.1.2.1">^</ci><ci id="A1.SS1.p1.11.m6.1.1.2.2.cmml" xref="A1.SS1.p1.11.m6.1.1.2.2">𝜌</ci></apply><ci id="A1.SS1.p1.11.m6.1.1.3.cmml" xref="A1.SS1.p1.11.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.11.m6.1c">\widehat{\rho}_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.11.m6.1d">over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> can be estimated optimally in a data-driven fashion by risk minimization between the true (unknown) covariance <math alttext="\mathbf{C}_{j}" class="ltx_Math" display="inline" id="A1.SS1.p1.12.m7.1"><semantics id="A1.SS1.p1.12.m7.1a"><msub id="A1.SS1.p1.12.m7.1.1" xref="A1.SS1.p1.12.m7.1.1.cmml"><mi id="A1.SS1.p1.12.m7.1.1.2" mathsize="144%" xref="A1.SS1.p1.12.m7.1.1.2.cmml">𝐂</mi><mi id="A1.SS1.p1.12.m7.1.1.3" mathsize="144%" xref="A1.SS1.p1.12.m7.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.12.m7.1b"><apply id="A1.SS1.p1.12.m7.1.1.cmml" xref="A1.SS1.p1.12.m7.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.12.m7.1.1.1.cmml" xref="A1.SS1.p1.12.m7.1.1">subscript</csymbol><ci id="A1.SS1.p1.12.m7.1.1.2.cmml" xref="A1.SS1.p1.12.m7.1.1.2">𝐂</ci><ci id="A1.SS1.p1.12.m7.1.1.3.cmml" xref="A1.SS1.p1.12.m7.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.12.m7.1c">\mathbf{C}_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.12.m7.1d">bold_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and its shrunk counterpart <math alttext="\widehat{\mathbf{C}}_{j}" class="ltx_Math" display="inline" id="A1.SS1.p1.13.m8.1"><semantics id="A1.SS1.p1.13.m8.1a"><msub id="A1.SS1.p1.13.m8.1.1" xref="A1.SS1.p1.13.m8.1.1.cmml"><mover accent="true" id="A1.SS1.p1.13.m8.1.1.2" xref="A1.SS1.p1.13.m8.1.1.2.cmml"><mi id="A1.SS1.p1.13.m8.1.1.2.2" mathsize="144%" xref="A1.SS1.p1.13.m8.1.1.2.2.cmml">𝐂</mi><mo id="A1.SS1.p1.13.m8.1.1.2.1" mathsize="144%" xref="A1.SS1.p1.13.m8.1.1.2.1.cmml">^</mo></mover><mi id="A1.SS1.p1.13.m8.1.1.3" mathsize="144%" xref="A1.SS1.p1.13.m8.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.13.m8.1b"><apply id="A1.SS1.p1.13.m8.1.1.cmml" xref="A1.SS1.p1.13.m8.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.13.m8.1.1.1.cmml" xref="A1.SS1.p1.13.m8.1.1">subscript</csymbol><apply id="A1.SS1.p1.13.m8.1.1.2.cmml" xref="A1.SS1.p1.13.m8.1.1.2"><ci id="A1.SS1.p1.13.m8.1.1.2.1.cmml" xref="A1.SS1.p1.13.m8.1.1.2.1">^</ci><ci id="A1.SS1.p1.13.m8.1.1.2.2.cmml" xref="A1.SS1.p1.13.m8.1.1.2.2">𝐂</ci></apply><ci id="A1.SS1.p1.13.m8.1.1.3.cmml" xref="A1.SS1.p1.13.m8.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.13.m8.1c">\widehat{\mathbf{C}}_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.13.m8.1d">over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. This leads to the closed-form expression <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib29" title="">29</a>]</cite>: <table class="ltx_equation ltx_eqn_table" id="A1.E24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widehat{\rho}_{j}=\frac{\operatorname{tr}(\widehat{\mathbf{S}}^{2}_{j})+% \operatorname{tr}^{2}(\widehat{\mathbf{S}}_{j})-2\operatorname{tr}(\widehat{% \mathbf{S}}_{j}\circ\widehat{\mathbf{S}}_{j})}{(T+1)(\operatorname{tr}(% \widehat{\mathbf{S}}_{j}^{2})-\operatorname{tr}(\widehat{\mathbf{S}}_{j}\circ% \widehat{\mathbf{S}}_{j}))}\,," class="ltx_Math" display="block" id="A1.E24.m1.11"><semantics id="A1.E24.m1.11a"><mrow id="A1.E24.m1.11.11.1" xref="A1.E24.m1.11.11.1.1.cmml"><mrow id="A1.E24.m1.11.11.1.1" xref="A1.E24.m1.11.11.1.1.cmml"><msub id="A1.E24.m1.11.11.1.1.2" 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over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) - 2 roman_tr ( over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) end_ARG start_ARG ( italic_T + 1 ) ( roman_tr ( over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - roman_tr ( over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ over^ start_ARG bold_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(24)</td> </tr></tbody> </table> where <math alttext="\circ" class="ltx_Math" display="inline" id="A1.SS1.p1.14.m1.1"><semantics id="A1.SS1.p1.14.m1.1a"><mo id="A1.SS1.p1.14.m1.1.1" mathsize="144%" 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Characterize Exoplanets in Direct Imaging Observations">4.1</a>. To obtain a full-frame detection map, <math alttext="\widehat{\alpha}" class="ltx_Math" display="inline" id="A1.SS2.p1.1.m1.1"><semantics id="A1.SS2.p1.1.m1.1a"><mover accent="true" id="A1.SS2.p1.1.m1.1.1" xref="A1.SS2.p1.1.m1.1.1.cmml"><mi id="A1.SS2.p1.1.m1.1.1.2" mathsize="144%" xref="A1.SS2.p1.1.m1.1.1.2.cmml">α</mi><mo id="A1.SS2.p1.1.m1.1.1.1" mathsize="144%" xref="A1.SS2.p1.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.1.m1.1b"><apply id="A1.SS2.p1.1.m1.1.1.cmml" xref="A1.SS2.p1.1.m1.1.1"><ci id="A1.SS2.p1.1.m1.1.1.1.cmml" xref="A1.SS2.p1.1.m1.1.1.1">^</ci><ci id="A1.SS2.p1.1.m1.1.1.2.cmml" xref="A1.SS2.p1.1.m1.1.1.2">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.1.m1.1c">\widehat{\alpha}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.1.m1.1d">over^ start_ARG italic_α end_ARG</annotation></semantics></math> and <math alttext="\widehat{\sigma}_{\alpha}" class="ltx_Math" 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xref="A1.SS2.p1.2.m2.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.2.m2.1c">\widehat{\sigma}_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.2.m2.1d">over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> must be computed for all spatial coordinates, which can be formalized by: <table class="ltx_equation ltx_eqn_table" id="A1.E25"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall\boldsymbol{x}_{i}\in\mathcal{G}_{p},\ \widehat{\alpha}_{i}=% \operatorname*{arg\,max}_{\alpha_{i}}\ell(\alpha_{i},\boldsymbol{x}_{i})\,." class="ltx_Math" display="block" id="A1.E25.m1.1"><semantics id="A1.E25.m1.1a"><mrow id="A1.E25.m1.1.1.1"><mrow id="A1.E25.m1.1.1.1.1.2" xref="A1.E25.m1.1.1.1.1.3.cmml"><mrow id="A1.E25.m1.1.1.1.1.1.1" 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xref="A1.E25.m1.1.1.1.1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A1.E25.m1.1.1.1.1.2.2.2.2.2.2.1.cmml" xref="A1.E25.m1.1.1.1.1.2.2.2.2.2.2">subscript</csymbol><ci id="A1.E25.m1.1.1.1.1.2.2.2.2.2.2.2.cmml" xref="A1.E25.m1.1.1.1.1.2.2.2.2.2.2.2">𝒙</ci><ci id="A1.E25.m1.1.1.1.1.2.2.2.2.2.2.3.cmml" xref="A1.E25.m1.1.1.1.1.2.2.2.2.2.2.3">𝑖</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E25.m1.1c">\forall\boldsymbol{x}_{i}\in\mathcal{G}_{p},\ \widehat{\alpha}_{i}=% \operatorname*{arg\,max}_{\alpha_{i}}\ell(\alpha_{i},\boldsymbol{x}_{i})\,.</annotation><annotation encoding="application/x-llamapun" id="A1.E25.m1.1d">∀ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_G start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , over^ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_ℓ ( italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(25)</td> </tr></tbody> </table> Note that it corresponds to solve independent optimization problems at each location <math alttext="i" class="ltx_Math" display="inline" id="A1.SS2.p1.3.m1.1"><semantics id="A1.SS2.p1.3.m1.1a"><mi id="A1.SS2.p1.3.m1.1.1" mathsize="144%" xref="A1.SS2.p1.3.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.3.m1.1b"><ci id="A1.SS2.p1.3.m1.1.1.cmml" xref="A1.SS2.p1.3.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.3.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.3.m1.1d">italic_i</annotation></semantics></math>. To efficiently solve Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E25" style="font-size:144%;" title="Equation 25 ‣ A.2 Dense detection map computation details ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">25</a>), we employ a fast approximation to compute the terms <math alttext="a(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="A1.SS2.p1.4.m2.1"><semantics id="A1.SS2.p1.4.m2.1a"><mrow id="A1.SS2.p1.4.m2.1.1" xref="A1.SS2.p1.4.m2.1.1.cmml"><mi id="A1.SS2.p1.4.m2.1.1.3" mathsize="144%" xref="A1.SS2.p1.4.m2.1.1.3.cmml">a</mi><mo id="A1.SS2.p1.4.m2.1.1.2" xref="A1.SS2.p1.4.m2.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.p1.4.m2.1.1.1.1" xref="A1.SS2.p1.4.m2.1.1.1.1.1.cmml"><mo id="A1.SS2.p1.4.m2.1.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.SS2.p1.4.m2.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.p1.4.m2.1.1.1.1.1" xref="A1.SS2.p1.4.m2.1.1.1.1.1.cmml"><mi id="A1.SS2.p1.4.m2.1.1.1.1.1.2" mathsize="144%" xref="A1.SS2.p1.4.m2.1.1.1.1.1.2.cmml">𝒙</mi><mi id="A1.SS2.p1.4.m2.1.1.1.1.1.3" mathsize="144%" xref="A1.SS2.p1.4.m2.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.SS2.p1.4.m2.1.1.1.1.3" maxsize="144%" minsize="144%" xref="A1.SS2.p1.4.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.4.m2.1b"><apply id="A1.SS2.p1.4.m2.1.1.cmml" xref="A1.SS2.p1.4.m2.1.1"><times id="A1.SS2.p1.4.m2.1.1.2.cmml" xref="A1.SS2.p1.4.m2.1.1.2"></times><ci id="A1.SS2.p1.4.m2.1.1.3.cmml" xref="A1.SS2.p1.4.m2.1.1.3">𝑎</ci><apply id="A1.SS2.p1.4.m2.1.1.1.1.1.cmml" xref="A1.SS2.p1.4.m2.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.4.m2.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.4.m2.1.1.1.1">subscript</csymbol><ci id="A1.SS2.p1.4.m2.1.1.1.1.1.2.cmml" xref="A1.SS2.p1.4.m2.1.1.1.1.1.2">𝒙</ci><ci id="A1.SS2.p1.4.m2.1.1.1.1.1.3.cmml" xref="A1.SS2.p1.4.m2.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.4.m2.1c">a(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.4.m2.1d">italic_a ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="b_{t}(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="A1.SS2.p1.5.m3.1"><semantics id="A1.SS2.p1.5.m3.1a"><mrow id="A1.SS2.p1.5.m3.1.1" xref="A1.SS2.p1.5.m3.1.1.cmml"><msub id="A1.SS2.p1.5.m3.1.1.3" xref="A1.SS2.p1.5.m3.1.1.3.cmml"><mi id="A1.SS2.p1.5.m3.1.1.3.2" mathsize="144%" xref="A1.SS2.p1.5.m3.1.1.3.2.cmml">b</mi><mi id="A1.SS2.p1.5.m3.1.1.3.3" mathsize="144%" xref="A1.SS2.p1.5.m3.1.1.3.3.cmml">t</mi></msub><mo id="A1.SS2.p1.5.m3.1.1.2" xref="A1.SS2.p1.5.m3.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.p1.5.m3.1.1.1.1" xref="A1.SS2.p1.5.m3.1.1.1.1.1.cmml"><mo id="A1.SS2.p1.5.m3.1.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.SS2.p1.5.m3.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.p1.5.m3.1.1.1.1.1" xref="A1.SS2.p1.5.m3.1.1.1.1.1.cmml"><mi id="A1.SS2.p1.5.m3.1.1.1.1.1.2" mathsize="144%" xref="A1.SS2.p1.5.m3.1.1.1.1.1.2.cmml">𝒙</mi><mi id="A1.SS2.p1.5.m3.1.1.1.1.1.3" mathsize="144%" xref="A1.SS2.p1.5.m3.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.SS2.p1.5.m3.1.1.1.1.3" maxsize="144%" minsize="144%" xref="A1.SS2.p1.5.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.5.m3.1b"><apply id="A1.SS2.p1.5.m3.1.1.cmml" xref="A1.SS2.p1.5.m3.1.1"><times id="A1.SS2.p1.5.m3.1.1.2.cmml" xref="A1.SS2.p1.5.m3.1.1.2"></times><apply id="A1.SS2.p1.5.m3.1.1.3.cmml" xref="A1.SS2.p1.5.m3.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.p1.5.m3.1.1.3.1.cmml" xref="A1.SS2.p1.5.m3.1.1.3">subscript</csymbol><ci id="A1.SS2.p1.5.m3.1.1.3.2.cmml" xref="A1.SS2.p1.5.m3.1.1.3.2">𝑏</ci><ci id="A1.SS2.p1.5.m3.1.1.3.3.cmml" xref="A1.SS2.p1.5.m3.1.1.3.3">𝑡</ci></apply><apply id="A1.SS2.p1.5.m3.1.1.1.1.1.cmml" xref="A1.SS2.p1.5.m3.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.5.m3.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.5.m3.1.1.1.1">subscript</csymbol><ci id="A1.SS2.p1.5.m3.1.1.1.1.1.2.cmml" xref="A1.SS2.p1.5.m3.1.1.1.1.1.2">𝒙</ci><ci id="A1.SS2.p1.5.m3.1.1.1.1.1.3.cmml" xref="A1.SS2.p1.5.m3.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.5.m3.1c">b_{t}(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.5.m3.1d">italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> at each time step and for every initial position. First, the terms <math alttext="a_{t}" class="ltx_Math" display="inline" id="A1.SS2.p1.6.m4.1"><semantics id="A1.SS2.p1.6.m4.1a"><msub id="A1.SS2.p1.6.m4.1.1" xref="A1.SS2.p1.6.m4.1.1.cmml"><mi id="A1.SS2.p1.6.m4.1.1.2" mathsize="144%" xref="A1.SS2.p1.6.m4.1.1.2.cmml">a</mi><mi id="A1.SS2.p1.6.m4.1.1.3" mathsize="144%" xref="A1.SS2.p1.6.m4.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.6.m4.1b"><apply id="A1.SS2.p1.6.m4.1.1.cmml" xref="A1.SS2.p1.6.m4.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.6.m4.1.1.1.cmml" xref="A1.SS2.p1.6.m4.1.1">subscript</csymbol><ci id="A1.SS2.p1.6.m4.1.1.2.cmml" xref="A1.SS2.p1.6.m4.1.1.2">𝑎</ci><ci id="A1.SS2.p1.6.m4.1.1.3.cmml" xref="A1.SS2.p1.6.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.6.m4.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.6.m4.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="b_{t}" class="ltx_Math" display="inline" id="A1.SS2.p1.7.m5.1"><semantics id="A1.SS2.p1.7.m5.1a"><msub id="A1.SS2.p1.7.m5.1.1" xref="A1.SS2.p1.7.m5.1.1.cmml"><mi id="A1.SS2.p1.7.m5.1.1.2" mathsize="144%" xref="A1.SS2.p1.7.m5.1.1.2.cmml">b</mi><mi id="A1.SS2.p1.7.m5.1.1.3" mathsize="144%" xref="A1.SS2.p1.7.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.7.m5.1b"><apply id="A1.SS2.p1.7.m5.1.1.cmml" xref="A1.SS2.p1.7.m5.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.7.m5.1.1.1.cmml" xref="A1.SS2.p1.7.m5.1.1">subscript</csymbol><ci id="A1.SS2.p1.7.m5.1.1.2.cmml" xref="A1.SS2.p1.7.m5.1.1.2">𝑏</ci><ci id="A1.SS2.p1.7.m5.1.1.3.cmml" xref="A1.SS2.p1.7.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.7.m5.1c">b_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.7.m5.1d">italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> are computed for all positions in the grid <math alttext="\mathcal{G}_{p}" class="ltx_Math" display="inline" id="A1.SS2.p1.8.m6.1"><semantics id="A1.SS2.p1.8.m6.1a"><msub id="A1.SS2.p1.8.m6.1.1" xref="A1.SS2.p1.8.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.p1.8.m6.1.1.2" mathsize="144%" xref="A1.SS2.p1.8.m6.1.1.2.cmml">𝒢</mi><mi id="A1.SS2.p1.8.m6.1.1.3" mathsize="144%" xref="A1.SS2.p1.8.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.8.m6.1b"><apply id="A1.SS2.p1.8.m6.1.1.cmml" xref="A1.SS2.p1.8.m6.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.8.m6.1.1.1.cmml" xref="A1.SS2.p1.8.m6.1.1">subscript</csymbol><ci id="A1.SS2.p1.8.m6.1.1.2.cmml" xref="A1.SS2.p1.8.m6.1.1.2">𝒢</ci><ci id="A1.SS2.p1.8.m6.1.1.3.cmml" xref="A1.SS2.p1.8.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.8.m6.1c">\mathcal{G}_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.8.m6.1d">caligraphic_G start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> of spatial pixels. In the following, we note <math alttext="b_{ti}=b_{t}(\boldsymbol{x}_{i})" class="ltx_Math" display="inline" id="A1.SS2.p1.9.m7.1"><semantics id="A1.SS2.p1.9.m7.1a"><mrow id="A1.SS2.p1.9.m7.1.1" xref="A1.SS2.p1.9.m7.1.1.cmml"><msub id="A1.SS2.p1.9.m7.1.1.3" xref="A1.SS2.p1.9.m7.1.1.3.cmml"><mi id="A1.SS2.p1.9.m7.1.1.3.2" mathsize="144%" xref="A1.SS2.p1.9.m7.1.1.3.2.cmml">b</mi><mrow id="A1.SS2.p1.9.m7.1.1.3.3" xref="A1.SS2.p1.9.m7.1.1.3.3.cmml"><mi id="A1.SS2.p1.9.m7.1.1.3.3.2" mathsize="144%" xref="A1.SS2.p1.9.m7.1.1.3.3.2.cmml">t</mi><mo id="A1.SS2.p1.9.m7.1.1.3.3.1" xref="A1.SS2.p1.9.m7.1.1.3.3.1.cmml">⁢</mo><mi id="A1.SS2.p1.9.m7.1.1.3.3.3" mathsize="144%" xref="A1.SS2.p1.9.m7.1.1.3.3.3.cmml">i</mi></mrow></msub><mo id="A1.SS2.p1.9.m7.1.1.2" mathsize="144%" xref="A1.SS2.p1.9.m7.1.1.2.cmml">=</mo><mrow id="A1.SS2.p1.9.m7.1.1.1" xref="A1.SS2.p1.9.m7.1.1.1.cmml"><msub id="A1.SS2.p1.9.m7.1.1.1.3" xref="A1.SS2.p1.9.m7.1.1.1.3.cmml"><mi id="A1.SS2.p1.9.m7.1.1.1.3.2" mathsize="144%" xref="A1.SS2.p1.9.m7.1.1.1.3.2.cmml">b</mi><mi id="A1.SS2.p1.9.m7.1.1.1.3.3" mathsize="144%" xref="A1.SS2.p1.9.m7.1.1.1.3.3.cmml">t</mi></msub><mo id="A1.SS2.p1.9.m7.1.1.1.2" xref="A1.SS2.p1.9.m7.1.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.p1.9.m7.1.1.1.1.1" xref="A1.SS2.p1.9.m7.1.1.1.1.1.1.cmml"><mo id="A1.SS2.p1.9.m7.1.1.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.SS2.p1.9.m7.1.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.p1.9.m7.1.1.1.1.1.1" xref="A1.SS2.p1.9.m7.1.1.1.1.1.1.cmml"><mi id="A1.SS2.p1.9.m7.1.1.1.1.1.1.2" mathsize="144%" xref="A1.SS2.p1.9.m7.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="A1.SS2.p1.9.m7.1.1.1.1.1.1.3" mathsize="144%" xref="A1.SS2.p1.9.m7.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.SS2.p1.9.m7.1.1.1.1.1.3" maxsize="144%" minsize="144%" xref="A1.SS2.p1.9.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.9.m7.1b"><apply id="A1.SS2.p1.9.m7.1.1.cmml" xref="A1.SS2.p1.9.m7.1.1"><eq id="A1.SS2.p1.9.m7.1.1.2.cmml" xref="A1.SS2.p1.9.m7.1.1.2"></eq><apply id="A1.SS2.p1.9.m7.1.1.3.cmml" xref="A1.SS2.p1.9.m7.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.p1.9.m7.1.1.3.1.cmml" xref="A1.SS2.p1.9.m7.1.1.3">subscript</csymbol><ci id="A1.SS2.p1.9.m7.1.1.3.2.cmml" xref="A1.SS2.p1.9.m7.1.1.3.2">𝑏</ci><apply id="A1.SS2.p1.9.m7.1.1.3.3.cmml" xref="A1.SS2.p1.9.m7.1.1.3.3"><times id="A1.SS2.p1.9.m7.1.1.3.3.1.cmml" xref="A1.SS2.p1.9.m7.1.1.3.3.1"></times><ci id="A1.SS2.p1.9.m7.1.1.3.3.2.cmml" xref="A1.SS2.p1.9.m7.1.1.3.3.2">𝑡</ci><ci id="A1.SS2.p1.9.m7.1.1.3.3.3.cmml" xref="A1.SS2.p1.9.m7.1.1.3.3.3">𝑖</ci></apply></apply><apply id="A1.SS2.p1.9.m7.1.1.1.cmml" xref="A1.SS2.p1.9.m7.1.1.1"><times id="A1.SS2.p1.9.m7.1.1.1.2.cmml" xref="A1.SS2.p1.9.m7.1.1.1.2"></times><apply id="A1.SS2.p1.9.m7.1.1.1.3.cmml" xref="A1.SS2.p1.9.m7.1.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.p1.9.m7.1.1.1.3.1.cmml" xref="A1.SS2.p1.9.m7.1.1.1.3">subscript</csymbol><ci id="A1.SS2.p1.9.m7.1.1.1.3.2.cmml" xref="A1.SS2.p1.9.m7.1.1.1.3.2">𝑏</ci><ci id="A1.SS2.p1.9.m7.1.1.1.3.3.cmml" xref="A1.SS2.p1.9.m7.1.1.1.3.3">𝑡</ci></apply><apply id="A1.SS2.p1.9.m7.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.9.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.9.m7.1.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.9.m7.1.1.1.1.1">subscript</csymbol><ci id="A1.SS2.p1.9.m7.1.1.1.1.1.1.2.cmml" xref="A1.SS2.p1.9.m7.1.1.1.1.1.1.2">𝒙</ci><ci id="A1.SS2.p1.9.m7.1.1.1.1.1.1.3.cmml" xref="A1.SS2.p1.9.m7.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.9.m7.1c">b_{ti}=b_{t}(\boldsymbol{x}_{i})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.9.m7.1d">italic_b start_POSTSUBSCRIPT italic_t italic_i end_POSTSUBSCRIPT = italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="a_{i}=a(\boldsymbol{x}_{i})" class="ltx_Math" display="inline" id="A1.SS2.p1.10.m8.1"><semantics id="A1.SS2.p1.10.m8.1a"><mrow id="A1.SS2.p1.10.m8.1.1" xref="A1.SS2.p1.10.m8.1.1.cmml"><msub id="A1.SS2.p1.10.m8.1.1.3" xref="A1.SS2.p1.10.m8.1.1.3.cmml"><mi id="A1.SS2.p1.10.m8.1.1.3.2" mathsize="144%" xref="A1.SS2.p1.10.m8.1.1.3.2.cmml">a</mi><mi id="A1.SS2.p1.10.m8.1.1.3.3" mathsize="144%" xref="A1.SS2.p1.10.m8.1.1.3.3.cmml">i</mi></msub><mo id="A1.SS2.p1.10.m8.1.1.2" mathsize="144%" xref="A1.SS2.p1.10.m8.1.1.2.cmml">=</mo><mrow id="A1.SS2.p1.10.m8.1.1.1" xref="A1.SS2.p1.10.m8.1.1.1.cmml"><mi id="A1.SS2.p1.10.m8.1.1.1.3" mathsize="144%" xref="A1.SS2.p1.10.m8.1.1.1.3.cmml">a</mi><mo id="A1.SS2.p1.10.m8.1.1.1.2" xref="A1.SS2.p1.10.m8.1.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.p1.10.m8.1.1.1.1.1" xref="A1.SS2.p1.10.m8.1.1.1.1.1.1.cmml"><mo id="A1.SS2.p1.10.m8.1.1.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.SS2.p1.10.m8.1.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.p1.10.m8.1.1.1.1.1.1" xref="A1.SS2.p1.10.m8.1.1.1.1.1.1.cmml"><mi id="A1.SS2.p1.10.m8.1.1.1.1.1.1.2" mathsize="144%" xref="A1.SS2.p1.10.m8.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="A1.SS2.p1.10.m8.1.1.1.1.1.1.3" mathsize="144%" xref="A1.SS2.p1.10.m8.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.SS2.p1.10.m8.1.1.1.1.1.3" maxsize="144%" minsize="144%" xref="A1.SS2.p1.10.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.10.m8.1b"><apply id="A1.SS2.p1.10.m8.1.1.cmml" xref="A1.SS2.p1.10.m8.1.1"><eq id="A1.SS2.p1.10.m8.1.1.2.cmml" xref="A1.SS2.p1.10.m8.1.1.2"></eq><apply id="A1.SS2.p1.10.m8.1.1.3.cmml" xref="A1.SS2.p1.10.m8.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.p1.10.m8.1.1.3.1.cmml" xref="A1.SS2.p1.10.m8.1.1.3">subscript</csymbol><ci id="A1.SS2.p1.10.m8.1.1.3.2.cmml" xref="A1.SS2.p1.10.m8.1.1.3.2">𝑎</ci><ci id="A1.SS2.p1.10.m8.1.1.3.3.cmml" xref="A1.SS2.p1.10.m8.1.1.3.3">𝑖</ci></apply><apply id="A1.SS2.p1.10.m8.1.1.1.cmml" xref="A1.SS2.p1.10.m8.1.1.1"><times id="A1.SS2.p1.10.m8.1.1.1.2.cmml" xref="A1.SS2.p1.10.m8.1.1.1.2"></times><ci id="A1.SS2.p1.10.m8.1.1.1.3.cmml" xref="A1.SS2.p1.10.m8.1.1.1.3">𝑎</ci><apply id="A1.SS2.p1.10.m8.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.10.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.10.m8.1.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.10.m8.1.1.1.1.1">subscript</csymbol><ci id="A1.SS2.p1.10.m8.1.1.1.1.1.1.2.cmml" xref="A1.SS2.p1.10.m8.1.1.1.1.1.1.2">𝒙</ci><ci id="A1.SS2.p1.10.m8.1.1.1.1.1.1.3.cmml" xref="A1.SS2.p1.10.m8.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.10.m8.1c">a_{i}=a(\boldsymbol{x}_{i})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.10.m8.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_a ( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\boldsymbol{h}_{ji}=\boldsymbol{h}^{(j)}(\boldsymbol{x}_{i})" class="ltx_Math" display="inline" id="A1.SS2.p1.11.m9.2"><semantics id="A1.SS2.p1.11.m9.2a"><mrow id="A1.SS2.p1.11.m9.2.2" xref="A1.SS2.p1.11.m9.2.2.cmml"><msub id="A1.SS2.p1.11.m9.2.2.3" xref="A1.SS2.p1.11.m9.2.2.3.cmml"><mi id="A1.SS2.p1.11.m9.2.2.3.2" mathsize="144%" xref="A1.SS2.p1.11.m9.2.2.3.2.cmml">𝒉</mi><mrow id="A1.SS2.p1.11.m9.2.2.3.3" xref="A1.SS2.p1.11.m9.2.2.3.3.cmml"><mi id="A1.SS2.p1.11.m9.2.2.3.3.2" mathsize="144%" xref="A1.SS2.p1.11.m9.2.2.3.3.2.cmml">j</mi><mo id="A1.SS2.p1.11.m9.2.2.3.3.1" xref="A1.SS2.p1.11.m9.2.2.3.3.1.cmml">⁢</mo><mi id="A1.SS2.p1.11.m9.2.2.3.3.3" mathsize="144%" xref="A1.SS2.p1.11.m9.2.2.3.3.3.cmml">i</mi></mrow></msub><mo id="A1.SS2.p1.11.m9.2.2.2" mathsize="144%" xref="A1.SS2.p1.11.m9.2.2.2.cmml">=</mo><mrow id="A1.SS2.p1.11.m9.2.2.1" xref="A1.SS2.p1.11.m9.2.2.1.cmml"><msup id="A1.SS2.p1.11.m9.2.2.1.3" xref="A1.SS2.p1.11.m9.2.2.1.3.cmml"><mi id="A1.SS2.p1.11.m9.2.2.1.3.2" mathsize="144%" xref="A1.SS2.p1.11.m9.2.2.1.3.2.cmml">𝒉</mi><mrow id="A1.SS2.p1.11.m9.1.1.1.3" xref="A1.SS2.p1.11.m9.2.2.1.3.cmml"><mo id="A1.SS2.p1.11.m9.1.1.1.3.1" maxsize="144%" minsize="144%" xref="A1.SS2.p1.11.m9.2.2.1.3.cmml">(</mo><mi id="A1.SS2.p1.11.m9.1.1.1.1" mathsize="144%" xref="A1.SS2.p1.11.m9.1.1.1.1.cmml">j</mi><mo id="A1.SS2.p1.11.m9.1.1.1.3.2" maxsize="144%" minsize="144%" xref="A1.SS2.p1.11.m9.2.2.1.3.cmml">)</mo></mrow></msup><mo id="A1.SS2.p1.11.m9.2.2.1.2" xref="A1.SS2.p1.11.m9.2.2.1.2.cmml">⁢</mo><mrow id="A1.SS2.p1.11.m9.2.2.1.1.1" xref="A1.SS2.p1.11.m9.2.2.1.1.1.1.cmml"><mo id="A1.SS2.p1.11.m9.2.2.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.SS2.p1.11.m9.2.2.1.1.1.1.cmml">(</mo><msub id="A1.SS2.p1.11.m9.2.2.1.1.1.1" xref="A1.SS2.p1.11.m9.2.2.1.1.1.1.cmml"><mi id="A1.SS2.p1.11.m9.2.2.1.1.1.1.2" mathsize="144%" xref="A1.SS2.p1.11.m9.2.2.1.1.1.1.2.cmml">𝒙</mi><mi id="A1.SS2.p1.11.m9.2.2.1.1.1.1.3" mathsize="144%" xref="A1.SS2.p1.11.m9.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.SS2.p1.11.m9.2.2.1.1.1.3" maxsize="144%" minsize="144%" xref="A1.SS2.p1.11.m9.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.11.m9.2b"><apply id="A1.SS2.p1.11.m9.2.2.cmml" xref="A1.SS2.p1.11.m9.2.2"><eq id="A1.SS2.p1.11.m9.2.2.2.cmml" xref="A1.SS2.p1.11.m9.2.2.2"></eq><apply id="A1.SS2.p1.11.m9.2.2.3.cmml" xref="A1.SS2.p1.11.m9.2.2.3"><csymbol cd="ambiguous" id="A1.SS2.p1.11.m9.2.2.3.1.cmml" xref="A1.SS2.p1.11.m9.2.2.3">subscript</csymbol><ci id="A1.SS2.p1.11.m9.2.2.3.2.cmml" xref="A1.SS2.p1.11.m9.2.2.3.2">𝒉</ci><apply id="A1.SS2.p1.11.m9.2.2.3.3.cmml" xref="A1.SS2.p1.11.m9.2.2.3.3"><times id="A1.SS2.p1.11.m9.2.2.3.3.1.cmml" xref="A1.SS2.p1.11.m9.2.2.3.3.1"></times><ci id="A1.SS2.p1.11.m9.2.2.3.3.2.cmml" xref="A1.SS2.p1.11.m9.2.2.3.3.2">𝑗</ci><ci id="A1.SS2.p1.11.m9.2.2.3.3.3.cmml" xref="A1.SS2.p1.11.m9.2.2.3.3.3">𝑖</ci></apply></apply><apply id="A1.SS2.p1.11.m9.2.2.1.cmml" xref="A1.SS2.p1.11.m9.2.2.1"><times id="A1.SS2.p1.11.m9.2.2.1.2.cmml" xref="A1.SS2.p1.11.m9.2.2.1.2"></times><apply id="A1.SS2.p1.11.m9.2.2.1.3.cmml" xref="A1.SS2.p1.11.m9.2.2.1.3"><csymbol cd="ambiguous" id="A1.SS2.p1.11.m9.2.2.1.3.1.cmml" xref="A1.SS2.p1.11.m9.2.2.1.3">superscript</csymbol><ci id="A1.SS2.p1.11.m9.2.2.1.3.2.cmml" xref="A1.SS2.p1.11.m9.2.2.1.3.2">𝒉</ci><ci id="A1.SS2.p1.11.m9.1.1.1.1.cmml" xref="A1.SS2.p1.11.m9.1.1.1.1">𝑗</ci></apply><apply id="A1.SS2.p1.11.m9.2.2.1.1.1.1.cmml" xref="A1.SS2.p1.11.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.11.m9.2.2.1.1.1.1.1.cmml" xref="A1.SS2.p1.11.m9.2.2.1.1.1">subscript</csymbol><ci id="A1.SS2.p1.11.m9.2.2.1.1.1.1.2.cmml" xref="A1.SS2.p1.11.m9.2.2.1.1.1.1.2">𝒙</ci><ci id="A1.SS2.p1.11.m9.2.2.1.1.1.1.3.cmml" xref="A1.SS2.p1.11.m9.2.2.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.11.m9.2c">\boldsymbol{h}_{ji}=\boldsymbol{h}^{(j)}(\boldsymbol{x}_{i})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.11.m9.2d">bold_italic_h start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT = bold_italic_h start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> for all <math alttext="\boldsymbol{x}_{i}" class="ltx_Math" display="inline" id="A1.SS2.p1.12.m10.1"><semantics id="A1.SS2.p1.12.m10.1a"><msub id="A1.SS2.p1.12.m10.1.1" xref="A1.SS2.p1.12.m10.1.1.cmml"><mi id="A1.SS2.p1.12.m10.1.1.2" mathsize="144%" xref="A1.SS2.p1.12.m10.1.1.2.cmml">𝒙</mi><mi id="A1.SS2.p1.12.m10.1.1.3" mathsize="144%" xref="A1.SS2.p1.12.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.12.m10.1b"><apply id="A1.SS2.p1.12.m10.1.1.cmml" xref="A1.SS2.p1.12.m10.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.12.m10.1.1.1.cmml" xref="A1.SS2.p1.12.m10.1.1">subscript</csymbol><ci id="A1.SS2.p1.12.m10.1.1.2.cmml" xref="A1.SS2.p1.12.m10.1.1.2">𝒙</ci><ci id="A1.SS2.p1.12.m10.1.1.3.cmml" xref="A1.SS2.p1.12.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.12.m10.1c">\boldsymbol{x}_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.12.m10.1d">bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathcal{G}_{p}" class="ltx_Math" display="inline" id="A1.SS2.p1.13.m11.1"><semantics id="A1.SS2.p1.13.m11.1a"><msub id="A1.SS2.p1.13.m11.1.1" xref="A1.SS2.p1.13.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS2.p1.13.m11.1.1.2" mathsize="144%" xref="A1.SS2.p1.13.m11.1.1.2.cmml">𝒢</mi><mi id="A1.SS2.p1.13.m11.1.1.3" mathsize="144%" xref="A1.SS2.p1.13.m11.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.13.m11.1b"><apply id="A1.SS2.p1.13.m11.1.1.cmml" xref="A1.SS2.p1.13.m11.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.13.m11.1.1.1.cmml" xref="A1.SS2.p1.13.m11.1.1">subscript</csymbol><ci id="A1.SS2.p1.13.m11.1.1.2.cmml" xref="A1.SS2.p1.13.m11.1.1.2">𝒢</ci><ci id="A1.SS2.p1.13.m11.1.1.3.cmml" xref="A1.SS2.p1.13.m11.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.13.m11.1c">\mathcal{G}_{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.13.m11.1d">caligraphic_G start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, such that: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx3"> <tbody id="A1.E26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall(i,t),b_{it}" class="ltx_Math" display="inline" id="A1.E26.m1.4"><semantics id="A1.E26.m1.4a"><mrow id="A1.E26.m1.4.4.2" xref="A1.E26.m1.4.4.3.cmml"><mrow id="A1.E26.m1.3.3.1.1" xref="A1.E26.m1.3.3.1.1.cmml"><mo id="A1.E26.m1.3.3.1.1.1" mathsize="144%" xref="A1.E26.m1.3.3.1.1.1.cmml">∀</mo><mrow id="A1.E26.m1.3.3.1.1.2.2" xref="A1.E26.m1.3.3.1.1.2.1.cmml"><mo id="A1.E26.m1.3.3.1.1.2.2.1" maxsize="144%" minsize="144%" xref="A1.E26.m1.3.3.1.1.2.1.cmml">(</mo><mi id="A1.E26.m1.1.1" mathsize="144%" xref="A1.E26.m1.1.1.cmml">i</mi><mo id="A1.E26.m1.3.3.1.1.2.2.2" mathsize="144%" xref="A1.E26.m1.3.3.1.1.2.1.cmml">,</mo><mi id="A1.E26.m1.2.2" mathsize="144%" xref="A1.E26.m1.2.2.cmml">t</mi><mo id="A1.E26.m1.3.3.1.1.2.2.3" maxsize="144%" minsize="144%" xref="A1.E26.m1.3.3.1.1.2.1.cmml">)</mo></mrow></mrow><mo id="A1.E26.m1.4.4.2.3" mathsize="144%" xref="A1.E26.m1.4.4.3.cmml">,</mo><msub id="A1.E26.m1.4.4.2.2" xref="A1.E26.m1.4.4.2.2.cmml"><mi id="A1.E26.m1.4.4.2.2.2" mathsize="144%" xref="A1.E26.m1.4.4.2.2.2.cmml">b</mi><mrow id="A1.E26.m1.4.4.2.2.3" xref="A1.E26.m1.4.4.2.2.3.cmml"><mi id="A1.E26.m1.4.4.2.2.3.2" mathsize="144%" xref="A1.E26.m1.4.4.2.2.3.2.cmml">i</mi><mo id="A1.E26.m1.4.4.2.2.3.1" xref="A1.E26.m1.4.4.2.2.3.1.cmml">⁢</mo><mi id="A1.E26.m1.4.4.2.2.3.3" mathsize="144%" xref="A1.E26.m1.4.4.2.2.3.3.cmml">t</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.E26.m1.4b"><list id="A1.E26.m1.4.4.3.cmml" xref="A1.E26.m1.4.4.2"><apply id="A1.E26.m1.3.3.1.1.cmml" xref="A1.E26.m1.3.3.1.1"><csymbol cd="latexml" id="A1.E26.m1.3.3.1.1.1.cmml" xref="A1.E26.m1.3.3.1.1.1">for-all</csymbol><interval closure="open" id="A1.E26.m1.3.3.1.1.2.1.cmml" xref="A1.E26.m1.3.3.1.1.2.2"><ci id="A1.E26.m1.1.1.cmml" xref="A1.E26.m1.1.1">𝑖</ci><ci id="A1.E26.m1.2.2.cmml" xref="A1.E26.m1.2.2">𝑡</ci></interval></apply><apply id="A1.E26.m1.4.4.2.2.cmml" xref="A1.E26.m1.4.4.2.2"><csymbol cd="ambiguous" id="A1.E26.m1.4.4.2.2.1.cmml" xref="A1.E26.m1.4.4.2.2">subscript</csymbol><ci id="A1.E26.m1.4.4.2.2.2.cmml" xref="A1.E26.m1.4.4.2.2.2">𝑏</ci><apply id="A1.E26.m1.4.4.2.2.3.cmml" xref="A1.E26.m1.4.4.2.2.3"><times id="A1.E26.m1.4.4.2.2.3.1.cmml" xref="A1.E26.m1.4.4.2.2.3.1"></times><ci id="A1.E26.m1.4.4.2.2.3.2.cmml" xref="A1.E26.m1.4.4.2.2.3.2">𝑖</ci><ci id="A1.E26.m1.4.4.2.2.3.3.cmml" xref="A1.E26.m1.4.4.2.2.3.3">𝑡</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A1.E26.m1.4c">\displaystyle\forall(i,t),b_{it}</annotation><annotation encoding="application/x-llamapun" id="A1.E26.m1.4d">∀ ( italic_i , italic_t ) , italic_b start_POSTSUBSCRIPT italic_i italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{j\in S(\boldsymbol{x}_{i})}w_{j}\,\boldsymbol{h}_{ji}^{% \top}\,\widehat{\mathbf{C}}_{j}^{-1}\,\left(\boldsymbol{y}_{t}^{(j)}-\widehat{% \boldsymbol{m}}_{j}\right)\,," class="ltx_Math" display="inline" id="A1.E26.m2.3"><semantics id="A1.E26.m2.3a"><mrow id="A1.E26.m2.3.3.1" xref="A1.E26.m2.3.3.1.1.cmml"><mrow id="A1.E26.m2.3.3.1.1" xref="A1.E26.m2.3.3.1.1.cmml"><mi id="A1.E26.m2.3.3.1.1.3" xref="A1.E26.m2.3.3.1.1.3.cmml"></mi><mo id="A1.E26.m2.3.3.1.1.2" mathsize="144%" xref="A1.E26.m2.3.3.1.1.2.cmml">=</mo><mrow id="A1.E26.m2.3.3.1.1.1" xref="A1.E26.m2.3.3.1.1.1.cmml"><mstyle displaystyle="true" id="A1.E26.m2.3.3.1.1.1.2" xref="A1.E26.m2.3.3.1.1.1.2.cmml"><munder id="A1.E26.m2.3.3.1.1.1.2a" xref="A1.E26.m2.3.3.1.1.1.2.cmml"><mo id="A1.E26.m2.3.3.1.1.1.2.2" maxsize="144%" minsize="144%" movablelimits="false" stretchy="true" xref="A1.E26.m2.3.3.1.1.1.2.2.cmml">∑</mo><mrow id="A1.E26.m2.1.1.1" xref="A1.E26.m2.1.1.1.cmml"><mi id="A1.E26.m2.1.1.1.3" mathsize="144%" xref="A1.E26.m2.1.1.1.3.cmml">j</mi><mo id="A1.E26.m2.1.1.1.2" mathsize="144%" xref="A1.E26.m2.1.1.1.2.cmml">∈</mo><mrow id="A1.E26.m2.1.1.1.1" xref="A1.E26.m2.1.1.1.1.cmml"><mi id="A1.E26.m2.1.1.1.1.3" mathsize="144%" xref="A1.E26.m2.1.1.1.1.3.cmml">S</mi><mo id="A1.E26.m2.1.1.1.1.2" xref="A1.E26.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.E26.m2.1.1.1.1.1.1" xref="A1.E26.m2.1.1.1.1.1.1.1.cmml"><mo id="A1.E26.m2.1.1.1.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.E26.m2.1.1.1.1.1.1.1.cmml">(</mo><msub id="A1.E26.m2.1.1.1.1.1.1.1" xref="A1.E26.m2.1.1.1.1.1.1.1.cmml"><mi id="A1.E26.m2.1.1.1.1.1.1.1.2" mathsize="144%" xref="A1.E26.m2.1.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="A1.E26.m2.1.1.1.1.1.1.1.3" mathsize="144%" xref="A1.E26.m2.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.E26.m2.1.1.1.1.1.1.3" maxsize="144%" minsize="144%" xref="A1.E26.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="A1.E26.m2.3.3.1.1.1.1" xref="A1.E26.m2.3.3.1.1.1.1.cmml"><msub id="A1.E26.m2.3.3.1.1.1.1.3" xref="A1.E26.m2.3.3.1.1.1.1.3.cmml"><mi id="A1.E26.m2.3.3.1.1.1.1.3.2" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.3.2.cmml">w</mi><mi id="A1.E26.m2.3.3.1.1.1.1.3.3" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.3.3.cmml">j</mi></msub><mo id="A1.E26.m2.3.3.1.1.1.1.2" xref="A1.E26.m2.3.3.1.1.1.1.2.cmml">⁢</mo><msubsup id="A1.E26.m2.3.3.1.1.1.1.4" xref="A1.E26.m2.3.3.1.1.1.1.4.cmml"><mi id="A1.E26.m2.3.3.1.1.1.1.4.2.2" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.4.2.2.cmml">𝒉</mi><mrow id="A1.E26.m2.3.3.1.1.1.1.4.2.3" xref="A1.E26.m2.3.3.1.1.1.1.4.2.3.cmml"><mi id="A1.E26.m2.3.3.1.1.1.1.4.2.3.2" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.4.2.3.2.cmml">j</mi><mo id="A1.E26.m2.3.3.1.1.1.1.4.2.3.1" xref="A1.E26.m2.3.3.1.1.1.1.4.2.3.1.cmml">⁢</mo><mi id="A1.E26.m2.3.3.1.1.1.1.4.2.3.3" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.4.2.3.3.cmml">i</mi></mrow><mo id="A1.E26.m2.3.3.1.1.1.1.4.3" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.4.3.cmml">⊤</mo></msubsup><mo id="A1.E26.m2.3.3.1.1.1.1.2a" xref="A1.E26.m2.3.3.1.1.1.1.2.cmml">⁢</mo><msubsup id="A1.E26.m2.3.3.1.1.1.1.5" xref="A1.E26.m2.3.3.1.1.1.1.5.cmml"><mover accent="true" id="A1.E26.m2.3.3.1.1.1.1.5.2.2" xref="A1.E26.m2.3.3.1.1.1.1.5.2.2.cmml"><mi id="A1.E26.m2.3.3.1.1.1.1.5.2.2.2" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.5.2.2.2.cmml">𝐂</mi><mo id="A1.E26.m2.3.3.1.1.1.1.5.2.2.1" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.5.2.2.1.cmml">^</mo></mover><mi id="A1.E26.m2.3.3.1.1.1.1.5.2.3" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.5.2.3.cmml">j</mi><mrow id="A1.E26.m2.3.3.1.1.1.1.5.3" xref="A1.E26.m2.3.3.1.1.1.1.5.3.cmml"><mo id="A1.E26.m2.3.3.1.1.1.1.5.3a" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.5.3.cmml">−</mo><mn id="A1.E26.m2.3.3.1.1.1.1.5.3.2" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.5.3.2.cmml">1</mn></mrow></msubsup><mo id="A1.E26.m2.3.3.1.1.1.1.2b" xref="A1.E26.m2.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.E26.m2.3.3.1.1.1.1.1.1" xref="A1.E26.m2.3.3.1.1.1.1.1.1.1.cmml"><mo id="A1.E26.m2.3.3.1.1.1.1.1.1.2" xref="A1.E26.m2.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.E26.m2.3.3.1.1.1.1.1.1.1" xref="A1.E26.m2.3.3.1.1.1.1.1.1.1.cmml"><msubsup id="A1.E26.m2.3.3.1.1.1.1.1.1.1.2" xref="A1.E26.m2.3.3.1.1.1.1.1.1.1.2.cmml"><mi id="A1.E26.m2.3.3.1.1.1.1.1.1.1.2.2.2" mathsize="144%" xref="A1.E26.m2.3.3.1.1.1.1.1.1.1.2.2.2.cmml">𝒚</mi><mi id="A1.E26.m2.3.3.1.1.1.1.1.1.1.2.2.3" mathsize="144%" 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id="A1.E27.m1.1.1.1.1.2.cmml" xref="A1.E27.m1.1.1.1.1.2">𝑖</ci></apply><apply id="A1.E27.m1.2.2.2.2.cmml" xref="A1.E27.m1.2.2.2.2"><csymbol cd="ambiguous" id="A1.E27.m1.2.2.2.2.1.cmml" xref="A1.E27.m1.2.2.2.2">subscript</csymbol><ci id="A1.E27.m1.2.2.2.2.2.cmml" xref="A1.E27.m1.2.2.2.2.2">𝑎</ci><ci id="A1.E27.m1.2.2.2.2.3.cmml" xref="A1.E27.m1.2.2.2.2.3">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A1.E27.m1.2c">\displaystyle\forall i,a_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.E27.m1.2d">∀ italic_i , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{j\in S(\boldsymbol{x}_{i})}w_{j}\,\boldsymbol{h}_{ji}^{% \top}\,\widehat{\mathbf{C}}_{j}^{-1}\,\boldsymbol{h}_{ji}\,." class="ltx_Math" display="inline" id="A1.E27.m2.2"><semantics id="A1.E27.m2.2a"><mrow id="A1.E27.m2.2.2.1" 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start_POSTSUBSCRIPT italic_j ∈ italic_S ( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_italic_h start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT bold_italic_h start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(27)</td> </tr></tbody> </table> With our convolutional approach, these terms can be computed efficiently by Cholesky factorization of the precision matrix, as detailed in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.SS3" style="font-size:144%;" title="A.3 Fast computation of 𝒃_𝑡 and 𝒂 ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.3</a>. Let <math alttext="\boldsymbol{b}_{t}" class="ltx_Math" display="inline" id="A1.SS2.p1.14.m1.1"><semantics id="A1.SS2.p1.14.m1.1a"><msub id="A1.SS2.p1.14.m1.1.1" xref="A1.SS2.p1.14.m1.1.1.cmml"><mi id="A1.SS2.p1.14.m1.1.1.2" mathsize="144%" xref="A1.SS2.p1.14.m1.1.1.2.cmml">𝒃</mi><mi id="A1.SS2.p1.14.m1.1.1.3" mathsize="144%" xref="A1.SS2.p1.14.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.14.m1.1b"><apply id="A1.SS2.p1.14.m1.1.1.cmml" xref="A1.SS2.p1.14.m1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.14.m1.1.1.1.cmml" xref="A1.SS2.p1.14.m1.1.1">subscript</csymbol><ci id="A1.SS2.p1.14.m1.1.1.2.cmml" xref="A1.SS2.p1.14.m1.1.1.2">𝒃</ci><ci id="A1.SS2.p1.14.m1.1.1.3.cmml" xref="A1.SS2.p1.14.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.14.m1.1c">\boldsymbol{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.14.m1.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="A1.SS2.p1.15.m2.1"><semantics id="A1.SS2.p1.15.m2.1a"><msup id="A1.SS2.p1.15.m2.1.1" xref="A1.SS2.p1.15.m2.1.1.cmml"><mi id="A1.SS2.p1.15.m2.1.1.2" mathsize="144%" xref="A1.SS2.p1.15.m2.1.1.2.cmml">ℝ</mi><mrow id="A1.SS2.p1.15.m2.1.1.3" xref="A1.SS2.p1.15.m2.1.1.3.cmml"><mi id="A1.SS2.p1.15.m2.1.1.3.2" mathsize="144%" xref="A1.SS2.p1.15.m2.1.1.3.2.cmml">H</mi><mo id="A1.SS2.p1.15.m2.1.1.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS2.p1.15.m2.1.1.3.1.cmml">×</mo><mi id="A1.SS2.p1.15.m2.1.1.3.3" mathsize="144%" xref="A1.SS2.p1.15.m2.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.15.m2.1b"><apply id="A1.SS2.p1.15.m2.1.1.cmml" xref="A1.SS2.p1.15.m2.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.15.m2.1.1.1.cmml" xref="A1.SS2.p1.15.m2.1.1">superscript</csymbol><ci id="A1.SS2.p1.15.m2.1.1.2.cmml" xref="A1.SS2.p1.15.m2.1.1.2">ℝ</ci><apply id="A1.SS2.p1.15.m2.1.1.3.cmml" xref="A1.SS2.p1.15.m2.1.1.3"><times id="A1.SS2.p1.15.m2.1.1.3.1.cmml" xref="A1.SS2.p1.15.m2.1.1.3.1"></times><ci id="A1.SS2.p1.15.m2.1.1.3.2.cmml" xref="A1.SS2.p1.15.m2.1.1.3.2">𝐻</ci><ci id="A1.SS2.p1.15.m2.1.1.3.3.cmml" xref="A1.SS2.p1.15.m2.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.15.m2.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.15.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{a}" class="ltx_Math" display="inline" id="A1.SS2.p1.16.m3.1"><semantics id="A1.SS2.p1.16.m3.1a"><mi id="A1.SS2.p1.16.m3.1.1" mathsize="144%" xref="A1.SS2.p1.16.m3.1.1.cmml">𝒂</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.16.m3.1b"><ci id="A1.SS2.p1.16.m3.1.1.cmml" xref="A1.SS2.p1.16.m3.1.1">𝒂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.16.m3.1c">\boldsymbol{a}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.16.m3.1d">bold_italic_a</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="A1.SS2.p1.17.m4.1"><semantics id="A1.SS2.p1.17.m4.1a"><msup id="A1.SS2.p1.17.m4.1.1" xref="A1.SS2.p1.17.m4.1.1.cmml"><mi id="A1.SS2.p1.17.m4.1.1.2" mathsize="144%" xref="A1.SS2.p1.17.m4.1.1.2.cmml">ℝ</mi><mrow id="A1.SS2.p1.17.m4.1.1.3" xref="A1.SS2.p1.17.m4.1.1.3.cmml"><mi id="A1.SS2.p1.17.m4.1.1.3.2" mathsize="144%" xref="A1.SS2.p1.17.m4.1.1.3.2.cmml">H</mi><mo id="A1.SS2.p1.17.m4.1.1.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS2.p1.17.m4.1.1.3.1.cmml">×</mo><mi id="A1.SS2.p1.17.m4.1.1.3.3" mathsize="144%" xref="A1.SS2.p1.17.m4.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.17.m4.1b"><apply id="A1.SS2.p1.17.m4.1.1.cmml" xref="A1.SS2.p1.17.m4.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.17.m4.1.1.1.cmml" xref="A1.SS2.p1.17.m4.1.1">superscript</csymbol><ci id="A1.SS2.p1.17.m4.1.1.2.cmml" xref="A1.SS2.p1.17.m4.1.1.2">ℝ</ci><apply id="A1.SS2.p1.17.m4.1.1.3.cmml" xref="A1.SS2.p1.17.m4.1.1.3"><times id="A1.SS2.p1.17.m4.1.1.3.1.cmml" xref="A1.SS2.p1.17.m4.1.1.3.1"></times><ci id="A1.SS2.p1.17.m4.1.1.3.2.cmml" xref="A1.SS2.p1.17.m4.1.1.3.2">𝐻</ci><ci id="A1.SS2.p1.17.m4.1.1.3.3.cmml" xref="A1.SS2.p1.17.m4.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.17.m4.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.17.m4.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> represent the resulting quantities. Then, <math alttext="b_{t}(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="A1.SS2.p1.18.m5.1"><semantics id="A1.SS2.p1.18.m5.1a"><mrow id="A1.SS2.p1.18.m5.1.1" xref="A1.SS2.p1.18.m5.1.1.cmml"><msub id="A1.SS2.p1.18.m5.1.1.3" xref="A1.SS2.p1.18.m5.1.1.3.cmml"><mi id="A1.SS2.p1.18.m5.1.1.3.2" mathsize="144%" xref="A1.SS2.p1.18.m5.1.1.3.2.cmml">b</mi><mi id="A1.SS2.p1.18.m5.1.1.3.3" mathsize="144%" xref="A1.SS2.p1.18.m5.1.1.3.3.cmml">t</mi></msub><mo id="A1.SS2.p1.18.m5.1.1.2" xref="A1.SS2.p1.18.m5.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.p1.18.m5.1.1.1.1" xref="A1.SS2.p1.18.m5.1.1.1.1.1.cmml"><mo id="A1.SS2.p1.18.m5.1.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.SS2.p1.18.m5.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.p1.18.m5.1.1.1.1.1" xref="A1.SS2.p1.18.m5.1.1.1.1.1.cmml"><mi id="A1.SS2.p1.18.m5.1.1.1.1.1.2" mathsize="144%" xref="A1.SS2.p1.18.m5.1.1.1.1.1.2.cmml">𝒙</mi><mi id="A1.SS2.p1.18.m5.1.1.1.1.1.3" mathsize="144%" xref="A1.SS2.p1.18.m5.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.SS2.p1.18.m5.1.1.1.1.3" maxsize="144%" minsize="144%" xref="A1.SS2.p1.18.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.18.m5.1b"><apply id="A1.SS2.p1.18.m5.1.1.cmml" xref="A1.SS2.p1.18.m5.1.1"><times id="A1.SS2.p1.18.m5.1.1.2.cmml" xref="A1.SS2.p1.18.m5.1.1.2"></times><apply id="A1.SS2.p1.18.m5.1.1.3.cmml" xref="A1.SS2.p1.18.m5.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.p1.18.m5.1.1.3.1.cmml" xref="A1.SS2.p1.18.m5.1.1.3">subscript</csymbol><ci id="A1.SS2.p1.18.m5.1.1.3.2.cmml" xref="A1.SS2.p1.18.m5.1.1.3.2">𝑏</ci><ci id="A1.SS2.p1.18.m5.1.1.3.3.cmml" xref="A1.SS2.p1.18.m5.1.1.3.3">𝑡</ci></apply><apply id="A1.SS2.p1.18.m5.1.1.1.1.1.cmml" xref="A1.SS2.p1.18.m5.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.18.m5.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.18.m5.1.1.1.1">subscript</csymbol><ci id="A1.SS2.p1.18.m5.1.1.1.1.1.2.cmml" xref="A1.SS2.p1.18.m5.1.1.1.1.1.2">𝒙</ci><ci id="A1.SS2.p1.18.m5.1.1.1.1.1.3.cmml" xref="A1.SS2.p1.18.m5.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.18.m5.1c">b_{t}(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.18.m5.1d">italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="a(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="A1.SS2.p1.19.m6.1"><semantics id="A1.SS2.p1.19.m6.1a"><mrow id="A1.SS2.p1.19.m6.1.1" xref="A1.SS2.p1.19.m6.1.1.cmml"><mi id="A1.SS2.p1.19.m6.1.1.3" mathsize="144%" xref="A1.SS2.p1.19.m6.1.1.3.cmml">a</mi><mo id="A1.SS2.p1.19.m6.1.1.2" xref="A1.SS2.p1.19.m6.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.p1.19.m6.1.1.1.1" xref="A1.SS2.p1.19.m6.1.1.1.1.1.cmml"><mo id="A1.SS2.p1.19.m6.1.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.SS2.p1.19.m6.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.p1.19.m6.1.1.1.1.1" xref="A1.SS2.p1.19.m6.1.1.1.1.1.cmml"><mi id="A1.SS2.p1.19.m6.1.1.1.1.1.2" mathsize="144%" xref="A1.SS2.p1.19.m6.1.1.1.1.1.2.cmml">𝒙</mi><mi id="A1.SS2.p1.19.m6.1.1.1.1.1.3" mathsize="144%" xref="A1.SS2.p1.19.m6.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.SS2.p1.19.m6.1.1.1.1.3" maxsize="144%" minsize="144%" xref="A1.SS2.p1.19.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.19.m6.1b"><apply id="A1.SS2.p1.19.m6.1.1.cmml" xref="A1.SS2.p1.19.m6.1.1"><times id="A1.SS2.p1.19.m6.1.1.2.cmml" xref="A1.SS2.p1.19.m6.1.1.2"></times><ci id="A1.SS2.p1.19.m6.1.1.3.cmml" xref="A1.SS2.p1.19.m6.1.1.3">𝑎</ci><apply id="A1.SS2.p1.19.m6.1.1.1.1.1.cmml" xref="A1.SS2.p1.19.m6.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.19.m6.1.1.1.1.1.1.cmml" xref="A1.SS2.p1.19.m6.1.1.1.1">subscript</csymbol><ci id="A1.SS2.p1.19.m6.1.1.1.1.1.2.cmml" xref="A1.SS2.p1.19.m6.1.1.1.1.1.2">𝒙</ci><ci id="A1.SS2.p1.19.m6.1.1.1.1.1.3.cmml" xref="A1.SS2.p1.19.m6.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.19.m6.1c">a(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.19.m6.1d">italic_a ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> can be approximated by spatial interpolation of <math alttext="\boldsymbol{b}_{t}" class="ltx_Math" display="inline" id="A1.SS2.p1.20.m7.1"><semantics id="A1.SS2.p1.20.m7.1a"><msub id="A1.SS2.p1.20.m7.1.1" xref="A1.SS2.p1.20.m7.1.1.cmml"><mi id="A1.SS2.p1.20.m7.1.1.2" mathsize="144%" xref="A1.SS2.p1.20.m7.1.1.2.cmml">𝒃</mi><mi id="A1.SS2.p1.20.m7.1.1.3" mathsize="144%" xref="A1.SS2.p1.20.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.20.m7.1b"><apply id="A1.SS2.p1.20.m7.1.1.cmml" xref="A1.SS2.p1.20.m7.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.20.m7.1.1.1.cmml" xref="A1.SS2.p1.20.m7.1.1">subscript</csymbol><ci id="A1.SS2.p1.20.m7.1.1.2.cmml" xref="A1.SS2.p1.20.m7.1.1.2">𝒃</ci><ci id="A1.SS2.p1.20.m7.1.1.3.cmml" xref="A1.SS2.p1.20.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.20.m7.1c">\boldsymbol{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.20.m7.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{a}" class="ltx_Math" display="inline" id="A1.SS2.p1.21.m8.1"><semantics id="A1.SS2.p1.21.m8.1a"><mi id="A1.SS2.p1.21.m8.1.1" mathsize="144%" xref="A1.SS2.p1.21.m8.1.1.cmml">𝒂</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.21.m8.1b"><ci id="A1.SS2.p1.21.m8.1.1.cmml" xref="A1.SS2.p1.21.m8.1.1">𝒂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.21.m8.1c">\boldsymbol{a}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.21.m8.1d">bold_italic_a</annotation></semantics></math>. However, the trajectories of all exoplanets in the observation are governed by a common parallactic rotation vector <math alttext="\boldsymbol{\phi}" class="ltx_Math" display="inline" id="A1.SS2.p1.22.m9.1"><semantics id="A1.SS2.p1.22.m9.1a"><mi class="ltx_mathvariant_bold-italic" id="A1.SS2.p1.22.m9.1.1" mathsize="144%" mathvariant="bold-italic" xref="A1.SS2.p1.22.m9.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.22.m9.1b"><ci id="A1.SS2.p1.22.m9.1.1.cmml" xref="A1.SS2.p1.22.m9.1.1">bold-italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.22.m9.1c">\boldsymbol{\phi}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.22.m9.1d">bold_italic_ϕ</annotation></semantics></math> in <math alttext="\mathbb{R}^{T}" class="ltx_Math" display="inline" id="A1.SS2.p1.23.m10.1"><semantics id="A1.SS2.p1.23.m10.1a"><msup id="A1.SS2.p1.23.m10.1.1" xref="A1.SS2.p1.23.m10.1.1.cmml"><mi id="A1.SS2.p1.23.m10.1.1.2" mathsize="144%" xref="A1.SS2.p1.23.m10.1.1.2.cmml">ℝ</mi><mi id="A1.SS2.p1.23.m10.1.1.3" mathsize="144%" xref="A1.SS2.p1.23.m10.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.23.m10.1b"><apply id="A1.SS2.p1.23.m10.1.1.cmml" xref="A1.SS2.p1.23.m10.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.23.m10.1.1.1.cmml" xref="A1.SS2.p1.23.m10.1.1">superscript</csymbol><ci id="A1.SS2.p1.23.m10.1.1.2.cmml" xref="A1.SS2.p1.23.m10.1.1.2">ℝ</ci><ci id="A1.SS2.p1.23.m10.1.1.3.cmml" xref="A1.SS2.p1.23.m10.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.23.m10.1c">\mathbb{R}^{T}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.23.m10.1d">blackboard_R start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>. So in practice, this interpolation can be computed efficiently for each time-step <math alttext="t" class="ltx_Math" display="inline" id="A1.SS2.p1.24.m11.1"><semantics id="A1.SS2.p1.24.m11.1a"><mi id="A1.SS2.p1.24.m11.1.1" mathsize="144%" xref="A1.SS2.p1.24.m11.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.24.m11.1b"><ci id="A1.SS2.p1.24.m11.1.1.cmml" xref="A1.SS2.p1.24.m11.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.24.m11.1c">t</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.24.m11.1d">italic_t</annotation></semantics></math>, by rotating the frames <math alttext="\boldsymbol{a}" class="ltx_Math" display="inline" id="A1.SS2.p1.25.m12.1"><semantics id="A1.SS2.p1.25.m12.1a"><mi id="A1.SS2.p1.25.m12.1.1" mathsize="144%" xref="A1.SS2.p1.25.m12.1.1.cmml">𝒂</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.25.m12.1b"><ci id="A1.SS2.p1.25.m12.1.1.cmml" xref="A1.SS2.p1.25.m12.1.1">𝒂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.25.m12.1c">\boldsymbol{a}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.25.m12.1d">bold_italic_a</annotation></semantics></math> and <math alttext="\boldsymbol{b}_{t}" class="ltx_Math" display="inline" id="A1.SS2.p1.26.m13.1"><semantics id="A1.SS2.p1.26.m13.1a"><msub id="A1.SS2.p1.26.m13.1.1" xref="A1.SS2.p1.26.m13.1.1.cmml"><mi id="A1.SS2.p1.26.m13.1.1.2" mathsize="144%" xref="A1.SS2.p1.26.m13.1.1.2.cmml">𝒃</mi><mi id="A1.SS2.p1.26.m13.1.1.3" mathsize="144%" xref="A1.SS2.p1.26.m13.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.26.m13.1b"><apply id="A1.SS2.p1.26.m13.1.1.cmml" xref="A1.SS2.p1.26.m13.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.26.m13.1.1.1.cmml" xref="A1.SS2.p1.26.m13.1.1">subscript</csymbol><ci id="A1.SS2.p1.26.m13.1.1.2.cmml" xref="A1.SS2.p1.26.m13.1.1.2">𝒃</ci><ci id="A1.SS2.p1.26.m13.1.1.3.cmml" xref="A1.SS2.p1.26.m13.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.26.m13.1c">\boldsymbol{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.26.m13.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> by an angle <math alttext="-\phi_{t}" class="ltx_Math" display="inline" id="A1.SS2.p1.27.m14.1"><semantics id="A1.SS2.p1.27.m14.1a"><mrow id="A1.SS2.p1.27.m14.1.1" xref="A1.SS2.p1.27.m14.1.1.cmml"><mo id="A1.SS2.p1.27.m14.1.1a" mathsize="144%" xref="A1.SS2.p1.27.m14.1.1.cmml">−</mo><msub id="A1.SS2.p1.27.m14.1.1.2" xref="A1.SS2.p1.27.m14.1.1.2.cmml"><mi id="A1.SS2.p1.27.m14.1.1.2.2" mathsize="144%" xref="A1.SS2.p1.27.m14.1.1.2.2.cmml">ϕ</mi><mi id="A1.SS2.p1.27.m14.1.1.2.3" mathsize="144%" xref="A1.SS2.p1.27.m14.1.1.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.27.m14.1b"><apply id="A1.SS2.p1.27.m14.1.1.cmml" xref="A1.SS2.p1.27.m14.1.1"><minus id="A1.SS2.p1.27.m14.1.1.1.cmml" xref="A1.SS2.p1.27.m14.1.1"></minus><apply id="A1.SS2.p1.27.m14.1.1.2.cmml" xref="A1.SS2.p1.27.m14.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.p1.27.m14.1.1.2.1.cmml" xref="A1.SS2.p1.27.m14.1.1.2">subscript</csymbol><ci id="A1.SS2.p1.27.m14.1.1.2.2.cmml" xref="A1.SS2.p1.27.m14.1.1.2.2">italic-ϕ</ci><ci id="A1.SS2.p1.27.m14.1.1.2.3.cmml" xref="A1.SS2.p1.27.m14.1.1.2.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.27.m14.1c">-\phi_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.27.m14.1d">- italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, a dense detection map <math alttext="\widehat{\boldsymbol{\gamma}}" class="ltx_Math" display="inline" id="A1.SS2.p1.28.m15.1"><semantics id="A1.SS2.p1.28.m15.1a"><mover accent="true" id="A1.SS2.p1.28.m15.1.1" xref="A1.SS2.p1.28.m15.1.1.cmml"><mi id="A1.SS2.p1.28.m15.1.1.2" mathsize="144%" xref="A1.SS2.p1.28.m15.1.1.2.cmml">𝜸</mi><mo id="A1.SS2.p1.28.m15.1.1.1" mathsize="144%" xref="A1.SS2.p1.28.m15.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.28.m15.1b"><apply id="A1.SS2.p1.28.m15.1.1.cmml" xref="A1.SS2.p1.28.m15.1.1"><ci id="A1.SS2.p1.28.m15.1.1.1.cmml" xref="A1.SS2.p1.28.m15.1.1.1">^</ci><ci id="A1.SS2.p1.28.m15.1.1.2.cmml" xref="A1.SS2.p1.28.m15.1.1.2">𝜸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.28.m15.1c">\widehat{\boldsymbol{\gamma}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.28.m15.1d">over^ start_ARG bold_italic_γ end_ARG</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="A1.SS2.p1.29.m16.1"><semantics id="A1.SS2.p1.29.m16.1a"><msup id="A1.SS2.p1.29.m16.1.1" xref="A1.SS2.p1.29.m16.1.1.cmml"><mi id="A1.SS2.p1.29.m16.1.1.2" mathsize="144%" xref="A1.SS2.p1.29.m16.1.1.2.cmml">ℝ</mi><mrow id="A1.SS2.p1.29.m16.1.1.3" xref="A1.SS2.p1.29.m16.1.1.3.cmml"><mi id="A1.SS2.p1.29.m16.1.1.3.2" mathsize="144%" xref="A1.SS2.p1.29.m16.1.1.3.2.cmml">H</mi><mo id="A1.SS2.p1.29.m16.1.1.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS2.p1.29.m16.1.1.3.1.cmml">×</mo><mi id="A1.SS2.p1.29.m16.1.1.3.3" mathsize="144%" xref="A1.SS2.p1.29.m16.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.29.m16.1b"><apply id="A1.SS2.p1.29.m16.1.1.cmml" xref="A1.SS2.p1.29.m16.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.29.m16.1.1.1.cmml" xref="A1.SS2.p1.29.m16.1.1">superscript</csymbol><ci id="A1.SS2.p1.29.m16.1.1.2.cmml" xref="A1.SS2.p1.29.m16.1.1.2">ℝ</ci><apply id="A1.SS2.p1.29.m16.1.1.3.cmml" xref="A1.SS2.p1.29.m16.1.1.3"><times id="A1.SS2.p1.29.m16.1.1.3.1.cmml" xref="A1.SS2.p1.29.m16.1.1.3.1"></times><ci id="A1.SS2.p1.29.m16.1.1.3.2.cmml" xref="A1.SS2.p1.29.m16.1.1.3.2">𝐻</ci><ci id="A1.SS2.p1.29.m16.1.1.3.3.cmml" xref="A1.SS2.p1.29.m16.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.29.m16.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.29.m16.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> can be obtained with: <table class="ltx_equation ltx_eqn_table" id="A1.E28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widehat{\boldsymbol{\gamma}}=\frac{\widehat{\boldsymbol{\alpha}}}{\widehat{% \boldsymbol{\sigma}}}=\frac{\sum_{t}\mathbf{R}_{-\phi_{t}}(\boldsymbol{b}_{t})% }{\sqrt{\sum_{t}\mathbf{R}_{-\phi_{t}}(\boldsymbol{a})}}\,," class="ltx_Math" display="block" id="A1.E28.m1.3"><semantics id="A1.E28.m1.3a"><mrow id="A1.E28.m1.3.3.1" xref="A1.E28.m1.3.3.1.1.cmml"><mrow id="A1.E28.m1.3.3.1.1" xref="A1.E28.m1.3.3.1.1.cmml"><mover accent="true" id="A1.E28.m1.3.3.1.1.2" xref="A1.E28.m1.3.3.1.1.2.cmml"><mi id="A1.E28.m1.3.3.1.1.2.2" mathsize="144%" xref="A1.E28.m1.3.3.1.1.2.2.cmml">𝜸</mi><mo id="A1.E28.m1.3.3.1.1.2.1" mathsize="144%" xref="A1.E28.m1.3.3.1.1.2.1.cmml">^</mo></mover><mo id="A1.E28.m1.3.3.1.1.3" mathsize="144%" xref="A1.E28.m1.3.3.1.1.3.cmml">=</mo><mfrac id="A1.E28.m1.3.3.1.1.4" xref="A1.E28.m1.3.3.1.1.4.cmml"><mover accent="true" id="A1.E28.m1.3.3.1.1.4.2" xref="A1.E28.m1.3.3.1.1.4.2.cmml"><mi id="A1.E28.m1.3.3.1.1.4.2.2" mathsize="144%" xref="A1.E28.m1.3.3.1.1.4.2.2.cmml">𝜶</mi><mo id="A1.E28.m1.3.3.1.1.4.2.1" mathsize="144%" xref="A1.E28.m1.3.3.1.1.4.2.1.cmml">^</mo></mover><mover accent="true" id="A1.E28.m1.3.3.1.1.4.3" xref="A1.E28.m1.3.3.1.1.4.3.cmml"><mi id="A1.E28.m1.3.3.1.1.4.3.2" mathsize="144%" xref="A1.E28.m1.3.3.1.1.4.3.2.cmml">𝝈</mi><mo id="A1.E28.m1.3.3.1.1.4.3.1" mathsize="144%" xref="A1.E28.m1.3.3.1.1.4.3.1.cmml">^</mo></mover></mfrac><mo id="A1.E28.m1.3.3.1.1.5" mathsize="144%" xref="A1.E28.m1.3.3.1.1.5.cmml">=</mo><mfrac id="A1.E28.m1.2.2" xref="A1.E28.m1.2.2.cmml"><mrow id="A1.E28.m1.1.1.1" xref="A1.E28.m1.1.1.1.cmml"><msub id="A1.E28.m1.1.1.1.2" xref="A1.E28.m1.1.1.1.2.cmml"><mo id="A1.E28.m1.1.1.1.2.2" maxsize="144%" minsize="144%" stretchy="true" xref="A1.E28.m1.1.1.1.2.2.cmml">∑</mo><mi id="A1.E28.m1.1.1.1.2.3" mathsize="144%" xref="A1.E28.m1.1.1.1.2.3.cmml">t</mi></msub><mrow id="A1.E28.m1.1.1.1.1" xref="A1.E28.m1.1.1.1.1.cmml"><msub id="A1.E28.m1.1.1.1.1.3" xref="A1.E28.m1.1.1.1.1.3.cmml"><mi id="A1.E28.m1.1.1.1.1.3.2" mathsize="144%" xref="A1.E28.m1.1.1.1.1.3.2.cmml">𝐑</mi><mrow id="A1.E28.m1.1.1.1.1.3.3" xref="A1.E28.m1.1.1.1.1.3.3.cmml"><mo id="A1.E28.m1.1.1.1.1.3.3a" mathsize="144%" xref="A1.E28.m1.1.1.1.1.3.3.cmml">−</mo><msub id="A1.E28.m1.1.1.1.1.3.3.2" xref="A1.E28.m1.1.1.1.1.3.3.2.cmml"><mi id="A1.E28.m1.1.1.1.1.3.3.2.2" mathsize="144%" xref="A1.E28.m1.1.1.1.1.3.3.2.2.cmml">ϕ</mi><mi id="A1.E28.m1.1.1.1.1.3.3.2.3" mathsize="144%" xref="A1.E28.m1.1.1.1.1.3.3.2.3.cmml">t</mi></msub></mrow></msub><mo id="A1.E28.m1.1.1.1.1.2" xref="A1.E28.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.E28.m1.1.1.1.1.1.1" xref="A1.E28.m1.1.1.1.1.1.1.1.cmml"><mo id="A1.E28.m1.1.1.1.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.E28.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="A1.E28.m1.1.1.1.1.1.1.1" xref="A1.E28.m1.1.1.1.1.1.1.1.cmml"><mi id="A1.E28.m1.1.1.1.1.1.1.1.2" mathsize="144%" xref="A1.E28.m1.1.1.1.1.1.1.1.2.cmml">𝒃</mi><mi id="A1.E28.m1.1.1.1.1.1.1.1.3" mathsize="144%" xref="A1.E28.m1.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.E28.m1.1.1.1.1.1.1.3" maxsize="144%" 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xref="A1.SS2.p1.30.m1.1.1.3.3"><csymbol cd="ambiguous" id="A1.SS2.p1.30.m1.1.1.3.3.1.cmml" xref="A1.SS2.p1.30.m1.1.1.3.3">superscript</csymbol><ci id="A1.SS2.p1.30.m1.1.1.3.3.2.cmml" xref="A1.SS2.p1.30.m1.1.1.3.3.2">ℝ</ci><apply id="A1.SS2.p1.30.m1.1.1.3.3.3.cmml" xref="A1.SS2.p1.30.m1.1.1.3.3.3"><times id="A1.SS2.p1.30.m1.1.1.3.3.3.1.cmml" xref="A1.SS2.p1.30.m1.1.1.3.3.3.1"></times><ci id="A1.SS2.p1.30.m1.1.1.3.3.3.2.cmml" xref="A1.SS2.p1.30.m1.1.1.3.3.3.2">𝐻</ci><ci id="A1.SS2.p1.30.m1.1.1.3.3.3.3.cmml" xref="A1.SS2.p1.30.m1.1.1.3.3.3.3">𝑊</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.30.m1.1c">\mathbf{R}_{-\phi_{t}}:\mathbb{R}^{H\times W}\to\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.30.m1.1d">bold_R start_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT : blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> is the frame rotation operator with angle <math alttext="-\phi_{t}" class="ltx_Math" display="inline" id="A1.SS2.p1.31.m2.1"><semantics id="A1.SS2.p1.31.m2.1a"><mrow id="A1.SS2.p1.31.m2.1.1" xref="A1.SS2.p1.31.m2.1.1.cmml"><mo id="A1.SS2.p1.31.m2.1.1a" mathsize="144%" xref="A1.SS2.p1.31.m2.1.1.cmml">−</mo><msub id="A1.SS2.p1.31.m2.1.1.2" xref="A1.SS2.p1.31.m2.1.1.2.cmml"><mi id="A1.SS2.p1.31.m2.1.1.2.2" mathsize="144%" xref="A1.SS2.p1.31.m2.1.1.2.2.cmml">ϕ</mi><mi id="A1.SS2.p1.31.m2.1.1.2.3" mathsize="144%" xref="A1.SS2.p1.31.m2.1.1.2.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.31.m2.1b"><apply id="A1.SS2.p1.31.m2.1.1.cmml" xref="A1.SS2.p1.31.m2.1.1"><minus id="A1.SS2.p1.31.m2.1.1.1.cmml" xref="A1.SS2.p1.31.m2.1.1"></minus><apply id="A1.SS2.p1.31.m2.1.1.2.cmml" xref="A1.SS2.p1.31.m2.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.p1.31.m2.1.1.2.1.cmml" xref="A1.SS2.p1.31.m2.1.1.2">subscript</csymbol><ci id="A1.SS2.p1.31.m2.1.1.2.2.cmml" xref="A1.SS2.p1.31.m2.1.1.2.2">italic-ϕ</ci><ci id="A1.SS2.p1.31.m2.1.1.2.3.cmml" xref="A1.SS2.p1.31.m2.1.1.2.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.31.m2.1c">-\phi_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.31.m2.1d">- italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\widehat{\boldsymbol{\alpha}},\widehat{\boldsymbol{\sigma}}" class="ltx_Math" display="inline" id="A1.SS2.p1.32.m3.2"><semantics id="A1.SS2.p1.32.m3.2a"><mrow id="A1.SS2.p1.32.m3.2.3.2" xref="A1.SS2.p1.32.m3.2.3.1.cmml"><mover accent="true" id="A1.SS2.p1.32.m3.1.1" xref="A1.SS2.p1.32.m3.1.1.cmml"><mi id="A1.SS2.p1.32.m3.1.1.2" mathsize="144%" xref="A1.SS2.p1.32.m3.1.1.2.cmml">𝜶</mi><mo id="A1.SS2.p1.32.m3.1.1.1" mathsize="144%" xref="A1.SS2.p1.32.m3.1.1.1.cmml">^</mo></mover><mo id="A1.SS2.p1.32.m3.2.3.2.1" mathsize="144%" xref="A1.SS2.p1.32.m3.2.3.1.cmml">,</mo><mover accent="true" id="A1.SS2.p1.32.m3.2.2" xref="A1.SS2.p1.32.m3.2.2.cmml"><mi id="A1.SS2.p1.32.m3.2.2.2" mathsize="144%" xref="A1.SS2.p1.32.m3.2.2.2.cmml">𝝈</mi><mo id="A1.SS2.p1.32.m3.2.2.1" mathsize="144%" xref="A1.SS2.p1.32.m3.2.2.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.32.m3.2b"><list id="A1.SS2.p1.32.m3.2.3.1.cmml" xref="A1.SS2.p1.32.m3.2.3.2"><apply id="A1.SS2.p1.32.m3.1.1.cmml" xref="A1.SS2.p1.32.m3.1.1"><ci id="A1.SS2.p1.32.m3.1.1.1.cmml" xref="A1.SS2.p1.32.m3.1.1.1">^</ci><ci id="A1.SS2.p1.32.m3.1.1.2.cmml" xref="A1.SS2.p1.32.m3.1.1.2">𝜶</ci></apply><apply id="A1.SS2.p1.32.m3.2.2.cmml" xref="A1.SS2.p1.32.m3.2.2"><ci id="A1.SS2.p1.32.m3.2.2.1.cmml" xref="A1.SS2.p1.32.m3.2.2.1">^</ci><ci id="A1.SS2.p1.32.m3.2.2.2.cmml" xref="A1.SS2.p1.32.m3.2.2.2">𝝈</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.32.m3.2c">\widehat{\boldsymbol{\alpha}},\widehat{\boldsymbol{\sigma}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.32.m3.2d">over^ start_ARG bold_italic_α end_ARG , over^ start_ARG bold_italic_σ end_ARG</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="A1.SS2.p1.33.m4.1"><semantics id="A1.SS2.p1.33.m4.1a"><msup id="A1.SS2.p1.33.m4.1.1" xref="A1.SS2.p1.33.m4.1.1.cmml"><mi id="A1.SS2.p1.33.m4.1.1.2" mathsize="144%" xref="A1.SS2.p1.33.m4.1.1.2.cmml">ℝ</mi><mrow id="A1.SS2.p1.33.m4.1.1.3" xref="A1.SS2.p1.33.m4.1.1.3.cmml"><mi id="A1.SS2.p1.33.m4.1.1.3.2" mathsize="144%" xref="A1.SS2.p1.33.m4.1.1.3.2.cmml">H</mi><mo id="A1.SS2.p1.33.m4.1.1.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS2.p1.33.m4.1.1.3.1.cmml">×</mo><mi id="A1.SS2.p1.33.m4.1.1.3.3" mathsize="144%" xref="A1.SS2.p1.33.m4.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS2.p1.33.m4.1b"><apply id="A1.SS2.p1.33.m4.1.1.cmml" xref="A1.SS2.p1.33.m4.1.1"><csymbol cd="ambiguous" id="A1.SS2.p1.33.m4.1.1.1.cmml" xref="A1.SS2.p1.33.m4.1.1">superscript</csymbol><ci id="A1.SS2.p1.33.m4.1.1.2.cmml" xref="A1.SS2.p1.33.m4.1.1.2">ℝ</ci><apply id="A1.SS2.p1.33.m4.1.1.3.cmml" xref="A1.SS2.p1.33.m4.1.1.3"><times id="A1.SS2.p1.33.m4.1.1.3.1.cmml" xref="A1.SS2.p1.33.m4.1.1.3.1"></times><ci id="A1.SS2.p1.33.m4.1.1.3.2.cmml" xref="A1.SS2.p1.33.m4.1.1.3.2">𝐻</ci><ci id="A1.SS2.p1.33.m4.1.1.3.3.cmml" xref="A1.SS2.p1.33.m4.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.33.m4.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p1.33.m4.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> such that: <table class="ltx_equation ltx_eqn_table" id="A1.E29"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widehat{\boldsymbol{\alpha}}=\frac{\sum_{t}\mathbf{R}_{-\phi_{t}}(\boldsymbol% {b}_{t})}{\sum_{t}\mathbf{R}_{-\phi_{t}}(\boldsymbol{a})}~{}~{}~{}\text{and}~{% }~{}~{}\widehat{\boldsymbol{\sigma}}=\frac{1}{\sqrt{\sum_{t}\mathbf{R}_{-\phi_% {t}}(\boldsymbol{a})}}\,." class="ltx_Math" display="block" id="A1.E29.m1.5"><semantics id="A1.E29.m1.5a"><mrow id="A1.E29.m1.5.5.1"><mrow id="A1.E29.m1.5.5.1.1.2" xref="A1.E29.m1.5.5.1.1.3.cmml"><mrow id="A1.E29.m1.5.5.1.1.1.1" xref="A1.E29.m1.5.5.1.1.1.1.cmml"><mover accent="true" id="A1.E29.m1.5.5.1.1.1.1.2" xref="A1.E29.m1.5.5.1.1.1.1.2.cmml"><mi id="A1.E29.m1.5.5.1.1.1.1.2.2" mathsize="144%" xref="A1.E29.m1.5.5.1.1.1.1.2.2.cmml">𝜶</mi><mo id="A1.E29.m1.5.5.1.1.1.1.2.1" mathsize="144%" xref="A1.E29.m1.5.5.1.1.1.1.2.1.cmml">^</mo></mover><mo id="A1.E29.m1.5.5.1.1.1.1.1" mathsize="144%" xref="A1.E29.m1.5.5.1.1.1.1.1.cmml">=</mo><mrow id="A1.E29.m1.5.5.1.1.1.1.3.2" xref="A1.E29.m1.5.5.1.1.1.1.3.1.cmml"><mfrac id="A1.E29.m1.2.2" xref="A1.E29.m1.2.2.cmml"><mrow id="A1.E29.m1.1.1.1" xref="A1.E29.m1.1.1.1.cmml"><msub id="A1.E29.m1.1.1.1.2" xref="A1.E29.m1.1.1.1.2.cmml"><mo id="A1.E29.m1.1.1.1.2.2" maxsize="144%" minsize="144%" stretchy="true" xref="A1.E29.m1.1.1.1.2.2.cmml">∑</mo><mi id="A1.E29.m1.1.1.1.2.3" mathsize="144%" xref="A1.E29.m1.1.1.1.2.3.cmml">t</mi></msub><mrow id="A1.E29.m1.1.1.1.1" xref="A1.E29.m1.1.1.1.1.cmml"><msub id="A1.E29.m1.1.1.1.1.3" xref="A1.E29.m1.1.1.1.1.3.cmml"><mi id="A1.E29.m1.1.1.1.1.3.2" mathsize="144%" xref="A1.E29.m1.1.1.1.1.3.2.cmml">𝐑</mi><mrow id="A1.E29.m1.1.1.1.1.3.3" xref="A1.E29.m1.1.1.1.1.3.3.cmml"><mo id="A1.E29.m1.1.1.1.1.3.3a" mathsize="144%" xref="A1.E29.m1.1.1.1.1.3.3.cmml">−</mo><msub id="A1.E29.m1.1.1.1.1.3.3.2" xref="A1.E29.m1.1.1.1.1.3.3.2.cmml"><mi id="A1.E29.m1.1.1.1.1.3.3.2.2" mathsize="144%" xref="A1.E29.m1.1.1.1.1.3.3.2.2.cmml">ϕ</mi><mi id="A1.E29.m1.1.1.1.1.3.3.2.3" mathsize="144%" xref="A1.E29.m1.1.1.1.1.3.3.2.3.cmml">t</mi></msub></mrow></msub><mo id="A1.E29.m1.1.1.1.1.2" xref="A1.E29.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.E29.m1.1.1.1.1.1.1" xref="A1.E29.m1.1.1.1.1.1.1.1.cmml"><mo id="A1.E29.m1.1.1.1.1.1.1.2" maxsize="144%" minsize="144%" xref="A1.E29.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="A1.E29.m1.1.1.1.1.1.1.1" xref="A1.E29.m1.1.1.1.1.1.1.1.cmml"><mi id="A1.E29.m1.1.1.1.1.1.1.1.2" mathsize="144%" xref="A1.E29.m1.1.1.1.1.1.1.1.2.cmml">𝒃</mi><mi id="A1.E29.m1.1.1.1.1.1.1.1.3" mathsize="144%" xref="A1.E29.m1.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A1.E29.m1.1.1.1.1.1.1.3" maxsize="144%" minsize="144%" xref="A1.E29.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mrow id="A1.E29.m1.2.2.2" xref="A1.E29.m1.2.2.2.cmml"><msub id="A1.E29.m1.2.2.2.2" xref="A1.E29.m1.2.2.2.2.cmml"><mo id="A1.E29.m1.2.2.2.2.2" maxsize="144%" minsize="144%" stretchy="true" 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id="A1.E29.m1.5c">\widehat{\boldsymbol{\alpha}}=\frac{\sum_{t}\mathbf{R}_{-\phi_{t}}(\boldsymbol% {b}_{t})}{\sum_{t}\mathbf{R}_{-\phi_{t}}(\boldsymbol{a})}~{}~{}~{}\text{and}~{% }~{}~{}\widehat{\boldsymbol{\sigma}}=\frac{1}{\sqrt{\sum_{t}\mathbf{R}_{-\phi_% {t}}(\boldsymbol{a})}}\,.</annotation><annotation encoding="application/x-llamapun" id="A1.E29.m1.5d">over^ start_ARG bold_italic_α end_ARG = divide start_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT bold_R start_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT bold_R start_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_a ) end_ARG and over^ start_ARG bold_italic_σ end_ARG = divide start_ARG 1 end_ARG start_ARG square-root start_ARG ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT bold_R start_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_a ) end_ARG end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(29)</td> </tr></tbody> </table> </div> </section> <section class="ltx_subsection" id="A1.SS3"> <h3 class="ltx_title ltx_title_subsection" style="font-size:144%;"> A.3 Fast computation of <math alttext="\boldsymbol{b}_{t}" class="ltx_Math" display="inline" id="A1.SS3.1.m1.1"><semantics id="A1.SS3.1.m1.1b"><msub id="A1.SS3.1.m1.1.1" xref="A1.SS3.1.m1.1.1.cmml"><mi id="A1.SS3.1.m1.1.1.2" xref="A1.SS3.1.m1.1.1.2.cmml">𝒃</mi><mi id="A1.SS3.1.m1.1.1.3" xref="A1.SS3.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS3.1.m1.1c"><apply id="A1.SS3.1.m1.1.1.cmml" xref="A1.SS3.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS3.1.m1.1.1.1.cmml" xref="A1.SS3.1.m1.1.1">subscript</csymbol><ci id="A1.SS3.1.m1.1.1.2.cmml" xref="A1.SS3.1.m1.1.1.2">𝒃</ci><ci id="A1.SS3.1.m1.1.1.3.cmml" xref="A1.SS3.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.1.m1.1d">\boldsymbol{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.1.m1.1e">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{a}" class="ltx_Math" display="inline" id="A1.SS3.2.m2.1"><semantics id="A1.SS3.2.m2.1b"><mi id="A1.SS3.2.m2.1.1" xref="A1.SS3.2.m2.1.1.cmml">𝒂</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.2.m2.1c"><ci id="A1.SS3.2.m2.1.1.cmml" xref="A1.SS3.2.m2.1.1">𝒂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.2.m2.1d">\boldsymbol{a}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.2.m2.1e">bold_italic_a</annotation></semantics></math> </h3> <div class="ltx_para" id="A1.SS3.p1"> In this section, we detail an efficient strategy to derive a dense two-dimensional detection map <math alttext="\widehat{\boldsymbol{\gamma}}" class="ltx_Math" display="inline" id="A1.SS3.p1.1.m1.1"><semantics id="A1.SS3.p1.1.m1.1a"><mover accent="true" id="A1.SS3.p1.1.m1.1.1" xref="A1.SS3.p1.1.m1.1.1.cmml"><mi id="A1.SS3.p1.1.m1.1.1.2" mathsize="144%" xref="A1.SS3.p1.1.m1.1.1.2.cmml">𝜸</mi><mo id="A1.SS3.p1.1.m1.1.1.1" mathsize="144%" xref="A1.SS3.p1.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.1.m1.1b"><apply id="A1.SS3.p1.1.m1.1.1.cmml" xref="A1.SS3.p1.1.m1.1.1"><ci id="A1.SS3.p1.1.m1.1.1.1.cmml" xref="A1.SS3.p1.1.m1.1.1.1">^</ci><ci id="A1.SS3.p1.1.m1.1.1.2.cmml" xref="A1.SS3.p1.1.m1.1.1.2">𝜸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.1.m1.1c">\widehat{\boldsymbol{\gamma}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.1.m1.1d">over^ start_ARG bold_italic_γ end_ARG</annotation></semantics></math> serving as the final quantity to detect exoplanets, see Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1" style="font-size:144%;" title="4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.1</a>. </div> <div class="ltx_para ltx_noindent" id="A1.SS3.p2"> The terms <math alttext="\boldsymbol{b}_{t}" class="ltx_Math" display="inline" id="A1.SS3.p2.1.m1.1"><semantics id="A1.SS3.p2.1.m1.1a"><msub id="A1.SS3.p2.1.m1.1.1" xref="A1.SS3.p2.1.m1.1.1.cmml"><mi id="A1.SS3.p2.1.m1.1.1.2" mathsize="144%" xref="A1.SS3.p2.1.m1.1.1.2.cmml">𝒃</mi><mi id="A1.SS3.p2.1.m1.1.1.3" mathsize="144%" xref="A1.SS3.p2.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.1.m1.1b"><apply id="A1.SS3.p2.1.m1.1.1.cmml" xref="A1.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS3.p2.1.m1.1.1.1.cmml" xref="A1.SS3.p2.1.m1.1.1">subscript</csymbol><ci id="A1.SS3.p2.1.m1.1.1.2.cmml" xref="A1.SS3.p2.1.m1.1.1.2">𝒃</ci><ci id="A1.SS3.p2.1.m1.1.1.3.cmml" xref="A1.SS3.p2.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.1.m1.1c">\boldsymbol{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.1.m1.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{a}" class="ltx_Math" display="inline" id="A1.SS3.p2.2.m2.1"><semantics id="A1.SS3.p2.2.m2.1a"><mi id="A1.SS3.p2.2.m2.1.1" mathsize="144%" xref="A1.SS3.p2.2.m2.1.1.cmml">𝒂</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.2.m2.1b"><ci id="A1.SS3.p2.2.m2.1.1.cmml" xref="A1.SS3.p2.2.m2.1.1">𝒂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.2.m2.1c">\boldsymbol{a}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.2.m2.1d">bold_italic_a</annotation></semantics></math> in <math alttext="\mathbb{R}^{H\times W}" class="ltx_Math" display="inline" id="A1.SS3.p2.3.m3.1"><semantics id="A1.SS3.p2.3.m3.1a"><msup id="A1.SS3.p2.3.m3.1.1" xref="A1.SS3.p2.3.m3.1.1.cmml"><mi id="A1.SS3.p2.3.m3.1.1.2" mathsize="144%" xref="A1.SS3.p2.3.m3.1.1.2.cmml">ℝ</mi><mrow id="A1.SS3.p2.3.m3.1.1.3" xref="A1.SS3.p2.3.m3.1.1.3.cmml"><mi id="A1.SS3.p2.3.m3.1.1.3.2" mathsize="144%" xref="A1.SS3.p2.3.m3.1.1.3.2.cmml">H</mi><mo id="A1.SS3.p2.3.m3.1.1.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS3.p2.3.m3.1.1.3.1.cmml">×</mo><mi id="A1.SS3.p2.3.m3.1.1.3.3" mathsize="144%" xref="A1.SS3.p2.3.m3.1.1.3.3.cmml">W</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.3.m3.1b"><apply id="A1.SS3.p2.3.m3.1.1.cmml" xref="A1.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS3.p2.3.m3.1.1.1.cmml" xref="A1.SS3.p2.3.m3.1.1">superscript</csymbol><ci id="A1.SS3.p2.3.m3.1.1.2.cmml" xref="A1.SS3.p2.3.m3.1.1.2">ℝ</ci><apply id="A1.SS3.p2.3.m3.1.1.3.cmml" xref="A1.SS3.p2.3.m3.1.1.3"><times id="A1.SS3.p2.3.m3.1.1.3.1.cmml" xref="A1.SS3.p2.3.m3.1.1.3.1"></times><ci id="A1.SS3.p2.3.m3.1.1.3.2.cmml" xref="A1.SS3.p2.3.m3.1.1.3.2">𝐻</ci><ci id="A1.SS3.p2.3.m3.1.1.3.3.cmml" xref="A1.SS3.p2.3.m3.1.1.3.3">𝑊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.3.m3.1c">\mathbb{R}^{H\times W}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_H × italic_W end_POSTSUPERSCRIPT</annotation></semantics></math> involved in <math alttext="\widehat{\boldsymbol{\gamma}}" class="ltx_Math" display="inline" id="A1.SS3.p2.4.m4.1"><semantics id="A1.SS3.p2.4.m4.1a"><mover accent="true" id="A1.SS3.p2.4.m4.1.1" xref="A1.SS3.p2.4.m4.1.1.cmml"><mi id="A1.SS3.p2.4.m4.1.1.2" mathsize="144%" xref="A1.SS3.p2.4.m4.1.1.2.cmml">𝜸</mi><mo id="A1.SS3.p2.4.m4.1.1.1" mathsize="144%" xref="A1.SS3.p2.4.m4.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.4.m4.1b"><apply id="A1.SS3.p2.4.m4.1.1.cmml" xref="A1.SS3.p2.4.m4.1.1"><ci id="A1.SS3.p2.4.m4.1.1.1.cmml" xref="A1.SS3.p2.4.m4.1.1.1">^</ci><ci id="A1.SS3.p2.4.m4.1.1.2.cmml" xref="A1.SS3.p2.4.m4.1.1.2">𝜸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.4.m4.1c">\widehat{\boldsymbol{\gamma}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.4.m4.1d">over^ start_ARG bold_italic_γ end_ARG</annotation></semantics></math> (see Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E26" style="font-size:144%;" title="Equation 26 ‣ A.2 Dense detection map computation details ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">26</a>)–(<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E28" style="font-size:144%;" title="Equation 28 ‣ A.2 Dense detection map computation details ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">28</a>)) can be computed efficiently through: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx4"> <tbody id="A1.E30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{b}_{t}" class="ltx_Math" display="inline" id="A1.E30.m1.1"><semantics id="A1.E30.m1.1a"><msub id="A1.E30.m1.1.1" xref="A1.E30.m1.1.1.cmml"><mi id="A1.E30.m1.1.1.2" mathsize="144%" xref="A1.E30.m1.1.1.2.cmml">𝒃</mi><mi id="A1.E30.m1.1.1.3" mathsize="144%" xref="A1.E30.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.E30.m1.1b"><apply id="A1.E30.m1.1.1.cmml" xref="A1.E30.m1.1.1"><csymbol cd="ambiguous" id="A1.E30.m1.1.1.1.cmml" xref="A1.E30.m1.1.1">subscript</csymbol><ci id="A1.E30.m1.1.1.2.cmml" xref="A1.E30.m1.1.1.2">𝒃</ci><ci id="A1.E30.m1.1.1.3.cmml" xref="A1.E30.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E30.m1.1c">\displaystyle\boldsymbol{b}_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.E30.m1.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="A1.E30.m2.2.2.1.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E30.m2.2c">\displaystyle=\sum_{j}w_{j}\mathbf{Q}_{j}(\boldsymbol{b}^{(j)}_{t})\,,</annotation><annotation encoding="application/x-llamapun" id="A1.E30.m2.2d">= ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( bold_italic_b start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(30)</td> </tr></tbody> <tbody id="A1.E31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math 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xref="A1.E31.m2.2.2.1.1.2.cmml">=</mo><mrow id="A1.E31.m2.2.2.1.1.1" xref="A1.E31.m2.2.2.1.1.1.cmml"><mstyle displaystyle="true" id="A1.E31.m2.2.2.1.1.1.2" xref="A1.E31.m2.2.2.1.1.1.2.cmml"><munder id="A1.E31.m2.2.2.1.1.1.2a" xref="A1.E31.m2.2.2.1.1.1.2.cmml"><mo id="A1.E31.m2.2.2.1.1.1.2.2" maxsize="144%" minsize="144%" movablelimits="false" stretchy="true" xref="A1.E31.m2.2.2.1.1.1.2.2.cmml">∑</mo><mi id="A1.E31.m2.2.2.1.1.1.2.3" mathsize="144%" xref="A1.E31.m2.2.2.1.1.1.2.3.cmml">j</mi></munder></mstyle><mrow id="A1.E31.m2.2.2.1.1.1.1" xref="A1.E31.m2.2.2.1.1.1.1.cmml"><msub id="A1.E31.m2.2.2.1.1.1.1.3" xref="A1.E31.m2.2.2.1.1.1.1.3.cmml"><mi id="A1.E31.m2.2.2.1.1.1.1.3.2" mathsize="144%" xref="A1.E31.m2.2.2.1.1.1.1.3.2.cmml">w</mi><mi id="A1.E31.m2.2.2.1.1.1.1.3.3" mathsize="144%" xref="A1.E31.m2.2.2.1.1.1.1.3.3.cmml">j</mi></msub><mo id="A1.E31.m2.2.2.1.1.1.1.2" xref="A1.E31.m2.2.2.1.1.1.1.2.cmml">⁢</mo><msub id="A1.E31.m2.2.2.1.1.1.1.4" xref="A1.E31.m2.2.2.1.1.1.1.4.cmml"><mi 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id="A1.E31.m2.2.2.1.1.1.1.cmml" xref="A1.E31.m2.2.2.1.1.1.1"><times id="A1.E31.m2.2.2.1.1.1.1.2.cmml" xref="A1.E31.m2.2.2.1.1.1.1.2"></times><apply id="A1.E31.m2.2.2.1.1.1.1.3.cmml" xref="A1.E31.m2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.E31.m2.2.2.1.1.1.1.3.1.cmml" xref="A1.E31.m2.2.2.1.1.1.1.3">subscript</csymbol><ci id="A1.E31.m2.2.2.1.1.1.1.3.2.cmml" xref="A1.E31.m2.2.2.1.1.1.1.3.2">𝑤</ci><ci id="A1.E31.m2.2.2.1.1.1.1.3.3.cmml" xref="A1.E31.m2.2.2.1.1.1.1.3.3">𝑗</ci></apply><apply id="A1.E31.m2.2.2.1.1.1.1.4.cmml" xref="A1.E31.m2.2.2.1.1.1.1.4"><csymbol cd="ambiguous" id="A1.E31.m2.2.2.1.1.1.1.4.1.cmml" xref="A1.E31.m2.2.2.1.1.1.1.4">subscript</csymbol><ci id="A1.E31.m2.2.2.1.1.1.1.4.2.cmml" xref="A1.E31.m2.2.2.1.1.1.1.4.2">𝐐</ci><ci id="A1.E31.m2.2.2.1.1.1.1.4.3.cmml" xref="A1.E31.m2.2.2.1.1.1.1.4.3">𝑗</ci></apply><apply id="A1.E31.m2.2.2.1.1.1.1.1.1.1.cmml" xref="A1.E31.m2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.E31.m2.2.2.1.1.1.1.1.1.1.1.cmml" xref="A1.E31.m2.2.2.1.1.1.1.1.1">superscript</csymbol><ci id="A1.E31.m2.2.2.1.1.1.1.1.1.1.2.cmml" xref="A1.E31.m2.2.2.1.1.1.1.1.1.1.2">𝒂</ci><ci id="A1.E31.m2.1.1.1.1.cmml" xref="A1.E31.m2.1.1.1.1">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E31.m2.2c">\displaystyle=\sum_{j}w_{j}\mathbf{Q}_{j}(\boldsymbol{a}^{(j)})\,,</annotation><annotation encoding="application/x-llamapun" id="A1.E31.m2.2d">= ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( bold_italic_a start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(31)</td> </tr></tbody> </table> where <math alttext="\mathbf{Q}_{j}" class="ltx_Math" display="inline" 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alttext="j" class="ltx_Math" display="inline" id="A1.SS3.p2.6.m2.1"><semantics id="A1.SS3.p2.6.m2.1a"><mi id="A1.SS3.p2.6.m2.1.1" mathsize="144%" xref="A1.SS3.p2.6.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.6.m2.1b"><ci id="A1.SS3.p2.6.m2.1.1.cmml" xref="A1.SS3.p2.6.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.6.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.6.m2.1d">italic_j</annotation></semantics></math> in the correct position in the image, and <math alttext="\boldsymbol{b}_{t}^{(j)}" class="ltx_Math" display="inline" id="A1.SS3.p2.7.m3.1"><semantics id="A1.SS3.p2.7.m3.1a"><msubsup id="A1.SS3.p2.7.m3.1.2" xref="A1.SS3.p2.7.m3.1.2.cmml"><mi id="A1.SS3.p2.7.m3.1.2.2.2" mathsize="144%" xref="A1.SS3.p2.7.m3.1.2.2.2.cmml">𝒃</mi><mi id="A1.SS3.p2.7.m3.1.2.2.3" mathsize="144%" xref="A1.SS3.p2.7.m3.1.2.2.3.cmml">t</mi><mrow id="A1.SS3.p2.7.m3.1.1.1.3" xref="A1.SS3.p2.7.m3.1.2.cmml"><mo id="A1.SS3.p2.7.m3.1.1.1.3.1" maxsize="144%" minsize="144%" xref="A1.SS3.p2.7.m3.1.2.cmml">(</mo><mi id="A1.SS3.p2.7.m3.1.1.1.1" mathsize="144%" xref="A1.SS3.p2.7.m3.1.1.1.1.cmml">j</mi><mo id="A1.SS3.p2.7.m3.1.1.1.3.2" maxsize="144%" minsize="144%" xref="A1.SS3.p2.7.m3.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.7.m3.1b"><apply id="A1.SS3.p2.7.m3.1.2.cmml" xref="A1.SS3.p2.7.m3.1.2"><csymbol cd="ambiguous" id="A1.SS3.p2.7.m3.1.2.1.cmml" xref="A1.SS3.p2.7.m3.1.2">superscript</csymbol><apply id="A1.SS3.p2.7.m3.1.2.2.cmml" xref="A1.SS3.p2.7.m3.1.2"><csymbol cd="ambiguous" id="A1.SS3.p2.7.m3.1.2.2.1.cmml" xref="A1.SS3.p2.7.m3.1.2">subscript</csymbol><ci id="A1.SS3.p2.7.m3.1.2.2.2.cmml" xref="A1.SS3.p2.7.m3.1.2.2.2">𝒃</ci><ci id="A1.SS3.p2.7.m3.1.2.2.3.cmml" xref="A1.SS3.p2.7.m3.1.2.2.3">𝑡</ci></apply><ci id="A1.SS3.p2.7.m3.1.1.1.1.cmml" xref="A1.SS3.p2.7.m3.1.1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.7.m3.1c">\boldsymbol{b}_{t}^{(j)}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.7.m3.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\boldsymbol{a}^{(j)}" class="ltx_Math" display="inline" id="A1.SS3.p2.8.m4.1"><semantics id="A1.SS3.p2.8.m4.1a"><msup id="A1.SS3.p2.8.m4.1.2" xref="A1.SS3.p2.8.m4.1.2.cmml"><mi id="A1.SS3.p2.8.m4.1.2.2" mathsize="144%" xref="A1.SS3.p2.8.m4.1.2.2.cmml">𝒂</mi><mrow id="A1.SS3.p2.8.m4.1.1.1.3" xref="A1.SS3.p2.8.m4.1.2.cmml"><mo id="A1.SS3.p2.8.m4.1.1.1.3.1" maxsize="144%" minsize="144%" xref="A1.SS3.p2.8.m4.1.2.cmml">(</mo><mi id="A1.SS3.p2.8.m4.1.1.1.1" mathsize="144%" xref="A1.SS3.p2.8.m4.1.1.1.1.cmml">j</mi><mo id="A1.SS3.p2.8.m4.1.1.1.3.2" maxsize="144%" minsize="144%" xref="A1.SS3.p2.8.m4.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.8.m4.1b"><apply id="A1.SS3.p2.8.m4.1.2.cmml" xref="A1.SS3.p2.8.m4.1.2"><csymbol cd="ambiguous" id="A1.SS3.p2.8.m4.1.2.1.cmml" xref="A1.SS3.p2.8.m4.1.2">superscript</csymbol><ci id="A1.SS3.p2.8.m4.1.2.2.cmml" xref="A1.SS3.p2.8.m4.1.2.2">𝒂</ci><ci id="A1.SS3.p2.8.m4.1.1.1.1.cmml" xref="A1.SS3.p2.8.m4.1.1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.8.m4.1c">\boldsymbol{a}^{(j)}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.8.m4.1d">bold_italic_a start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT</annotation></semantics></math> are in <math alttext="\mathbb{R}^{p}" class="ltx_Math" display="inline" id="A1.SS3.p2.9.m5.1"><semantics id="A1.SS3.p2.9.m5.1a"><msup id="A1.SS3.p2.9.m5.1.1" xref="A1.SS3.p2.9.m5.1.1.cmml"><mi id="A1.SS3.p2.9.m5.1.1.2" mathsize="144%" xref="A1.SS3.p2.9.m5.1.1.2.cmml">ℝ</mi><mi id="A1.SS3.p2.9.m5.1.1.3" mathsize="144%" xref="A1.SS3.p2.9.m5.1.1.3.cmml">p</mi></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.9.m5.1b"><apply id="A1.SS3.p2.9.m5.1.1.cmml" xref="A1.SS3.p2.9.m5.1.1"><csymbol cd="ambiguous" id="A1.SS3.p2.9.m5.1.1.1.cmml" xref="A1.SS3.p2.9.m5.1.1">superscript</csymbol><ci id="A1.SS3.p2.9.m5.1.1.2.cmml" xref="A1.SS3.p2.9.m5.1.1.2">ℝ</ci><ci id="A1.SS3.p2.9.m5.1.1.3.cmml" xref="A1.SS3.p2.9.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.9.m5.1c">\mathbb{R}^{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.9.m5.1d">blackboard_R start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math>. The individual terms <math alttext="\boldsymbol{a}^{(j)}" class="ltx_Math" display="inline" id="A1.SS3.p2.10.m6.1"><semantics id="A1.SS3.p2.10.m6.1a"><msup id="A1.SS3.p2.10.m6.1.2" xref="A1.SS3.p2.10.m6.1.2.cmml"><mi id="A1.SS3.p2.10.m6.1.2.2" mathsize="144%" xref="A1.SS3.p2.10.m6.1.2.2.cmml">𝒂</mi><mrow id="A1.SS3.p2.10.m6.1.1.1.3" xref="A1.SS3.p2.10.m6.1.2.cmml"><mo id="A1.SS3.p2.10.m6.1.1.1.3.1" maxsize="144%" minsize="144%" xref="A1.SS3.p2.10.m6.1.2.cmml">(</mo><mi id="A1.SS3.p2.10.m6.1.1.1.1" mathsize="144%" xref="A1.SS3.p2.10.m6.1.1.1.1.cmml">j</mi><mo id="A1.SS3.p2.10.m6.1.1.1.3.2" maxsize="144%" minsize="144%" xref="A1.SS3.p2.10.m6.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.10.m6.1b"><apply id="A1.SS3.p2.10.m6.1.2.cmml" xref="A1.SS3.p2.10.m6.1.2"><csymbol cd="ambiguous" id="A1.SS3.p2.10.m6.1.2.1.cmml" xref="A1.SS3.p2.10.m6.1.2">superscript</csymbol><ci id="A1.SS3.p2.10.m6.1.2.2.cmml" xref="A1.SS3.p2.10.m6.1.2.2">𝒂</ci><ci id="A1.SS3.p2.10.m6.1.1.1.1.cmml" xref="A1.SS3.p2.10.m6.1.1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.10.m6.1c">\boldsymbol{a}^{(j)}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.10.m6.1d">bold_italic_a start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{b}_{t}^{(j)}" class="ltx_Math" display="inline" id="A1.SS3.p2.11.m7.1"><semantics id="A1.SS3.p2.11.m7.1a"><msubsup id="A1.SS3.p2.11.m7.1.2" xref="A1.SS3.p2.11.m7.1.2.cmml"><mi id="A1.SS3.p2.11.m7.1.2.2.2" mathsize="144%" xref="A1.SS3.p2.11.m7.1.2.2.2.cmml">𝒃</mi><mi id="A1.SS3.p2.11.m7.1.2.2.3" mathsize="144%" xref="A1.SS3.p2.11.m7.1.2.2.3.cmml">t</mi><mrow id="A1.SS3.p2.11.m7.1.1.1.3" xref="A1.SS3.p2.11.m7.1.2.cmml"><mo id="A1.SS3.p2.11.m7.1.1.1.3.1" maxsize="144%" minsize="144%" xref="A1.SS3.p2.11.m7.1.2.cmml">(</mo><mi id="A1.SS3.p2.11.m7.1.1.1.1" mathsize="144%" xref="A1.SS3.p2.11.m7.1.1.1.1.cmml">j</mi><mo id="A1.SS3.p2.11.m7.1.1.1.3.2" maxsize="144%" minsize="144%" xref="A1.SS3.p2.11.m7.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.11.m7.1b"><apply id="A1.SS3.p2.11.m7.1.2.cmml" xref="A1.SS3.p2.11.m7.1.2"><csymbol cd="ambiguous" id="A1.SS3.p2.11.m7.1.2.1.cmml" xref="A1.SS3.p2.11.m7.1.2">superscript</csymbol><apply id="A1.SS3.p2.11.m7.1.2.2.cmml" xref="A1.SS3.p2.11.m7.1.2"><csymbol cd="ambiguous" id="A1.SS3.p2.11.m7.1.2.2.1.cmml" xref="A1.SS3.p2.11.m7.1.2">subscript</csymbol><ci id="A1.SS3.p2.11.m7.1.2.2.2.cmml" xref="A1.SS3.p2.11.m7.1.2.2.2">𝒃</ci><ci id="A1.SS3.p2.11.m7.1.2.2.3.cmml" xref="A1.SS3.p2.11.m7.1.2.2.3">𝑡</ci></apply><ci id="A1.SS3.p2.11.m7.1.1.1.1.cmml" xref="A1.SS3.p2.11.m7.1.1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.11.m7.1c">\boldsymbol{b}_{t}^{(j)}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.11.m7.1d">bold_italic_b start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT</annotation></semantics></math> are given by: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A4.EGx5"> <tbody id="A1.E32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{b}^{(j)}_{t}" class="ltx_Math" display="inline" id="A1.E32.m1.1"><semantics id="A1.E32.m1.1a"><msubsup id="A1.E32.m1.1.2" xref="A1.E32.m1.1.2.cmml"><mi id="A1.E32.m1.1.2.2.2" mathsize="144%" xref="A1.E32.m1.1.2.2.2.cmml">𝒃</mi><mi id="A1.E32.m1.1.2.3" mathsize="144%" xref="A1.E32.m1.1.2.3.cmml">t</mi><mrow id="A1.E32.m1.1.1.1.3" xref="A1.E32.m1.1.2.cmml"><mo id="A1.E32.m1.1.1.1.3.1" maxsize="144%" minsize="144%" xref="A1.E32.m1.1.2.cmml">(</mo><mi id="A1.E32.m1.1.1.1.1" mathsize="144%" xref="A1.E32.m1.1.1.1.1.cmml">j</mi><mo id="A1.E32.m1.1.1.1.3.2" maxsize="144%" minsize="144%" xref="A1.E32.m1.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="A1.E32.m1.1b"><apply id="A1.E32.m1.1.2.cmml" xref="A1.E32.m1.1.2"><csymbol cd="ambiguous" id="A1.E32.m1.1.2.1.cmml" xref="A1.E32.m1.1.2">subscript</csymbol><apply id="A1.E32.m1.1.2.2.cmml" xref="A1.E32.m1.1.2"><csymbol cd="ambiguous" id="A1.E32.m1.1.2.2.1.cmml" xref="A1.E32.m1.1.2">superscript</csymbol><ci id="A1.E32.m1.1.2.2.2.cmml" xref="A1.E32.m1.1.2.2.2">𝒃</ci><ci id="A1.E32.m1.1.1.1.1.cmml" xref="A1.E32.m1.1.1.1.1">𝑗</ci></apply><ci id="A1.E32.m1.1.2.3.cmml" xref="A1.E32.m1.1.2.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E32.m1.1c">\displaystyle\boldsymbol{b}^{(j)}_{t}</annotation><annotation encoding="application/x-llamapun" id="A1.E32.m1.1d">bold_italic_b start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathbf{P}^{\top}\widehat{\mathbf{C}}_{j}^{-1}\,\left(% \boldsymbol{y}_{t}^{(j)}-\widehat{\boldsymbol{m}}_{j}\right)\,," class="ltx_Math" display="inline" id="A1.E32.m2.2"><semantics id="A1.E32.m2.2a"><mrow id="A1.E32.m2.2.2.1" xref="A1.E32.m2.2.2.1.1.cmml"><mrow id="A1.E32.m2.2.2.1.1" xref="A1.E32.m2.2.2.1.1.cmml"><mi id="A1.E32.m2.2.2.1.1.3" xref="A1.E32.m2.2.2.1.1.3.cmml"></mi><mo id="A1.E32.m2.2.2.1.1.2" mathsize="144%" xref="A1.E32.m2.2.2.1.1.2.cmml">=</mo><mrow id="A1.E32.m2.2.2.1.1.1" xref="A1.E32.m2.2.2.1.1.1.cmml"><msup id="A1.E32.m2.2.2.1.1.1.3" xref="A1.E32.m2.2.2.1.1.1.3.cmml"><mi id="A1.E32.m2.2.2.1.1.1.3.2" mathsize="144%" xref="A1.E32.m2.2.2.1.1.1.3.2.cmml">𝐏</mi><mo id="A1.E32.m2.2.2.1.1.1.3.3" mathsize="144%" xref="A1.E32.m2.2.2.1.1.1.3.3.cmml">⊤</mo></msup><mo id="A1.E32.m2.2.2.1.1.1.2" xref="A1.E32.m2.2.2.1.1.1.2.cmml">⁢</mo><msubsup 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xref="A1.E32.m2.2.2.1.1.1.1.1.1.cmml"><msubsup id="A1.E32.m2.2.2.1.1.1.1.1.1.2" xref="A1.E32.m2.2.2.1.1.1.1.1.1.2.cmml"><mi id="A1.E32.m2.2.2.1.1.1.1.1.1.2.2.2" mathsize="144%" xref="A1.E32.m2.2.2.1.1.1.1.1.1.2.2.2.cmml">𝒚</mi><mi id="A1.E32.m2.2.2.1.1.1.1.1.1.2.2.3" mathsize="144%" xref="A1.E32.m2.2.2.1.1.1.1.1.1.2.2.3.cmml">t</mi><mrow id="A1.E32.m2.1.1.1.3" xref="A1.E32.m2.2.2.1.1.1.1.1.1.2.cmml"><mo id="A1.E32.m2.1.1.1.3.1" maxsize="144%" minsize="144%" xref="A1.E32.m2.2.2.1.1.1.1.1.1.2.cmml">(</mo><mi id="A1.E32.m2.1.1.1.1" mathsize="144%" xref="A1.E32.m2.1.1.1.1.cmml">j</mi><mo id="A1.E32.m2.1.1.1.3.2" maxsize="144%" minsize="144%" xref="A1.E32.m2.2.2.1.1.1.1.1.1.2.cmml">)</mo></mrow></msubsup><mo id="A1.E32.m2.2.2.1.1.1.1.1.1.1" mathsize="144%" xref="A1.E32.m2.2.2.1.1.1.1.1.1.1.cmml">−</mo><msub id="A1.E32.m2.2.2.1.1.1.1.1.1.3" xref="A1.E32.m2.2.2.1.1.1.1.1.1.3.cmml"><mover accent="true" id="A1.E32.m2.2.2.1.1.1.1.1.1.3.2" xref="A1.E32.m2.2.2.1.1.1.1.1.1.3.2.cmml"><mi 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encoding="application/x-llamapun" id="A1.E32.m2.2d">= bold_P start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT - over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(32)</td> </tr></tbody> <tbody id="A1.E33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{a}^{(j)}" class="ltx_Math" display="inline" id="A1.E33.m1.1"><semantics id="A1.E33.m1.1a"><msup id="A1.E33.m1.1.2" xref="A1.E33.m1.1.2.cmml"><mi 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xref="A1.E33.m2.1.1.1.1.1.1.1.1.4">𝐏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E33.m2.1c">\displaystyle=\text{diag}\left(\mathbf{P}^{\top}\widehat{\mathbf{C}}_{j}^{-1}% \mathbf{P}\right)\,,</annotation><annotation encoding="application/x-llamapun" id="A1.E33.m2.1d">= diag ( bold_P start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT bold_P ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(33)</td> </tr></tbody> </table> where <math alttext="\mathbf{P}" class="ltx_Math" display="inline" id="A1.SS3.p2.12.m1.1"><semantics id="A1.SS3.p2.12.m1.1a"><mi id="A1.SS3.p2.12.m1.1.1" mathsize="144%" xref="A1.SS3.p2.12.m1.1.1.cmml">𝐏</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.12.m1.1b"><ci id="A1.SS3.p2.12.m1.1.1.cmml" xref="A1.SS3.p2.12.m1.1.1">𝐏</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.12.m1.1c">\mathbf{P}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.12.m1.1d">bold_P</annotation></semantics></math> in <math alttext="\mathbb{R}^{p\times p}" class="ltx_Math" display="inline" id="A1.SS3.p2.13.m2.1"><semantics id="A1.SS3.p2.13.m2.1a"><msup id="A1.SS3.p2.13.m2.1.1" xref="A1.SS3.p2.13.m2.1.1.cmml"><mi id="A1.SS3.p2.13.m2.1.1.2" mathsize="144%" xref="A1.SS3.p2.13.m2.1.1.2.cmml">ℝ</mi><mrow id="A1.SS3.p2.13.m2.1.1.3" xref="A1.SS3.p2.13.m2.1.1.3.cmml"><mi id="A1.SS3.p2.13.m2.1.1.3.2" mathsize="144%" xref="A1.SS3.p2.13.m2.1.1.3.2.cmml">p</mi><mo id="A1.SS3.p2.13.m2.1.1.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS3.p2.13.m2.1.1.3.1.cmml">×</mo><mi id="A1.SS3.p2.13.m2.1.1.3.3" mathsize="144%" xref="A1.SS3.p2.13.m2.1.1.3.3.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.13.m2.1b"><apply id="A1.SS3.p2.13.m2.1.1.cmml" xref="A1.SS3.p2.13.m2.1.1"><csymbol cd="ambiguous" id="A1.SS3.p2.13.m2.1.1.1.cmml" xref="A1.SS3.p2.13.m2.1.1">superscript</csymbol><ci id="A1.SS3.p2.13.m2.1.1.2.cmml" xref="A1.SS3.p2.13.m2.1.1.2">ℝ</ci><apply id="A1.SS3.p2.13.m2.1.1.3.cmml" xref="A1.SS3.p2.13.m2.1.1.3"><times id="A1.SS3.p2.13.m2.1.1.3.1.cmml" xref="A1.SS3.p2.13.m2.1.1.3.1"></times><ci id="A1.SS3.p2.13.m2.1.1.3.2.cmml" xref="A1.SS3.p2.13.m2.1.1.3.2">𝑝</ci><ci id="A1.SS3.p2.13.m2.1.1.3.3.cmml" xref="A1.SS3.p2.13.m2.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.13.m2.1c">\mathbb{R}^{p\times p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.13.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_p × italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> is a circulant matrix, with each column representing the flattened PSF centered on a different pixel. Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E32" style="font-size:144%;" title="Equation 32 ‣ A.3 Fast computation of 𝒃_𝑡 and 𝒂 ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">32</a>) can be implemented efficiently with a convolution operation. Implementing Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E33" style="font-size:144%;" title="Equation 33 ‣ A.3 Fast computation of 𝒃_𝑡 and 𝒂 ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">33</a>) directly is however computationally challenging as it requires storing the matrix <math alttext="\mathbf{P}" class="ltx_Math" display="inline" id="A1.SS3.p2.14.m3.1"><semantics id="A1.SS3.p2.14.m3.1a"><mi id="A1.SS3.p2.14.m3.1.1" mathsize="144%" xref="A1.SS3.p2.14.m3.1.1.cmml">𝐏</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.14.m3.1b"><ci id="A1.SS3.p2.14.m3.1.1.cmml" xref="A1.SS3.p2.14.m3.1.1">𝐏</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.14.m3.1c">\mathbf{P}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.14.m3.1d">bold_P</annotation></semantics></math>, which is highly inefficient, especially for large patches. To circumvent this issue, we propose to decompose <math alttext="\boldsymbol{a}^{(j)}" class="ltx_Math" display="inline" id="A1.SS3.p2.15.m4.1"><semantics id="A1.SS3.p2.15.m4.1a"><msup id="A1.SS3.p2.15.m4.1.2" xref="A1.SS3.p2.15.m4.1.2.cmml"><mi id="A1.SS3.p2.15.m4.1.2.2" mathsize="144%" xref="A1.SS3.p2.15.m4.1.2.2.cmml">𝒂</mi><mrow id="A1.SS3.p2.15.m4.1.1.1.3" xref="A1.SS3.p2.15.m4.1.2.cmml"><mo id="A1.SS3.p2.15.m4.1.1.1.3.1" maxsize="144%" minsize="144%" xref="A1.SS3.p2.15.m4.1.2.cmml">(</mo><mi id="A1.SS3.p2.15.m4.1.1.1.1" mathsize="144%" xref="A1.SS3.p2.15.m4.1.1.1.1.cmml">j</mi><mo id="A1.SS3.p2.15.m4.1.1.1.3.2" maxsize="144%" minsize="144%" xref="A1.SS3.p2.15.m4.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.15.m4.1b"><apply id="A1.SS3.p2.15.m4.1.2.cmml" xref="A1.SS3.p2.15.m4.1.2"><csymbol cd="ambiguous" id="A1.SS3.p2.15.m4.1.2.1.cmml" xref="A1.SS3.p2.15.m4.1.2">superscript</csymbol><ci id="A1.SS3.p2.15.m4.1.2.2.cmml" xref="A1.SS3.p2.15.m4.1.2.2">𝒂</ci><ci id="A1.SS3.p2.15.m4.1.1.1.1.cmml" xref="A1.SS3.p2.15.m4.1.1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.15.m4.1c">\boldsymbol{a}^{(j)}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.15.m4.1d">bold_italic_a start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT</annotation></semantics></math> as: <table class="ltx_equation ltx_eqn_table" id="A1.E34"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\boldsymbol{a}^{(j)}=\mathbf{N}(\widehat{\mathbf{L}}_{j}\mathbf{P})\,," class="ltx_Math" display="block" id="A1.E34.m1.2"><semantics id="A1.E34.m1.2a"><mrow id="A1.E34.m1.2.2.1" xref="A1.E34.m1.2.2.1.1.cmml"><mrow id="A1.E34.m1.2.2.1.1" xref="A1.E34.m1.2.2.1.1.cmml"><msup id="A1.E34.m1.2.2.1.1.3" xref="A1.E34.m1.2.2.1.1.3.cmml"><mi id="A1.E34.m1.2.2.1.1.3.2" 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xref="A1.E34.m1.2.2.1.1.1.1.1.1.2">subscript</csymbol><apply id="A1.E34.m1.2.2.1.1.1.1.1.1.2.2.cmml" xref="A1.E34.m1.2.2.1.1.1.1.1.1.2.2"><ci id="A1.E34.m1.2.2.1.1.1.1.1.1.2.2.1.cmml" xref="A1.E34.m1.2.2.1.1.1.1.1.1.2.2.1">^</ci><ci id="A1.E34.m1.2.2.1.1.1.1.1.1.2.2.2.cmml" xref="A1.E34.m1.2.2.1.1.1.1.1.1.2.2.2">𝐋</ci></apply><ci id="A1.E34.m1.2.2.1.1.1.1.1.1.2.3.cmml" xref="A1.E34.m1.2.2.1.1.1.1.1.1.2.3">𝑗</ci></apply><ci id="A1.E34.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A1.E34.m1.2.2.1.1.1.1.1.1.3">𝐏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E34.m1.2c">\boldsymbol{a}^{(j)}=\mathbf{N}(\widehat{\mathbf{L}}_{j}\mathbf{P})\,,</annotation><annotation encoding="application/x-llamapun" id="A1.E34.m1.2d">bold_italic_a start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT = bold_N ( over^ start_ARG bold_L end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_P ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(34)</td> </tr></tbody> </table> where <math alttext="\mathbf{N}:\mathbb{R}^{p\times p}\to\mathbb{R}^{p}" class="ltx_Math" display="inline" id="A1.SS3.p2.16.m1.1"><semantics id="A1.SS3.p2.16.m1.1a"><mrow id="A1.SS3.p2.16.m1.1.1" xref="A1.SS3.p2.16.m1.1.1.cmml"><mi id="A1.SS3.p2.16.m1.1.1.2" mathsize="144%" xref="A1.SS3.p2.16.m1.1.1.2.cmml">𝐍</mi><mo id="A1.SS3.p2.16.m1.1.1.1" lspace="0.278em" mathsize="144%" rspace="0.278em" xref="A1.SS3.p2.16.m1.1.1.1.cmml">:</mo><mrow id="A1.SS3.p2.16.m1.1.1.3" xref="A1.SS3.p2.16.m1.1.1.3.cmml"><msup id="A1.SS3.p2.16.m1.1.1.3.2" xref="A1.SS3.p2.16.m1.1.1.3.2.cmml"><mi id="A1.SS3.p2.16.m1.1.1.3.2.2" mathsize="144%" xref="A1.SS3.p2.16.m1.1.1.3.2.2.cmml">ℝ</mi><mrow id="A1.SS3.p2.16.m1.1.1.3.2.3" xref="A1.SS3.p2.16.m1.1.1.3.2.3.cmml"><mi id="A1.SS3.p2.16.m1.1.1.3.2.3.2" mathsize="144%" xref="A1.SS3.p2.16.m1.1.1.3.2.3.2.cmml">p</mi><mo id="A1.SS3.p2.16.m1.1.1.3.2.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS3.p2.16.m1.1.1.3.2.3.1.cmml">×</mo><mi id="A1.SS3.p2.16.m1.1.1.3.2.3.3" mathsize="144%" xref="A1.SS3.p2.16.m1.1.1.3.2.3.3.cmml">p</mi></mrow></msup><mo id="A1.SS3.p2.16.m1.1.1.3.1" mathsize="144%" stretchy="false" xref="A1.SS3.p2.16.m1.1.1.3.1.cmml">→</mo><msup id="A1.SS3.p2.16.m1.1.1.3.3" xref="A1.SS3.p2.16.m1.1.1.3.3.cmml"><mi id="A1.SS3.p2.16.m1.1.1.3.3.2" mathsize="144%" xref="A1.SS3.p2.16.m1.1.1.3.3.2.cmml">ℝ</mi><mi id="A1.SS3.p2.16.m1.1.1.3.3.3" mathsize="144%" xref="A1.SS3.p2.16.m1.1.1.3.3.3.cmml">p</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.16.m1.1b"><apply id="A1.SS3.p2.16.m1.1.1.cmml" xref="A1.SS3.p2.16.m1.1.1"><ci id="A1.SS3.p2.16.m1.1.1.1.cmml" xref="A1.SS3.p2.16.m1.1.1.1">:</ci><ci id="A1.SS3.p2.16.m1.1.1.2.cmml" xref="A1.SS3.p2.16.m1.1.1.2">𝐍</ci><apply id="A1.SS3.p2.16.m1.1.1.3.cmml" xref="A1.SS3.p2.16.m1.1.1.3"><ci id="A1.SS3.p2.16.m1.1.1.3.1.cmml" xref="A1.SS3.p2.16.m1.1.1.3.1">→</ci><apply id="A1.SS3.p2.16.m1.1.1.3.2.cmml" xref="A1.SS3.p2.16.m1.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS3.p2.16.m1.1.1.3.2.1.cmml" xref="A1.SS3.p2.16.m1.1.1.3.2">superscript</csymbol><ci id="A1.SS3.p2.16.m1.1.1.3.2.2.cmml" xref="A1.SS3.p2.16.m1.1.1.3.2.2">ℝ</ci><apply id="A1.SS3.p2.16.m1.1.1.3.2.3.cmml" xref="A1.SS3.p2.16.m1.1.1.3.2.3"><times id="A1.SS3.p2.16.m1.1.1.3.2.3.1.cmml" xref="A1.SS3.p2.16.m1.1.1.3.2.3.1"></times><ci id="A1.SS3.p2.16.m1.1.1.3.2.3.2.cmml" xref="A1.SS3.p2.16.m1.1.1.3.2.3.2">𝑝</ci><ci id="A1.SS3.p2.16.m1.1.1.3.2.3.3.cmml" xref="A1.SS3.p2.16.m1.1.1.3.2.3.3">𝑝</ci></apply></apply><apply id="A1.SS3.p2.16.m1.1.1.3.3.cmml" xref="A1.SS3.p2.16.m1.1.1.3.3"><csymbol cd="ambiguous" id="A1.SS3.p2.16.m1.1.1.3.3.1.cmml" xref="A1.SS3.p2.16.m1.1.1.3.3">superscript</csymbol><ci id="A1.SS3.p2.16.m1.1.1.3.3.2.cmml" xref="A1.SS3.p2.16.m1.1.1.3.3.2">ℝ</ci><ci id="A1.SS3.p2.16.m1.1.1.3.3.3.cmml" xref="A1.SS3.p2.16.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.16.m1.1c">\mathbf{N}:\mathbb{R}^{p\times p}\to\mathbb{R}^{p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.16.m1.1d">bold_N : blackboard_R start_POSTSUPERSCRIPT italic_p × italic_p end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> is the operator that computes the column-wise squared <math alttext="\ell_{2}" class="ltx_Math" display="inline" id="A1.SS3.p2.17.m2.1"><semantics id="A1.SS3.p2.17.m2.1a"><msub id="A1.SS3.p2.17.m2.1.1" xref="A1.SS3.p2.17.m2.1.1.cmml"><mi id="A1.SS3.p2.17.m2.1.1.2" mathsize="144%" mathvariant="normal" xref="A1.SS3.p2.17.m2.1.1.2.cmml">ℓ</mi><mn id="A1.SS3.p2.17.m2.1.1.3" mathsize="144%" xref="A1.SS3.p2.17.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.17.m2.1b"><apply id="A1.SS3.p2.17.m2.1.1.cmml" xref="A1.SS3.p2.17.m2.1.1"><csymbol cd="ambiguous" id="A1.SS3.p2.17.m2.1.1.1.cmml" xref="A1.SS3.p2.17.m2.1.1">subscript</csymbol><ci id="A1.SS3.p2.17.m2.1.1.2.cmml" xref="A1.SS3.p2.17.m2.1.1.2">ℓ</ci><cn id="A1.SS3.p2.17.m2.1.1.3.cmml" type="integer" xref="A1.SS3.p2.17.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.17.m2.1c">\ell_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.17.m2.1d">roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>–norm, and the matrix <math alttext="\widehat{\mathbf{L}}_{j}" class="ltx_Math" display="inline" id="A1.SS3.p2.18.m3.1"><semantics id="A1.SS3.p2.18.m3.1a"><msub id="A1.SS3.p2.18.m3.1.1" xref="A1.SS3.p2.18.m3.1.1.cmml"><mover accent="true" id="A1.SS3.p2.18.m3.1.1.2" xref="A1.SS3.p2.18.m3.1.1.2.cmml"><mi id="A1.SS3.p2.18.m3.1.1.2.2" mathsize="144%" xref="A1.SS3.p2.18.m3.1.1.2.2.cmml">𝐋</mi><mo id="A1.SS3.p2.18.m3.1.1.2.1" mathsize="144%" xref="A1.SS3.p2.18.m3.1.1.2.1.cmml">^</mo></mover><mi id="A1.SS3.p2.18.m3.1.1.3" mathsize="144%" xref="A1.SS3.p2.18.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.18.m3.1b"><apply id="A1.SS3.p2.18.m3.1.1.cmml" xref="A1.SS3.p2.18.m3.1.1"><csymbol cd="ambiguous" id="A1.SS3.p2.18.m3.1.1.1.cmml" xref="A1.SS3.p2.18.m3.1.1">subscript</csymbol><apply id="A1.SS3.p2.18.m3.1.1.2.cmml" xref="A1.SS3.p2.18.m3.1.1.2"><ci id="A1.SS3.p2.18.m3.1.1.2.1.cmml" xref="A1.SS3.p2.18.m3.1.1.2.1">^</ci><ci id="A1.SS3.p2.18.m3.1.1.2.2.cmml" xref="A1.SS3.p2.18.m3.1.1.2.2">𝐋</ci></apply><ci id="A1.SS3.p2.18.m3.1.1.3.cmml" xref="A1.SS3.p2.18.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.18.m3.1c">\widehat{\mathbf{L}}_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.18.m3.1d">over^ start_ARG bold_L end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{p\times p}" class="ltx_Math" display="inline" id="A1.SS3.p2.19.m4.1"><semantics id="A1.SS3.p2.19.m4.1a"><msup id="A1.SS3.p2.19.m4.1.1" xref="A1.SS3.p2.19.m4.1.1.cmml"><mi id="A1.SS3.p2.19.m4.1.1.2" mathsize="144%" xref="A1.SS3.p2.19.m4.1.1.2.cmml">ℝ</mi><mrow id="A1.SS3.p2.19.m4.1.1.3" xref="A1.SS3.p2.19.m4.1.1.3.cmml"><mi id="A1.SS3.p2.19.m4.1.1.3.2" mathsize="144%" xref="A1.SS3.p2.19.m4.1.1.3.2.cmml">p</mi><mo id="A1.SS3.p2.19.m4.1.1.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS3.p2.19.m4.1.1.3.1.cmml">×</mo><mi id="A1.SS3.p2.19.m4.1.1.3.3" mathsize="144%" xref="A1.SS3.p2.19.m4.1.1.3.3.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.19.m4.1b"><apply id="A1.SS3.p2.19.m4.1.1.cmml" xref="A1.SS3.p2.19.m4.1.1"><csymbol cd="ambiguous" id="A1.SS3.p2.19.m4.1.1.1.cmml" xref="A1.SS3.p2.19.m4.1.1">superscript</csymbol><ci id="A1.SS3.p2.19.m4.1.1.2.cmml" xref="A1.SS3.p2.19.m4.1.1.2">ℝ</ci><apply id="A1.SS3.p2.19.m4.1.1.3.cmml" xref="A1.SS3.p2.19.m4.1.1.3"><times id="A1.SS3.p2.19.m4.1.1.3.1.cmml" xref="A1.SS3.p2.19.m4.1.1.3.1"></times><ci id="A1.SS3.p2.19.m4.1.1.3.2.cmml" xref="A1.SS3.p2.19.m4.1.1.3.2">𝑝</ci><ci id="A1.SS3.p2.19.m4.1.1.3.3.cmml" xref="A1.SS3.p2.19.m4.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.19.m4.1c">\mathbb{R}^{p\times p}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.19.m4.1d">blackboard_R start_POSTSUPERSCRIPT italic_p × italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> is the lower triangular matrix obtained via Cholesky decomposition of the precision matrix (i.e., <math alttext="\widehat{\mathbf{C}}_{j}^{-1}=\widehat{\mathbf{L}}_{j}\widehat{\mathbf{L}}_{j}% ^{\top}" class="ltx_Math" display="inline" id="A1.SS3.p2.20.m5.1"><semantics id="A1.SS3.p2.20.m5.1a"><mrow id="A1.SS3.p2.20.m5.1.1" xref="A1.SS3.p2.20.m5.1.1.cmml"><msubsup id="A1.SS3.p2.20.m5.1.1.2" xref="A1.SS3.p2.20.m5.1.1.2.cmml"><mover accent="true" id="A1.SS3.p2.20.m5.1.1.2.2.2" xref="A1.SS3.p2.20.m5.1.1.2.2.2.cmml"><mi id="A1.SS3.p2.20.m5.1.1.2.2.2.2" mathsize="144%" xref="A1.SS3.p2.20.m5.1.1.2.2.2.2.cmml">𝐂</mi><mo id="A1.SS3.p2.20.m5.1.1.2.2.2.1" mathsize="144%" xref="A1.SS3.p2.20.m5.1.1.2.2.2.1.cmml">^</mo></mover><mi id="A1.SS3.p2.20.m5.1.1.2.2.3" mathsize="144%" xref="A1.SS3.p2.20.m5.1.1.2.2.3.cmml">j</mi><mrow id="A1.SS3.p2.20.m5.1.1.2.3" xref="A1.SS3.p2.20.m5.1.1.2.3.cmml"><mo id="A1.SS3.p2.20.m5.1.1.2.3a" mathsize="144%" xref="A1.SS3.p2.20.m5.1.1.2.3.cmml">−</mo><mn id="A1.SS3.p2.20.m5.1.1.2.3.2" mathsize="144%" xref="A1.SS3.p2.20.m5.1.1.2.3.2.cmml">1</mn></mrow></msubsup><mo id="A1.SS3.p2.20.m5.1.1.1" mathsize="144%" xref="A1.SS3.p2.20.m5.1.1.1.cmml">=</mo><mrow id="A1.SS3.p2.20.m5.1.1.3" xref="A1.SS3.p2.20.m5.1.1.3.cmml"><msub id="A1.SS3.p2.20.m5.1.1.3.2" xref="A1.SS3.p2.20.m5.1.1.3.2.cmml"><mover accent="true" id="A1.SS3.p2.20.m5.1.1.3.2.2" 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xref="A1.SS3.p2.20.m5.1.1.3.3.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.20.m5.1b"><apply id="A1.SS3.p2.20.m5.1.1.cmml" xref="A1.SS3.p2.20.m5.1.1"><eq id="A1.SS3.p2.20.m5.1.1.1.cmml" xref="A1.SS3.p2.20.m5.1.1.1"></eq><apply id="A1.SS3.p2.20.m5.1.1.2.cmml" xref="A1.SS3.p2.20.m5.1.1.2"><csymbol cd="ambiguous" id="A1.SS3.p2.20.m5.1.1.2.1.cmml" xref="A1.SS3.p2.20.m5.1.1.2">superscript</csymbol><apply id="A1.SS3.p2.20.m5.1.1.2.2.cmml" xref="A1.SS3.p2.20.m5.1.1.2"><csymbol cd="ambiguous" id="A1.SS3.p2.20.m5.1.1.2.2.1.cmml" xref="A1.SS3.p2.20.m5.1.1.2">subscript</csymbol><apply id="A1.SS3.p2.20.m5.1.1.2.2.2.cmml" xref="A1.SS3.p2.20.m5.1.1.2.2.2"><ci id="A1.SS3.p2.20.m5.1.1.2.2.2.1.cmml" xref="A1.SS3.p2.20.m5.1.1.2.2.2.1">^</ci><ci id="A1.SS3.p2.20.m5.1.1.2.2.2.2.cmml" xref="A1.SS3.p2.20.m5.1.1.2.2.2.2">𝐂</ci></apply><ci id="A1.SS3.p2.20.m5.1.1.2.2.3.cmml" xref="A1.SS3.p2.20.m5.1.1.2.2.3">𝑗</ci></apply><apply id="A1.SS3.p2.20.m5.1.1.2.3.cmml" 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id="A1.SS3.p2.20.m5.1d">over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = over^ start_ARG bold_L end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT over^ start_ARG bold_L end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>). The formulation of Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E34" style="font-size:144%;" title="Equation 34 ‣ A.3 Fast computation of 𝒃_𝑡 and 𝒂 ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">34</a>) allows implementing the matrix multiplication with <math alttext="\mathbf{P}" class="ltx_Math" display="inline" id="A1.SS3.p2.21.m6.1"><semantics id="A1.SS3.p2.21.m6.1a"><mi id="A1.SS3.p2.21.m6.1.1" mathsize="144%" xref="A1.SS3.p2.21.m6.1.1.cmml">𝐏</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.p2.21.m6.1b"><ci id="A1.SS3.p2.21.m6.1.1.cmml" xref="A1.SS3.p2.21.m6.1.1">𝐏</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p2.21.m6.1c">\mathbf{P}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p2.21.m6.1d">bold_P</annotation></semantics></math> efficiently as a convolution operation. </div> </section> <section class="ltx_subsection" id="A1.SS4"> <h3 class="ltx_title ltx_title_subsection" style="font-size:144%;"> A.4 Iterative optimization scheme for astrometry</h3> <div class="ltx_para ltx_noindent" id="A1.SS4.p1"> We detail in this section the characterization procedure mentioned in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS1" style="font-size:144%;" title="4.1 Convolutional statistical model ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">4.1</a> for precisely estimating the flux <math alttext="\alpha" class="ltx_Math" display="inline" id="A1.SS4.p1.1.m1.1"><semantics id="A1.SS4.p1.1.m1.1a"><mi id="A1.SS4.p1.1.m1.1.1" mathsize="144%" xref="A1.SS4.p1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A1.SS4.p1.1.m1.1b"><ci id="A1.SS4.p1.1.m1.1.1.cmml" xref="A1.SS4.p1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS4.p1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.1.m1.1d">italic_α</annotation></semantics></math> and the two-dimensional sub-pixel position <math alttext="\boldsymbol{x}_{0}" class="ltx_Math" display="inline" id="A1.SS4.p1.2.m2.1"><semantics id="A1.SS4.p1.2.m2.1a"><msub id="A1.SS4.p1.2.m2.1.1" xref="A1.SS4.p1.2.m2.1.1.cmml"><mi id="A1.SS4.p1.2.m2.1.1.2" mathsize="144%" xref="A1.SS4.p1.2.m2.1.1.2.cmml">𝒙</mi><mn id="A1.SS4.p1.2.m2.1.1.3" mathsize="144%" xref="A1.SS4.p1.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS4.p1.2.m2.1b"><apply id="A1.SS4.p1.2.m2.1.1.cmml" xref="A1.SS4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS4.p1.2.m2.1.1.1.cmml" xref="A1.SS4.p1.2.m2.1.1">subscript</csymbol><ci id="A1.SS4.p1.2.m2.1.1.2.cmml" xref="A1.SS4.p1.2.m2.1.1.2">𝒙</ci><cn id="A1.SS4.p1.2.m2.1.1.3.cmml" type="integer" xref="A1.SS4.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS4.p1.2.m2.1c">\boldsymbol{x}_{0}</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.2.m2.1d">bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> of an exoplanet. This procedure is described by <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#alg1" style="font-size:144%;" title="In A.4 Iterative optimization scheme for astrometry ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Algorithm 1</a> in pseudo-code form, where <math alttext="\boldsymbol{z}=[\alpha,\boldsymbol{x}_{0}]" class="ltx_Math" display="inline" id="A1.SS4.p1.3.m3.2"><semantics id="A1.SS4.p1.3.m3.2a"><mrow id="A1.SS4.p1.3.m3.2.2" xref="A1.SS4.p1.3.m3.2.2.cmml"><mi id="A1.SS4.p1.3.m3.2.2.3" mathsize="144%" xref="A1.SS4.p1.3.m3.2.2.3.cmml">𝒛</mi><mo id="A1.SS4.p1.3.m3.2.2.2" mathsize="144%" xref="A1.SS4.p1.3.m3.2.2.2.cmml">=</mo><mrow id="A1.SS4.p1.3.m3.2.2.1.1" xref="A1.SS4.p1.3.m3.2.2.1.2.cmml"><mo id="A1.SS4.p1.3.m3.2.2.1.1.2" maxsize="144%" minsize="144%" xref="A1.SS4.p1.3.m3.2.2.1.2.cmml">[</mo><mi id="A1.SS4.p1.3.m3.1.1" mathsize="144%" xref="A1.SS4.p1.3.m3.1.1.cmml">α</mi><mo id="A1.SS4.p1.3.m3.2.2.1.1.3" mathsize="144%" xref="A1.SS4.p1.3.m3.2.2.1.2.cmml">,</mo><msub id="A1.SS4.p1.3.m3.2.2.1.1.1" xref="A1.SS4.p1.3.m3.2.2.1.1.1.cmml"><mi id="A1.SS4.p1.3.m3.2.2.1.1.1.2" mathsize="144%" xref="A1.SS4.p1.3.m3.2.2.1.1.1.2.cmml">𝒙</mi><mn id="A1.SS4.p1.3.m3.2.2.1.1.1.3" mathsize="144%" xref="A1.SS4.p1.3.m3.2.2.1.1.1.3.cmml">0</mn></msub><mo id="A1.SS4.p1.3.m3.2.2.1.1.4" maxsize="144%" minsize="144%" xref="A1.SS4.p1.3.m3.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS4.p1.3.m3.2b"><apply id="A1.SS4.p1.3.m3.2.2.cmml" xref="A1.SS4.p1.3.m3.2.2"><eq id="A1.SS4.p1.3.m3.2.2.2.cmml" xref="A1.SS4.p1.3.m3.2.2.2"></eq><ci id="A1.SS4.p1.3.m3.2.2.3.cmml" xref="A1.SS4.p1.3.m3.2.2.3">𝒛</ci><interval closure="closed" id="A1.SS4.p1.3.m3.2.2.1.2.cmml" xref="A1.SS4.p1.3.m3.2.2.1.1"><ci id="A1.SS4.p1.3.m3.1.1.cmml" xref="A1.SS4.p1.3.m3.1.1">𝛼</ci><apply id="A1.SS4.p1.3.m3.2.2.1.1.1.cmml" xref="A1.SS4.p1.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.SS4.p1.3.m3.2.2.1.1.1.1.cmml" xref="A1.SS4.p1.3.m3.2.2.1.1.1">subscript</csymbol><ci id="A1.SS4.p1.3.m3.2.2.1.1.1.2.cmml" xref="A1.SS4.p1.3.m3.2.2.1.1.1.2">𝒙</ci><cn id="A1.SS4.p1.3.m3.2.2.1.1.1.3.cmml" type="integer" xref="A1.SS4.p1.3.m3.2.2.1.1.1.3">0</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS4.p1.3.m3.2c">\boldsymbol{z}=[\alpha,\boldsymbol{x}_{0}]</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.3.m3.2d">bold_italic_z = [ italic_α , bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math> in <math alttext="\mathbb{R}^{3}" class="ltx_Math" display="inline" id="A1.SS4.p1.4.m4.1"><semantics id="A1.SS4.p1.4.m4.1a"><msup id="A1.SS4.p1.4.m4.1.1" xref="A1.SS4.p1.4.m4.1.1.cmml"><mi id="A1.SS4.p1.4.m4.1.1.2" mathsize="144%" xref="A1.SS4.p1.4.m4.1.1.2.cmml">ℝ</mi><mn id="A1.SS4.p1.4.m4.1.1.3" mathsize="144%" xref="A1.SS4.p1.4.m4.1.1.3.cmml">3</mn></msup><annotation-xml encoding="MathML-Content" id="A1.SS4.p1.4.m4.1b"><apply id="A1.SS4.p1.4.m4.1.1.cmml" xref="A1.SS4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A1.SS4.p1.4.m4.1.1.1.cmml" xref="A1.SS4.p1.4.m4.1.1">superscript</csymbol><ci id="A1.SS4.p1.4.m4.1.1.2.cmml" xref="A1.SS4.p1.4.m4.1.1.2">ℝ</ci><cn id="A1.SS4.p1.4.m4.1.1.3.cmml" type="integer" xref="A1.SS4.p1.4.m4.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS4.p1.4.m4.1c">\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.4.m4.1d">blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> is the variable to be refined. The position is initialized to the coordinates of the pixel with the highest detection score from the detection step, while the flux is initialized using the value extracted from the flux map <math alttext="\boldsymbol{\alpha}" class="ltx_Math" display="inline" id="A1.SS4.p1.5.m5.1"><semantics id="A1.SS4.p1.5.m5.1a"><mi id="A1.SS4.p1.5.m5.1.1" mathsize="144%" xref="A1.SS4.p1.5.m5.1.1.cmml">𝜶</mi><annotation-xml encoding="MathML-Content" id="A1.SS4.p1.5.m5.1b"><ci id="A1.SS4.p1.5.m5.1.1.cmml" xref="A1.SS4.p1.5.m5.1.1">𝜶</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS4.p1.5.m5.1c">\boldsymbol{\alpha}</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.5.m5.1d">bold_italic_α</annotation></semantics></math> at that location. We decompose the involved loss function <math alttext="\ell" class="ltx_Math" display="inline" id="A1.SS4.p1.6.m6.1"><semantics id="A1.SS4.p1.6.m6.1a"><mi id="A1.SS4.p1.6.m6.1.1" mathsize="144%" mathvariant="normal" xref="A1.SS4.p1.6.m6.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="A1.SS4.p1.6.m6.1b"><ci id="A1.SS4.p1.6.m6.1.1.cmml" xref="A1.SS4.p1.6.m6.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS4.p1.6.m6.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.6.m6.1d">roman_ℓ</annotation></semantics></math> as: <table class="ltx_equation ltx_eqn_table" id="A1.E35"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ell(\boldsymbol{z},\{\boldsymbol{m}_{j},\mathbf{C}_{j}\}_{j})=\sum_{t}\sum_{j% \in S(\boldsymbol{x}_{t})}w_{j}\ell_{j}(\boldsymbol{z},\boldsymbol{m}_{j},% \mathbf{C}_{j})\,," 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id="A1.E35.m1.4.4.1.1.3.2.2.2.2.2.1.cmml" xref="A1.E35.m1.4.4.1.1.3.2.2.2.2.2">subscript</csymbol><ci id="A1.E35.m1.4.4.1.1.3.2.2.2.2.2.2.cmml" xref="A1.E35.m1.4.4.1.1.3.2.2.2.2.2.2">𝐂</ci><ci id="A1.E35.m1.4.4.1.1.3.2.2.2.2.2.3.cmml" xref="A1.E35.m1.4.4.1.1.3.2.2.2.2.2.3">𝑗</ci></apply></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E35.m1.4c">\ell(\boldsymbol{z},\{\boldsymbol{m}_{j},\mathbf{C}_{j}\}_{j})=\sum_{t}\sum_{j% \in S(\boldsymbol{x}_{t})}w_{j}\ell_{j}(\boldsymbol{z},\boldsymbol{m}_{j},% \mathbf{C}_{j})\,,</annotation><annotation encoding="application/x-llamapun" id="A1.E35.m1.4d">roman_ℓ ( bold_italic_z , { bold_italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_j ∈ italic_S ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( bold_italic_z , bold_italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(35)</td> </tr></tbody> </table> where: <table class="ltx_equation ltx_eqn_table" id="A1.E36"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ell_{j}([\alpha,\boldsymbol{x}_{0}],\boldsymbol{m}_{j},\mathbf{C}_{j})=\\ \hfill\left\lVert\mathbf{L}_{j}\left(\boldsymbol{y}_{t}^{(j)}-\alpha% \boldsymbol{h}^{(j)}(r(\boldsymbol{x}_{0},\phi_{t}))-\boldsymbol{m}_{j}\right)% \right\rVert_{2}^{2}\,," class="ltx_Math" 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id="A1.SS4.p1.7.m1.1c">\mathbf{C}_{j}^{-1}=\mathbf{L}_{j}\mathbf{L}_{j}^{\top}</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.7.m1.1d">bold_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = bold_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. We also define the feasible set <math alttext="Z" class="ltx_Math" display="inline" id="A1.SS4.p1.8.m2.1"><semantics id="A1.SS4.p1.8.m2.1a"><mi id="A1.SS4.p1.8.m2.1.1" mathsize="144%" xref="A1.SS4.p1.8.m2.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="A1.SS4.p1.8.m2.1b"><ci id="A1.SS4.p1.8.m2.1.1.cmml" xref="A1.SS4.p1.8.m2.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS4.p1.8.m2.1c">Z</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.8.m2.1d">italic_Z</annotation></semantics></math> such that <math alttext="\alpha>0" class="ltx_Math" display="inline" id="A1.SS4.p1.9.m3.1"><semantics id="A1.SS4.p1.9.m3.1a"><mrow id="A1.SS4.p1.9.m3.1.1" xref="A1.SS4.p1.9.m3.1.1.cmml"><mi id="A1.SS4.p1.9.m3.1.1.2" mathsize="144%" xref="A1.SS4.p1.9.m3.1.1.2.cmml">α</mi><mo id="A1.SS4.p1.9.m3.1.1.1" mathsize="144%" xref="A1.SS4.p1.9.m3.1.1.1.cmml">></mo><mn id="A1.SS4.p1.9.m3.1.1.3" mathsize="144%" xref="A1.SS4.p1.9.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS4.p1.9.m3.1b"><apply id="A1.SS4.p1.9.m3.1.1.cmml" xref="A1.SS4.p1.9.m3.1.1"><gt id="A1.SS4.p1.9.m3.1.1.1.cmml" xref="A1.SS4.p1.9.m3.1.1.1"></gt><ci id="A1.SS4.p1.9.m3.1.1.2.cmml" xref="A1.SS4.p1.9.m3.1.1.2">𝛼</ci><cn id="A1.SS4.p1.9.m3.1.1.3.cmml" type="integer" xref="A1.SS4.p1.9.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS4.p1.9.m3.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.9.m3.1d">italic_α > 0</annotation></semantics></math> (enforcing a non-negativity constraint on the exoplanet flux) and the estimated position remains within a fixed radius of the initial estimate. We denote by <math alttext="\text{Proj}_{Z}" class="ltx_Math" display="inline" id="A1.SS4.p1.10.m4.1"><semantics id="A1.SS4.p1.10.m4.1a"><msub id="A1.SS4.p1.10.m4.1.1" xref="A1.SS4.p1.10.m4.1.1.cmml"><mtext id="A1.SS4.p1.10.m4.1.1.2" mathsize="144%" xref="A1.SS4.p1.10.m4.1.1.2a.cmml">Proj</mtext><mi id="A1.SS4.p1.10.m4.1.1.3" mathsize="144%" xref="A1.SS4.p1.10.m4.1.1.3.cmml">Z</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS4.p1.10.m4.1b"><apply id="A1.SS4.p1.10.m4.1.1.cmml" xref="A1.SS4.p1.10.m4.1.1"><csymbol cd="ambiguous" id="A1.SS4.p1.10.m4.1.1.1.cmml" xref="A1.SS4.p1.10.m4.1.1">subscript</csymbol><ci id="A1.SS4.p1.10.m4.1.1.2a.cmml" xref="A1.SS4.p1.10.m4.1.1.2"><mtext id="A1.SS4.p1.10.m4.1.1.2.cmml" mathsize="144%" xref="A1.SS4.p1.10.m4.1.1.2">Proj</mtext></ci><ci id="A1.SS4.p1.10.m4.1.1.3.cmml" xref="A1.SS4.p1.10.m4.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS4.p1.10.m4.1c">\text{Proj}_{Z}</annotation><annotation encoding="application/x-llamapun" id="A1.SS4.p1.10.m4.1d">Proj start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT</annotation></semantics></math> the projection associated with it. In <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#alg1" style="font-size:144%;" title="In A.4 Iterative optimization scheme for astrometry ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Algorithm 1</a>, we also denote by Estim(.) the procedure described in Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.Ex2" style="font-size:144%;" title="A.1 Estimation of the statistical parameters ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.1</a>)–(<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E24" style="font-size:144%;" title="Equation 24 ‣ A.1 Estimation of the statistical parameters ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">24</a>) to estimate the statistical parameters of the nuisance component. </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <figcaption class="ltx_caption" style="font-size:144%;">Algorithm 1 Iterative procedure for flux and position estimation.</figcaption> <div class="ltx_listing ltx_listing" id="alg1.5"> <div class="ltx_listingline" id="alg1.l1"> <math alttext="\boldsymbol{y}\in\mathbb{R}^{T\times H\times W}" class="ltx_Math" display="inline" id="alg1.l1.m1.1"><semantics id="alg1.l1.m1.1a"><mrow id="alg1.l1.m1.1.1" xref="alg1.l1.m1.1.1.cmml"><mi id="alg1.l1.m1.1.1.2" mathsize="144%" xref="alg1.l1.m1.1.1.2.cmml">𝒚</mi><mo id="alg1.l1.m1.1.1.1" mathsize="144%" xref="alg1.l1.m1.1.1.1.cmml">∈</mo><msup id="alg1.l1.m1.1.1.3" xref="alg1.l1.m1.1.1.3.cmml"><mi id="alg1.l1.m1.1.1.3.2" mathsize="144%" xref="alg1.l1.m1.1.1.3.2.cmml">ℝ</mi><mrow id="alg1.l1.m1.1.1.3.3" xref="alg1.l1.m1.1.1.3.3.cmml"><mi id="alg1.l1.m1.1.1.3.3.2" mathsize="144%" xref="alg1.l1.m1.1.1.3.3.2.cmml">T</mi><mo id="alg1.l1.m1.1.1.3.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="alg1.l1.m1.1.1.3.3.1.cmml">×</mo><mi id="alg1.l1.m1.1.1.3.3.3" mathsize="144%" xref="alg1.l1.m1.1.1.3.3.3.cmml">H</mi><mo id="alg1.l1.m1.1.1.3.3.1a" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="alg1.l1.m1.1.1.3.3.1.cmml">×</mo><mi id="alg1.l1.m1.1.1.3.3.4" mathsize="144%" xref="alg1.l1.m1.1.1.3.3.4.cmml">W</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m1.1b"><apply id="alg1.l1.m1.1.1.cmml" xref="alg1.l1.m1.1.1"><in id="alg1.l1.m1.1.1.1.cmml" xref="alg1.l1.m1.1.1.1"></in><ci id="alg1.l1.m1.1.1.2.cmml" xref="alg1.l1.m1.1.1.2">𝒚</ci><apply id="alg1.l1.m1.1.1.3.cmml" xref="alg1.l1.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l1.m1.1.1.3.1.cmml" xref="alg1.l1.m1.1.1.3">superscript</csymbol><ci id="alg1.l1.m1.1.1.3.2.cmml" xref="alg1.l1.m1.1.1.3.2">ℝ</ci><apply id="alg1.l1.m1.1.1.3.3.cmml" xref="alg1.l1.m1.1.1.3.3"><times id="alg1.l1.m1.1.1.3.3.1.cmml" xref="alg1.l1.m1.1.1.3.3.1"></times><ci id="alg1.l1.m1.1.1.3.3.2.cmml" xref="alg1.l1.m1.1.1.3.3.2">𝑇</ci><ci id="alg1.l1.m1.1.1.3.3.3.cmml" xref="alg1.l1.m1.1.1.3.3.3">𝐻</ci><ci id="alg1.l1.m1.1.1.3.3.4.cmml" xref="alg1.l1.m1.1.1.3.3.4">𝑊</ci></apply></apply></apply></annotation-xml><annotation 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id="alg1.l1.m2.1.1.1.cmml" xref="alg1.l1.m2.1.1.1"></in><ci id="alg1.l1.m2.1.1.2.cmml" xref="alg1.l1.m2.1.1.2">𝒛</ci><apply id="alg1.l1.m2.1.1.3.cmml" xref="alg1.l1.m2.1.1.3"><csymbol cd="ambiguous" id="alg1.l1.m2.1.1.3.1.cmml" xref="alg1.l1.m2.1.1.3">superscript</csymbol><ci id="alg1.l1.m2.1.1.3.2.cmml" xref="alg1.l1.m2.1.1.3.2">ℝ</ci><cn id="alg1.l1.m2.1.1.3.3.cmml" type="integer" xref="alg1.l1.m2.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m2.1c">\boldsymbol{z}\in\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m2.1d">bold_italic_z ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l2"> for <math alttext="l=0\cdots L-1" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><mrow id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml"><mi id="alg1.l2.m1.1.1.2" mathsize="144%" xref="alg1.l2.m1.1.1.2.cmml">l</mi><mo id="alg1.l2.m1.1.1.1" mathsize="144%" xref="alg1.l2.m1.1.1.1.cmml">=</mo><mrow id="alg1.l2.m1.1.1.3" xref="alg1.l2.m1.1.1.3.cmml"><mrow id="alg1.l2.m1.1.1.3.2" xref="alg1.l2.m1.1.1.3.2.cmml"><mn id="alg1.l2.m1.1.1.3.2.2" mathsize="144%" xref="alg1.l2.m1.1.1.3.2.2.cmml">0</mn><mo id="alg1.l2.m1.1.1.3.2.1" xref="alg1.l2.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l2.m1.1.1.3.2.3" mathsize="144%" mathvariant="normal" xref="alg1.l2.m1.1.1.3.2.3.cmml">⋯</mi><mo id="alg1.l2.m1.1.1.3.2.1a" xref="alg1.l2.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l2.m1.1.1.3.2.4" mathsize="144%" xref="alg1.l2.m1.1.1.3.2.4.cmml">L</mi></mrow><mo id="alg1.l2.m1.1.1.3.1" mathsize="144%" xref="alg1.l2.m1.1.1.3.1.cmml">−</mo><mn id="alg1.l2.m1.1.1.3.3" mathsize="144%" xref="alg1.l2.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><apply id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1"><eq id="alg1.l2.m1.1.1.1.cmml" xref="alg1.l2.m1.1.1.1"></eq><ci id="alg1.l2.m1.1.1.2.cmml" xref="alg1.l2.m1.1.1.2">𝑙</ci><apply id="alg1.l2.m1.1.1.3.cmml" xref="alg1.l2.m1.1.1.3"><minus id="alg1.l2.m1.1.1.3.1.cmml" xref="alg1.l2.m1.1.1.3.1"></minus><apply id="alg1.l2.m1.1.1.3.2.cmml" xref="alg1.l2.m1.1.1.3.2"><times id="alg1.l2.m1.1.1.3.2.1.cmml" xref="alg1.l2.m1.1.1.3.2.1"></times><cn id="alg1.l2.m1.1.1.3.2.2.cmml" type="integer" xref="alg1.l2.m1.1.1.3.2.2">0</cn><ci id="alg1.l2.m1.1.1.3.2.3.cmml" xref="alg1.l2.m1.1.1.3.2.3">⋯</ci><ci id="alg1.l2.m1.1.1.3.2.4.cmml" xref="alg1.l2.m1.1.1.3.2.4">𝐿</ci></apply><cn id="alg1.l2.m1.1.1.3.3.cmml" type="integer" xref="alg1.l2.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">l=0\cdots L-1</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">italic_l = 0 ⋯ italic_L - 1</annotation></semantics></math> do </div> <div class="ltx_listingline" id="alg1.l3"> <math alttext="\alpha,\boldsymbol{x}_{0}\leftarrow\boldsymbol{z}" class="ltx_Math" display="inline" id="alg1.l3.m1.2"><semantics id="alg1.l3.m1.2a"><mrow id="alg1.l3.m1.2.2" xref="alg1.l3.m1.2.2.cmml"><mrow id="alg1.l3.m1.2.2.1.1" xref="alg1.l3.m1.2.2.1.2.cmml"><mi id="alg1.l3.m1.1.1" mathsize="144%" xref="alg1.l3.m1.1.1.cmml">α</mi><mo id="alg1.l3.m1.2.2.1.1.2" mathsize="144%" xref="alg1.l3.m1.2.2.1.2.cmml">,</mo><msub id="alg1.l3.m1.2.2.1.1.1" xref="alg1.l3.m1.2.2.1.1.1.cmml"><mi id="alg1.l3.m1.2.2.1.1.1.2" mathsize="144%" xref="alg1.l3.m1.2.2.1.1.1.2.cmml">𝒙</mi><mn id="alg1.l3.m1.2.2.1.1.1.3" mathsize="144%" xref="alg1.l3.m1.2.2.1.1.1.3.cmml">0</mn></msub></mrow><mo id="alg1.l3.m1.2.2.2" mathsize="144%" stretchy="false" xref="alg1.l3.m1.2.2.2.cmml">←</mo><mi id="alg1.l3.m1.2.2.3" mathsize="144%" xref="alg1.l3.m1.2.2.3.cmml">𝒛</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.2b"><apply id="alg1.l3.m1.2.2.cmml" xref="alg1.l3.m1.2.2"><ci id="alg1.l3.m1.2.2.2.cmml" xref="alg1.l3.m1.2.2.2">←</ci><list id="alg1.l3.m1.2.2.1.2.cmml" xref="alg1.l3.m1.2.2.1.1"><ci id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1">𝛼</ci><apply id="alg1.l3.m1.2.2.1.1.1.cmml" xref="alg1.l3.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.l3.m1.2.2.1.1.1.1.cmml" xref="alg1.l3.m1.2.2.1.1.1">subscript</csymbol><ci id="alg1.l3.m1.2.2.1.1.1.2.cmml" xref="alg1.l3.m1.2.2.1.1.1.2">𝒙</ci><cn id="alg1.l3.m1.2.2.1.1.1.3.cmml" type="integer" xref="alg1.l3.m1.2.2.1.1.1.3">0</cn></apply></list><ci id="alg1.l3.m1.2.2.3.cmml" xref="alg1.l3.m1.2.2.3">𝒛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.2c">\alpha,\boldsymbol{x}_{0}\leftarrow\boldsymbol{z}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.2d">italic_α , bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ← bold_italic_z</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l4"> for <math alttext="t=0\cdots T-1" class="ltx_Math" display="inline" id="alg1.l4.m1.1"><semantics id="alg1.l4.m1.1a"><mrow id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml"><mi id="alg1.l4.m1.1.1.2" mathsize="144%" xref="alg1.l4.m1.1.1.2.cmml">t</mi><mo id="alg1.l4.m1.1.1.1" mathsize="144%" xref="alg1.l4.m1.1.1.1.cmml">=</mo><mrow id="alg1.l4.m1.1.1.3" xref="alg1.l4.m1.1.1.3.cmml"><mrow id="alg1.l4.m1.1.1.3.2" xref="alg1.l4.m1.1.1.3.2.cmml"><mn id="alg1.l4.m1.1.1.3.2.2" mathsize="144%" xref="alg1.l4.m1.1.1.3.2.2.cmml">0</mn><mo id="alg1.l4.m1.1.1.3.2.1" xref="alg1.l4.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l4.m1.1.1.3.2.3" mathsize="144%" mathvariant="normal" xref="alg1.l4.m1.1.1.3.2.3.cmml">⋯</mi><mo id="alg1.l4.m1.1.1.3.2.1a" xref="alg1.l4.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l4.m1.1.1.3.2.4" mathsize="144%" xref="alg1.l4.m1.1.1.3.2.4.cmml">T</mi></mrow><mo id="alg1.l4.m1.1.1.3.1" mathsize="144%" xref="alg1.l4.m1.1.1.3.1.cmml">−</mo><mn id="alg1.l4.m1.1.1.3.3" mathsize="144%" xref="alg1.l4.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.1b"><apply id="alg1.l4.m1.1.1.cmml" xref="alg1.l4.m1.1.1"><eq id="alg1.l4.m1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1"></eq><ci id="alg1.l4.m1.1.1.2.cmml" xref="alg1.l4.m1.1.1.2">𝑡</ci><apply id="alg1.l4.m1.1.1.3.cmml" xref="alg1.l4.m1.1.1.3"><minus id="alg1.l4.m1.1.1.3.1.cmml" xref="alg1.l4.m1.1.1.3.1"></minus><apply id="alg1.l4.m1.1.1.3.2.cmml" xref="alg1.l4.m1.1.1.3.2"><times id="alg1.l4.m1.1.1.3.2.1.cmml" xref="alg1.l4.m1.1.1.3.2.1"></times><cn id="alg1.l4.m1.1.1.3.2.2.cmml" type="integer" xref="alg1.l4.m1.1.1.3.2.2">0</cn><ci id="alg1.l4.m1.1.1.3.2.3.cmml" xref="alg1.l4.m1.1.1.3.2.3">⋯</ci><ci id="alg1.l4.m1.1.1.3.2.4.cmml" xref="alg1.l4.m1.1.1.3.2.4">𝑇</ci></apply><cn id="alg1.l4.m1.1.1.3.3.cmml" type="integer" xref="alg1.l4.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.1c">t=0\cdots T-1</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.1d">italic_t = 0 ⋯ italic_T - 1</annotation></semantics></math> do <math alttext="\triangleright" class="ltx_Math" display="inline" id="alg1.l4.1.m1.1"><semantics id="alg1.l4.1.m1.1a"><mo id="alg1.l4.1.m1.1.1" xref="alg1.l4.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg1.l4.1.m1.1b"><ci id="alg1.l4.1.m1.1.1.cmml" xref="alg1.l4.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.1.m1.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.1.m1.1d">▷</annotation></semantics></math> Update nuisance estimate </div> <div class="ltx_listingline" id="alg1.l5"> <math alttext="\boldsymbol{x}_{t}\leftarrow r(\boldsymbol{x}_{0},\phi_{t})" class="ltx_Math" display="inline" id="alg1.l5.m1.2"><semantics id="alg1.l5.m1.2a"><mrow id="alg1.l5.m1.2.2" xref="alg1.l5.m1.2.2.cmml"><msub id="alg1.l5.m1.2.2.4" xref="alg1.l5.m1.2.2.4.cmml"><mi id="alg1.l5.m1.2.2.4.2" mathsize="144%" xref="alg1.l5.m1.2.2.4.2.cmml">𝒙</mi><mi id="alg1.l5.m1.2.2.4.3" mathsize="144%" xref="alg1.l5.m1.2.2.4.3.cmml">t</mi></msub><mo id="alg1.l5.m1.2.2.3" mathsize="144%" stretchy="false" xref="alg1.l5.m1.2.2.3.cmml">←</mo><mrow id="alg1.l5.m1.2.2.2" xref="alg1.l5.m1.2.2.2.cmml"><mi id="alg1.l5.m1.2.2.2.4" mathsize="144%" xref="alg1.l5.m1.2.2.2.4.cmml">r</mi><mo id="alg1.l5.m1.2.2.2.3" xref="alg1.l5.m1.2.2.2.3.cmml">⁢</mo><mrow id="alg1.l5.m1.2.2.2.2.2" xref="alg1.l5.m1.2.2.2.2.3.cmml"><mo id="alg1.l5.m1.2.2.2.2.2.3" maxsize="144%" minsize="144%" xref="alg1.l5.m1.2.2.2.2.3.cmml">(</mo><msub id="alg1.l5.m1.1.1.1.1.1.1" xref="alg1.l5.m1.1.1.1.1.1.1.cmml"><mi id="alg1.l5.m1.1.1.1.1.1.1.2" mathsize="144%" xref="alg1.l5.m1.1.1.1.1.1.1.2.cmml">𝒙</mi><mn id="alg1.l5.m1.1.1.1.1.1.1.3" mathsize="144%" xref="alg1.l5.m1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="alg1.l5.m1.2.2.2.2.2.4" mathsize="144%" xref="alg1.l5.m1.2.2.2.2.3.cmml">,</mo><msub id="alg1.l5.m1.2.2.2.2.2.2" xref="alg1.l5.m1.2.2.2.2.2.2.cmml"><mi id="alg1.l5.m1.2.2.2.2.2.2.2" mathsize="144%" xref="alg1.l5.m1.2.2.2.2.2.2.2.cmml">ϕ</mi><mi id="alg1.l5.m1.2.2.2.2.2.2.3" mathsize="144%" xref="alg1.l5.m1.2.2.2.2.2.2.3.cmml">t</mi></msub><mo id="alg1.l5.m1.2.2.2.2.2.5" maxsize="144%" minsize="144%" xref="alg1.l5.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.2b"><apply id="alg1.l5.m1.2.2.cmml" xref="alg1.l5.m1.2.2"><ci id="alg1.l5.m1.2.2.3.cmml" xref="alg1.l5.m1.2.2.3">←</ci><apply id="alg1.l5.m1.2.2.4.cmml" xref="alg1.l5.m1.2.2.4"><csymbol cd="ambiguous" id="alg1.l5.m1.2.2.4.1.cmml" xref="alg1.l5.m1.2.2.4">subscript</csymbol><ci id="alg1.l5.m1.2.2.4.2.cmml" xref="alg1.l5.m1.2.2.4.2">𝒙</ci><ci id="alg1.l5.m1.2.2.4.3.cmml" xref="alg1.l5.m1.2.2.4.3">𝑡</ci></apply><apply id="alg1.l5.m1.2.2.2.cmml" xref="alg1.l5.m1.2.2.2"><times id="alg1.l5.m1.2.2.2.3.cmml" xref="alg1.l5.m1.2.2.2.3"></times><ci id="alg1.l5.m1.2.2.2.4.cmml" xref="alg1.l5.m1.2.2.2.4">𝑟</ci><interval closure="open" id="alg1.l5.m1.2.2.2.2.3.cmml" xref="alg1.l5.m1.2.2.2.2.2"><apply id="alg1.l5.m1.1.1.1.1.1.1.cmml" xref="alg1.l5.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l5.m1.1.1.1.1.1.1.1.cmml" xref="alg1.l5.m1.1.1.1.1.1.1">subscript</csymbol><ci id="alg1.l5.m1.1.1.1.1.1.1.2.cmml" xref="alg1.l5.m1.1.1.1.1.1.1.2">𝒙</ci><cn id="alg1.l5.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="alg1.l5.m1.1.1.1.1.1.1.3">0</cn></apply><apply id="alg1.l5.m1.2.2.2.2.2.2.cmml" xref="alg1.l5.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="alg1.l5.m1.2.2.2.2.2.2.1.cmml" xref="alg1.l5.m1.2.2.2.2.2.2">subscript</csymbol><ci id="alg1.l5.m1.2.2.2.2.2.2.2.cmml" xref="alg1.l5.m1.2.2.2.2.2.2.2">italic-ϕ</ci><ci id="alg1.l5.m1.2.2.2.2.2.2.3.cmml" xref="alg1.l5.m1.2.2.2.2.2.2.3">𝑡</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.2c">\boldsymbol{x}_{t}\leftarrow r(\boldsymbol{x}_{0},\phi_{t})</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.2d">bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ← italic_r ( bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l6"> <math alttext="\boldsymbol{s}_{t}\leftarrow\boldsymbol{y}_{t}-\boldsymbol{h}(\boldsymbol{x}_{% t})" class="ltx_Math" display="inline" id="alg1.l6.m1.1"><semantics id="alg1.l6.m1.1a"><mrow id="alg1.l6.m1.1.1" xref="alg1.l6.m1.1.1.cmml"><msub id="alg1.l6.m1.1.1.3" xref="alg1.l6.m1.1.1.3.cmml"><mi id="alg1.l6.m1.1.1.3.2" mathsize="144%" xref="alg1.l6.m1.1.1.3.2.cmml">𝒔</mi><mi id="alg1.l6.m1.1.1.3.3" mathsize="144%" xref="alg1.l6.m1.1.1.3.3.cmml">t</mi></msub><mo id="alg1.l6.m1.1.1.2" mathsize="144%" stretchy="false" xref="alg1.l6.m1.1.1.2.cmml">←</mo><mrow id="alg1.l6.m1.1.1.1" xref="alg1.l6.m1.1.1.1.cmml"><msub id="alg1.l6.m1.1.1.1.3" xref="alg1.l6.m1.1.1.1.3.cmml"><mi id="alg1.l6.m1.1.1.1.3.2" mathsize="144%" xref="alg1.l6.m1.1.1.1.3.2.cmml">𝒚</mi><mi id="alg1.l6.m1.1.1.1.3.3" mathsize="144%" xref="alg1.l6.m1.1.1.1.3.3.cmml">t</mi></msub><mo id="alg1.l6.m1.1.1.1.2" mathsize="144%" xref="alg1.l6.m1.1.1.1.2.cmml">−</mo><mrow id="alg1.l6.m1.1.1.1.1" xref="alg1.l6.m1.1.1.1.1.cmml"><mi id="alg1.l6.m1.1.1.1.1.3" mathsize="144%" xref="alg1.l6.m1.1.1.1.1.3.cmml">𝒉</mi><mo id="alg1.l6.m1.1.1.1.1.2" xref="alg1.l6.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="alg1.l6.m1.1.1.1.1.1.1" xref="alg1.l6.m1.1.1.1.1.1.1.1.cmml"><mo id="alg1.l6.m1.1.1.1.1.1.1.2" maxsize="144%" minsize="144%" xref="alg1.l6.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="alg1.l6.m1.1.1.1.1.1.1.1" xref="alg1.l6.m1.1.1.1.1.1.1.1.cmml"><mi id="alg1.l6.m1.1.1.1.1.1.1.1.2" mathsize="144%" xref="alg1.l6.m1.1.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="alg1.l6.m1.1.1.1.1.1.1.1.3" mathsize="144%" xref="alg1.l6.m1.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="alg1.l6.m1.1.1.1.1.1.1.3" maxsize="144%" minsize="144%" xref="alg1.l6.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m1.1b"><apply id="alg1.l6.m1.1.1.cmml" xref="alg1.l6.m1.1.1"><ci id="alg1.l6.m1.1.1.2.cmml" xref="alg1.l6.m1.1.1.2">←</ci><apply id="alg1.l6.m1.1.1.3.cmml" xref="alg1.l6.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.3.1.cmml" xref="alg1.l6.m1.1.1.3">subscript</csymbol><ci id="alg1.l6.m1.1.1.3.2.cmml" xref="alg1.l6.m1.1.1.3.2">𝒔</ci><ci id="alg1.l6.m1.1.1.3.3.cmml" xref="alg1.l6.m1.1.1.3.3">𝑡</ci></apply><apply id="alg1.l6.m1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1"><minus id="alg1.l6.m1.1.1.1.2.cmml" xref="alg1.l6.m1.1.1.1.2"></minus><apply id="alg1.l6.m1.1.1.1.3.cmml" xref="alg1.l6.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.1.3.1.cmml" xref="alg1.l6.m1.1.1.1.3">subscript</csymbol><ci id="alg1.l6.m1.1.1.1.3.2.cmml" xref="alg1.l6.m1.1.1.1.3.2">𝒚</ci><ci id="alg1.l6.m1.1.1.1.3.3.cmml" xref="alg1.l6.m1.1.1.1.3.3">𝑡</ci></apply><apply id="alg1.l6.m1.1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1.1"><times id="alg1.l6.m1.1.1.1.1.2.cmml" xref="alg1.l6.m1.1.1.1.1.2"></times><ci id="alg1.l6.m1.1.1.1.1.3.cmml" xref="alg1.l6.m1.1.1.1.1.3">𝒉</ci><apply id="alg1.l6.m1.1.1.1.1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.1.1.1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1.1.1.1">subscript</csymbol><ci id="alg1.l6.m1.1.1.1.1.1.1.1.2.cmml" xref="alg1.l6.m1.1.1.1.1.1.1.1.2">𝒙</ci><ci id="alg1.l6.m1.1.1.1.1.1.1.1.3.cmml" xref="alg1.l6.m1.1.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.1c">\boldsymbol{s}_{t}\leftarrow\boldsymbol{y}_{t}-\boldsymbol{h}(\boldsymbol{x}_{% t})</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m1.1d">bold_italic_s start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ← bold_italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - bold_italic_h ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l7"> end for </div> <div class="ltx_listingline" id="alg1.l8"> <math alttext="\boldsymbol{g},\mathbf{H}\leftarrow\mathbf{0}_{\mathbb{R}^{3}},\mathbf{0}_{% \mathbb{R}^{3\times 3}}" class="ltx_Math" display="inline" id="alg1.l8.m1.4"><semantics id="alg1.l8.m1.4a"><mrow id="alg1.l8.m1.4.4.2" xref="alg1.l8.m1.4.4.3.cmml"><mrow id="alg1.l8.m1.3.3.1.1" xref="alg1.l8.m1.3.3.1.1.cmml"><mrow id="alg1.l8.m1.3.3.1.1.2.2" xref="alg1.l8.m1.3.3.1.1.2.1.cmml"><mi id="alg1.l8.m1.1.1" mathsize="144%" xref="alg1.l8.m1.1.1.cmml">𝒈</mi><mo id="alg1.l8.m1.3.3.1.1.2.2.1" mathsize="144%" xref="alg1.l8.m1.3.3.1.1.2.1.cmml">,</mo><mi id="alg1.l8.m1.2.2" mathsize="144%" xref="alg1.l8.m1.2.2.cmml">𝐇</mi></mrow><mo id="alg1.l8.m1.3.3.1.1.1" mathsize="144%" stretchy="false" xref="alg1.l8.m1.3.3.1.1.1.cmml">←</mo><msub id="alg1.l8.m1.3.3.1.1.3" xref="alg1.l8.m1.3.3.1.1.3.cmml"><mn id="alg1.l8.m1.3.3.1.1.3.2" mathsize="144%" xref="alg1.l8.m1.3.3.1.1.3.2.cmml">𝟎</mn><msup id="alg1.l8.m1.3.3.1.1.3.3" xref="alg1.l8.m1.3.3.1.1.3.3.cmml"><mi id="alg1.l8.m1.3.3.1.1.3.3.2" mathsize="144%" xref="alg1.l8.m1.3.3.1.1.3.3.2.cmml">ℝ</mi><mn id="alg1.l8.m1.3.3.1.1.3.3.3" mathsize="144%" xref="alg1.l8.m1.3.3.1.1.3.3.3.cmml">3</mn></msup></msub></mrow><mo id="alg1.l8.m1.4.4.2.3" mathsize="144%" xref="alg1.l8.m1.4.4.3a.cmml">,</mo><msub id="alg1.l8.m1.4.4.2.2" xref="alg1.l8.m1.4.4.2.2.cmml"><mn id="alg1.l8.m1.4.4.2.2.2" mathsize="144%" xref="alg1.l8.m1.4.4.2.2.2.cmml">𝟎</mn><msup id="alg1.l8.m1.4.4.2.2.3" xref="alg1.l8.m1.4.4.2.2.3.cmml"><mi id="alg1.l8.m1.4.4.2.2.3.2" mathsize="144%" xref="alg1.l8.m1.4.4.2.2.3.2.cmml">ℝ</mi><mrow id="alg1.l8.m1.4.4.2.2.3.3" xref="alg1.l8.m1.4.4.2.2.3.3.cmml"><mn id="alg1.l8.m1.4.4.2.2.3.3.2" mathsize="144%" xref="alg1.l8.m1.4.4.2.2.3.3.2.cmml">3</mn><mo id="alg1.l8.m1.4.4.2.2.3.3.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="alg1.l8.m1.4.4.2.2.3.3.1.cmml">×</mo><mn id="alg1.l8.m1.4.4.2.2.3.3.3" mathsize="144%" xref="alg1.l8.m1.4.4.2.2.3.3.3.cmml">3</mn></mrow></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.l8.m1.4b"><apply id="alg1.l8.m1.4.4.3.cmml" xref="alg1.l8.m1.4.4.2"><csymbol cd="ambiguous" id="alg1.l8.m1.4.4.3a.cmml" xref="alg1.l8.m1.4.4.2.3">formulae-sequence</csymbol><apply id="alg1.l8.m1.3.3.1.1.cmml" xref="alg1.l8.m1.3.3.1.1"><ci id="alg1.l8.m1.3.3.1.1.1.cmml" xref="alg1.l8.m1.3.3.1.1.1">←</ci><list id="alg1.l8.m1.3.3.1.1.2.1.cmml" xref="alg1.l8.m1.3.3.1.1.2.2"><ci id="alg1.l8.m1.1.1.cmml" xref="alg1.l8.m1.1.1">𝒈</ci><ci id="alg1.l8.m1.2.2.cmml" xref="alg1.l8.m1.2.2">𝐇</ci></list><apply id="alg1.l8.m1.3.3.1.1.3.cmml" xref="alg1.l8.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="alg1.l8.m1.3.3.1.1.3.1.cmml" xref="alg1.l8.m1.3.3.1.1.3">subscript</csymbol><cn id="alg1.l8.m1.3.3.1.1.3.2.cmml" type="integer" xref="alg1.l8.m1.3.3.1.1.3.2">0</cn><apply id="alg1.l8.m1.3.3.1.1.3.3.cmml" xref="alg1.l8.m1.3.3.1.1.3.3"><csymbol cd="ambiguous" id="alg1.l8.m1.3.3.1.1.3.3.1.cmml" xref="alg1.l8.m1.3.3.1.1.3.3">superscript</csymbol><ci id="alg1.l8.m1.3.3.1.1.3.3.2.cmml" xref="alg1.l8.m1.3.3.1.1.3.3.2">ℝ</ci><cn id="alg1.l8.m1.3.3.1.1.3.3.3.cmml" type="integer" xref="alg1.l8.m1.3.3.1.1.3.3.3">3</cn></apply></apply></apply><apply id="alg1.l8.m1.4.4.2.2.cmml" xref="alg1.l8.m1.4.4.2.2"><csymbol cd="ambiguous" id="alg1.l8.m1.4.4.2.2.1.cmml" xref="alg1.l8.m1.4.4.2.2">subscript</csymbol><cn id="alg1.l8.m1.4.4.2.2.2.cmml" type="integer" xref="alg1.l8.m1.4.4.2.2.2">0</cn><apply id="alg1.l8.m1.4.4.2.2.3.cmml" xref="alg1.l8.m1.4.4.2.2.3"><csymbol cd="ambiguous" id="alg1.l8.m1.4.4.2.2.3.1.cmml" xref="alg1.l8.m1.4.4.2.2.3">superscript</csymbol><ci id="alg1.l8.m1.4.4.2.2.3.2.cmml" xref="alg1.l8.m1.4.4.2.2.3.2">ℝ</ci><apply id="alg1.l8.m1.4.4.2.2.3.3.cmml" xref="alg1.l8.m1.4.4.2.2.3.3"><times id="alg1.l8.m1.4.4.2.2.3.3.1.cmml" xref="alg1.l8.m1.4.4.2.2.3.3.1"></times><cn id="alg1.l8.m1.4.4.2.2.3.3.2.cmml" type="integer" xref="alg1.l8.m1.4.4.2.2.3.3.2">3</cn><cn id="alg1.l8.m1.4.4.2.2.3.3.3.cmml" type="integer" xref="alg1.l8.m1.4.4.2.2.3.3.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m1.4c">\boldsymbol{g},\mathbf{H}\leftarrow\mathbf{0}_{\mathbb{R}^{3}},\mathbf{0}_{% \mathbb{R}^{3\times 3}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l8.m1.4d">bold_italic_g , bold_H ← bold_0 start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , bold_0 start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 3 × 3 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l9"> for <math alttext="t=0\cdots T-1" class="ltx_Math" display="inline" id="alg1.l9.m1.1"><semantics id="alg1.l9.m1.1a"><mrow id="alg1.l9.m1.1.1" xref="alg1.l9.m1.1.1.cmml"><mi id="alg1.l9.m1.1.1.2" mathsize="144%" xref="alg1.l9.m1.1.1.2.cmml">t</mi><mo id="alg1.l9.m1.1.1.1" mathsize="144%" xref="alg1.l9.m1.1.1.1.cmml">=</mo><mrow id="alg1.l9.m1.1.1.3" xref="alg1.l9.m1.1.1.3.cmml"><mrow id="alg1.l9.m1.1.1.3.2" xref="alg1.l9.m1.1.1.3.2.cmml"><mn id="alg1.l9.m1.1.1.3.2.2" mathsize="144%" xref="alg1.l9.m1.1.1.3.2.2.cmml">0</mn><mo id="alg1.l9.m1.1.1.3.2.1" xref="alg1.l9.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l9.m1.1.1.3.2.3" mathsize="144%" mathvariant="normal" xref="alg1.l9.m1.1.1.3.2.3.cmml">⋯</mi><mo id="alg1.l9.m1.1.1.3.2.1a" xref="alg1.l9.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l9.m1.1.1.3.2.4" mathsize="144%" xref="alg1.l9.m1.1.1.3.2.4.cmml">T</mi></mrow><mo id="alg1.l9.m1.1.1.3.1" mathsize="144%" xref="alg1.l9.m1.1.1.3.1.cmml">−</mo><mn id="alg1.l9.m1.1.1.3.3" mathsize="144%" xref="alg1.l9.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l9.m1.1b"><apply id="alg1.l9.m1.1.1.cmml" xref="alg1.l9.m1.1.1"><eq id="alg1.l9.m1.1.1.1.cmml" xref="alg1.l9.m1.1.1.1"></eq><ci id="alg1.l9.m1.1.1.2.cmml" xref="alg1.l9.m1.1.1.2">𝑡</ci><apply id="alg1.l9.m1.1.1.3.cmml" xref="alg1.l9.m1.1.1.3"><minus id="alg1.l9.m1.1.1.3.1.cmml" xref="alg1.l9.m1.1.1.3.1"></minus><apply id="alg1.l9.m1.1.1.3.2.cmml" xref="alg1.l9.m1.1.1.3.2"><times id="alg1.l9.m1.1.1.3.2.1.cmml" xref="alg1.l9.m1.1.1.3.2.1"></times><cn id="alg1.l9.m1.1.1.3.2.2.cmml" type="integer" xref="alg1.l9.m1.1.1.3.2.2">0</cn><ci id="alg1.l9.m1.1.1.3.2.3.cmml" xref="alg1.l9.m1.1.1.3.2.3">⋯</ci><ci id="alg1.l9.m1.1.1.3.2.4.cmml" xref="alg1.l9.m1.1.1.3.2.4">𝑇</ci></apply><cn id="alg1.l9.m1.1.1.3.3.cmml" type="integer" xref="alg1.l9.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m1.1c">t=0\cdots T-1</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m1.1d">italic_t = 0 ⋯ italic_T - 1</annotation></semantics></math> do <math alttext="\triangleright" class="ltx_Math" display="inline" id="alg1.l9.1.m1.1"><semantics id="alg1.l9.1.m1.1a"><mo id="alg1.l9.1.m1.1.1" xref="alg1.l9.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg1.l9.1.m1.1b"><ci id="alg1.l9.1.m1.1.1.cmml" xref="alg1.l9.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.1.m1.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.1.m1.1d">▷</annotation></semantics></math> Gradient and Hessian </div> <div class="ltx_listingline" id="alg1.l10"> for <math alttext="j\in S(\boldsymbol{x}_{t})" class="ltx_Math" display="inline" id="alg1.l10.m1.1"><semantics id="alg1.l10.m1.1a"><mrow id="alg1.l10.m1.1.1" xref="alg1.l10.m1.1.1.cmml"><mi id="alg1.l10.m1.1.1.3" mathsize="144%" xref="alg1.l10.m1.1.1.3.cmml">j</mi><mo id="alg1.l10.m1.1.1.2" mathsize="144%" xref="alg1.l10.m1.1.1.2.cmml">∈</mo><mrow id="alg1.l10.m1.1.1.1" xref="alg1.l10.m1.1.1.1.cmml"><mi id="alg1.l10.m1.1.1.1.3" mathsize="144%" xref="alg1.l10.m1.1.1.1.3.cmml">S</mi><mo id="alg1.l10.m1.1.1.1.2" xref="alg1.l10.m1.1.1.1.2.cmml">⁢</mo><mrow id="alg1.l10.m1.1.1.1.1.1" xref="alg1.l10.m1.1.1.1.1.1.1.cmml"><mo id="alg1.l10.m1.1.1.1.1.1.2" maxsize="144%" minsize="144%" xref="alg1.l10.m1.1.1.1.1.1.1.cmml">(</mo><msub id="alg1.l10.m1.1.1.1.1.1.1" xref="alg1.l10.m1.1.1.1.1.1.1.cmml"><mi id="alg1.l10.m1.1.1.1.1.1.1.2" mathsize="144%" xref="alg1.l10.m1.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="alg1.l10.m1.1.1.1.1.1.1.3" mathsize="144%" xref="alg1.l10.m1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="alg1.l10.m1.1.1.1.1.1.3" maxsize="144%" minsize="144%" xref="alg1.l10.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m1.1b"><apply id="alg1.l10.m1.1.1.cmml" xref="alg1.l10.m1.1.1"><in id="alg1.l10.m1.1.1.2.cmml" xref="alg1.l10.m1.1.1.2"></in><ci id="alg1.l10.m1.1.1.3.cmml" xref="alg1.l10.m1.1.1.3">𝑗</ci><apply id="alg1.l10.m1.1.1.1.cmml" xref="alg1.l10.m1.1.1.1"><times id="alg1.l10.m1.1.1.1.2.cmml" xref="alg1.l10.m1.1.1.1.2"></times><ci id="alg1.l10.m1.1.1.1.3.cmml" xref="alg1.l10.m1.1.1.1.3">𝑆</ci><apply id="alg1.l10.m1.1.1.1.1.1.1.cmml" xref="alg1.l10.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l10.m1.1.1.1.1.1.1.1.cmml" xref="alg1.l10.m1.1.1.1.1.1">subscript</csymbol><ci id="alg1.l10.m1.1.1.1.1.1.1.2.cmml" xref="alg1.l10.m1.1.1.1.1.1.1.2">𝒙</ci><ci id="alg1.l10.m1.1.1.1.1.1.1.3.cmml" xref="alg1.l10.m1.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m1.1c">j\in S(\boldsymbol{x}_{t})</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.m1.1d">italic_j ∈ italic_S ( bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> do </div> <div class="ltx_listingline" id="alg1.l11"> <math alttext="\widehat{\boldsymbol{m}}_{j},\widehat{\mathbf{C}}_{j}\leftarrow\text{Estim}(\{% \boldsymbol{s}^{(j)}_{t}\}_{t})" class="ltx_Math" display="inline" id="alg1.l11.m1.4"><semantics id="alg1.l11.m1.4a"><mrow id="alg1.l11.m1.4.4" xref="alg1.l11.m1.4.4.cmml"><mrow id="alg1.l11.m1.3.3.2.2" xref="alg1.l11.m1.3.3.2.3.cmml"><msub id="alg1.l11.m1.2.2.1.1.1" xref="alg1.l11.m1.2.2.1.1.1.cmml"><mover accent="true" id="alg1.l11.m1.2.2.1.1.1.2" xref="alg1.l11.m1.2.2.1.1.1.2.cmml"><mi id="alg1.l11.m1.2.2.1.1.1.2.2" mathsize="144%" xref="alg1.l11.m1.2.2.1.1.1.2.2.cmml">𝒎</mi><mo id="alg1.l11.m1.2.2.1.1.1.2.1" mathsize="144%" xref="alg1.l11.m1.2.2.1.1.1.2.1.cmml">^</mo></mover><mi id="alg1.l11.m1.2.2.1.1.1.3" mathsize="144%" xref="alg1.l11.m1.2.2.1.1.1.3.cmml">j</mi></msub><mo id="alg1.l11.m1.3.3.2.2.3" mathsize="144%" xref="alg1.l11.m1.3.3.2.3.cmml">,</mo><msub id="alg1.l11.m1.3.3.2.2.2" xref="alg1.l11.m1.3.3.2.2.2.cmml"><mover accent="true" id="alg1.l11.m1.3.3.2.2.2.2" xref="alg1.l11.m1.3.3.2.2.2.2.cmml"><mi id="alg1.l11.m1.3.3.2.2.2.2.2" mathsize="144%" xref="alg1.l11.m1.3.3.2.2.2.2.2.cmml">𝐂</mi><mo id="alg1.l11.m1.3.3.2.2.2.2.1" mathsize="144%" xref="alg1.l11.m1.3.3.2.2.2.2.1.cmml">^</mo></mover><mi id="alg1.l11.m1.3.3.2.2.2.3" mathsize="144%" xref="alg1.l11.m1.3.3.2.2.2.3.cmml">j</mi></msub></mrow><mo id="alg1.l11.m1.4.4.4" mathsize="144%" stretchy="false" xref="alg1.l11.m1.4.4.4.cmml">←</mo><mrow id="alg1.l11.m1.4.4.3" xref="alg1.l11.m1.4.4.3.cmml"><mtext id="alg1.l11.m1.4.4.3.3" mathsize="144%" xref="alg1.l11.m1.4.4.3.3a.cmml">Estim</mtext><mo id="alg1.l11.m1.4.4.3.2" xref="alg1.l11.m1.4.4.3.2.cmml">⁢</mo><mrow id="alg1.l11.m1.4.4.3.1.1" xref="alg1.l11.m1.4.4.3.1.1.1.cmml"><mo id="alg1.l11.m1.4.4.3.1.1.2" maxsize="144%" minsize="144%" xref="alg1.l11.m1.4.4.3.1.1.1.cmml">(</mo><msub id="alg1.l11.m1.4.4.3.1.1.1" xref="alg1.l11.m1.4.4.3.1.1.1.cmml"><mrow id="alg1.l11.m1.4.4.3.1.1.1.1.1" xref="alg1.l11.m1.4.4.3.1.1.1.1.2.cmml"><mo id="alg1.l11.m1.4.4.3.1.1.1.1.1.2" maxsize="144%" minsize="144%" xref="alg1.l11.m1.4.4.3.1.1.1.1.2.cmml">{</mo><msubsup id="alg1.l11.m1.4.4.3.1.1.1.1.1.1" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1.cmml"><mi id="alg1.l11.m1.4.4.3.1.1.1.1.1.1.2.2" mathsize="144%" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1.2.2.cmml">𝒔</mi><mi id="alg1.l11.m1.4.4.3.1.1.1.1.1.1.3" mathsize="144%" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1.3.cmml">t</mi><mrow id="alg1.l11.m1.1.1.1.3" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1.cmml"><mo id="alg1.l11.m1.1.1.1.3.1" maxsize="144%" minsize="144%" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1.cmml">(</mo><mi id="alg1.l11.m1.1.1.1.1" mathsize="144%" xref="alg1.l11.m1.1.1.1.1.cmml">j</mi><mo id="alg1.l11.m1.1.1.1.3.2" maxsize="144%" minsize="144%" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="alg1.l11.m1.4.4.3.1.1.1.1.1.3" maxsize="144%" minsize="144%" xref="alg1.l11.m1.4.4.3.1.1.1.1.2.cmml">}</mo></mrow><mi id="alg1.l11.m1.4.4.3.1.1.1.3" mathsize="144%" xref="alg1.l11.m1.4.4.3.1.1.1.3.cmml">t</mi></msub><mo id="alg1.l11.m1.4.4.3.1.1.3" maxsize="144%" minsize="144%" xref="alg1.l11.m1.4.4.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l11.m1.4b"><apply id="alg1.l11.m1.4.4.cmml" xref="alg1.l11.m1.4.4"><ci id="alg1.l11.m1.4.4.4.cmml" xref="alg1.l11.m1.4.4.4">←</ci><list id="alg1.l11.m1.3.3.2.3.cmml" xref="alg1.l11.m1.3.3.2.2"><apply id="alg1.l11.m1.2.2.1.1.1.cmml" xref="alg1.l11.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.l11.m1.2.2.1.1.1.1.cmml" xref="alg1.l11.m1.2.2.1.1.1">subscript</csymbol><apply id="alg1.l11.m1.2.2.1.1.1.2.cmml" xref="alg1.l11.m1.2.2.1.1.1.2"><ci id="alg1.l11.m1.2.2.1.1.1.2.1.cmml" xref="alg1.l11.m1.2.2.1.1.1.2.1">^</ci><ci id="alg1.l11.m1.2.2.1.1.1.2.2.cmml" xref="alg1.l11.m1.2.2.1.1.1.2.2">𝒎</ci></apply><ci id="alg1.l11.m1.2.2.1.1.1.3.cmml" xref="alg1.l11.m1.2.2.1.1.1.3">𝑗</ci></apply><apply id="alg1.l11.m1.3.3.2.2.2.cmml" xref="alg1.l11.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="alg1.l11.m1.3.3.2.2.2.1.cmml" xref="alg1.l11.m1.3.3.2.2.2">subscript</csymbol><apply id="alg1.l11.m1.3.3.2.2.2.2.cmml" xref="alg1.l11.m1.3.3.2.2.2.2"><ci id="alg1.l11.m1.3.3.2.2.2.2.1.cmml" xref="alg1.l11.m1.3.3.2.2.2.2.1">^</ci><ci id="alg1.l11.m1.3.3.2.2.2.2.2.cmml" xref="alg1.l11.m1.3.3.2.2.2.2.2">𝐂</ci></apply><ci id="alg1.l11.m1.3.3.2.2.2.3.cmml" xref="alg1.l11.m1.3.3.2.2.2.3">𝑗</ci></apply></list><apply id="alg1.l11.m1.4.4.3.cmml" xref="alg1.l11.m1.4.4.3"><times id="alg1.l11.m1.4.4.3.2.cmml" xref="alg1.l11.m1.4.4.3.2"></times><ci id="alg1.l11.m1.4.4.3.3a.cmml" xref="alg1.l11.m1.4.4.3.3"><mtext id="alg1.l11.m1.4.4.3.3.cmml" mathsize="144%" xref="alg1.l11.m1.4.4.3.3">Estim</mtext></ci><apply id="alg1.l11.m1.4.4.3.1.1.1.cmml" xref="alg1.l11.m1.4.4.3.1.1"><csymbol cd="ambiguous" id="alg1.l11.m1.4.4.3.1.1.1.2.cmml" xref="alg1.l11.m1.4.4.3.1.1">subscript</csymbol><set id="alg1.l11.m1.4.4.3.1.1.1.1.2.cmml" xref="alg1.l11.m1.4.4.3.1.1.1.1.1"><apply id="alg1.l11.m1.4.4.3.1.1.1.1.1.1.cmml" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l11.m1.4.4.3.1.1.1.1.1.1.1.cmml" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1">subscript</csymbol><apply id="alg1.l11.m1.4.4.3.1.1.1.1.1.1.2.cmml" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l11.m1.4.4.3.1.1.1.1.1.1.2.1.cmml" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1">superscript</csymbol><ci id="alg1.l11.m1.4.4.3.1.1.1.1.1.1.2.2.cmml" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1.2.2">𝒔</ci><ci id="alg1.l11.m1.1.1.1.1.cmml" xref="alg1.l11.m1.1.1.1.1">𝑗</ci></apply><ci id="alg1.l11.m1.4.4.3.1.1.1.1.1.1.3.cmml" xref="alg1.l11.m1.4.4.3.1.1.1.1.1.1.3">𝑡</ci></apply></set><ci id="alg1.l11.m1.4.4.3.1.1.1.3.cmml" xref="alg1.l11.m1.4.4.3.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.m1.4c">\widehat{\boldsymbol{m}}_{j},\widehat{\mathbf{C}}_{j}\leftarrow\text{Estim}(\{% \boldsymbol{s}^{(j)}_{t}\}_{t})</annotation><annotation encoding="application/x-llamapun" id="alg1.l11.m1.4d">over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ← Estim ( { bold_italic_s start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> <math alttext="\triangleright" class="ltx_Math" display="inline" id="alg1.l11.1.m1.1"><semantics id="alg1.l11.1.m1.1a"><mo id="alg1.l11.1.m1.1.1" xref="alg1.l11.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg1.l11.1.m1.1b"><ci id="alg1.l11.1.m1.1.1.cmml" xref="alg1.l11.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.1.m1.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg1.l11.1.m1.1d">▷</annotation></semantics></math> Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.Ex2" title="A.1 Estimation of the statistical parameters ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">A.1</a>)–(<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E24" title="Equation 24 ‣ A.1 Estimation of the statistical parameters ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">24</a>) </div> <div class="ltx_listingline" id="alg1.l12"> <math alttext="\boldsymbol{g}\leftarrow\boldsymbol{g}+w_{j}\frac{\partial\ell_{j}}{\partial% \boldsymbol{z}}(\boldsymbol{z},\widehat{\boldsymbol{m}}_{j},\widehat{\mathbf{C% }}_{j})" class="ltx_Math" display="inline" id="alg1.l12.m1.3"><semantics id="alg1.l12.m1.3a"><mrow id="alg1.l12.m1.3.3" xref="alg1.l12.m1.3.3.cmml"><mi id="alg1.l12.m1.3.3.4" 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xref="alg1.l13.m1.3.3.2.2.2.2.2.3">𝑗</ci></apply></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l13.m1.3c">\mathbf{H}\leftarrow\mathbf{H}+w_{j}\frac{\partial^{2}\ell_{j}}{\partial% \boldsymbol{z}^{2}}(\boldsymbol{z},\widehat{\boldsymbol{m}}_{j},\widehat{% \mathbf{C}}_{j})</annotation><annotation encoding="application/x-llamapun" id="alg1.l13.m1.3d">bold_H ← bold_H + italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG ∂ bold_italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( bold_italic_z , over^ start_ARG bold_italic_m end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , over^ start_ARG bold_C end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l14"> end for </div> <div class="ltx_listingline" id="alg1.l15"> end for </div> <div class="ltx_listingline" id="alg1.l16"> <math alttext="\boldsymbol{z}\leftarrow\text{Proj}_{Z}(\boldsymbol{z}-\mathbf{H}^{-1}% \boldsymbol{g}" class="ltx_math_unparsed" display="inline" id="alg1.l16.m1.1"><semantics id="alg1.l16.m1.1a"><mrow id="alg1.l16.m1.1b"><mi id="alg1.l16.m1.1.1" mathsize="144%">𝒛</mi><mo id="alg1.l16.m1.1.2" mathsize="144%" stretchy="false">←</mo><msub id="alg1.l16.m1.1.3"><mtext id="alg1.l16.m1.1.3.2" mathsize="144%">Proj</mtext><mi id="alg1.l16.m1.1.3.3" mathsize="144%">Z</mi></msub><mrow id="alg1.l16.m1.1.4"><mo id="alg1.l16.m1.1.4.1" maxsize="144%" minsize="144%">(</mo><mi id="alg1.l16.m1.1.4.2" mathsize="144%">𝒛</mi><mo id="alg1.l16.m1.1.4.3" mathsize="144%">−</mo><msup id="alg1.l16.m1.1.4.4"><mi id="alg1.l16.m1.1.4.4.2" mathsize="144%">𝐇</mi><mrow id="alg1.l16.m1.1.4.4.3"><mo id="alg1.l16.m1.1.4.4.3a" mathsize="144%">−</mo><mn id="alg1.l16.m1.1.4.4.3.2" mathsize="144%">1</mn></mrow></msup><mi id="alg1.l16.m1.1.4.5" mathsize="144%">𝒈</mi></mrow></mrow><annotation encoding="application/x-tex" id="alg1.l16.m1.1c">\boldsymbol{z}\leftarrow\text{Proj}_{Z}(\boldsymbol{z}-\mathbf{H}^{-1}% \boldsymbol{g}</annotation><annotation encoding="application/x-llamapun" id="alg1.l16.m1.1d">bold_italic_z ← Proj start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ( bold_italic_z - bold_H start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT bold_italic_g</annotation></semantics></math>) <math alttext="\triangleright" class="ltx_Math" display="inline" id="alg1.l16.1.m1.1"><semantics id="alg1.l16.1.m1.1a"><mo id="alg1.l16.1.m1.1.1" xref="alg1.l16.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg1.l16.1.m1.1b"><ci id="alg1.l16.1.m1.1.1.cmml" xref="alg1.l16.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l16.1.m1.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg1.l16.1.m1.1d">▷</annotation></semantics></math> Projection on feasible set </div> <div class="ltx_listingline" id="alg1.l17"> end for </div> <div class="ltx_listingline" id="alg1.l18"> return <math alttext="\boldsymbol{z}" class="ltx_Math" display="inline" id="alg1.l18.m1.1"><semantics id="alg1.l18.m1.1a"><mi id="alg1.l18.m1.1.1" mathsize="144%" xref="alg1.l18.m1.1.1.cmml">𝒛</mi><annotation-xml encoding="MathML-Content" id="alg1.l18.m1.1b"><ci id="alg1.l18.m1.1.1.cmml" xref="alg1.l18.m1.1.1">𝒛</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l18.m1.1c">\boldsymbol{z}</annotation><annotation encoding="application/x-llamapun" id="alg1.l18.m1.1d">bold_italic_z</annotation></semantics></math> </div> </div> </figure> </section> <section class="ltx_subsection" id="A1.SS5"> <h3 class="ltx_title ltx_title_subsection" style="font-size:144%;"> A.5 Prior information on the exoplanet spectrum</h3> <div class="ltx_para" id="A1.SS5.p1"> The channel-wise detection maps <math alttext="\widetilde{\boldsymbol{\gamma}}_{c}=\widetilde{\boldsymbol{b}}_{c}/\sqrt{% 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xref="A1.SS5.p1.1.m1.1.1.3.3.2.3">𝑐</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS5.p1.1.m1.1c">\widetilde{\boldsymbol{\gamma}}_{c}=\widetilde{\boldsymbol{b}}_{c}/\sqrt{% \widetilde{\boldsymbol{a}}_{c}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS5.p1.1.m1.1d">over~ start_ARG bold_italic_γ end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = over~ start_ARG bold_italic_b end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT / square-root start_ARG over~ start_ARG bold_italic_a end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> are combined into a single detection map by averaging <math alttext="\widetilde{\boldsymbol{a}}_{c}" class="ltx_Math" display="inline" id="A1.SS5.p1.2.m2.1"><semantics id="A1.SS5.p1.2.m2.1a"><msub id="A1.SS5.p1.2.m2.1.1" xref="A1.SS5.p1.2.m2.1.1.cmml"><mover accent="true" id="A1.SS5.p1.2.m2.1.1.2" xref="A1.SS5.p1.2.m2.1.1.2.cmml"><mi id="A1.SS5.p1.2.m2.1.1.2.2" mathsize="144%" xref="A1.SS5.p1.2.m2.1.1.2.2.cmml">𝒂</mi><mo id="A1.SS5.p1.2.m2.1.1.2.1" mathsize="144%" xref="A1.SS5.p1.2.m2.1.1.2.1.cmml">~</mo></mover><mi id="A1.SS5.p1.2.m2.1.1.3" mathsize="144%" xref="A1.SS5.p1.2.m2.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS5.p1.2.m2.1b"><apply id="A1.SS5.p1.2.m2.1.1.cmml" xref="A1.SS5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS5.p1.2.m2.1.1.1.cmml" xref="A1.SS5.p1.2.m2.1.1">subscript</csymbol><apply id="A1.SS5.p1.2.m2.1.1.2.cmml" xref="A1.SS5.p1.2.m2.1.1.2"><ci id="A1.SS5.p1.2.m2.1.1.2.1.cmml" xref="A1.SS5.p1.2.m2.1.1.2.1">~</ci><ci id="A1.SS5.p1.2.m2.1.1.2.2.cmml" xref="A1.SS5.p1.2.m2.1.1.2.2">𝒂</ci></apply><ci id="A1.SS5.p1.2.m2.1.1.3.cmml" xref="A1.SS5.p1.2.m2.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS5.p1.2.m2.1c">\widetilde{\boldsymbol{a}}_{c}</annotation><annotation encoding="application/x-llamapun" id="A1.SS5.p1.2.m2.1d">over~ start_ARG bold_italic_a end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\widetilde{\boldsymbol{b}}_{c}" class="ltx_Math" display="inline" id="A1.SS5.p1.3.m3.1"><semantics id="A1.SS5.p1.3.m3.1a"><msub id="A1.SS5.p1.3.m3.1.1" xref="A1.SS5.p1.3.m3.1.1.cmml"><mover accent="true" id="A1.SS5.p1.3.m3.1.1.2" xref="A1.SS5.p1.3.m3.1.1.2.cmml"><mi id="A1.SS5.p1.3.m3.1.1.2.2" mathsize="144%" xref="A1.SS5.p1.3.m3.1.1.2.2.cmml">𝒃</mi><mo id="A1.SS5.p1.3.m3.1.1.2.1" mathsize="144%" xref="A1.SS5.p1.3.m3.1.1.2.1.cmml">~</mo></mover><mi id="A1.SS5.p1.3.m3.1.1.3" mathsize="144%" xref="A1.SS5.p1.3.m3.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS5.p1.3.m3.1b"><apply id="A1.SS5.p1.3.m3.1.1.cmml" xref="A1.SS5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS5.p1.3.m3.1.1.1.cmml" xref="A1.SS5.p1.3.m3.1.1">subscript</csymbol><apply id="A1.SS5.p1.3.m3.1.1.2.cmml" xref="A1.SS5.p1.3.m3.1.1.2"><ci id="A1.SS5.p1.3.m3.1.1.2.1.cmml" xref="A1.SS5.p1.3.m3.1.1.2.1">~</ci><ci id="A1.SS5.p1.3.m3.1.1.2.2.cmml" xref="A1.SS5.p1.3.m3.1.1.2.2">𝒃</ci></apply><ci id="A1.SS5.p1.3.m3.1.1.3.cmml" xref="A1.SS5.p1.3.m3.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS5.p1.3.m3.1c">\widetilde{\boldsymbol{b}}_{c}</annotation><annotation encoding="application/x-llamapun" id="A1.SS5.p1.3.m3.1d">over~ start_ARG bold_italic_b end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> with weights <math alttext="e_{c}" class="ltx_Math" display="inline" id="A1.SS5.p1.4.m4.1"><semantics id="A1.SS5.p1.4.m4.1a"><msub id="A1.SS5.p1.4.m4.1.1" xref="A1.SS5.p1.4.m4.1.1.cmml"><mi id="A1.SS5.p1.4.m4.1.1.2" mathsize="144%" xref="A1.SS5.p1.4.m4.1.1.2.cmml">e</mi><mi id="A1.SS5.p1.4.m4.1.1.3" mathsize="144%" xref="A1.SS5.p1.4.m4.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS5.p1.4.m4.1b"><apply id="A1.SS5.p1.4.m4.1.1.cmml" xref="A1.SS5.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A1.SS5.p1.4.m4.1.1.1.cmml" xref="A1.SS5.p1.4.m4.1.1">subscript</csymbol><ci id="A1.SS5.p1.4.m4.1.1.2.cmml" xref="A1.SS5.p1.4.m4.1.1.2">𝑒</ci><ci id="A1.SS5.p1.4.m4.1.1.3.cmml" xref="A1.SS5.p1.4.m4.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS5.p1.4.m4.1c">e_{c}</annotation><annotation encoding="application/x-llamapun" id="A1.SS5.p1.4.m4.1d">italic_e start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathbb{R}^{+}" class="ltx_Math" display="inline" id="A1.SS5.p1.5.m5.1"><semantics id="A1.SS5.p1.5.m5.1a"><msup id="A1.SS5.p1.5.m5.1.1" xref="A1.SS5.p1.5.m5.1.1.cmml"><mi id="A1.SS5.p1.5.m5.1.1.2" mathsize="144%" xref="A1.SS5.p1.5.m5.1.1.2.cmml">ℝ</mi><mo id="A1.SS5.p1.5.m5.1.1.3" mathsize="144%" xref="A1.SS5.p1.5.m5.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS5.p1.5.m5.1b"><apply id="A1.SS5.p1.5.m5.1.1.cmml" xref="A1.SS5.p1.5.m5.1.1"><csymbol cd="ambiguous" id="A1.SS5.p1.5.m5.1.1.1.cmml" xref="A1.SS5.p1.5.m5.1.1">superscript</csymbol><ci id="A1.SS5.p1.5.m5.1.1.2.cmml" xref="A1.SS5.p1.5.m5.1.1.2">ℝ</ci><plus id="A1.SS5.p1.5.m5.1.1.3.cmml" xref="A1.SS5.p1.5.m5.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS5.p1.5.m5.1c">\mathbb{R}^{+}</annotation><annotation encoding="application/x-llamapun" id="A1.SS5.p1.5.m5.1d">blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> satisfying <math alttext="\sum_{c}e_{c}=1" class="ltx_Math" display="inline" id="A1.SS5.p1.6.m6.1"><semantics id="A1.SS5.p1.6.m6.1a"><mrow id="A1.SS5.p1.6.m6.1.1" xref="A1.SS5.p1.6.m6.1.1.cmml"><mrow id="A1.SS5.p1.6.m6.1.1.2" xref="A1.SS5.p1.6.m6.1.1.2.cmml"><msub id="A1.SS5.p1.6.m6.1.1.2.1" xref="A1.SS5.p1.6.m6.1.1.2.1.cmml"><mo id="A1.SS5.p1.6.m6.1.1.2.1.2" maxsize="144%" minsize="144%" stretchy="true" xref="A1.SS5.p1.6.m6.1.1.2.1.2.cmml">∑</mo><mi id="A1.SS5.p1.6.m6.1.1.2.1.3" mathsize="144%" xref="A1.SS5.p1.6.m6.1.1.2.1.3.cmml">c</mi></msub><msub id="A1.SS5.p1.6.m6.1.1.2.2" xref="A1.SS5.p1.6.m6.1.1.2.2.cmml"><mi id="A1.SS5.p1.6.m6.1.1.2.2.2" mathsize="144%" xref="A1.SS5.p1.6.m6.1.1.2.2.2.cmml">e</mi><mi id="A1.SS5.p1.6.m6.1.1.2.2.3" mathsize="144%" xref="A1.SS5.p1.6.m6.1.1.2.2.3.cmml">c</mi></msub></mrow><mo id="A1.SS5.p1.6.m6.1.1.1" mathsize="144%" xref="A1.SS5.p1.6.m6.1.1.1.cmml">=</mo><mn id="A1.SS5.p1.6.m6.1.1.3" mathsize="144%" xref="A1.SS5.p1.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS5.p1.6.m6.1b"><apply id="A1.SS5.p1.6.m6.1.1.cmml" xref="A1.SS5.p1.6.m6.1.1"><eq id="A1.SS5.p1.6.m6.1.1.1.cmml" xref="A1.SS5.p1.6.m6.1.1.1"></eq><apply id="A1.SS5.p1.6.m6.1.1.2.cmml" xref="A1.SS5.p1.6.m6.1.1.2"><apply id="A1.SS5.p1.6.m6.1.1.2.1.cmml" xref="A1.SS5.p1.6.m6.1.1.2.1"><csymbol cd="ambiguous" id="A1.SS5.p1.6.m6.1.1.2.1.1.cmml" xref="A1.SS5.p1.6.m6.1.1.2.1">subscript</csymbol><sum id="A1.SS5.p1.6.m6.1.1.2.1.2.cmml" xref="A1.SS5.p1.6.m6.1.1.2.1.2"></sum><ci id="A1.SS5.p1.6.m6.1.1.2.1.3.cmml" xref="A1.SS5.p1.6.m6.1.1.2.1.3">𝑐</ci></apply><apply id="A1.SS5.p1.6.m6.1.1.2.2.cmml" xref="A1.SS5.p1.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="A1.SS5.p1.6.m6.1.1.2.2.1.cmml" xref="A1.SS5.p1.6.m6.1.1.2.2">subscript</csymbol><ci id="A1.SS5.p1.6.m6.1.1.2.2.2.cmml" xref="A1.SS5.p1.6.m6.1.1.2.2.2">𝑒</ci><ci id="A1.SS5.p1.6.m6.1.1.2.2.3.cmml" xref="A1.SS5.p1.6.m6.1.1.2.2.3">𝑐</ci></apply></apply><cn id="A1.SS5.p1.6.m6.1.1.3.cmml" type="integer" xref="A1.SS5.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS5.p1.6.m6.1c">\sum_{c}e_{c}=1</annotation><annotation encoding="application/x-llamapun" id="A1.SS5.p1.6.m6.1d">∑ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 1</annotation></semantics></math> such that: <table class="ltx_equation ltx_eqn_table" id="A1.E37"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\boldsymbol{a}}_{C}=\sum_{c}e_{c}\,\widetilde{\boldsymbol{a}}_{c}\,% \quad\text{and}\quad\widetilde{\boldsymbol{b}}_{C}=\sum_{c}e_{c}\,\widetilde{% \boldsymbol{b}}_{c}\,." class="ltx_Math" display="block" id="A1.E37.m1.2"><semantics id="A1.E37.m1.2a"><mrow id="A1.E37.m1.2.2.1"><mrow id="A1.E37.m1.2.2.1.1.2" xref="A1.E37.m1.2.2.1.1.3.cmml"><mrow id="A1.E37.m1.2.2.1.1.1.1" xref="A1.E37.m1.2.2.1.1.1.1.cmml"><msub id="A1.E37.m1.2.2.1.1.1.1.3" xref="A1.E37.m1.2.2.1.1.1.1.3.cmml"><mover accent="true" id="A1.E37.m1.2.2.1.1.1.1.3.2" xref="A1.E37.m1.2.2.1.1.1.1.3.2.cmml"><mi id="A1.E37.m1.2.2.1.1.1.1.3.2.2" mathsize="144%" xref="A1.E37.m1.2.2.1.1.1.1.3.2.2.cmml">𝒂</mi><mo id="A1.E37.m1.2.2.1.1.1.1.3.2.1" mathsize="144%" xref="A1.E37.m1.2.2.1.1.1.1.3.2.1.cmml">~</mo></mover><mi id="A1.E37.m1.2.2.1.1.1.1.3.3" mathsize="144%" 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end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT over~ start_ARG bold_italic_b end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(37)</td> </tr></tbody> </table> The final detection score, spectrally combined over <math alttext="C" class="ltx_Math" display="inline" id="A1.SS5.p1.7.m1.1"><semantics id="A1.SS5.p1.7.m1.1a"><mi id="A1.SS5.p1.7.m1.1.1" mathsize="144%" xref="A1.SS5.p1.7.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="A1.SS5.p1.7.m1.1b"><ci id="A1.SS5.p1.7.m1.1.1.cmml" xref="A1.SS5.p1.7.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS5.p1.7.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="A1.SS5.p1.7.m1.1d">italic_C</annotation></semantics></math> 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The weights <math alttext="e_{c}" class="ltx_Math" display="inline" id="A1.SS5.p1.10.m4.1"><semantics id="A1.SS5.p1.10.m4.1a"><msub id="A1.SS5.p1.10.m4.1.1" xref="A1.SS5.p1.10.m4.1.1.cmml"><mi id="A1.SS5.p1.10.m4.1.1.2" mathsize="144%" xref="A1.SS5.p1.10.m4.1.1.2.cmml">e</mi><mi id="A1.SS5.p1.10.m4.1.1.3" mathsize="144%" xref="A1.SS5.p1.10.m4.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS5.p1.10.m4.1b"><apply id="A1.SS5.p1.10.m4.1.1.cmml" xref="A1.SS5.p1.10.m4.1.1"><csymbol cd="ambiguous" id="A1.SS5.p1.10.m4.1.1.1.cmml" xref="A1.SS5.p1.10.m4.1.1">subscript</csymbol><ci id="A1.SS5.p1.10.m4.1.1.2.cmml" xref="A1.SS5.p1.10.m4.1.1.2">𝑒</ci><ci id="A1.SS5.p1.10.m4.1.1.3.cmml" xref="A1.SS5.p1.10.m4.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS5.p1.10.m4.1c">e_{c}</annotation><annotation encoding="application/x-llamapun" id="A1.SS5.p1.10.m4.1d">italic_e start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> reflect prior assumptions about the exoplanet spectrum, and can be used (if available) to boost the detection of exoplanets having similar spectrum, see e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib12" title="">12</a>]</cite>. For all experiments and baselines, we assumed a non-informative flat prior (i.e., <math alttext="e_{c}=1/C" class="ltx_Math" display="inline" id="A1.SS5.p1.11.m5.1"><semantics id="A1.SS5.p1.11.m5.1a"><mrow id="A1.SS5.p1.11.m5.1.1" xref="A1.SS5.p1.11.m5.1.1.cmml"><msub id="A1.SS5.p1.11.m5.1.1.2" xref="A1.SS5.p1.11.m5.1.1.2.cmml"><mi id="A1.SS5.p1.11.m5.1.1.2.2" mathsize="144%" xref="A1.SS5.p1.11.m5.1.1.2.2.cmml">e</mi><mi id="A1.SS5.p1.11.m5.1.1.2.3" mathsize="144%" xref="A1.SS5.p1.11.m5.1.1.2.3.cmml">c</mi></msub><mo id="A1.SS5.p1.11.m5.1.1.1" mathsize="144%" xref="A1.SS5.p1.11.m5.1.1.1.cmml">=</mo><mrow id="A1.SS5.p1.11.m5.1.1.3" xref="A1.SS5.p1.11.m5.1.1.3.cmml"><mn id="A1.SS5.p1.11.m5.1.1.3.2" mathsize="144%" xref="A1.SS5.p1.11.m5.1.1.3.2.cmml">1</mn><mo id="A1.SS5.p1.11.m5.1.1.3.1" maxsize="144%" minsize="144%" stretchy="true" symmetric="true" xref="A1.SS5.p1.11.m5.1.1.3.1.cmml">/</mo><mi id="A1.SS5.p1.11.m5.1.1.3.3" mathsize="144%" xref="A1.SS5.p1.11.m5.1.1.3.3.cmml">C</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS5.p1.11.m5.1b"><apply id="A1.SS5.p1.11.m5.1.1.cmml" xref="A1.SS5.p1.11.m5.1.1"><eq id="A1.SS5.p1.11.m5.1.1.1.cmml" xref="A1.SS5.p1.11.m5.1.1.1"></eq><apply id="A1.SS5.p1.11.m5.1.1.2.cmml" xref="A1.SS5.p1.11.m5.1.1.2"><csymbol cd="ambiguous" id="A1.SS5.p1.11.m5.1.1.2.1.cmml" xref="A1.SS5.p1.11.m5.1.1.2">subscript</csymbol><ci id="A1.SS5.p1.11.m5.1.1.2.2.cmml" xref="A1.SS5.p1.11.m5.1.1.2.2">𝑒</ci><ci id="A1.SS5.p1.11.m5.1.1.2.3.cmml" xref="A1.SS5.p1.11.m5.1.1.2.3">𝑐</ci></apply><apply id="A1.SS5.p1.11.m5.1.1.3.cmml" xref="A1.SS5.p1.11.m5.1.1.3"><divide id="A1.SS5.p1.11.m5.1.1.3.1.cmml" xref="A1.SS5.p1.11.m5.1.1.3.1"></divide><cn id="A1.SS5.p1.11.m5.1.1.3.2.cmml" type="integer" xref="A1.SS5.p1.11.m5.1.1.3.2">1</cn><ci id="A1.SS5.p1.11.m5.1.1.3.3.cmml" xref="A1.SS5.p1.11.m5.1.1.3.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS5.p1.11.m5.1c">e_{c}=1/C</annotation><annotation encoding="application/x-llamapun" id="A1.SS5.p1.11.m5.1d">italic_e start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 1 / italic_C</annotation></semantics></math>). If additional information about the exoplanet’s spectrum is available, the prior can be adjusted accordingly. </div> </section> <section class="ltx_subsection" id="A1.SS6"> <h3 class="ltx_title ltx_title_subsection" style="font-size:144%;"> A.6 Calibration procedure</h3> <figure class="ltx_figure" id="A1.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="364" id="A1.F6.g1" src="x6.png" width="380"/> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">Figure F: Calibrated PFA versus test PFA: the calibration procedure allows obtaining a reliable approximation of the PFA, except in the extreme low-PFA regime (high detection score under <math alttext="\mathcal{H}_{0}" class="ltx_Math" display="inline" id="A1.F6.2.1.m1.1"><semantics id="A1.F6.2.1.m1.1b"><msub id="A1.F6.2.1.m1.1.1" xref="A1.F6.2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.F6.2.1.m1.1.1.2" xref="A1.F6.2.1.m1.1.1.2.cmml">ℋ</mi><mn id="A1.F6.2.1.m1.1.1.3" xref="A1.F6.2.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A1.F6.2.1.m1.1c"><apply id="A1.F6.2.1.m1.1.1.cmml" xref="A1.F6.2.1.m1.1.1"><csymbol cd="ambiguous" id="A1.F6.2.1.m1.1.1.1.cmml" xref="A1.F6.2.1.m1.1.1">subscript</csymbol><ci id="A1.F6.2.1.m1.1.1.2.cmml" xref="A1.F6.2.1.m1.1.1.2">ℋ</ci><cn id="A1.F6.2.1.m1.1.1.3.cmml" type="integer" xref="A1.F6.2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F6.2.1.m1.1d">\mathcal{H}_{0}</annotation><annotation encoding="application/x-llamapun" id="A1.F6.2.1.m1.1e">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>) where samples are exceedingly rare.</figcaption> </figure> <div class="ltx_para" id="A1.SS6.p1"> This section outlines the calibration procedure for the full model described in <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#S4.SS3" style="font-size:144%;" title="4.3 End-to-end trainable approach ‣ 4 Proposed method ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Sec. 4.3</a>, following the approach proposed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#bib.bib5" title="">5</a>]</cite>, which we briefly summarize here. </div> <figure class="ltx_figure" id="A1.F7"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_3"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="A1.F7.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="951" id="A1.F7.sf1.g1" src="x7.png" width="951"/> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">(a) Distribution of the number of frames per observation</figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_3"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="A1.F7.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="951" id="A1.F7.sf2.g1" src="x8.png" width="951"/> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">(b) Repartition of the filters used</figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_3"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="A1.F7.sf3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="951" id="A1.F7.sf3.g1" src="x9.png" width="951"/> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">(c) Distribution of the parallactic rotation per observation</figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">Figure G: Additional details on the F150 dataset</figcaption> </figure> <div class="ltx_para ltx_noindent" id="A1.SS6.p2"> We aim to calibrate the predicted probability of false alarm (PFA), a critical metric for astronomers to quantify detection uncertainty. 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Temporal shuffling of the frames and parallactic rotation reversal are applied to augment the dataset and ensure no real source is present. These custom data-augmentations disrupt the temporal coherence of any (potentially unknown) real exoplanets, preventing them from biasing the training and calibration processes. 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xref="A1.SS6.p2.6.m6.1.1.cmml"><mi id="A1.SS6.p2.6.m6.1.1.2" mathsize="144%" xref="A1.SS6.p2.6.m6.1.1.2.cmml">n</mi><mtext id="A1.SS6.p2.6.m6.1.1.3" mathsize="144%" xref="A1.SS6.p2.6.m6.1.1.3a.cmml">calib</mtext></msub><annotation-xml encoding="MathML-Content" id="A1.SS6.p2.6.m6.1b"><apply id="A1.SS6.p2.6.m6.1.1.cmml" xref="A1.SS6.p2.6.m6.1.1"><csymbol cd="ambiguous" id="A1.SS6.p2.6.m6.1.1.1.cmml" xref="A1.SS6.p2.6.m6.1.1">subscript</csymbol><ci id="A1.SS6.p2.6.m6.1.1.2.cmml" xref="A1.SS6.p2.6.m6.1.1.2">𝑛</ci><ci id="A1.SS6.p2.6.m6.1.1.3a.cmml" xref="A1.SS6.p2.6.m6.1.1.3"><mtext id="A1.SS6.p2.6.m6.1.1.3.cmml" mathsize="101%" xref="A1.SS6.p2.6.m6.1.1.3">calib</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS6.p2.6.m6.1c">n_{\text{calib}}</annotation><annotation encoding="application/x-llamapun" id="A1.SS6.p2.6.m6.1d">italic_n start_POSTSUBSCRIPT calib end_POSTSUBSCRIPT</annotation></semantics></math>, and compute the empirical cumulative distribution function (eCDF): <table class="ltx_equation ltx_eqn_table" id="A1.E38"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall\,\tau\in\mathbb{R},\quad\widehat{F}_{\widetilde{\boldsymbol{\gamma}}|% \mathcal{H}_{0}}(\tau):=\frac{1}{n_{\text{calib}}}\sum_{i=0}^{n_{\text{calib}}% -1}{\mathbf{1}}_{\widetilde{\boldsymbol{\gamma}}_{\text{calib},i}\leq\tau}\,." class="ltx_Math" display="block" id="A1.E38.m1.4"><semantics id="A1.E38.m1.4a"><mrow id="A1.E38.m1.4.4.1"><mrow id="A1.E38.m1.4.4.1.1.2" xref="A1.E38.m1.4.4.1.1.3.cmml"><mrow id="A1.E38.m1.4.4.1.1.1.1" xref="A1.E38.m1.4.4.1.1.1.1.cmml"><mrow id="A1.E38.m1.4.4.1.1.1.1.2" xref="A1.E38.m1.4.4.1.1.1.1.2.cmml"><mo id="A1.E38.m1.4.4.1.1.1.1.2.1" mathsize="144%" rspace="0.337em" xref="A1.E38.m1.4.4.1.1.1.1.2.1.cmml">∀</mo><mi id="A1.E38.m1.4.4.1.1.1.1.2.2" mathsize="144%" xref="A1.E38.m1.4.4.1.1.1.1.2.2.cmml">τ</mi></mrow><mo id="A1.E38.m1.4.4.1.1.1.1.1" mathsize="144%" xref="A1.E38.m1.4.4.1.1.1.1.1.cmml">∈</mo><mi id="A1.E38.m1.4.4.1.1.1.1.3" mathsize="144%" xref="A1.E38.m1.4.4.1.1.1.1.3.cmml">ℝ</mi></mrow><mo id="A1.E38.m1.4.4.1.1.2.3" mathsize="144%" rspace="1.607em" xref="A1.E38.m1.4.4.1.1.3a.cmml">,</mo><mrow id="A1.E38.m1.4.4.1.1.2.2" xref="A1.E38.m1.4.4.1.1.2.2.cmml"><mrow id="A1.E38.m1.4.4.1.1.2.2.2" xref="A1.E38.m1.4.4.1.1.2.2.2.cmml"><msub id="A1.E38.m1.4.4.1.1.2.2.2.2" xref="A1.E38.m1.4.4.1.1.2.2.2.2.cmml"><mover accent="true" id="A1.E38.m1.4.4.1.1.2.2.2.2.2" xref="A1.E38.m1.4.4.1.1.2.2.2.2.2.cmml"><mi id="A1.E38.m1.4.4.1.1.2.2.2.2.2.2" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.2.2.2.2.cmml">F</mi><mo id="A1.E38.m1.4.4.1.1.2.2.2.2.2.1" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.2.2.2.1.cmml">^</mo></mover><mrow id="A1.E38.m1.4.4.1.1.2.2.2.2.3" xref="A1.E38.m1.4.4.1.1.2.2.2.2.3.cmml"><mover accent="true" id="A1.E38.m1.4.4.1.1.2.2.2.2.3.2" xref="A1.E38.m1.4.4.1.1.2.2.2.2.3.2.cmml"><mi id="A1.E38.m1.4.4.1.1.2.2.2.2.3.2.2" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.2.2.3.2.2.cmml">𝜸</mi><mo id="A1.E38.m1.4.4.1.1.2.2.2.2.3.2.1" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.2.2.3.2.1.cmml">~</mo></mover><mo fence="false" id="A1.E38.m1.4.4.1.1.2.2.2.2.3.1" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.2.2.3.1.cmml">|</mo><msub id="A1.E38.m1.4.4.1.1.2.2.2.2.3.3" xref="A1.E38.m1.4.4.1.1.2.2.2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.E38.m1.4.4.1.1.2.2.2.2.3.3.2" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.2.2.3.3.2.cmml">ℋ</mi><mn id="A1.E38.m1.4.4.1.1.2.2.2.2.3.3.3" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.2.2.3.3.3.cmml">0</mn></msub></mrow></msub><mo id="A1.E38.m1.4.4.1.1.2.2.2.1" xref="A1.E38.m1.4.4.1.1.2.2.2.1.cmml">⁢</mo><mrow id="A1.E38.m1.4.4.1.1.2.2.2.3.2" xref="A1.E38.m1.4.4.1.1.2.2.2.cmml"><mo id="A1.E38.m1.4.4.1.1.2.2.2.3.2.1" maxsize="144%" minsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.2.cmml">(</mo><mi id="A1.E38.m1.3.3" mathsize="144%" xref="A1.E38.m1.3.3.cmml">τ</mi><mo id="A1.E38.m1.4.4.1.1.2.2.2.3.2.2" maxsize="144%" minsize="144%" rspace="0.278em" xref="A1.E38.m1.4.4.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="A1.E38.m1.4.4.1.1.2.2.1" mathsize="144%" rspace="0.278em" xref="A1.E38.m1.4.4.1.1.2.2.1.cmml">:=</mo><mrow id="A1.E38.m1.4.4.1.1.2.2.3" xref="A1.E38.m1.4.4.1.1.2.2.3.cmml"><mfrac id="A1.E38.m1.4.4.1.1.2.2.3.2" xref="A1.E38.m1.4.4.1.1.2.2.3.2.cmml"><mn id="A1.E38.m1.4.4.1.1.2.2.3.2.2" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.3.2.2.cmml">1</mn><msub id="A1.E38.m1.4.4.1.1.2.2.3.2.3" xref="A1.E38.m1.4.4.1.1.2.2.3.2.3.cmml"><mi id="A1.E38.m1.4.4.1.1.2.2.3.2.3.2" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.3.2.3.2.cmml">n</mi><mtext id="A1.E38.m1.4.4.1.1.2.2.3.2.3.3" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.3.2.3.3a.cmml">calib</mtext></msub></mfrac><mo id="A1.E38.m1.4.4.1.1.2.2.3.1" xref="A1.E38.m1.4.4.1.1.2.2.3.1.cmml">⁢</mo><mrow id="A1.E38.m1.4.4.1.1.2.2.3.3" 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id="A1.E38.m1.4.4.1.1.2.2.3.3.1.3.2.3" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.3.3.1.3.2.3a.cmml">calib</mtext></msub><mo id="A1.E38.m1.4.4.1.1.2.2.3.3.1.3.1" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.3.3.1.3.1.cmml">−</mo><mn id="A1.E38.m1.4.4.1.1.2.2.3.3.1.3.3" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.3.3.1.3.3.cmml">1</mn></mrow></munderover><msub id="A1.E38.m1.4.4.1.1.2.2.3.3.2" xref="A1.E38.m1.4.4.1.1.2.2.3.3.2.cmml"><mn id="A1.E38.m1.4.4.1.1.2.2.3.3.2.2" mathsize="144%" xref="A1.E38.m1.4.4.1.1.2.2.3.3.2.2.cmml">𝟏</mn><mrow id="A1.E38.m1.2.2.2" xref="A1.E38.m1.2.2.2.cmml"><msub id="A1.E38.m1.2.2.2.4" xref="A1.E38.m1.2.2.2.4.cmml"><mover accent="true" id="A1.E38.m1.2.2.2.4.2" xref="A1.E38.m1.2.2.2.4.2.cmml"><mi id="A1.E38.m1.2.2.2.4.2.2" mathsize="144%" xref="A1.E38.m1.2.2.2.4.2.2.cmml">𝜸</mi><mo id="A1.E38.m1.2.2.2.4.2.1" mathsize="144%" xref="A1.E38.m1.2.2.2.4.2.1.cmml">~</mo></mover><mrow id="A1.E38.m1.2.2.2.2.2.4" xref="A1.E38.m1.2.2.2.2.2.3.cmml"><mtext id="A1.E38.m1.1.1.1.1.1.1" mathsize="144%" xref="A1.E38.m1.1.1.1.1.1.1a.cmml">calib</mtext><mo id="A1.E38.m1.2.2.2.2.2.4.1" mathsize="144%" xref="A1.E38.m1.2.2.2.2.2.3.cmml">,</mo><mi id="A1.E38.m1.2.2.2.2.2.2" mathsize="144%" xref="A1.E38.m1.2.2.2.2.2.2.cmml">i</mi></mrow></msub><mo id="A1.E38.m1.2.2.2.3" mathsize="144%" xref="A1.E38.m1.2.2.2.3.cmml">≤</mo><mi id="A1.E38.m1.2.2.2.5" mathsize="144%" xref="A1.E38.m1.2.2.2.5.cmml">τ</mi></mrow></msub></mrow></mrow></mrow></mrow><mo id="A1.E38.m1.4.4.1.2" lspace="0em" mathsize="144%">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.E38.m1.4b"><apply id="A1.E38.m1.4.4.1.1.3.cmml" xref="A1.E38.m1.4.4.1.1.2"><csymbol cd="ambiguous" id="A1.E38.m1.4.4.1.1.3a.cmml" xref="A1.E38.m1.4.4.1.1.2.3">formulae-sequence</csymbol><apply id="A1.E38.m1.4.4.1.1.1.1.cmml" xref="A1.E38.m1.4.4.1.1.1.1"><in id="A1.E38.m1.4.4.1.1.1.1.1.cmml" xref="A1.E38.m1.4.4.1.1.1.1.1"></in><apply id="A1.E38.m1.4.4.1.1.1.1.2.cmml" 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xref="A1.E38.m1.2.2.2.5">𝜏</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E38.m1.4c">\forall\,\tau\in\mathbb{R},\quad\widehat{F}_{\widetilde{\boldsymbol{\gamma}}|% \mathcal{H}_{0}}(\tau):=\frac{1}{n_{\text{calib}}}\sum_{i=0}^{n_{\text{calib}}% -1}{\mathbf{1}}_{\widetilde{\boldsymbol{\gamma}}_{\text{calib},i}\leq\tau}\,.</annotation><annotation encoding="application/x-llamapun" id="A1.E38.m1.4d">∀ italic_τ ∈ blackboard_R , over^ start_ARG italic_F end_ARG start_POSTSUBSCRIPT over~ start_ARG bold_italic_γ end_ARG | caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_τ ) := divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT calib end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT calib end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT bold_1 start_POSTSUBSCRIPT over~ start_ARG bold_italic_γ end_ARG start_POSTSUBSCRIPT calib , italic_i end_POSTSUBSCRIPT ≤ italic_τ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(38)</td> </tr></tbody> </table> where <math alttext="\mathbf{1}" class="ltx_Math" display="inline" id="A1.SS6.p2.7.m1.1"><semantics id="A1.SS6.p2.7.m1.1a"><mn id="A1.SS6.p2.7.m1.1.1" mathsize="144%" xref="A1.SS6.p2.7.m1.1.1.cmml">𝟏</mn><annotation-xml encoding="MathML-Content" id="A1.SS6.p2.7.m1.1b"><cn id="A1.SS6.p2.7.m1.1.1.cmml" type="integer" xref="A1.SS6.p2.7.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.SS6.p2.7.m1.1c">\mathbf{1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS6.p2.7.m1.1d">bold_1</annotation></semantics></math> is the indicator function. The estimated probability of false alarm is related to the eCDF through: <table class="ltx_equation ltx_eqn_table" id="A1.E39"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall\,\tau\in\mathbb{R},\quad\widehat{\text{PFA}}(\tau)=\widehat{\mathbb{P}% }(\widetilde{\boldsymbol{\gamma}}>\tau|\mathcal{H}_{0})=1-\widehat{F}_{% \widetilde{\boldsymbol{\gamma}}|\mathcal{H}_{0}}(\tau)\,." class="ltx_Math" display="block" id="A1.E39.m1.3"><semantics id="A1.E39.m1.3a"><mrow id="A1.E39.m1.3.3.1"><mrow id="A1.E39.m1.3.3.1.1.2" xref="A1.E39.m1.3.3.1.1.3.cmml"><mrow id="A1.E39.m1.3.3.1.1.1.1" xref="A1.E39.m1.3.3.1.1.1.1.cmml"><mrow id="A1.E39.m1.3.3.1.1.1.1.2" xref="A1.E39.m1.3.3.1.1.1.1.2.cmml"><mo id="A1.E39.m1.3.3.1.1.1.1.2.1" mathsize="144%" rspace="0.337em" xref="A1.E39.m1.3.3.1.1.1.1.2.1.cmml">∀</mo><mi id="A1.E39.m1.3.3.1.1.1.1.2.2" mathsize="144%" 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caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_τ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(39)</td> </tr></tbody> </table> In <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.F6" style="font-size:144%;" title="In A.6 Calibration procedure ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Fig. F</a>, we evaluate the PFA across a range of detection thresholds using the calibration set and apply the same thresholds to the test set for comparison. The results show that the calibrated PFA closely matches the test PFA, except in the extremely low-PFA regime. This slight discrepancy is expected, as high detection scores under <math alttext="\mathcal{H}_{0}" class="ltx_Math" display="inline" id="A1.SS6.p2.8.m1.1"><semantics id="A1.SS6.p2.8.m1.1a"><msub id="A1.SS6.p2.8.m1.1.1" xref="A1.SS6.p2.8.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.SS6.p2.8.m1.1.1.2" mathsize="144%" xref="A1.SS6.p2.8.m1.1.1.2.cmml">ℋ</mi><mn id="A1.SS6.p2.8.m1.1.1.3" mathsize="144%" xref="A1.SS6.p2.8.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS6.p2.8.m1.1b"><apply id="A1.SS6.p2.8.m1.1.1.cmml" xref="A1.SS6.p2.8.m1.1.1"><csymbol cd="ambiguous" id="A1.SS6.p2.8.m1.1.1.1.cmml" xref="A1.SS6.p2.8.m1.1.1">subscript</csymbol><ci id="A1.SS6.p2.8.m1.1.1.2.cmml" xref="A1.SS6.p2.8.m1.1.1.2">ℋ</ci><cn id="A1.SS6.p2.8.m1.1.1.3.cmml" type="integer" xref="A1.SS6.p2.8.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS6.p2.8.m1.1c">\mathcal{H}_{0}</annotation><annotation encoding="application/x-llamapun" id="A1.SS6.p2.8.m1.1d">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> are exceedingly rare, leading to increased noise in PFA estimation. In this regime, our estimate is nevertheless conservative, i.e., the experienced PFA is slightly smaller than predicted one. </div> </section> <section class="ltx_subsection" id="A1.SS7"> <h3 class="ltx_title ltx_title_subsection" style="font-size:144%;"> A.7 Implementation details</h3> <div class="ltx_para" id="A1.SS7.p1"> Our model implementation and training are based on the PyTorch framework. We optimize the neural network weights using the Adam optimizer with a learning rate of <math alttext="5\times 10^{-4}" class="ltx_Math" display="inline" id="A1.SS7.p1.1.m1.1"><semantics id="A1.SS7.p1.1.m1.1a"><mrow id="A1.SS7.p1.1.m1.1.1" xref="A1.SS7.p1.1.m1.1.1.cmml"><mn id="A1.SS7.p1.1.m1.1.1.2" mathsize="144%" xref="A1.SS7.p1.1.m1.1.1.2.cmml">5</mn><mo id="A1.SS7.p1.1.m1.1.1.1" lspace="0.222em" mathsize="144%" rspace="0.222em" xref="A1.SS7.p1.1.m1.1.1.1.cmml">×</mo><msup id="A1.SS7.p1.1.m1.1.1.3" xref="A1.SS7.p1.1.m1.1.1.3.cmml"><mn id="A1.SS7.p1.1.m1.1.1.3.2" mathsize="144%" xref="A1.SS7.p1.1.m1.1.1.3.2.cmml">10</mn><mrow id="A1.SS7.p1.1.m1.1.1.3.3" xref="A1.SS7.p1.1.m1.1.1.3.3.cmml"><mo id="A1.SS7.p1.1.m1.1.1.3.3a" mathsize="144%" xref="A1.SS7.p1.1.m1.1.1.3.3.cmml">−</mo><mn id="A1.SS7.p1.1.m1.1.1.3.3.2" mathsize="144%" xref="A1.SS7.p1.1.m1.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS7.p1.1.m1.1b"><apply id="A1.SS7.p1.1.m1.1.1.cmml" xref="A1.SS7.p1.1.m1.1.1"><times id="A1.SS7.p1.1.m1.1.1.1.cmml" xref="A1.SS7.p1.1.m1.1.1.1"></times><cn id="A1.SS7.p1.1.m1.1.1.2.cmml" type="integer" xref="A1.SS7.p1.1.m1.1.1.2">5</cn><apply id="A1.SS7.p1.1.m1.1.1.3.cmml" xref="A1.SS7.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A1.SS7.p1.1.m1.1.1.3.1.cmml" xref="A1.SS7.p1.1.m1.1.1.3">superscript</csymbol><cn id="A1.SS7.p1.1.m1.1.1.3.2.cmml" type="integer" xref="A1.SS7.p1.1.m1.1.1.3.2">10</cn><apply id="A1.SS7.p1.1.m1.1.1.3.3.cmml" xref="A1.SS7.p1.1.m1.1.1.3.3"><minus id="A1.SS7.p1.1.m1.1.1.3.3.1.cmml" xref="A1.SS7.p1.1.m1.1.1.3.3"></minus><cn id="A1.SS7.p1.1.m1.1.1.3.3.2.cmml" type="integer" xref="A1.SS7.p1.1.m1.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS7.p1.1.m1.1c">5\times 10^{-4}</annotation><annotation encoding="application/x-llamapun" id="A1.SS7.p1.1.m1.1d">5 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>, a batch size of 16, and 50,000 iterations. We apply data augmentation techniques, including random <math alttext="90\degree" class="ltx_Math" display="inline" id="A1.SS7.p1.2.m2.1"><semantics id="A1.SS7.p1.2.m2.1a"><mrow id="A1.SS7.p1.2.m2.1.1" xref="A1.SS7.p1.2.m2.1.1.cmml"><mn id="A1.SS7.p1.2.m2.1.1.2" mathsize="144%" xref="A1.SS7.p1.2.m2.1.1.2.cmml">90</mn><mo id="A1.SS7.p1.2.m2.1.1.1" xref="A1.SS7.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="A1.SS7.p1.2.m2.1.1.3" mathsize="144%" mathvariant="normal" xref="A1.SS7.p1.2.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS7.p1.2.m2.1b"><apply id="A1.SS7.p1.2.m2.1.1.cmml" xref="A1.SS7.p1.2.m2.1.1"><times id="A1.SS7.p1.2.m2.1.1.1.cmml" xref="A1.SS7.p1.2.m2.1.1.1"></times><cn id="A1.SS7.p1.2.m2.1.1.2.cmml" type="integer" xref="A1.SS7.p1.2.m2.1.1.2">90</cn><ci id="A1.SS7.p1.2.m2.1.1.3.cmml" xref="A1.SS7.p1.2.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS7.p1.2.m2.1c">90\degree</annotation><annotation encoding="application/x-llamapun" id="A1.SS7.p1.2.m2.1d">90 °</annotation></semantics></math> rotations, vertical/horizontal flipping, and frame shuffling. Bicubic interpolation is used for spectral and exoplanet alignment. For ADI, the model is trained on 64 frames, while for ASDI, data cubes have 32 frames with two channels each. All evaluations and tests are done with 64 frames in both ADI and ASDI. To ensure accurate results, pixels containing real known exoplanets are masked out during training before the temporal summation in <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.E29" style="font-size:144%;" title="In A.2 Dense detection map computation details ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">Eq. 29</a>. The loss is calculated on the final output for pixels with at least 8 valid frames, unaffected by real exoplanets. </div> </section> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix" style="font-size:144%;"> Appendix B Dataset description</h2> <div class="ltx_para" id="A2.p1"> SHINE-F150 is a subset of the SPHERE large survey, with 322 data cubes (<math alttext="20,462" class="ltx_Math" display="inline" id="A2.p1.1.m1.2"><semantics id="A2.p1.1.m1.2a"><mrow id="A2.p1.1.m1.2.3.2" xref="A2.p1.1.m1.2.3.1.cmml"><mn id="A2.p1.1.m1.1.1" mathsize="144%" xref="A2.p1.1.m1.1.1.cmml">20</mn><mo id="A2.p1.1.m1.2.3.2.1" mathsize="144%" xref="A2.p1.1.m1.2.3.1.cmml">,</mo><mn id="A2.p1.1.m1.2.2" mathsize="144%" xref="A2.p1.1.m1.2.2.cmml">462</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.1.m1.2b"><list id="A2.p1.1.m1.2.3.1.cmml" xref="A2.p1.1.m1.2.3.2"><cn id="A2.p1.1.m1.1.1.cmml" type="integer" xref="A2.p1.1.m1.1.1">20</cn><cn id="A2.p1.1.m1.2.2.cmml" type="integer" xref="A2.p1.1.m1.2.2">462</cn></list></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.1.m1.2c">20,462</annotation><annotation encoding="application/x-llamapun" id="A2.p1.1.m1.2d">20 , 462</annotation></semantics></math> images in total) from 150 stars. The spatial resolution is 12 milliarcseconds/pixel. The spectral resolution <math alttext="\lambda/\Delta_{\lambda}" class="ltx_Math" display="inline" id="A2.p1.2.m2.1"><semantics id="A2.p1.2.m2.1a"><mrow id="A2.p1.2.m2.1.1" xref="A2.p1.2.m2.1.1.cmml"><mi id="A2.p1.2.m2.1.1.2" mathsize="144%" xref="A2.p1.2.m2.1.1.2.cmml">λ</mi><mo id="A2.p1.2.m2.1.1.1" maxsize="144%" minsize="144%" stretchy="true" symmetric="true" xref="A2.p1.2.m2.1.1.1.cmml">/</mo><msub id="A2.p1.2.m2.1.1.3" xref="A2.p1.2.m2.1.1.3.cmml"><mi id="A2.p1.2.m2.1.1.3.2" mathsize="144%" mathvariant="normal" xref="A2.p1.2.m2.1.1.3.2.cmml">Δ</mi><mi id="A2.p1.2.m2.1.1.3.3" mathsize="144%" xref="A2.p1.2.m2.1.1.3.3.cmml">λ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.2.m2.1b"><apply id="A2.p1.2.m2.1.1.cmml" xref="A2.p1.2.m2.1.1"><divide id="A2.p1.2.m2.1.1.1.cmml" xref="A2.p1.2.m2.1.1.1"></divide><ci id="A2.p1.2.m2.1.1.2.cmml" xref="A2.p1.2.m2.1.1.2">𝜆</ci><apply id="A2.p1.2.m2.1.1.3.cmml" xref="A2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A2.p1.2.m2.1.1.3.1.cmml" xref="A2.p1.2.m2.1.1.3">subscript</csymbol><ci id="A2.p1.2.m2.1.1.3.2.cmml" xref="A2.p1.2.m2.1.1.3.2">Δ</ci><ci id="A2.p1.2.m2.1.1.3.3.cmml" xref="A2.p1.2.m2.1.1.3.3">𝜆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.2.m2.1c">\lambda/\Delta_{\lambda}</annotation><annotation encoding="application/x-llamapun" id="A2.p1.2.m2.1d">italic_λ / roman_Δ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT</annotation></semantics></math> is 20. Detailed description is in [39]. We followed [5] to exclude corrupted data, leaving 220 datasets, with 165 used for training. The main statistics of the F150 dataset are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A1.F7" style="font-size:144%;" title="Figure G ‣ A.6 Calibration procedure ‣ Appendix A Technical details ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">G</a>. </div> </section> <section class="ltx_appendix" id="A3"> <h2 class="ltx_title ltx_title_appendix" style="font-size:144%;"> Appendix C Comparison of state-of-the-art methods</h2> <figure class="ltx_table" id="A3.T4"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="A3.T4.2" style="width:216.8pt;height:59.3pt;vertical-align:-0.4pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A3.T4.2.1"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A3.T4.2.1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" id="A3.T4.2.1.1.1.1">Method</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A3.T4.2.1.1.1.2">nuisance model</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A3.T4.2.1.1.1.3">P1</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A3.T4.2.1.1.1.4">P2</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A3.T4.2.1.1.1.5">P3</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A3.T4.2.1.1.1.6">P4</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A3.T4.2.1.1.1.7">P5</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A3.T4.2.1.1.1.8">P6</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A3.T4.2.1.1.1.9">P7</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A3.T4.2.1.2.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="A3.T4.2.1.2.1.1">PACO</th> <td class="ltx_td ltx_align_center ltx_border_t" id="A3.T4.2.1.2.1.2">statistical on data patches</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A3.T4.2.1.2.1.3">X</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A3.T4.2.1.2.1.4">✓</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A3.T4.2.1.2.1.5">✓</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A3.T4.2.1.2.1.6">✓</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A3.T4.2.1.2.1.7">X</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A3.T4.2.1.2.1.8">✓</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A3.T4.2.1.2.1.9">✓</td> </tr> <tr class="ltx_tr" id="A3.T4.2.1.3.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="A3.T4.2.1.3.2.1">SODINN</th> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.3.2.2">learned on PCA residuals</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.3.2.3">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.3.2.4">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.3.2.5">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.3.2.6">✓✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.3.2.7">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.3.2.8">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.3.2.9">X</td> </tr> <tr class="ltx_tr" id="A3.T4.2.1.4.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="A3.T4.2.1.4.3.1">deep PACO</th> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.4.3.2">learned after PACO whitening</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.4.3.3">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.4.3.4">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.4.3.5">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.4.3.6">✓✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.4.3.7">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.4.3.8">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.4.3.9">X</td> </tr> <tr class="ltx_tr" id="A3.T4.2.1.5.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="A3.T4.2.1.5.4.1">Super-RDI</th> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.5.4.2">PCA</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.5.4.3">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.5.4.4">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.5.4.5">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.5.4.6">✓✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.5.4.7">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.5.4.8">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.5.4.9">✓</td> </tr> <tr class="ltx_tr" id="A3.T4.2.1.6.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="A3.T4.2.1.6.5.1">MWIN5-RB</th> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.6.5.2">learned on PCA residuals</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.6.5.3">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.6.5.4">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.6.5.5">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.6.5.6">✓✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.6.5.7">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.6.5.8">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.6.5.9">✓✓</td> </tr> <tr class="ltx_tr" id="A3.T4.2.1.7.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="A3.T4.2.1.7.6.1">ConStruct</th> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.7.6.2">generative, learned on data</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.7.6.3">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.7.6.4">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.7.6.5">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.7.6.6">✓✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.7.6.7">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.7.6.8">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.7.6.9">✓✓✓</td> </tr> <tr class="ltx_tr" id="A3.T4.2.1.8.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="A3.T4.2.1.8.7.1">MODEL&CO</th> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.8.7.2">statistical on learned features</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.8.7.3">✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.8.7.4">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.8.7.5">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.8.7.6">✓✓✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.8.7.7">✓✓</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.8.7.8">X</td> <td class="ltx_td ltx_align_center" id="A3.T4.2.1.8.7.9">✓✓✓</td> </tr> <tr class="ltx_tr" id="A3.T4.2.1.9.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_bb ltx_border_r" id="A3.T4.2.1.9.8.1">proposed</th> <td class="ltx_td ltx_align_center ltx_border_bb" id="A3.T4.2.1.9.8.2">multi-scale stat. on learned features</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A3.T4.2.1.9.8.3">✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A3.T4.2.1.9.8.4">✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A3.T4.2.1.9.8.5">✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A3.T4.2.1.9.8.6">✓✓✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A3.T4.2.1.9.8.7">✓✓✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A3.T4.2.1.9.8.8">✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A3.T4.2.1.9.8.9">✓✓✓</td> </tr> </tbody> </table> </div> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">Table D: Comparison of advanced exoplanet imaging algorithms based on key desirable properties. P1=observation-independence of the model, P2=physical interpretability, P3=multi-spectral, P4=detection sensitivity, P5=accuracy near the star, P6=position and flux estimation, P7=practicality on large-scale surveys.</figcaption> </figure> <div class="ltx_para" id="A3.p1"> Table <a class="ltx_ref" href="https://arxiv.org/html/2503.17117v1#A3.T4" style="font-size:144%;" title="Table D ‣ Appendix C Comparison of state-of-the-art methods ‣ A New Statistical Model of Star Speckles for Learning to Detect and Characterize Exoplanets in Direct Imaging Observations">D</a> compares the key properties of state-of-the-art algorithms for exoplanet detection via direct imaging. </div> </section> <section class="ltx_appendix" id="A4"> <h2 class="ltx_title ltx_title_appendix" style="font-size:144%;"> Appendix D Additional results</h2> <div class="ltx_para" id="A4.p1"> In this section we provide additional visual results, and show detection maps obtained with synthetic exoplanets. Consistent with the experimental findings in the main body of the paper, our method performs on par or better than baselines in ADI, and outperforms other approaches in ASDI. </div> <div class="ltx_pagination ltx_role_newpage"></div> <figure class="ltx_figure" id="A4.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="464" id="A4.F8.g1" src="x10.png" width="447"/> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">Figure H: Detection maps on observations of HD 102647 star with synthetic exoplanets.</figcaption> </figure> <figure class="ltx_figure" id="A4.F9"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="464" id="A4.F9.g1" src="x11.png" width="447"/> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">Figure I: Detection maps on observations of HD 188228 star with synthetic exoplanets.</figcaption> </figure> <figure class="ltx_figure" id="A4.F10"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="464" id="A4.F10.g1" src="x12.png" width="447"/> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">Figure J: Detection maps on observations of HD 206860 star with synthetic exoplanets.</figcaption> </figure> <figure class="ltx_figure" id="A4.F11"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="464" id="A4.F11.g1" src="x13.png" width="447"/> <figcaption class="ltx_caption ltx_centering" style="font-size:144%;">Figure K: Detection maps on observations of HD 216803 star with synthetic exoplanets.</figcaption> </figure> <div class="ltx_pagination ltx_role_newpage"></div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Fri Mar 21 12:55:39 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/">LaTeXML<img alt="Mascot Sammy" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAOCAYAAAD5YeaVAAAAAXNSR0IArs4c6QAAAAZiS0dEAP8A/wD/oL2nkwAAAAlwSFlzAAALEwAACxMBAJqcGAAAAAd0SU1FB9wKExQZLWTEaOUAAAAddEVYdENvbW1lbnQAQ3JlYXRlZCB3aXRoIFRoZSBHSU1Q72QlbgAAAdpJREFUKM9tkL+L2nAARz9fPZNCKFapUn8kyI0e4iRHSR1Kb8ng0lJw6FYHFwv2LwhOpcWxTjeUunYqOmqd6hEoRDhtDWdA8ApRYsSUCDHNt5ul13vz4w0vWCgUnnEc975arX6ORqN3VqtVZbfbTQC4uEHANM3jSqXymFI6yWazP2KxWAXAL9zCUa1Wy2tXVxheKA9YNoR8Pt+aTqe4FVVVvz05O6MBhqUIBGk8Hn8HAOVy+T+XLJfLS4ZhTiRJgqIoVBRFIoric47jPnmeB1mW/9rr9ZpSSn3Lsmir1fJZlqWlUonKsvwWwD8ymc/nXwVBeLjf7xEKhdBut9Hr9WgmkyGEkJwsy5eHG5vN5g0AKIoCAEgkEkin0wQAfN9/cXPdheu6P33fBwB4ngcAcByHJpPJl+fn54mD3Gg0NrquXxeLRQAAwzAYj8cwTZPwPH9/sVg8PXweDAauqqr2cDjEer1GJBLBZDJBs9mE4zjwfZ85lAGg2+06hmGgXq+j3+/DsixYlgVN03a9Xu8jgCNCyIegIAgx13Vfd7vdu+FweG8YRkjXdWy329+dTgeSJD3ieZ7RNO0VAXAPwDEAO5VKndi2fWrb9jWl9Esul6PZbDY9Go1OZ7PZ9z/lyuD3OozU2wAAAABJRU5ErkJggg=="/></a> </div></footer> </div> </body> </html>