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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>A generative approach for lensless imaging in low-light conditions</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/2501.03511v1/"/></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/2501.03511v1#S1" title="In A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S1.SS1" title="In 1 Introduction ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Motivation and Aim</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S1.SS2" title="In 1 Introduction ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Basic Idea and Contributions</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S2" title="In A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Problem analysis</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3" title="In A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Proposed Method</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.SS1" title="In 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Fisrt Stage</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.SS2" title="In 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Second Stage</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.SS2.SSS1" title="In 3.2 Second Stage ‣ 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2.1 </span>Conditional Diffusion Model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.SS2.SSS2" title="In 3.2 Second Stage ‣ 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2.2 </span>Processing structure</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.SS3" title="In 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Loss Function</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4" title="In A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Experiment and Results</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.SS1" title="In 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Dataset</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.SS2" title="In 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Impletmentation Details</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.SS3" title="In 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Quantitative Metrics</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.SS4" title="In 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.4 </span>Simulated Reconstruction</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.SS5" title="In 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.5 </span>Measured Reconstruction</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S5" title="In A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Discussion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S6" title="In A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Conclusion</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document"> <h1 class="ltx_title ltx_title_document">A generative approach for lensless imaging in low-light conditions</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">ZIYANG LIU </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><span class="ltx_ERROR undefined" id="id1.1.id1">\authormark</span>1 <span class="ltx_ERROR undefined" id="id2.2.id2">\orcidlink</span>0009-0008-0034-3755 TIANJIAO ZENG </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><span class="ltx_ERROR undefined" id="id3.1.id1">\authormark</span>2,* <span class="ltx_ERROR undefined" id="id4.2.id2">\orcidlink</span>0000-0002-6780-6100 XU ZHAN </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><span class="ltx_ERROR undefined" id="id5.1.id1">\authormark</span>1 <span class="ltx_ERROR undefined" id="id6.2.id2">\orcidlink</span>0000-0003-2816-9791 </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> XIAOLING ZHANG<span class="ltx_ERROR undefined" id="id7.1.id1">\authormark</span>1 <span class="ltx_ERROR undefined" id="id8.2.id2">\orcidlink</span>0000-0003-2343-3055 and EDMUND Y. LAM<span class="ltx_ERROR undefined" id="id9.3.id3">\authormark</span>3 <span class="ltx_ERROR undefined" id="id10.4.id4">\orcidlink</span>0000-0001-6268-950X </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_address"><span class="ltx_ERROR undefined" id="id11.5.id1">\authormark</span>1School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, China <br class="ltx_break"/><span class="ltx_ERROR undefined" id="id12.6.id2">\authormark</span>2School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu, China <br class="ltx_break"/><span class="ltx_ERROR undefined" id="id13.7.id3">\authormark</span>3Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam, Hong Kong SAR, China <br class="ltx_break"/> </span> <span class="ltx_contact ltx_role_email"><a href="mailto:*tzeng@uestc.edu.cn"><span class="ltx_ERROR undefined" id="id14.8.id1">\authormark</span>*tzeng@uestc.edu.cn</a> </span></span></span> </div> <span class="ltx_note ltx_note_frontmatter ltx_role_journal" id="id1"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">journal: </span>opticajournal</span></span></span><span class="ltx_note ltx_note_frontmatter ltx_role_articletype" id="id2"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">articletype: </span>Research Article</span></span></span> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\setaddedmarkup</span> <p class="ltx_p" id="p1.2"><span class="ltx_text" id="p1.2.1" style="color:#FF0000;">#1</span></p> </div> <div class="ltx_para" id="p2"> <span class="ltx_ERROR undefined" id="p2.1">{abstract*}</span> <p class="ltx_p" id="p2.2">Lensless imaging offers a lightweight, compact alternative to traditional lens-based systems, ideal for exploration in space-constrained environments. However, the absence of a focusing lens and limited lighting in such environments often result in low-light conditions, where the measurements suffer from complex noise interference due to insufficient capture of photons. This study presents a robust reconstruction method for high-quality imaging in low-light scenarios, employing two complementary perspectives: model-driven and data-driven. First, we apply a physic-model-driven perspective to reconstruct in the range space of the pseudo-inverse of the measurement model—as a first guidance to extract information in the noisy measurements. Then, we integrate a generative-model based perspective to suppress residual noises—as the second guidance to suppress noises in the initial noisy results. Specifically, a learnable Wiener filter-based module generates an initial, noisy reconstruction. Then, for fast and, more importantly, stable generation of the clear image from the noisy version, we implement a modified conditional generative diffusion module. This module converts the raw image into the latent wavelet domain for efficiency and uses modified bidirectional training processes for stabilization. Simulations and real-world experiments demonstrate substantial improvements in overall visual quality, advancing lensless imaging in challenging low-light environments<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>This manuscript has been accepted by Optics Express. © 2024 Optica Publishing Group. Users may use, reuse, and build upon the article, or use the article for text or data mining, so long as such uses are for non-commercial purposes and appropriate attribution is maintained. All other rights are reserved.</span></span></span>.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">While lens technology has significantly propelled the progress of imaging science, its inherent physical constraints pose bottlenecks for further miniaturization, lightweight design, and cost reduction <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib2" title="">2</a>]</cite>. The contradiction between these physical constraints imposed by optical lenses on traditional imaging device sizes and the pursuit of miniaturization and thinness has sparked the emergence of lensless imaging technology. Lensless imaging follows the new evolution of ground-breaking computational imaging techniques. Through computational imaging—a tight integration of the sensing system and computation to form images of interest—we can access information that was otherwise not possible. This approach has shown promising performance across diverse areas such as holographic imaging <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib3" title="">3</a>]</cite>, phase recovery <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib5" title="">5</a>]</cite>, fluorescence microscopy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib7" title="">7</a>]</cite>, high dynamic range (HDR) imaging <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib8" title="">8</a>]</cite>, underwater imaging <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib9" title="">9</a>]</cite>, etc.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Lensless imaging utilizes simple and inexpensive optical encoders to replace costly and complex lens assemblies, leveraging computational imaging techniques to reconstruct clear images from collected measurements <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib14" title="">14</a>]</cite>. In lensless imaging, reconstruction is crucial due to the significant difference between measured data and the original scene. Most techniques use regularization-based methods to solve underdetermined linear equations, optimizing fidelity and data prior terms. Simple cases may use Tikhonov regularization for closed-form solutions, while complex scenarios require iterative algorithms like the iterative shrinkage-thresholding algorithm (ISTA) or the alternating direction method of multipliers (ADMM), offering better quality but with higher computational costs and manual parameter tuning.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Despite advancements, traditional model-based methods often fall short due to imprecise modeling of data priors and limitations in handling real-world complexities. Deep learning has introduced neural networks as powerful inversion operators, directly mapping raw measurements to imaging scenes through data-driven learning <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib15" title="">15</a>]</cite>. For instance, Pan et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib16" title="">16</a>]</cite> developed a transformer-based end-to-end reconstruction network. However, these methods often overlook the forward physical model, leading to image artifacts and loss of fine details. To bridge this gap, hybrid methods combine traditional optimization with deep learning. Monakhova et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib17" title="">17</a>]</cite> introduced Le-ADMM-U, incorporating a neural network into an unrolled ADMM optimization loop, improving reconstruction by learning from data while maintaining optimization principles. Similarly, Khan et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib18" title="">18</a>]</cite> proposed FlatNet, which refines a learnable Tikhonov-based reconstruction through a GAN with perceptual loss, enhancing image quality. One key challenge in hybrid methods is model mismatch—the difference between the assumed forward model and the actual system—which can degrade image quality. In our previous work <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib19" title="">19</a>]</cite>, we quantified error accumulation from model mismatch and proposed a multi-stage information loss compensation method to improve reconstruction accuracy and stability. Following our work, Kingshott <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib20" title="">20</a>]</cite> introduced a learned optimization scheme through an unrolled primal-dual reconstruction method to reduce model error. Li <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib21" title="">21</a>]</cite> introduced a multi-scale Wiener deconvolution approach to recover lost information. Qian <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib22" title="">22</a>]</cite> integrated a deep denoising module into the iterative reconstruction process to minimize the model error. More recently, Cai <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib23" title="">23</a>]</cite> combined a spatially-variable learnable deconvolution method with a generative model for refinement reconstruction.</p> </div> <section class="ltx_subsection" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Motivation and Aim</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.1">Despite significant advances in reconstruction techniques, the performance of lensless imaging systems under low-light conditions remains an underexplored challenge. Without a focusing lens, these systems suffer from significant signal attenuation as light disperses through the mask, leading to reduced signal-to-noise ratios (SNR). This issue is further exacerbated by the small size of sensors, making high-quality imaging in resource-constrained or low-light environments particularly difficult. Most lensless imaging methods, such as those described in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib25" title="">25</a>]</cite>, adopt a two-stage network design. The first stage incorporates a forward physical model to recover low-frequency image content, followed by a post-processing network (e.g., a denoiser or generative model) to refine and enhance the image. While these approaches yield promising results under normal lighting conditions, their performance degrades significantly in low-light scenarios due to the following limitations:</p> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.1">Noise Characteristics: In low-light conditions, the measurements are heavily influenced by complex noise patterns, which differ from those in normal lighting. Current two-stage methods often employ a denoiser or generative model in the second stage, but these are not optimized for low-light noise characteristics, leading to suboptimal results.</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.1">Brightness Insufficiency: Low photon counts cause severely underexposed lensless measurements, posing challenges for existing network architectures in restoring brightness while maintaining fine image details and textures, often leading to unstable results and degraded reconstruction quality.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.1">Therefore, this study aims to develop a robust reconstruction framework specifically designed for low-light lensless imaging, balancing brightness restoration, noise suppression, and detail preservation by leveraging the strengths of both physics-driven and generative models.</p> </div> </section> <section class="ltx_subsection" id="S1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Basic Idea and Contributions</h3> <div class="ltx_para" id="S1.SS2.p1"> <p class="ltx_p" id="S1.SS2.p1.1">To enable lensless imaging in low-light conditions, our work builds upon two fundamental aspects: theoretical foundations and algorithmic methodologies.</p> </div> <div class="ltx_para" id="S1.SS2.p2"> <ul class="ltx_itemize" id="S1.I2"> <li class="ltx_item" id="S1.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I2.i1.p1"> <p class="ltx_p" id="S1.I2.i1.p1.1">On the theoretical front, we present the first comprehensive analysis of noise characteristics inherent in low-light lensless imaging, and propose a theoretical model that serves as a foundation for designing reconstruction methods and generating simulation data tailored for network training.</p> </div> </li> <li class="ltx_item" id="S1.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I2.i2.p1"> <p class="ltx_p" id="S1.I2.i2.p1.1">Algorithmically, we propose a novel multi-step diffusion model explicitly conditioned on low-light illumination and intricate noise components within a two-stage reconstruction framework. Unlike methods for well-illuminated conditions which overlooks the complexity of photon-limited noise, we leverage wavelet-domain decomposition to separate brightness and noise in the latent space, serving as conditions to target these issues directly, and employ multi-step diffusion process for superior noise suppression compared to one-step generative models. A bidirectional training strategy further ensures stability and robustness under challenging low-light scenarios.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S1.SS2.p3"> <p class="ltx_p" id="S1.SS2.p3.1">Specifically, we first analyze the forward measurement process of lensless imaging, examining each phase of data transition in detail based on the camera’s characteristics in low-light conditions. This analysis establishes a model that accounts for two key features of lensless imaging results: complex noise patterns and insufficient brightness. This also provides us with the tools needed for subsequent dataset construction for neural network training.</p> </div> <div class="ltx_para" id="S1.SS2.p4"> <p class="ltx_p" id="S1.SS2.p4.1">Secondly, we follow a two-stage framework, leveraging the forward measurement model as a strong prior to guide the initial reconstruction. This allows us to obtain partial information of the imaging scene in the range space of its adjoint pseudo-inverse. The transition from measurement space to image range space more prominently reveals the two low-light features mentioned above.</p> </div> <div class="ltx_para" id="S1.SS2.p5"> <p class="ltx_p" id="S1.SS2.p5.1">Third, we employ a diffusion model to refine the initial result, addressing the two low-light features through a conditional approach. We incorporate these features into the diffusion model’s generation process. Specifically, we decompose the initial result through wavelet transforms to separate brightness and noise information in the latent space, using these as conditions for the generative model. For the nullspace refinement, we refine the remaining texture information separately through a depth-separable convolutional neural network. This separation also allows the generation process to occur in a smaller latent space, enabling memory-efficient training and testing. Additionally, to address the increased instability of generation in underdetermined low-light conditions, we implement a bidirectional training strategy—incorporating both generation and diffusion processes—to stabilize the final imaging result.</p> </div> <div class="ltx_para" id="S1.SS2.p6"> <p class="ltx_p" id="S1.SS2.p6.1">To thoroughly evaluate the effectiveness of our proposed method, we conducted a series of both simulated and real-world experiments using a self-built, lensless camera within a carefully controlled lighting environment. For a comprehensive comparison, we employed both traditional non-learning-based approaches and cutting-edge learning-based models. The results are telling: conventional methods experience significant performance degradation, particularly under photon-limited conditions, where some even fail entirely. In contrast, our newly proposed method not only holds up but shines—quite literally. It demonstrates remarkable improvements in image brightness, superior noise reduction, and a clear enhancement in overall image quality.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Problem analysis</h2> <figure class="ltx_figure" id="S2.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="146" id="S2.F1.g1" src="extracted/6115207/figure/1.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>The measurement process in low-light conditions, illustrating the mixture of multiple noise types.</figcaption> </figure> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">In this section, we analyze the impact of low-light conditions on lensless imaging from the perspective of the measurement process <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib26" title="">26</a>]</cite>, as illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S2.F1" title="Figure 1 ‣ 2 Problem analysis ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">1</span></a>. Let’s consider an intensity distribution to be measured, denoted as <math alttext="x(i,j)" class="ltx_Math" display="inline" id="S2.p1.1.m1.2"><semantics id="S2.p1.1.m1.2a"><mrow id="S2.p1.1.m1.2.3" xref="S2.p1.1.m1.2.3.cmml"><mi id="S2.p1.1.m1.2.3.2" xref="S2.p1.1.m1.2.3.2.cmml">x</mi><mo id="S2.p1.1.m1.2.3.1" xref="S2.p1.1.m1.2.3.1.cmml">⁢</mo><mrow id="S2.p1.1.m1.2.3.3.2" xref="S2.p1.1.m1.2.3.3.1.cmml"><mo id="S2.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">i</mi><mo id="S2.p1.1.m1.2.3.3.2.2" xref="S2.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.p1.1.m1.2.2" xref="S2.p1.1.m1.2.2.cmml">j</mi><mo id="S2.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.2b"><apply id="S2.p1.1.m1.2.3.cmml" xref="S2.p1.1.m1.2.3"><times id="S2.p1.1.m1.2.3.1.cmml" xref="S2.p1.1.m1.2.3.1"></times><ci id="S2.p1.1.m1.2.3.2.cmml" xref="S2.p1.1.m1.2.3.2">𝑥</ci><interval closure="open" id="S2.p1.1.m1.2.3.3.1.cmml" xref="S2.p1.1.m1.2.3.3.2"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝑖</ci><ci id="S2.p1.1.m1.2.2.cmml" xref="S2.p1.1.m1.2.2">𝑗</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.2c">x(i,j)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.2d">italic_x ( italic_i , italic_j )</annotation></semantics></math>. This distribution undergoes a linear conversion to a photon distribution:</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="{b_{p}}(i,j)=K\times x(i,j)" class="ltx_Math" display="block" id="S2.E1.m1.4"><semantics id="S2.E1.m1.4a"><mrow id="S2.E1.m1.4.5" xref="S2.E1.m1.4.5.cmml"><mrow id="S2.E1.m1.4.5.2" xref="S2.E1.m1.4.5.2.cmml"><msub id="S2.E1.m1.4.5.2.2" xref="S2.E1.m1.4.5.2.2.cmml"><mi id="S2.E1.m1.4.5.2.2.2" xref="S2.E1.m1.4.5.2.2.2.cmml">b</mi><mi id="S2.E1.m1.4.5.2.2.3" xref="S2.E1.m1.4.5.2.2.3.cmml">p</mi></msub><mo id="S2.E1.m1.4.5.2.1" xref="S2.E1.m1.4.5.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.4.5.2.3.2" xref="S2.E1.m1.4.5.2.3.1.cmml"><mo id="S2.E1.m1.4.5.2.3.2.1" stretchy="false" xref="S2.E1.m1.4.5.2.3.1.cmml">(</mo><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">i</mi><mo id="S2.E1.m1.4.5.2.3.2.2" xref="S2.E1.m1.4.5.2.3.1.cmml">,</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">j</mi><mo id="S2.E1.m1.4.5.2.3.2.3" stretchy="false" xref="S2.E1.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.4.5.1" xref="S2.E1.m1.4.5.1.cmml">=</mo><mrow id="S2.E1.m1.4.5.3" xref="S2.E1.m1.4.5.3.cmml"><mrow id="S2.E1.m1.4.5.3.2" xref="S2.E1.m1.4.5.3.2.cmml"><mi id="S2.E1.m1.4.5.3.2.2" xref="S2.E1.m1.4.5.3.2.2.cmml">K</mi><mo id="S2.E1.m1.4.5.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.E1.m1.4.5.3.2.1.cmml">×</mo><mi id="S2.E1.m1.4.5.3.2.3" xref="S2.E1.m1.4.5.3.2.3.cmml">x</mi></mrow><mo id="S2.E1.m1.4.5.3.1" xref="S2.E1.m1.4.5.3.1.cmml">⁢</mo><mrow id="S2.E1.m1.4.5.3.3.2" xref="S2.E1.m1.4.5.3.3.1.cmml"><mo id="S2.E1.m1.4.5.3.3.2.1" stretchy="false" xref="S2.E1.m1.4.5.3.3.1.cmml">(</mo><mi id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml">i</mi><mo id="S2.E1.m1.4.5.3.3.2.2" xref="S2.E1.m1.4.5.3.3.1.cmml">,</mo><mi id="S2.E1.m1.4.4" xref="S2.E1.m1.4.4.cmml">j</mi><mo id="S2.E1.m1.4.5.3.3.2.3" stretchy="false" xref="S2.E1.m1.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.m1.4b"><apply id="S2.E1.m1.4.5.cmml" xref="S2.E1.m1.4.5"><eq id="S2.E1.m1.4.5.1.cmml" xref="S2.E1.m1.4.5.1"></eq><apply id="S2.E1.m1.4.5.2.cmml" xref="S2.E1.m1.4.5.2"><times id="S2.E1.m1.4.5.2.1.cmml" xref="S2.E1.m1.4.5.2.1"></times><apply id="S2.E1.m1.4.5.2.2.cmml" xref="S2.E1.m1.4.5.2.2"><csymbol cd="ambiguous" id="S2.E1.m1.4.5.2.2.1.cmml" xref="S2.E1.m1.4.5.2.2">subscript</csymbol><ci id="S2.E1.m1.4.5.2.2.2.cmml" xref="S2.E1.m1.4.5.2.2.2">𝑏</ci><ci id="S2.E1.m1.4.5.2.2.3.cmml" xref="S2.E1.m1.4.5.2.2.3">𝑝</ci></apply><interval closure="open" id="S2.E1.m1.4.5.2.3.1.cmml" xref="S2.E1.m1.4.5.2.3.2"><ci id="S2.E1.m1.1.1.cmml" xref="S2.E1.m1.1.1">𝑖</ci><ci id="S2.E1.m1.2.2.cmml" xref="S2.E1.m1.2.2">𝑗</ci></interval></apply><apply id="S2.E1.m1.4.5.3.cmml" xref="S2.E1.m1.4.5.3"><times id="S2.E1.m1.4.5.3.1.cmml" xref="S2.E1.m1.4.5.3.1"></times><apply id="S2.E1.m1.4.5.3.2.cmml" xref="S2.E1.m1.4.5.3.2"><times id="S2.E1.m1.4.5.3.2.1.cmml" xref="S2.E1.m1.4.5.3.2.1"></times><ci id="S2.E1.m1.4.5.3.2.2.cmml" xref="S2.E1.m1.4.5.3.2.2">𝐾</ci><ci id="S2.E1.m1.4.5.3.2.3.cmml" xref="S2.E1.m1.4.5.3.2.3">𝑥</ci></apply><interval closure="open" id="S2.E1.m1.4.5.3.3.1.cmml" xref="S2.E1.m1.4.5.3.3.2"><ci id="S2.E1.m1.3.3.cmml" xref="S2.E1.m1.3.3">𝑖</ci><ci id="S2.E1.m1.4.4.cmml" xref="S2.E1.m1.4.4">𝑗</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.4c">{b_{p}}(i,j)=K\times x(i,j)</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.4d">italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_i , italic_j ) = italic_K × italic_x ( italic_i , italic_j )</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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.2">where <math alttext="K" class="ltx_Math" display="inline" id="S2.p1.2.m1.1"><semantics id="S2.p1.2.m1.1a"><mi id="S2.p1.2.m1.1.1" xref="S2.p1.2.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m1.1b"><ci id="S2.p1.2.m1.1.1.cmml" xref="S2.p1.2.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m1.1c">K</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m1.1d">italic_K</annotation></semantics></math> represents the photon conversion efficiency. In low-light conditions, photon conversion efficiency decreases, resulting in a significant reduction in photon numbers. The process of photons reaching and being captured by the sensor follows a random Poisson process, introducing Poisson noise (also known as shot noise). This noise is amplified due to the reduced photon count. The captured photons are then linearly converted to electrons:</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="{b_{e}}(i,j)=\eta\times{\rm{Poisson}}({b_{p}}(i,j))" class="ltx_Math" display="block" id="S2.E2.m1.5"><semantics id="S2.E2.m1.5a"><mrow id="S2.E2.m1.5.5" xref="S2.E2.m1.5.5.cmml"><mrow id="S2.E2.m1.5.5.3" xref="S2.E2.m1.5.5.3.cmml"><msub id="S2.E2.m1.5.5.3.2" xref="S2.E2.m1.5.5.3.2.cmml"><mi id="S2.E2.m1.5.5.3.2.2" xref="S2.E2.m1.5.5.3.2.2.cmml">b</mi><mi id="S2.E2.m1.5.5.3.2.3" xref="S2.E2.m1.5.5.3.2.3.cmml">e</mi></msub><mo id="S2.E2.m1.5.5.3.1" xref="S2.E2.m1.5.5.3.1.cmml">⁢</mo><mrow id="S2.E2.m1.5.5.3.3.2" xref="S2.E2.m1.5.5.3.3.1.cmml"><mo id="S2.E2.m1.5.5.3.3.2.1" stretchy="false" xref="S2.E2.m1.5.5.3.3.1.cmml">(</mo><mi id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml">i</mi><mo 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start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_i , italic_j ) )</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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.5">where <math alttext="\eta" class="ltx_Math" display="inline" id="S2.p1.3.m1.1"><semantics id="S2.p1.3.m1.1a"><mi id="S2.p1.3.m1.1.1" xref="S2.p1.3.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.p1.3.m1.1b"><ci id="S2.p1.3.m1.1.1.cmml" xref="S2.p1.3.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m1.1d">italic_η</annotation></semantics></math> denotes the quantum efficiency, and <math alttext="{\rm{Poisson}}\left(\lambda\right)" class="ltx_Math" display="inline" id="S2.p1.4.m2.1"><semantics id="S2.p1.4.m2.1a"><mrow id="S2.p1.4.m2.1.2" xref="S2.p1.4.m2.1.2.cmml"><mi id="S2.p1.4.m2.1.2.2" xref="S2.p1.4.m2.1.2.2.cmml">Poisson</mi><mo id="S2.p1.4.m2.1.2.1" xref="S2.p1.4.m2.1.2.1.cmml">⁢</mo><mrow id="S2.p1.4.m2.1.2.3.2" xref="S2.p1.4.m2.1.2.cmml"><mo id="S2.p1.4.m2.1.2.3.2.1" xref="S2.p1.4.m2.1.2.cmml">(</mo><mi id="S2.p1.4.m2.1.1" xref="S2.p1.4.m2.1.1.cmml">λ</mi><mo id="S2.p1.4.m2.1.2.3.2.2" xref="S2.p1.4.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.4.m2.1b"><apply id="S2.p1.4.m2.1.2.cmml" xref="S2.p1.4.m2.1.2"><times id="S2.p1.4.m2.1.2.1.cmml" xref="S2.p1.4.m2.1.2.1"></times><ci id="S2.p1.4.m2.1.2.2.cmml" xref="S2.p1.4.m2.1.2.2">Poisson</ci><ci id="S2.p1.4.m2.1.1.cmml" xref="S2.p1.4.m2.1.1">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m2.1c">{\rm{Poisson}}\left(\lambda\right)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m2.1d">roman_Poisson ( italic_λ )</annotation></semantics></math> is an operator that samples a Poisson random variable with mean <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.p1.5.m3.1"><semantics id="S2.p1.5.m3.1a"><mi id="S2.p1.5.m3.1.1" xref="S2.p1.5.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.p1.5.m3.1b"><ci id="S2.p1.5.m3.1.1.cmml" xref="S2.p1.5.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m3.1d">italic_λ</annotation></semantics></math>. This process also introduces additive Gaussian noise, known as readout noise. The resulting electron distribution becomes:</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="{b_{r}}(i,j)={b_{e}}(i,j)+{n_{r}}" class="ltx_Math" display="block" id="S2.E3.m1.4"><semantics id="S2.E3.m1.4a"><mrow id="S2.E3.m1.4.5" xref="S2.E3.m1.4.5.cmml"><mrow id="S2.E3.m1.4.5.2" xref="S2.E3.m1.4.5.2.cmml"><msub id="S2.E3.m1.4.5.2.2" xref="S2.E3.m1.4.5.2.2.cmml"><mi id="S2.E3.m1.4.5.2.2.2" xref="S2.E3.m1.4.5.2.2.2.cmml">b</mi><mi id="S2.E3.m1.4.5.2.2.3" xref="S2.E3.m1.4.5.2.2.3.cmml">r</mi></msub><mo id="S2.E3.m1.4.5.2.1" xref="S2.E3.m1.4.5.2.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.5.2.3.2" xref="S2.E3.m1.4.5.2.3.1.cmml"><mo id="S2.E3.m1.4.5.2.3.2.1" stretchy="false" xref="S2.E3.m1.4.5.2.3.1.cmml">(</mo><mi id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml">i</mi><mo id="S2.E3.m1.4.5.2.3.2.2" xref="S2.E3.m1.4.5.2.3.1.cmml">,</mo><mi id="S2.E3.m1.2.2" xref="S2.E3.m1.2.2.cmml">j</mi><mo id="S2.E3.m1.4.5.2.3.2.3" stretchy="false" xref="S2.E3.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.4.5.1" xref="S2.E3.m1.4.5.1.cmml">=</mo><mrow id="S2.E3.m1.4.5.3" xref="S2.E3.m1.4.5.3.cmml"><mrow id="S2.E3.m1.4.5.3.2" xref="S2.E3.m1.4.5.3.2.cmml"><msub id="S2.E3.m1.4.5.3.2.2" xref="S2.E3.m1.4.5.3.2.2.cmml"><mi id="S2.E3.m1.4.5.3.2.2.2" xref="S2.E3.m1.4.5.3.2.2.2.cmml">b</mi><mi id="S2.E3.m1.4.5.3.2.2.3" xref="S2.E3.m1.4.5.3.2.2.3.cmml">e</mi></msub><mo id="S2.E3.m1.4.5.3.2.1" xref="S2.E3.m1.4.5.3.2.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.5.3.2.3.2" xref="S2.E3.m1.4.5.3.2.3.1.cmml"><mo id="S2.E3.m1.4.5.3.2.3.2.1" stretchy="false" xref="S2.E3.m1.4.5.3.2.3.1.cmml">(</mo><mi id="S2.E3.m1.3.3" xref="S2.E3.m1.3.3.cmml">i</mi><mo id="S2.E3.m1.4.5.3.2.3.2.2" xref="S2.E3.m1.4.5.3.2.3.1.cmml">,</mo><mi id="S2.E3.m1.4.4" xref="S2.E3.m1.4.4.cmml">j</mi><mo id="S2.E3.m1.4.5.3.2.3.2.3" stretchy="false" xref="S2.E3.m1.4.5.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.4.5.3.1" xref="S2.E3.m1.4.5.3.1.cmml">+</mo><msub id="S2.E3.m1.4.5.3.3" xref="S2.E3.m1.4.5.3.3.cmml"><mi id="S2.E3.m1.4.5.3.3.2" xref="S2.E3.m1.4.5.3.3.2.cmml">n</mi><mi id="S2.E3.m1.4.5.3.3.3" xref="S2.E3.m1.4.5.3.3.3.cmml">r</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.m1.4b"><apply id="S2.E3.m1.4.5.cmml" xref="S2.E3.m1.4.5"><eq id="S2.E3.m1.4.5.1.cmml" xref="S2.E3.m1.4.5.1"></eq><apply id="S2.E3.m1.4.5.2.cmml" xref="S2.E3.m1.4.5.2"><times id="S2.E3.m1.4.5.2.1.cmml" xref="S2.E3.m1.4.5.2.1"></times><apply id="S2.E3.m1.4.5.2.2.cmml" xref="S2.E3.m1.4.5.2.2"><csymbol cd="ambiguous" id="S2.E3.m1.4.5.2.2.1.cmml" xref="S2.E3.m1.4.5.2.2">subscript</csymbol><ci id="S2.E3.m1.4.5.2.2.2.cmml" xref="S2.E3.m1.4.5.2.2.2">𝑏</ci><ci id="S2.E3.m1.4.5.2.2.3.cmml" xref="S2.E3.m1.4.5.2.2.3">𝑟</ci></apply><interval closure="open" id="S2.E3.m1.4.5.2.3.1.cmml" xref="S2.E3.m1.4.5.2.3.2"><ci id="S2.E3.m1.1.1.cmml" xref="S2.E3.m1.1.1">𝑖</ci><ci id="S2.E3.m1.2.2.cmml" xref="S2.E3.m1.2.2">𝑗</ci></interval></apply><apply id="S2.E3.m1.4.5.3.cmml" xref="S2.E3.m1.4.5.3"><plus id="S2.E3.m1.4.5.3.1.cmml" xref="S2.E3.m1.4.5.3.1"></plus><apply id="S2.E3.m1.4.5.3.2.cmml" xref="S2.E3.m1.4.5.3.2"><times id="S2.E3.m1.4.5.3.2.1.cmml" xref="S2.E3.m1.4.5.3.2.1"></times><apply id="S2.E3.m1.4.5.3.2.2.cmml" xref="S2.E3.m1.4.5.3.2.2"><csymbol cd="ambiguous" id="S2.E3.m1.4.5.3.2.2.1.cmml" xref="S2.E3.m1.4.5.3.2.2">subscript</csymbol><ci id="S2.E3.m1.4.5.3.2.2.2.cmml" xref="S2.E3.m1.4.5.3.2.2.2">𝑏</ci><ci id="S2.E3.m1.4.5.3.2.2.3.cmml" xref="S2.E3.m1.4.5.3.2.2.3">𝑒</ci></apply><interval closure="open" id="S2.E3.m1.4.5.3.2.3.1.cmml" xref="S2.E3.m1.4.5.3.2.3.2"><ci id="S2.E3.m1.3.3.cmml" xref="S2.E3.m1.3.3">𝑖</ci><ci id="S2.E3.m1.4.4.cmml" xref="S2.E3.m1.4.4">𝑗</ci></interval></apply><apply id="S2.E3.m1.4.5.3.3.cmml" xref="S2.E3.m1.4.5.3.3"><csymbol cd="ambiguous" id="S2.E3.m1.4.5.3.3.1.cmml" xref="S2.E3.m1.4.5.3.3">subscript</csymbol><ci id="S2.E3.m1.4.5.3.3.2.cmml" xref="S2.E3.m1.4.5.3.3.2">𝑛</ci><ci id="S2.E3.m1.4.5.3.3.3.cmml" xref="S2.E3.m1.4.5.3.3.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.4c">{b_{r}}(i,j)={b_{e}}(i,j)+{n_{r}}</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.4d">italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_i , italic_j ) = italic_b start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_i , italic_j ) + italic_n start_POSTSUBSCRIPT italic_r 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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.7">where <math alttext="n_{r}\sim\mathcal{N}(0,\sigma^{2})" class="ltx_Math" display="inline" id="S2.p1.6.m1.2"><semantics id="S2.p1.6.m1.2a"><mrow id="S2.p1.6.m1.2.2" xref="S2.p1.6.m1.2.2.cmml"><msub id="S2.p1.6.m1.2.2.3" xref="S2.p1.6.m1.2.2.3.cmml"><mi id="S2.p1.6.m1.2.2.3.2" xref="S2.p1.6.m1.2.2.3.2.cmml">n</mi><mi id="S2.p1.6.m1.2.2.3.3" xref="S2.p1.6.m1.2.2.3.3.cmml">r</mi></msub><mo id="S2.p1.6.m1.2.2.2" xref="S2.p1.6.m1.2.2.2.cmml">∼</mo><mrow id="S2.p1.6.m1.2.2.1" xref="S2.p1.6.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p1.6.m1.2.2.1.3" xref="S2.p1.6.m1.2.2.1.3.cmml">𝒩</mi><mo id="S2.p1.6.m1.2.2.1.2" xref="S2.p1.6.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.p1.6.m1.2.2.1.1.1" xref="S2.p1.6.m1.2.2.1.1.2.cmml"><mo id="S2.p1.6.m1.2.2.1.1.1.2" stretchy="false" xref="S2.p1.6.m1.2.2.1.1.2.cmml">(</mo><mn id="S2.p1.6.m1.1.1" xref="S2.p1.6.m1.1.1.cmml">0</mn><mo id="S2.p1.6.m1.2.2.1.1.1.3" xref="S2.p1.6.m1.2.2.1.1.2.cmml">,</mo><msup id="S2.p1.6.m1.2.2.1.1.1.1" xref="S2.p1.6.m1.2.2.1.1.1.1.cmml"><mi id="S2.p1.6.m1.2.2.1.1.1.1.2" xref="S2.p1.6.m1.2.2.1.1.1.1.2.cmml">σ</mi><mn id="S2.p1.6.m1.2.2.1.1.1.1.3" xref="S2.p1.6.m1.2.2.1.1.1.1.3.cmml">2</mn></msup><mo id="S2.p1.6.m1.2.2.1.1.1.4" stretchy="false" xref="S2.p1.6.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.6.m1.2b"><apply id="S2.p1.6.m1.2.2.cmml" xref="S2.p1.6.m1.2.2"><csymbol cd="latexml" id="S2.p1.6.m1.2.2.2.cmml" xref="S2.p1.6.m1.2.2.2">similar-to</csymbol><apply id="S2.p1.6.m1.2.2.3.cmml" xref="S2.p1.6.m1.2.2.3"><csymbol cd="ambiguous" id="S2.p1.6.m1.2.2.3.1.cmml" xref="S2.p1.6.m1.2.2.3">subscript</csymbol><ci id="S2.p1.6.m1.2.2.3.2.cmml" xref="S2.p1.6.m1.2.2.3.2">𝑛</ci><ci id="S2.p1.6.m1.2.2.3.3.cmml" xref="S2.p1.6.m1.2.2.3.3">𝑟</ci></apply><apply id="S2.p1.6.m1.2.2.1.cmml" xref="S2.p1.6.m1.2.2.1"><times id="S2.p1.6.m1.2.2.1.2.cmml" xref="S2.p1.6.m1.2.2.1.2"></times><ci id="S2.p1.6.m1.2.2.1.3.cmml" xref="S2.p1.6.m1.2.2.1.3">𝒩</ci><interval closure="open" id="S2.p1.6.m1.2.2.1.1.2.cmml" xref="S2.p1.6.m1.2.2.1.1.1"><cn id="S2.p1.6.m1.1.1.cmml" type="integer" xref="S2.p1.6.m1.1.1">0</cn><apply id="S2.p1.6.m1.2.2.1.1.1.1.cmml" xref="S2.p1.6.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.6.m1.2.2.1.1.1.1.1.cmml" xref="S2.p1.6.m1.2.2.1.1.1.1">superscript</csymbol><ci id="S2.p1.6.m1.2.2.1.1.1.1.2.cmml" xref="S2.p1.6.m1.2.2.1.1.1.1.2">𝜎</ci><cn id="S2.p1.6.m1.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.p1.6.m1.2.2.1.1.1.1.3">2</cn></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m1.2c">n_{r}\sim\mathcal{N}(0,\sigma^{2})</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m1.2d">italic_n start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.p1.7.m2.1"><semantics id="S2.p1.7.m2.1a"><mi id="S2.p1.7.m2.1.1" xref="S2.p1.7.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.p1.7.m2.1b"><ci id="S2.p1.7.m2.1.1.cmml" xref="S2.p1.7.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m2.1d">italic_σ</annotation></semantics></math> represents the standard deviation of the readout noise. Subsequently, this electron distribution is digitized with a certain bias and quantized into the measured image:</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="{b_{a}}(i,j)=d\times{b_{r}}(i,j)+{b_{l}}" class="ltx_Math" display="block" id="S2.E4.m1.4"><semantics id="S2.E4.m1.4a"><mrow id="S2.E4.m1.4.5" xref="S2.E4.m1.4.5.cmml"><mrow id="S2.E4.m1.4.5.2" xref="S2.E4.m1.4.5.2.cmml"><msub id="S2.E4.m1.4.5.2.2" xref="S2.E4.m1.4.5.2.2.cmml"><mi id="S2.E4.m1.4.5.2.2.2" xref="S2.E4.m1.4.5.2.2.2.cmml">b</mi><mi id="S2.E4.m1.4.5.2.2.3" xref="S2.E4.m1.4.5.2.2.3.cmml">a</mi></msub><mo id="S2.E4.m1.4.5.2.1" xref="S2.E4.m1.4.5.2.1.cmml">⁢</mo><mrow id="S2.E4.m1.4.5.2.3.2" xref="S2.E4.m1.4.5.2.3.1.cmml"><mo id="S2.E4.m1.4.5.2.3.2.1" stretchy="false" xref="S2.E4.m1.4.5.2.3.1.cmml">(</mo><mi id="S2.E4.m1.1.1" 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italic_a end_POSTSUBSCRIPT ( italic_i , italic_j ) = italic_d × italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_i , italic_j ) + italic_b start_POSTSUBSCRIPT italic_l 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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.9">where <math alttext="d" class="ltx_Math" display="inline" id="S2.p1.8.m1.1"><semantics id="S2.p1.8.m1.1a"><mi id="S2.p1.8.m1.1.1" xref="S2.p1.8.m1.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S2.p1.8.m1.1b"><ci id="S2.p1.8.m1.1.1.cmml" xref="S2.p1.8.m1.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m1.1c">d</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m1.1d">italic_d</annotation></semantics></math> denotes the analogue-to-digital conversion operation, and <math alttext="{b_{l}}" class="ltx_Math" display="inline" id="S2.p1.9.m2.1"><semantics id="S2.p1.9.m2.1a"><msub id="S2.p1.9.m2.1.1" xref="S2.p1.9.m2.1.1.cmml"><mi id="S2.p1.9.m2.1.1.2" xref="S2.p1.9.m2.1.1.2.cmml">b</mi><mi id="S2.p1.9.m2.1.1.3" xref="S2.p1.9.m2.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.9.m2.1b"><apply id="S2.p1.9.m2.1.1.cmml" xref="S2.p1.9.m2.1.1"><csymbol cd="ambiguous" id="S2.p1.9.m2.1.1.1.cmml" xref="S2.p1.9.m2.1.1">subscript</csymbol><ci id="S2.p1.9.m2.1.1.2.cmml" xref="S2.p1.9.m2.1.1.2">𝑏</ci><ci id="S2.p1.9.m2.1.1.3.cmml" xref="S2.p1.9.m2.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m2.1c">{b_{l}}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m2.1d">italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> is the bias amount. The digital image is then quantized for storage, and the final captured image can be expressed as:</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="{b_{cap}}(i,j)={\rm{Quantize}}({b_{a}}(i,j))" class="ltx_Math" display="block" id="S2.E5.m1.5"><semantics id="S2.E5.m1.5a"><mrow id="S2.E5.m1.5.5" xref="S2.E5.m1.5.5.cmml"><mrow id="S2.E5.m1.5.5.3" xref="S2.E5.m1.5.5.3.cmml"><msub id="S2.E5.m1.5.5.3.2" xref="S2.E5.m1.5.5.3.2.cmml"><mi id="S2.E5.m1.5.5.3.2.2" xref="S2.E5.m1.5.5.3.2.2.cmml">b</mi><mrow id="S2.E5.m1.5.5.3.2.3" xref="S2.E5.m1.5.5.3.2.3.cmml"><mi id="S2.E5.m1.5.5.3.2.3.2" xref="S2.E5.m1.5.5.3.2.3.2.cmml">c</mi><mo id="S2.E5.m1.5.5.3.2.3.1" xref="S2.E5.m1.5.5.3.2.3.1.cmml">⁢</mo><mi id="S2.E5.m1.5.5.3.2.3.3" xref="S2.E5.m1.5.5.3.2.3.3.cmml">a</mi><mo id="S2.E5.m1.5.5.3.2.3.1a" 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encoding="application/x-tex" id="S2.E5.m1.5c">{b_{cap}}(i,j)={\rm{Quantize}}({b_{a}}(i,j))</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.5d">italic_b start_POSTSUBSCRIPT italic_c italic_a italic_p end_POSTSUBSCRIPT ( italic_i , italic_j ) = roman_Quantize ( italic_b start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_i , italic_j ) )</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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.10">where Quantize(·) denotes the quantization operation, which introduces additional uniform noise into the stored digital image.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">Throughout this process, multiple types of noise accumulate. In low-light conditions, the severe lack of photon capture and significant amplification of Poisson noise serve as initial sources that compound subsequent noise effects. This combination weakens the measured image quality, ultimately complicating the process of reconstructing and recovering lensless measurements in low-light conditions.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Proposed Method</h2> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="411" id="S3.F2.g1" src="extracted/6115207/figure/2.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Overview of the reconstruction pipeline for the proposed framework.</figcaption> </figure> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">Analysis from the previous section reveals that the dominant challenge is noise interference, which necessitates a reconstruction method resistant to such interference. In this context, we introduce our proposed method. We begin with a general explanation of our methodology and then delve into the specifics in the following subsections.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">Intuitively, the severe noise interference in the measured image degrades the information we can extract directly, making it challenging to use a network to map the relationship from the noisy image to the scene. Therefore, we employ a "closer-to-closer" strategy. We fully utilize the physics model as a prior to guide an initial noisy reconstruction, then progressively refine it to achieve a clear reconstruction through data-driven mapping. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.F2" title="Figure 2 ‣ 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">2</span></a> illustrates the entire framework of our method. We will now break it down into more technical details.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Fisrt Stage</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.4">In the first stage, we rely on the forward measurement model of lensless imaging as follows:</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="{\bf{b}}={\bf{H}}{\bf{x}}" class="ltx_Math" display="block" id="S3.E6.m1.1"><semantics id="S3.E6.m1.1a"><mrow id="S3.E6.m1.1.1" xref="S3.E6.m1.1.1.cmml"><mi id="S3.E6.m1.1.1.2" xref="S3.E6.m1.1.1.2.cmml">𝐛</mi><mo id="S3.E6.m1.1.1.1" xref="S3.E6.m1.1.1.1.cmml">=</mo><mi id="S3.E6.m1.1.1.3" xref="S3.E6.m1.1.1.3.cmml">𝐇𝐱</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.E6.m1.1b"><apply id="S3.E6.m1.1.1.cmml" xref="S3.E6.m1.1.1"><eq id="S3.E6.m1.1.1.1.cmml" xref="S3.E6.m1.1.1.1"></eq><ci id="S3.E6.m1.1.1.2.cmml" xref="S3.E6.m1.1.1.2">𝐛</ci><ci id="S3.E6.m1.1.1.3.cmml" xref="S3.E6.m1.1.1.3">𝐇𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E6.m1.1c">{\bf{b}}={\bf{H}}{\bf{x}}</annotation><annotation encoding="application/x-llamapun" id="S3.E6.m1.1d">bold_b = bold_Hx</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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p1.3">where <math alttext="{\bf{b}}" 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">𝐛</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">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">{\bf{b}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">bold_b</annotation></semantics></math> denotes the measurements collected by the sensor, <math alttext="{\bf{H}}" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mi id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml">𝐇</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">𝐇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">{\bf{H}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">bold_H</annotation></semantics></math> represents the forward measurement process of the system (the convolutional matrix of the system’s point-spread function (PSF) obtained through practical calibrations), and <math alttext="{\bf{x}}" 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">𝐱</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">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">{\bf{x}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">bold_x</annotation></semantics></math> denotes the measured scene.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.1">From this linear equation, we can see that partial information of the scene <math alttext="{\bf{x}}" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">{\bf{x}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">bold_x</annotation></semantics></math> lies in the range space of the adjoint operator of the forward measurement process:</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="{\bf{H^{+}b}}={\bf{H^{+}H}}{\bf{x}}=\bf{x^{+}}" class="ltx_Math" display="block" id="S3.E7.m1.1"><semantics id="S3.E7.m1.1a"><mrow id="S3.E7.m1.1.1" xref="S3.E7.m1.1.1.cmml"><mrow id="S3.E7.m1.1.1.2" xref="S3.E7.m1.1.1.2.cmml"><msup id="S3.E7.m1.1.1.2.2" xref="S3.E7.m1.1.1.2.2.cmml"><mi id="S3.E7.m1.1.1.2.2.2" xref="S3.E7.m1.1.1.2.2.2.cmml">𝐇</mi><mo id="S3.E7.m1.1.1.2.2.3" xref="S3.E7.m1.1.1.2.2.3.cmml">+</mo></msup><mo id="S3.E7.m1.1.1.2.1" xref="S3.E7.m1.1.1.2.1.cmml">⁢</mo><mi id="S3.E7.m1.1.1.2.3" xref="S3.E7.m1.1.1.2.3.cmml">𝐛</mi></mrow><mo id="S3.E7.m1.1.1.3" xref="S3.E7.m1.1.1.3.cmml">=</mo><mrow id="S3.E7.m1.1.1.4" xref="S3.E7.m1.1.1.4.cmml"><msup id="S3.E7.m1.1.1.4.2" xref="S3.E7.m1.1.1.4.2.cmml"><mi id="S3.E7.m1.1.1.4.2.2" xref="S3.E7.m1.1.1.4.2.2.cmml">𝐇</mi><mo id="S3.E7.m1.1.1.4.2.3" xref="S3.E7.m1.1.1.4.2.3.cmml">+</mo></msup><mo id="S3.E7.m1.1.1.4.1" xref="S3.E7.m1.1.1.4.1.cmml">⁢</mo><mi id="S3.E7.m1.1.1.4.3" xref="S3.E7.m1.1.1.4.3.cmml">𝐇𝐱</mi></mrow><mo id="S3.E7.m1.1.1.5" xref="S3.E7.m1.1.1.5.cmml">=</mo><msup id="S3.E7.m1.1.1.6" xref="S3.E7.m1.1.1.6.cmml"><mi id="S3.E7.m1.1.1.6.2" xref="S3.E7.m1.1.1.6.2.cmml">𝐱</mi><mo id="S3.E7.m1.1.1.6.3" xref="S3.E7.m1.1.1.6.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.E7.m1.1b"><apply id="S3.E7.m1.1.1.cmml" xref="S3.E7.m1.1.1"><and id="S3.E7.m1.1.1a.cmml" 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xref="S3.E7.m1.1.1.4.2.3"></plus></apply><ci id="S3.E7.m1.1.1.4.3.cmml" xref="S3.E7.m1.1.1.4.3">𝐇𝐱</ci></apply></apply><apply id="S3.E7.m1.1.1c.cmml" xref="S3.E7.m1.1.1"><eq id="S3.E7.m1.1.1.5.cmml" xref="S3.E7.m1.1.1.5"></eq><share href="https://arxiv.org/html/2501.03511v1#S3.E7.m1.1.1.4.cmml" id="S3.E7.m1.1.1d.cmml" xref="S3.E7.m1.1.1"></share><apply id="S3.E7.m1.1.1.6.cmml" xref="S3.E7.m1.1.1.6"><csymbol cd="ambiguous" id="S3.E7.m1.1.1.6.1.cmml" xref="S3.E7.m1.1.1.6">superscript</csymbol><ci id="S3.E7.m1.1.1.6.2.cmml" xref="S3.E7.m1.1.1.6.2">𝐱</ci><plus id="S3.E7.m1.1.1.6.3.cmml" xref="S3.E7.m1.1.1.6.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m1.1c">{\bf{H^{+}b}}={\bf{H^{+}H}}{\bf{x}}=\bf{x^{+}}</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m1.1d">bold_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT bold_b = bold_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT bold_Hx = bold_x start_POSTSUPERSCRIPT + 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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p2.5">where <math alttext="\bf{H^{+}}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m1.1"><semantics id="S3.SS1.p2.2.m1.1a"><msup id="S3.SS1.p2.2.m1.1.1" xref="S3.SS1.p2.2.m1.1.1.cmml"><mi id="S3.SS1.p2.2.m1.1.1.2" xref="S3.SS1.p2.2.m1.1.1.2.cmml">𝐇</mi><mo id="S3.SS1.p2.2.m1.1.1.3" xref="S3.SS1.p2.2.m1.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m1.1b"><apply id="S3.SS1.p2.2.m1.1.1.cmml" xref="S3.SS1.p2.2.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m1.1.1.1.cmml" xref="S3.SS1.p2.2.m1.1.1">superscript</csymbol><ci id="S3.SS1.p2.2.m1.1.1.2.cmml" xref="S3.SS1.p2.2.m1.1.1.2">𝐇</ci><plus id="S3.SS1.p2.2.m1.1.1.3.cmml" xref="S3.SS1.p2.2.m1.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m1.1c">\bf{H^{+}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m1.1d">bold_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is the adjoint operator of the forward measurement process. Considering the orthogonal decomposition <math alttext="\bf{x}=(\bf{H^{+}H})\bf{x}+(\bf{I}-(\bf{(H^{+}H)})\bf{x}=\bf{x^{+}+x^{-}}" class="ltx_math_unparsed" display="inline" id="S3.SS1.p2.3.m2.1"><semantics id="S3.SS1.p2.3.m2.1a"><mrow id="S3.SS1.p2.3.m2.1b"><mi id="S3.SS1.p2.3.m2.1.1">𝐱</mi><mo id="S3.SS1.p2.3.m2.1.2">=</mo><mrow id="S3.SS1.p2.3.m2.1.3"><mo id="S3.SS1.p2.3.m2.1.3.1" stretchy="false">(</mo><msup id="S3.SS1.p2.3.m2.1.3.2"><mi id="S3.SS1.p2.3.m2.1.3.2.2">𝐇</mi><mo id="S3.SS1.p2.3.m2.1.3.2.3">+</mo></msup><mi id="S3.SS1.p2.3.m2.1.3.3">𝐇</mi><mo id="S3.SS1.p2.3.m2.1.3.4" stretchy="false">)</mo></mrow><mi id="S3.SS1.p2.3.m2.1.4">𝐱</mi><mo id="S3.SS1.p2.3.m2.1.5">+</mo><mrow id="S3.SS1.p2.3.m2.1.6"><mo id="S3.SS1.p2.3.m2.1.6.1" stretchy="false">(</mo><mi id="S3.SS1.p2.3.m2.1.6.2">𝐈</mi><mo id="S3.SS1.p2.3.m2.1.6.3">−</mo><mrow id="S3.SS1.p2.3.m2.1.6.4"><mo id="S3.SS1.p2.3.m2.1.6.4.1" stretchy="false">(</mo><mrow id="S3.SS1.p2.3.m2.1.6.4.2"><mo id="S3.SS1.p2.3.m2.1.6.4.2.1" stretchy="false">(</mo><msup id="S3.SS1.p2.3.m2.1.6.4.2.2"><mi id="S3.SS1.p2.3.m2.1.6.4.2.2.2">𝐇</mi><mo id="S3.SS1.p2.3.m2.1.6.4.2.2.3">+</mo></msup><mi id="S3.SS1.p2.3.m2.1.6.4.2.3">𝐇</mi><mo id="S3.SS1.p2.3.m2.1.6.4.2.4" stretchy="false">)</mo></mrow><mo id="S3.SS1.p2.3.m2.1.6.4.3" stretchy="false">)</mo></mrow><mi id="S3.SS1.p2.3.m2.1.6.5">𝐱</mi><mo id="S3.SS1.p2.3.m2.1.6.6">=</mo><msup id="S3.SS1.p2.3.m2.1.6.7"><mi id="S3.SS1.p2.3.m2.1.6.7.2">𝐱</mi><mo id="S3.SS1.p2.3.m2.1.6.7.3">+</mo></msup><mo id="S3.SS1.p2.3.m2.1.6.8">+</mo><msup id="S3.SS1.p2.3.m2.1.6.9"><mi id="S3.SS1.p2.3.m2.1.6.9.2">𝐱</mi><mo id="S3.SS1.p2.3.m2.1.6.9.3">−</mo></msup></mrow></mrow><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m2.1c">\bf{x}=(\bf{H^{+}H})\bf{x}+(\bf{I}-(\bf{(H^{+}H)})\bf{x}=\bf{x^{+}+x^{-}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m2.1d">bold_x = ( bold_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT bold_H ) bold_x + ( bold_I - ( ( bold_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT bold_H ) ) bold_x = bold_x start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT + bold_x start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\bf{x^{+}}" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m3.1"><semantics id="S3.SS1.p2.4.m3.1a"><msup id="S3.SS1.p2.4.m3.1.1" xref="S3.SS1.p2.4.m3.1.1.cmml"><mi id="S3.SS1.p2.4.m3.1.1.2" xref="S3.SS1.p2.4.m3.1.1.2.cmml">𝐱</mi><mo id="S3.SS1.p2.4.m3.1.1.3" xref="S3.SS1.p2.4.m3.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m3.1b"><apply id="S3.SS1.p2.4.m3.1.1.cmml" xref="S3.SS1.p2.4.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.4.m3.1.1.1.cmml" xref="S3.SS1.p2.4.m3.1.1">superscript</csymbol><ci id="S3.SS1.p2.4.m3.1.1.2.cmml" xref="S3.SS1.p2.4.m3.1.1.2">𝐱</ci><plus id="S3.SS1.p2.4.m3.1.1.3.cmml" xref="S3.SS1.p2.4.m3.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m3.1c">\bf{x^{+}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m3.1d">bold_x start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> represents the range component of <math alttext="{\bf{x}}" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m4.1"><semantics id="S3.SS1.p2.5.m4.1a"><mi id="S3.SS1.p2.5.m4.1.1" xref="S3.SS1.p2.5.m4.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m4.1b"><ci id="S3.SS1.p2.5.m4.1.1.cmml" xref="S3.SS1.p2.5.m4.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.5.m4.1c">{\bf{x}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.5.m4.1d">bold_x</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.5">For fast computation, we turn to the direct inverse in the frequency domain, known as Wiener filtering:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E8"> 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ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p3.4">where <math alttext="\lambda" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mi id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><ci id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">italic_λ</annotation></semantics></math> is a noise-related factor (fixed in the experiment), <math alttext="{\cal F}" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.1b"><ci id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1c">{\cal F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1d">caligraphic_F</annotation></semantics></math> and <math alttext="{{\cal F}^{-1}}" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><msup id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.3.m3.1.1.2" xref="S3.SS1.p3.3.m3.1.1.2.cmml">ℱ</mi><mrow id="S3.SS1.p3.3.m3.1.1.3" xref="S3.SS1.p3.3.m3.1.1.3.cmml"><mo id="S3.SS1.p3.3.m3.1.1.3a" xref="S3.SS1.p3.3.m3.1.1.3.cmml">−</mo><mn id="S3.SS1.p3.3.m3.1.1.3.2" xref="S3.SS1.p3.3.m3.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><apply id="S3.SS1.p3.3.m3.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.3.m3.1.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1">superscript</csymbol><ci id="S3.SS1.p3.3.m3.1.1.2.cmml" xref="S3.SS1.p3.3.m3.1.1.2">ℱ</ci><apply id="S3.SS1.p3.3.m3.1.1.3.cmml" xref="S3.SS1.p3.3.m3.1.1.3"><minus id="S3.SS1.p3.3.m3.1.1.3.1.cmml" xref="S3.SS1.p3.3.m3.1.1.3"></minus><cn id="S3.SS1.p3.3.m3.1.1.3.2.cmml" type="integer" xref="S3.SS1.p3.3.m3.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">{{\cal F}^{-1}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> represent the Fourier transform and its inverse, respectively, and <math alttext="\bf{h}" class="ltx_Math" display="inline" id="S3.SS1.p3.4.m4.1"><semantics id="S3.SS1.p3.4.m4.1a"><mi id="S3.SS1.p3.4.m4.1.1" xref="S3.SS1.p3.4.m4.1.1.cmml">𝐡</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.4.m4.1b"><ci id="S3.SS1.p3.4.m4.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1">𝐡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.4.m4.1c">\bf{h}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.4.m4.1d">bold_h</annotation></semantics></math> denotes the PSF. Here, it is initialized with the calibrated one but is set as learnable.</p> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.4">This direct inverse provides partial information of <math alttext="{\bf{x}}" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.1"><semantics id="S3.SS1.p4.1.m1.1a"><mi id="S3.SS1.p4.1.m1.1.1" xref="S3.SS1.p4.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.1b"><ci id="S3.SS1.p4.1.m1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.1.m1.1c">{\bf{x}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.1.m1.1d">bold_x</annotation></semantics></math> embedded in the range space. However, as the measurement <math alttext="\bf{b}" class="ltx_Math" display="inline" id="S3.SS1.p4.2.m2.1"><semantics id="S3.SS1.p4.2.m2.1a"><mi id="S3.SS1.p4.2.m2.1.1" xref="S3.SS1.p4.2.m2.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.2.m2.1b"><ci id="S3.SS1.p4.2.m2.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.1c">\bf{b}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.1d">bold_b</annotation></semantics></math> is highly noisy, the obtained information of <math alttext="{\bf{x}}" class="ltx_Math" display="inline" id="S3.SS1.p4.3.m3.1"><semantics id="S3.SS1.p4.3.m3.1a"><mi id="S3.SS1.p4.3.m3.1.1" xref="S3.SS1.p4.3.m3.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.3.m3.1b"><ci id="S3.SS1.p4.3.m3.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.3.m3.1c">{\bf{x}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.3.m3.1d">bold_x</annotation></semantics></math> is still affected by noise, not entirely accurate, and the part of information <math alttext="\bf{x}-\bf{x^{+}}" class="ltx_Math" display="inline" id="S3.SS1.p4.4.m4.1"><semantics id="S3.SS1.p4.4.m4.1a"><mrow id="S3.SS1.p4.4.m4.1.1" xref="S3.SS1.p4.4.m4.1.1.cmml"><mi id="S3.SS1.p4.4.m4.1.1.2" xref="S3.SS1.p4.4.m4.1.1.2.cmml">𝐱</mi><mo id="S3.SS1.p4.4.m4.1.1.1" xref="S3.SS1.p4.4.m4.1.1.1.cmml">−</mo><msup id="S3.SS1.p4.4.m4.1.1.3" xref="S3.SS1.p4.4.m4.1.1.3.cmml"><mi id="S3.SS1.p4.4.m4.1.1.3.2" xref="S3.SS1.p4.4.m4.1.1.3.2.cmml">𝐱</mi><mo id="S3.SS1.p4.4.m4.1.1.3.3" xref="S3.SS1.p4.4.m4.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.4.m4.1b"><apply id="S3.SS1.p4.4.m4.1.1.cmml" xref="S3.SS1.p4.4.m4.1.1"><minus id="S3.SS1.p4.4.m4.1.1.1.cmml" xref="S3.SS1.p4.4.m4.1.1.1"></minus><ci id="S3.SS1.p4.4.m4.1.1.2.cmml" xref="S3.SS1.p4.4.m4.1.1.2">𝐱</ci><apply id="S3.SS1.p4.4.m4.1.1.3.cmml" xref="S3.SS1.p4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p4.4.m4.1.1.3.1.cmml" xref="S3.SS1.p4.4.m4.1.1.3">superscript</csymbol><ci id="S3.SS1.p4.4.m4.1.1.3.2.cmml" xref="S3.SS1.p4.4.m4.1.1.3.2">𝐱</ci><plus id="S3.SS1.p4.4.m4.1.1.3.3.cmml" xref="S3.SS1.p4.4.m4.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.4.m4.1c">\bf{x}-\bf{x^{+}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.4.m4.1d">bold_x - bold_x start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is still missing. Consequently, these initial results suffer from issues such as amplified noise, extremely low brightness, and poor readability, as seen in the experimental results. To address this, in the next stage, we adopt a diffusion generative model to suppress the noise and progressively generate the missing information.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Second Stage</h3> <section class="ltx_subsubsection" id="S3.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.2.1 </span>Conditional Diffusion Model</h4> <div class="ltx_para" id="S3.SS2.SSS1.p1"> <p class="ltx_p" id="S3.SS2.SSS1.p1.1">In the second stage, we implement a sophisticated data-driven diffusion generative model. This model’s core principle is to gradually generate the distribution of the target image <math alttext="\bf{x_{0}}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p1.1.m1.1"><semantics id="S3.SS2.SSS1.p1.1.m1.1a"><msub id="S3.SS2.SSS1.p1.1.m1.1.1" xref="S3.SS2.SSS1.p1.1.m1.1.1.cmml"><mi id="S3.SS2.SSS1.p1.1.m1.1.1.2" xref="S3.SS2.SSS1.p1.1.m1.1.1.2.cmml">𝐱</mi><mn id="S3.SS2.SSS1.p1.1.m1.1.1.3" xref="S3.SS2.SSS1.p1.1.m1.1.1.3.cmml">𝟎</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p1.1.m1.1b"><apply id="S3.SS2.SSS1.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p1.1.m1.1.1.1.cmml" xref="S3.SS2.SSS1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.SSS1.p1.1.m1.1.1.2.cmml" xref="S3.SS2.SSS1.p1.1.m1.1.1.2">𝐱</ci><cn id="S3.SS2.SSS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS2.SSS1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p1.1.m1.1c">\bf{x_{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p1.1.m1.1d">bold_x start_POSTSUBSCRIPT bold_0 end_POSTSUBSCRIPT</annotation></semantics></math> from noise, following a meticulously designed multiple-step Markov chain. The term "diffusion" aptly describes the inverse of the generation process, as noise is systematically introduced into the clear image—effectively diffusing it.</p> </div> <div class="ltx_para" id="S3.SS2.SSS1.p2"> <p class="ltx_p" id="S3.SS2.SSS1.p2.2">The relationship between adjacent images <math alttext="\bf{x_{t}}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p2.1.m1.1"><semantics id="S3.SS2.SSS1.p2.1.m1.1a"><msub id="S3.SS2.SSS1.p2.1.m1.1.1" xref="S3.SS2.SSS1.p2.1.m1.1.1.cmml"><mi id="S3.SS2.SSS1.p2.1.m1.1.1.2" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.cmml">𝐱</mi><mi id="S3.SS2.SSS1.p2.1.m1.1.1.3" xref="S3.SS2.SSS1.p2.1.m1.1.1.3.cmml">𝐭</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p2.1.m1.1b"><apply id="S3.SS2.SSS1.p2.1.m1.1.1.cmml" xref="S3.SS2.SSS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p2.1.m1.1.1.1.cmml" xref="S3.SS2.SSS1.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.SSS1.p2.1.m1.1.1.2.cmml" xref="S3.SS2.SSS1.p2.1.m1.1.1.2">𝐱</ci><ci id="S3.SS2.SSS1.p2.1.m1.1.1.3.cmml" xref="S3.SS2.SSS1.p2.1.m1.1.1.3">𝐭</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p2.1.m1.1c">\bf{x_{t}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p2.1.m1.1d">bold_x start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\bf{x_{t-1}}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p2.2.m2.1"><semantics id="S3.SS2.SSS1.p2.2.m2.1a"><msub id="S3.SS2.SSS1.p2.2.m2.1.1" xref="S3.SS2.SSS1.p2.2.m2.1.1.cmml"><mi id="S3.SS2.SSS1.p2.2.m2.1.1.2" xref="S3.SS2.SSS1.p2.2.m2.1.1.2.cmml">𝐱</mi><mrow id="S3.SS2.SSS1.p2.2.m2.1.1.3" xref="S3.SS2.SSS1.p2.2.m2.1.1.3.cmml"><mi id="S3.SS2.SSS1.p2.2.m2.1.1.3.2" xref="S3.SS2.SSS1.p2.2.m2.1.1.3.2.cmml">𝐭</mi><mo id="S3.SS2.SSS1.p2.2.m2.1.1.3.1" xref="S3.SS2.SSS1.p2.2.m2.1.1.3.1.cmml">−</mo><mn id="S3.SS2.SSS1.p2.2.m2.1.1.3.3" xref="S3.SS2.SSS1.p2.2.m2.1.1.3.3.cmml">𝟏</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p2.2.m2.1b"><apply id="S3.SS2.SSS1.p2.2.m2.1.1.cmml" xref="S3.SS2.SSS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p2.2.m2.1.1.1.cmml" xref="S3.SS2.SSS1.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.SSS1.p2.2.m2.1.1.2.cmml" xref="S3.SS2.SSS1.p2.2.m2.1.1.2">𝐱</ci><apply id="S3.SS2.SSS1.p2.2.m2.1.1.3.cmml" xref="S3.SS2.SSS1.p2.2.m2.1.1.3"><minus id="S3.SS2.SSS1.p2.2.m2.1.1.3.1.cmml" xref="S3.SS2.SSS1.p2.2.m2.1.1.3.1"></minus><ci id="S3.SS2.SSS1.p2.2.m2.1.1.3.2.cmml" xref="S3.SS2.SSS1.p2.2.m2.1.1.3.2">𝐭</ci><cn id="S3.SS2.SSS1.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S3.SS2.SSS1.p2.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p2.2.m2.1c">\bf{x_{t-1}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p2.2.m2.1d">bold_x start_POSTSUBSCRIPT bold_t - bold_1 end_POSTSUBSCRIPT</annotation></semantics></math> in this diffusion process can be mathematically expressed as follows <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib27" title="">27</a>]</cite>:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E9"> <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="q({\bf{x_{t}}}|{\bf{x_{t-1}}})={\cal N}(\bf{x_{t}};\sqrt{{\alpha_{t}}}{\bf{x_{% t-1}}},(1-{\alpha_{t}}){\bf{I}})" class="ltx_Math" display="block" id="S3.E9.m1.4"><semantics id="S3.E9.m1.4a"><mrow id="S3.E9.m1.4.4" xref="S3.E9.m1.4.4.cmml"><mrow id="S3.E9.m1.1.1.1" xref="S3.E9.m1.1.1.1.cmml"><mi id="S3.E9.m1.1.1.1.3" xref="S3.E9.m1.1.1.1.3.cmml">q</mi><mo id="S3.E9.m1.1.1.1.2" xref="S3.E9.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E9.m1.1.1.1.1.1" xref="S3.E9.m1.1.1.1.1.1.1.cmml"><mo id="S3.E9.m1.1.1.1.1.1.2" stretchy="false" xref="S3.E9.m1.1.1.1.1.1.1.cmml">(</mo><mrow 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ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.SSS1.p2.3">Here, <math alttext="\alpha\bf{{}_{t}}" class="ltx_math_unparsed" display="inline" id="S3.SS2.SSS1.p2.3.m1.1"><semantics id="S3.SS2.SSS1.p2.3.m1.1a"><mrow id="S3.SS2.SSS1.p2.3.m1.1b"><mi id="S3.SS2.SSS1.p2.3.m1.1.1">α</mi><msub id="S3.SS2.SSS1.p2.3.m1.1.2"><mi id="S3.SS2.SSS1.p2.3.m1.1.2a"></mi><mi id="S3.SS2.SSS1.p2.3.m1.1.2.1">𝐭</mi></msub></mrow><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p2.3.m1.1c">\alpha\bf{{}_{t}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p2.3.m1.1d">italic_α start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT</annotation></semantics></math> represents predefined diffusion parameters. As the steps are sufficiently close, we can approximate the added noise as Gaussian. Through successive accumulation steps, the final diffused image converges to a normal distribution.</p> </div> <div class="ltx_para" id="S3.SS2.SSS1.p3"> <p class="ltx_p" id="S3.SS2.SSS1.p3.3">Conversely, in the generation process, we can relate these two images using the Bayesian theorem:</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="{p_{\theta}}({\bf{x}_{t-1}}|{\bf{x_{t}}})={\cal N}({\bf{x_{t-1}}};{\mu_{t}}(% \bf{x_{t}},t),{\tilde{\beta}_{t}}{\bf{I}})" class="ltx_Math" display="block" id="S3.E10.m1.5"><semantics id="S3.E10.m1.5a"><mrow id="S3.E10.m1.5.5" xref="S3.E10.m1.5.5.cmml"><mrow id="S3.E10.m1.2.2.1" xref="S3.E10.m1.2.2.1.cmml"><msub id="S3.E10.m1.2.2.1.3" xref="S3.E10.m1.2.2.1.3.cmml"><mi id="S3.E10.m1.2.2.1.3.2" xref="S3.E10.m1.2.2.1.3.2.cmml">p</mi><mi id="S3.E10.m1.2.2.1.3.3" 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bold_x start_POSTSUBSCRIPT bold_t - bold_1 end_POSTSUBSCRIPT | bold_x start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT ) = caligraphic_N ( bold_x start_POSTSUBSCRIPT bold_t - bold_1 end_POSTSUBSCRIPT ; italic_μ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT , bold_t ) , over~ start_ARG italic_β end_ARG start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT bold_I )</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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.SSS1.p3.2">The mean <math alttext="\mu\bf{{}_{t}}" class="ltx_math_unparsed" display="inline" id="S3.SS2.SSS1.p3.1.m1.1"><semantics id="S3.SS2.SSS1.p3.1.m1.1a"><mrow id="S3.SS2.SSS1.p3.1.m1.1b"><mi id="S3.SS2.SSS1.p3.1.m1.1.1">μ</mi><msub id="S3.SS2.SSS1.p3.1.m1.1.2"><mi id="S3.SS2.SSS1.p3.1.m1.1.2a"></mi><mi id="S3.SS2.SSS1.p3.1.m1.1.2.1">𝐭</mi></msub></mrow><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p3.1.m1.1c">\mu\bf{{}_{t}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p3.1.m1.1d">italic_μ start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT</annotation></semantics></math> and variance <math alttext="\tilde{\beta}\bf{{}_{t}}" class="ltx_math_unparsed" display="inline" id="S3.SS2.SSS1.p3.2.m2.1"><semantics id="S3.SS2.SSS1.p3.2.m2.1a"><mrow id="S3.SS2.SSS1.p3.2.m2.1b"><mover accent="true" id="S3.SS2.SSS1.p3.2.m2.1.1"><mi id="S3.SS2.SSS1.p3.2.m2.1.1.2">β</mi><mo id="S3.SS2.SSS1.p3.2.m2.1.1.1">~</mo></mover><msub id="S3.SS2.SSS1.p3.2.m2.1.2"><mi id="S3.SS2.SSS1.p3.2.m2.1.2a"></mi><mi id="S3.SS2.SSS1.p3.2.m2.1.2.1">𝐭</mi></msub></mrow><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p3.2.m2.1c">\tilde{\beta}\bf{{}_{t}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p3.2.m2.1d">over~ start_ARG italic_β end_ARG start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT</annotation></semantics></math> in this equation are expressed as:</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="{\mu_{t}}(\bf{x_{t}},t)=\frac{1}{{\sqrt{1-{{\bar{\alpha}}_{t}}}}}(\bf{x_{t}}-% \frac{{{\beta_{t}}}}{{\sqrt{1-{{\bar{\alpha}}_{t}}}}}{\varepsilon_{\theta}(\bf% {x_{t}},t)}),{\tilde{\beta}_{t}}=\frac{{1-{{\bar{\alpha}}_{t-1}}}}{{1-{{\bar{% \alpha}}_{t}}}}{\beta_{t}}" class="ltx_Math" display="block" id="S3.E11.m1.4"><semantics id="S3.E11.m1.4a"><mrow id="S3.E11.m1.4.4.2" xref="S3.E11.m1.4.4.3.cmml"><mrow id="S3.E11.m1.3.3.1.1" xref="S3.E11.m1.3.3.1.1.cmml"><mrow id="S3.E11.m1.3.3.1.1.1" xref="S3.E11.m1.3.3.1.1.1.cmml"><msub id="S3.E11.m1.3.3.1.1.1.3" xref="S3.E11.m1.3.3.1.1.1.3.cmml"><mi id="S3.E11.m1.3.3.1.1.1.3.2" xref="S3.E11.m1.3.3.1.1.1.3.2.cmml">μ</mi><mi id="S3.E11.m1.3.3.1.1.1.3.3" xref="S3.E11.m1.3.3.1.1.1.3.3.cmml">t</mi></msub><mo id="S3.E11.m1.3.3.1.1.1.2" xref="S3.E11.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.E11.m1.3.3.1.1.1.1.1" xref="S3.E11.m1.3.3.1.1.1.1.2.cmml"><mo id="S3.E11.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.E11.m1.3.3.1.1.1.1.2.cmml">(</mo><msub id="S3.E11.m1.3.3.1.1.1.1.1.1" xref="S3.E11.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S3.E11.m1.3.3.1.1.1.1.1.1.2" xref="S3.E11.m1.3.3.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="S3.E11.m1.3.3.1.1.1.1.1.1.3" xref="S3.E11.m1.3.3.1.1.1.1.1.1.3.cmml">𝐭</mi></msub><mo id="S3.E11.m1.3.3.1.1.1.1.1.3" xref="S3.E11.m1.3.3.1.1.1.1.2.cmml">,</mo><mi id="S3.E11.m1.1.1" xref="S3.E11.m1.1.1.cmml">𝐭</mi><mo id="S3.E11.m1.3.3.1.1.1.1.1.4" stretchy="false" xref="S3.E11.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E11.m1.3.3.1.1.3" xref="S3.E11.m1.3.3.1.1.3.cmml">=</mo><mrow id="S3.E11.m1.3.3.1.1.2" xref="S3.E11.m1.3.3.1.1.2.cmml"><mfrac 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\frac{{{\beta_{t}}}}{{\sqrt{1-{{\bar{\alpha}}_{t}}}}}{\varepsilon_{\theta}(\bf% {x_{t}},t)}),{\tilde{\beta}_{t}}=\frac{{1-{{\bar{\alpha}}_{t-1}}}}{{1-{{\bar{% \alpha}}_{t}}}}{\beta_{t}}</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m1.4d">italic_μ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT , bold_t ) = divide start_ARG bold_1 end_ARG start_ARG square-root start_ARG bold_1 - over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT end_ARG end_ARG ( bold_x start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT - divide start_ARG italic_β start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG bold_1 - over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT end_ARG end_ARG italic_ε start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT , bold_t ) ) , over~ start_ARG italic_β end_ARG start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT = divide start_ARG bold_1 - over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT bold_t - bold_1 end_POSTSUBSCRIPT end_ARG start_ARG bold_1 - over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT end_ARG italic_β start_POSTSUBSCRIPT bold_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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS2.SSS1.p4"> <p class="ltx_p" id="S3.SS2.SSS1.p4.4">In these equations, <math alttext="{\beta\bf{{}_{t}}}=1-{\alpha\bf{{}_{t}}}" class="ltx_math_unparsed" display="inline" id="S3.SS2.SSS1.p4.1.m1.1"><semantics id="S3.SS2.SSS1.p4.1.m1.1a"><mrow id="S3.SS2.SSS1.p4.1.m1.1b"><mi id="S3.SS2.SSS1.p4.1.m1.1.1">β</mi><mmultiscripts id="S3.SS2.SSS1.p4.1.m1.1.2"><mo id="S3.SS2.SSS1.p4.1.m1.1.2.2">=</mo><mprescripts id="S3.SS2.SSS1.p4.1.m1.1.2a"></mprescripts><mi id="S3.SS2.SSS1.p4.1.m1.1.2.3">𝐭</mi><mrow id="S3.SS2.SSS1.p4.1.m1.1.2b"></mrow></mmultiscripts><mn id="S3.SS2.SSS1.p4.1.m1.1.3">1</mn><mo id="S3.SS2.SSS1.p4.1.m1.1.4">−</mo><mi id="S3.SS2.SSS1.p4.1.m1.1.5">α</mi><msub id="S3.SS2.SSS1.p4.1.m1.1.6"><mi id="S3.SS2.SSS1.p4.1.m1.1.6a"></mi><mi id="S3.SS2.SSS1.p4.1.m1.1.6.1">𝐭</mi></msub></mrow><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p4.1.m1.1c">{\beta\bf{{}_{t}}}=1-{\alpha\bf{{}_{t}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p4.1.m1.1d">italic_β start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT = 1 - italic_α start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT</annotation></semantics></math>, <math alttext="{\bar{\alpha}\bf{{}_{t}}}=\prod\nolimits_{1}^{T}{{\alpha\bf{{}_{t}}}}" class="ltx_math_unparsed" display="inline" id="S3.SS2.SSS1.p4.2.m2.1"><semantics id="S3.SS2.SSS1.p4.2.m2.1a"><mrow id="S3.SS2.SSS1.p4.2.m2.1b"><mover accent="true" id="S3.SS2.SSS1.p4.2.m2.1.1"><mi id="S3.SS2.SSS1.p4.2.m2.1.1.2">α</mi><mo id="S3.SS2.SSS1.p4.2.m2.1.1.1">¯</mo></mover><mmultiscripts id="S3.SS2.SSS1.p4.2.m2.1.2"><mo id="S3.SS2.SSS1.p4.2.m2.1.2.2" rspace="0.111em">=</mo><mprescripts id="S3.SS2.SSS1.p4.2.m2.1.2a"></mprescripts><mi id="S3.SS2.SSS1.p4.2.m2.1.2.3">𝐭</mi><mrow id="S3.SS2.SSS1.p4.2.m2.1.2b"></mrow></mmultiscripts><msubsup id="S3.SS2.SSS1.p4.2.m2.1.3"><mo id="S3.SS2.SSS1.p4.2.m2.1.3.2.2">∏</mo><mn id="S3.SS2.SSS1.p4.2.m2.1.3.2.3">1</mn><mi id="S3.SS2.SSS1.p4.2.m2.1.3.3">T</mi></msubsup><mi id="S3.SS2.SSS1.p4.2.m2.1.4">α</mi><msub id="S3.SS2.SSS1.p4.2.m2.1.5"><mi id="S3.SS2.SSS1.p4.2.m2.1.5a"></mi><mi id="S3.SS2.SSS1.p4.2.m2.1.5.1">𝐭</mi></msub></mrow><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p4.2.m2.1c">{\bar{\alpha}\bf{{}_{t}}}=\prod\nolimits_{1}^{T}{{\alpha\bf{{}_{t}}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p4.2.m2.1d">over¯ start_ARG italic_α end_ARG start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT = ∏ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_α start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT</annotation></semantics></math>, and <math alttext="T" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p4.3.m3.1"><semantics id="S3.SS2.SSS1.p4.3.m3.1a"><mi id="S3.SS2.SSS1.p4.3.m3.1.1" xref="S3.SS2.SSS1.p4.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p4.3.m3.1b"><ci id="S3.SS2.SSS1.p4.3.m3.1.1.cmml" xref="S3.SS2.SSS1.p4.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p4.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p4.3.m3.1d">italic_T</annotation></semantics></math> denotes the total number of generation steps. The term <math alttext="{\varepsilon_{\theta}}(\bf{x_{t}},t)" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p4.4.m4.2"><semantics id="S3.SS2.SSS1.p4.4.m4.2a"><mrow id="S3.SS2.SSS1.p4.4.m4.2.2" xref="S3.SS2.SSS1.p4.4.m4.2.2.cmml"><msub id="S3.SS2.SSS1.p4.4.m4.2.2.3" xref="S3.SS2.SSS1.p4.4.m4.2.2.3.cmml"><mi id="S3.SS2.SSS1.p4.4.m4.2.2.3.2" xref="S3.SS2.SSS1.p4.4.m4.2.2.3.2.cmml">ε</mi><mi id="S3.SS2.SSS1.p4.4.m4.2.2.3.3" xref="S3.SS2.SSS1.p4.4.m4.2.2.3.3.cmml">θ</mi></msub><mo id="S3.SS2.SSS1.p4.4.m4.2.2.2" xref="S3.SS2.SSS1.p4.4.m4.2.2.2.cmml">⁢</mo><mrow id="S3.SS2.SSS1.p4.4.m4.2.2.1.1" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.2.cmml"><mo id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.2" stretchy="false" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.2.cmml">(</mo><msub id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.cmml"><mi id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.2" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.2.cmml">𝐱</mi><mi id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.3" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.3.cmml">𝐭</mi></msub><mo id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.3" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.2.cmml">,</mo><mi id="S3.SS2.SSS1.p4.4.m4.1.1" xref="S3.SS2.SSS1.p4.4.m4.1.1.cmml">𝐭</mi><mo id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.4" stretchy="false" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p4.4.m4.2b"><apply id="S3.SS2.SSS1.p4.4.m4.2.2.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2"><times id="S3.SS2.SSS1.p4.4.m4.2.2.2.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.2"></times><apply id="S3.SS2.SSS1.p4.4.m4.2.2.3.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p4.4.m4.2.2.3.1.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.3">subscript</csymbol><ci id="S3.SS2.SSS1.p4.4.m4.2.2.3.2.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.3.2">𝜀</ci><ci id="S3.SS2.SSS1.p4.4.m4.2.2.3.3.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.3.3">𝜃</ci></apply><interval closure="open" id="S3.SS2.SSS1.p4.4.m4.2.2.1.2.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.1"><apply id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.1.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.2.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.2">𝐱</ci><ci id="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.3.cmml" xref="S3.SS2.SSS1.p4.4.m4.2.2.1.1.1.3">𝐭</ci></apply><ci id="S3.SS2.SSS1.p4.4.m4.1.1.cmml" xref="S3.SS2.SSS1.p4.4.m4.1.1">𝐭</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p4.4.m4.2c">{\varepsilon_{\theta}}(\bf{x_{t}},t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p4.4.m4.2d">italic_ε start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT , bold_t )</annotation></semantics></math> denotes the noise added during the diffusion process and is the key variable that the generation module must learn to predict. For a deeper dive into the intricacies of the diffusion model, we refer readers to the work by Ho et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib27" title="">27</a>]</cite>.</p> </div> <div class="ltx_para" id="S3.SS2.SSS1.p5"> <p class="ltx_p" id="S3.SS2.SSS1.p5.1">The original diffusion model described above is intended for general image generation tasks, where a high-quality realistic image can be generated. However, for our task, we have one important condition to consider: the generated image must adhere to the measured image using the lensless forward measurement model. In other words, the generation process must be guided, becoming less random, to produce the result we require. In this context, we design two techniques: one for the generation process itself and another for the training process.</p> </div> <div class="ltx_para" id="S3.SS2.SSS1.p6"> <p class="ltx_p" id="S3.SS2.SSS1.p6.3">To enhance the generation process, we introduce a conditional distribution approach, which is the LL Module in Fig <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.F2" title="Figure 2 ‣ 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">2</span></a>. By incorporating the initial reconstruction result as an additional input to the network, we effectively condition the generation on the context of the imaging scene and the measurement process noise. This can be formalized as:</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="{p_{\theta}}(\bf{x}_{t-1})|{\bf{x}_{t}},s)={\cal N}({\bf{x}_{t-1}};{\mu_{% \theta}}({\bf{x}_{t}},s,t),{\rm{}}{\tilde{\beta}_{t}}{\bf{I}})" class="ltx_math_unparsed" display="block" id="S3.E12.m1.1"><semantics id="S3.E12.m1.1a"><mrow id="S3.E12.m1.1b"><msub id="S3.E12.m1.1.2"><mi id="S3.E12.m1.1.2.2">p</mi><mi id="S3.E12.m1.1.2.3">θ</mi></msub><mrow id="S3.E12.m1.1.3"><mo id="S3.E12.m1.1.3.1" stretchy="false">(</mo><msub id="S3.E12.m1.1.3.2"><mi id="S3.E12.m1.1.3.2.2">𝐱</mi><mrow id="S3.E12.m1.1.3.2.3"><mi id="S3.E12.m1.1.3.2.3.2">𝐭</mi><mo id="S3.E12.m1.1.3.2.3.1">−</mo><mn id="S3.E12.m1.1.3.2.3.3">𝟏</mn></mrow></msub><mo id="S3.E12.m1.1.3.3" stretchy="false">)</mo></mrow><mo fence="false" id="S3.E12.m1.1.4" rspace="0.167em" stretchy="false">|</mo><msub id="S3.E12.m1.1.5"><mi id="S3.E12.m1.1.5.2">𝐱</mi><mi id="S3.E12.m1.1.5.3">𝐭</mi></msub><mo id="S3.E12.m1.1.6">,</mo><mi id="S3.E12.m1.1.1">𝐬</mi><mo id="S3.E12.m1.1.7" stretchy="false">)</mo><mo id="S3.E12.m1.1.8">=</mo><mi class="ltx_font_mathcaligraphic" id="S3.E12.m1.1.9">𝒩</mi><mo id="S3.E12.m1.1.10" stretchy="false">(</mo><msub id="S3.E12.m1.1.11"><mi id="S3.E12.m1.1.11.2">𝐱</mi><mrow id="S3.E12.m1.1.11.3"><mi id="S3.E12.m1.1.11.3.2">𝐭</mi><mo id="S3.E12.m1.1.11.3.1">−</mo><mn id="S3.E12.m1.1.11.3.3">𝟏</mn></mrow></msub><mo id="S3.E12.m1.1.12">;</mo><msub id="S3.E12.m1.1.13"><mi id="S3.E12.m1.1.13.2">μ</mi><mi id="S3.E12.m1.1.13.3">θ</mi></msub><mrow id="S3.E12.m1.1.14"><mo id="S3.E12.m1.1.14.1" stretchy="false">(</mo><msub id="S3.E12.m1.1.14.2"><mi id="S3.E12.m1.1.14.2.2">𝐱</mi><mi id="S3.E12.m1.1.14.2.3">𝐭</mi></msub><mo id="S3.E12.m1.1.14.3">,</mo><mi id="S3.E12.m1.1.14.4">𝐬</mi><mo id="S3.E12.m1.1.14.5">,</mo><mi id="S3.E12.m1.1.14.6">𝐭</mi><mo id="S3.E12.m1.1.14.7" stretchy="false">)</mo></mrow><mo id="S3.E12.m1.1.15">,</mo><msub id="S3.E12.m1.1.16"><mover accent="true" id="S3.E12.m1.1.16.2"><mi id="S3.E12.m1.1.16.2.2">β</mi><mo id="S3.E12.m1.1.16.2.1">~</mo></mover><mi id="S3.E12.m1.1.16.3">𝐭</mi></msub><mi id="S3.E12.m1.1.17">𝐈</mi><mo id="S3.E12.m1.1.18" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S3.E12.m1.1c">{p_{\theta}}(\bf{x}_{t-1})|{\bf{x}_{t}},s)={\cal N}({\bf{x}_{t-1}};{\mu_{% \theta}}({\bf{x}_{t}},s,t),{\rm{}}{\tilde{\beta}_{t}}{\bf{I}})</annotation><annotation encoding="application/x-llamapun" id="S3.E12.m1.1d">italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT bold_t - bold_1 end_POSTSUBSCRIPT ) | bold_x start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT , bold_s ) = caligraphic_N ( bold_x start_POSTSUBSCRIPT bold_t - bold_1 end_POSTSUBSCRIPT ; italic_μ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT , bold_s , bold_t ) , over~ start_ARG italic_β end_ARG start_POSTSUBSCRIPT bold_t end_POSTSUBSCRIPT bold_I )</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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.SSS1.p6.2">Here, <math alttext="\mathbf{s}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p6.1.m1.1"><semantics id="S3.SS2.SSS1.p6.1.m1.1a"><mi id="S3.SS2.SSS1.p6.1.m1.1.1" xref="S3.SS2.SSS1.p6.1.m1.1.1.cmml">𝐬</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p6.1.m1.1b"><ci id="S3.SS2.SSS1.p6.1.m1.1.1.cmml" xref="S3.SS2.SSS1.p6.1.m1.1.1">𝐬</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p6.1.m1.1c">\mathbf{s}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p6.1.m1.1d">bold_s</annotation></semantics></math> represents the initial reconstructed result from the first stage, encapsulating the low-light conditions. Consequently, the noise prediction task is reformulated as <math alttext="\varepsilon_{\theta}(\mathbf{x}\bf{{}_{t}},\mathbf{s},\bf{t})" class="ltx_math_unparsed" display="inline" id="S3.SS2.SSS1.p6.2.m2.1"><semantics id="S3.SS2.SSS1.p6.2.m2.1a"><mrow id="S3.SS2.SSS1.p6.2.m2.1b"><msub id="S3.SS2.SSS1.p6.2.m2.1.1"><mi id="S3.SS2.SSS1.p6.2.m2.1.1.2">ε</mi><mi id="S3.SS2.SSS1.p6.2.m2.1.1.3">θ</mi></msub><mrow id="S3.SS2.SSS1.p6.2.m2.1.2"><mo id="S3.SS2.SSS1.p6.2.m2.1.2.1" stretchy="false">(</mo><mi id="S3.SS2.SSS1.p6.2.m2.1.2.2">𝐱</mi><mmultiscripts id="S3.SS2.SSS1.p6.2.m2.1.2.3"><mo id="S3.SS2.SSS1.p6.2.m2.1.2.3.2">,</mo><mprescripts id="S3.SS2.SSS1.p6.2.m2.1.2.3a"></mprescripts><mi id="S3.SS2.SSS1.p6.2.m2.1.2.3.3">𝐭</mi><mrow id="S3.SS2.SSS1.p6.2.m2.1.2.3b"></mrow></mmultiscripts><mi id="S3.SS2.SSS1.p6.2.m2.1.2.4">𝐬</mi><mo id="S3.SS2.SSS1.p6.2.m2.1.2.5">,</mo><mi id="S3.SS2.SSS1.p6.2.m2.1.2.6">𝐭</mi><mo id="S3.SS2.SSS1.p6.2.m2.1.2.7" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p6.2.m2.1c">\varepsilon_{\theta}(\mathbf{x}\bf{{}_{t}},\mathbf{s},\bf{t})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p6.2.m2.1d">italic_ε start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( bold_x start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT , bold_s , bold_t )</annotation></semantics></math></p> </div> <div class="ltx_para" id="S3.SS2.SSS1.p7"> <p class="ltx_p" id="S3.SS2.SSS1.p7.1">Secondly, to mitigate potential instabilities during inference, we implement a comprehensive training regimen. This approach requires the network to execute both the forward diffusion process—where random Gaussian noise is systematically added to both the high-quality image and the conditioned initial reconstructed result under low-light guidance—and the reverse generation process. The latter involves continuous noise removal based on the neural network’s learned priors. During the testing phase, only the reverse generation process is employed, wherein the initial reconstructed result and a randomly Gaussian-distributed image undergo progressive denoising and enhancement, leveraging the network’s learned priors to yield the desired high-quality, realistic image.</p> </div> <div class="ltx_para" id="S3.SS2.SSS1.p8"> <p class="ltx_p" id="S3.SS2.SSS1.p8.6">Specifically, we first preprocess the preliminarily reconstructed image from the first stage. The Wavelet Transform can significantly reduce the spatial dimension of images without losing information. We utilize the Haar Discrete Wavelet Transform (DWT) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib28" title="">28</a>]</cite> to transform the preliminarily reconstructed image into a higher-dimensional wavelet domain. By decomposing the image, we obtain four smaller sub-bands: the low-frequency component, and the high-frequency components in the horizontal, vertical, and diagonal directions. This transformation can be expressed as:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E13"> <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="\begin{array}[]{l}\left\{{\bf{LL},\bf{LH},\bf{HL},\bf{HH}}\right\}={2{\rm{D-% DWT}}}\left\{\bf{x}\right\}\\ \hat{\bf{x}}=2{\rm{D-IDWT}}\left\{{\hat{\bf{LL}},\hat{\bf{LH}},\hat{\bf{HL}},% \hat{\bf{HH}}}\right\}\end{array}" class="ltx_Math" display="block" id="S3.E13.m1.9"><semantics id="S3.E13.m1.9a"><mtable displaystyle="true" id="S3.E13.m1.9.9" rowspacing="0pt" xref="S3.E13.m1.9.9.cmml"><mtr id="S3.E13.m1.9.9a" xref="S3.E13.m1.9.9.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.E13.m1.9.9b" xref="S3.E13.m1.9.9.cmml"><mrow id="S3.E13.m1.5.5.5.5.5" xref="S3.E13.m1.5.5.5.5.5.cmml"><mrow id="S3.E13.m1.5.5.5.5.5.7.2" xref="S3.E13.m1.5.5.5.5.5.7.1.cmml"><mo id="S3.E13.m1.5.5.5.5.5.7.2.1" 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id="S3.E13.m1.8.8.8.3.3.3.1.cmml" xref="S3.E13.m1.8.8.8.3.3.3.1">^</ci><ci id="S3.E13.m1.8.8.8.3.3.3.2.cmml" xref="S3.E13.m1.8.8.8.3.3.3.2">𝐇𝐋</ci></apply><apply id="S3.E13.m1.9.9.9.4.4.4.cmml" xref="S3.E13.m1.9.9.9.4.4.4"><ci id="S3.E13.m1.9.9.9.4.4.4.1.cmml" xref="S3.E13.m1.9.9.9.4.4.4.1">^</ci><ci id="S3.E13.m1.9.9.9.4.4.4.2.cmml" xref="S3.E13.m1.9.9.9.4.4.4.2">𝐇𝐇</ci></apply></set></apply></apply></apply></matrixrow></matrix></annotation-xml><annotation encoding="application/x-tex" id="S3.E13.m1.9c">\begin{array}[]{l}\left\{{\bf{LL},\bf{LH},\bf{HL},\bf{HH}}\right\}={2{\rm{D-% DWT}}}\left\{\bf{x}\right\}\\ \hat{\bf{x}}=2{\rm{D-IDWT}}\left\{{\hat{\bf{LL}},\hat{\bf{LH}},\hat{\bf{HL}},% \hat{\bf{HH}}}\right\}\end{array}</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m1.9d">start_ARRAY start_ROW start_CELL { bold_LL , bold_LH , bold_HL , bold_HH } = 2 roman_D - roman_DWT { bold_x } end_CELL end_ROW start_ROW start_CELL over^ start_ARG bold_x end_ARG = 2 roman_D - roman_IDWT { over^ start_ARG bold_LL end_ARG , over^ start_ARG bold_LH end_ARG , over^ start_ARG bold_HL end_ARG , over^ start_ARG bold_HH end_ARG } end_CELL end_ROW end_ARRAY</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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.SSS1.p8.5">where 2D-DWT and 2D-IDWT represent the 2D Discrete Wavelet Transform and the 2D Inverse Discrete Wavelet Transform, respectively. <math alttext="\bf{x}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p8.1.m1.1"><semantics id="S3.SS2.SSS1.p8.1.m1.1a"><mi id="S3.SS2.SSS1.p8.1.m1.1.1" xref="S3.SS2.SSS1.p8.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p8.1.m1.1b"><ci id="S3.SS2.SSS1.p8.1.m1.1.1.cmml" xref="S3.SS2.SSS1.p8.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p8.1.m1.1c">\bf{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p8.1.m1.1d">bold_x</annotation></semantics></math> denotes the input image, <math alttext="\bf{LL}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p8.2.m2.1"><semantics id="S3.SS2.SSS1.p8.2.m2.1a"><mi id="S3.SS2.SSS1.p8.2.m2.1.1" xref="S3.SS2.SSS1.p8.2.m2.1.1.cmml">𝐋𝐋</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p8.2.m2.1b"><ci id="S3.SS2.SSS1.p8.2.m2.1.1.cmml" xref="S3.SS2.SSS1.p8.2.m2.1.1">𝐋𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p8.2.m2.1c">\bf{LL}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p8.2.m2.1d">bold_LL</annotation></semantics></math> represents the low-frequency information, while <math alttext="\bf{LH}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p8.3.m3.1"><semantics id="S3.SS2.SSS1.p8.3.m3.1a"><mi id="S3.SS2.SSS1.p8.3.m3.1.1" xref="S3.SS2.SSS1.p8.3.m3.1.1.cmml">𝐋𝐇</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p8.3.m3.1b"><ci id="S3.SS2.SSS1.p8.3.m3.1.1.cmml" xref="S3.SS2.SSS1.p8.3.m3.1.1">𝐋𝐇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p8.3.m3.1c">\bf{LH}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p8.3.m3.1d">bold_LH</annotation></semantics></math>, <math alttext="\bf{HL}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p8.4.m4.1"><semantics id="S3.SS2.SSS1.p8.4.m4.1a"><mi id="S3.SS2.SSS1.p8.4.m4.1.1" xref="S3.SS2.SSS1.p8.4.m4.1.1.cmml">𝐇𝐋</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p8.4.m4.1b"><ci id="S3.SS2.SSS1.p8.4.m4.1.1.cmml" xref="S3.SS2.SSS1.p8.4.m4.1.1">𝐇𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p8.4.m4.1c">\bf{HL}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p8.4.m4.1d">bold_HL</annotation></semantics></math>, and <math alttext="\bf{HH}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p8.5.m5.1"><semantics id="S3.SS2.SSS1.p8.5.m5.1a"><mi id="S3.SS2.SSS1.p8.5.m5.1.1" xref="S3.SS2.SSS1.p8.5.m5.1.1.cmml">𝐇𝐇</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p8.5.m5.1b"><ci id="S3.SS2.SSS1.p8.5.m5.1.1.cmml" xref="S3.SS2.SSS1.p8.5.m5.1.1">𝐇𝐇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p8.5.m5.1c">\bf{HH}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p8.5.m5.1d">bold_HH</annotation></semantics></math> represent the high-frequency information in the vertical, horizontal, and diagonal directions, respectively. The hatted variables denote the corresponding reconstructed images. By applying the wavelet transform twice, we reduce the image resolution by a factor of four, lowering memory and computational demands while preserving key information for the diffusion model. This process decomposes the image into low- and high-frequency components. The low-frequency component retains global structural information, while the high-frequency component captures fine details. This separation allows the conditional diffusion model (LL module) to focus on low frequencies, enhancing brightness, reducing noise, and recovering basic contours. Meanwhile, the depthwise separable convolution network (HF module) targets high frequencies, enhancing textures and fine details.</p> </div> </section> <section class="ltx_subsubsection" id="S3.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.2.2 </span>Processing structure</h4> <div class="ltx_para" id="S3.SS2.SSS2.p1"> <p class="ltx_p" id="S3.SS2.SSS2.p1.1">In the second stage, the LL Module and HF Module are employed to further denoise and enhance the coarse reconstruction results from the first stage. Specifically, a wavelet transform is applied to decompose the initial reconstruction into low-frequency (LL) and high-frequency (HF, including HH, LH, and HL) components. The LL Module utilizes a conditional diffusion model to process the low-frequency sub-band <math alttext="\bf{LL}" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p1.1.m1.1"><semantics id="S3.SS2.SSS2.p1.1.m1.1a"><mi id="S3.SS2.SSS2.p1.1.m1.1.1" xref="S3.SS2.SSS2.p1.1.m1.1.1.cmml">𝐋𝐋</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p1.1.m1.1b"><ci id="S3.SS2.SSS2.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS2.p1.1.m1.1.1">𝐋𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p1.1.m1.1c">\bf{LL}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p1.1.m1.1d">bold_LL</annotation></semantics></math> extracted from the wavelet-transformed coarse result. By concatenating the low-frequency sub-band from the initial reconstruction with the corresponding sub-band from a normal-light reference image, the diffusion model is guided to generate a high-quality normal-light sub-band image from the noisy low-light input. Simultaneously, the HF Module leverages a depth-wise separable convolutional network to denoise and restore the high-frequency sub-bands. Through a cross-attention mechanism, the network enhances feature interactions among the high-frequency sub-bands (HH, LH, HL), ultimately producing optimized results. This two-module design ensures effective enhancement of both low-frequency and high-frequency information, significantly improving the overall quality of the reconstructed image.</p> </div> <div class="ltx_para" id="S3.SS2.SSS2.p2"> <p class="ltx_p" id="S3.SS2.SSS2.p2.1">We utilize a deep separable convolutional network within the HF Module, as illustrated in Fig <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.F2" title="Figure 2 ‣ 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">2</span></a>, to restore fine details and high-frequency information extracted from the wavelet transform sub-bands (HH, LH, HL). This module is designed to enhance image clarity and texture by effectively processing and fusing high-frequency components. Initially, depth-wise separable convolution is employed to preliminarily extract features from the input sub-bands. This approach processes each channel independently, significantly reducing computational complexity while preserving essential details. The extracted features then interact through a cross-attention mechanism, which captures correlations and complementary information across different frequency components. This step facilitates more accurate feature fusion. Following feature fusion, the features undergo further refinement through additional depth-wise separable convolution layers, enhancing feature representation and improving network robustness. The processed sub-bands (HH, LH, HL) are then output as optimized high-frequency feature maps. By integrating efficient convolution operations with an attention mechanism, this design effectively extracts and fuses high-frequency information, improving the image’s overall texture and detail quality.</p> </div> </section> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>Loss Function</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">The network first employs mean squared error (MSE) loss to constrain the forward diffusion process of the diffusion model, aiming to reduce the discrepancy between the predicted noise and the added noise, as shown in the following equation:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E14"> <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="{\cal L}_{1}={{\rm{E}}_{\bf{t},{x_{0}},{\varepsilon\bf{{}_{t}}}}}[{\left\|{% \varepsilon\bf{{}_{t}}-{\varepsilon_{\theta}}({x\bf{{}_{t}}},s,\bf{t})}\right% \|^{2}}]" class="ltx_math_unparsed" display="block" id="S3.E14.m1.2"><semantics id="S3.E14.m1.2a"><mrow id="S3.E14.m1.2b"><msub id="S3.E14.m1.2.3"><mi class="ltx_font_mathcaligraphic" id="S3.E14.m1.2.3.2">ℒ</mi><mn id="S3.E14.m1.2.3.3">1</mn></msub><mo id="S3.E14.m1.2.4">=</mo><msub id="S3.E14.m1.2.5"><mi id="S3.E14.m1.2.5.2" mathvariant="normal">E</mi><mrow id="S3.E14.m1.2.2.2"><mi id="S3.E14.m1.1.1.1.1">𝐭</mi><mo id="S3.E14.m1.2.2.2.3">,</mo><msub id="S3.E14.m1.2.2.2.4"><mi id="S3.E14.m1.2.2.2.4.2">𝐱</mi><mn id="S3.E14.m1.2.2.2.4.3">𝟎</mn></msub><mo id="S3.E14.m1.2.2.2.5">,</mo><mi id="S3.E14.m1.2.2.2.2">ε</mi><msub id="S3.E14.m1.2.2.2.6"><mi id="S3.E14.m1.2.2.2.6a"></mi><mi id="S3.E14.m1.2.2.2.6.1">𝐭</mi></msub></mrow></msub><mrow id="S3.E14.m1.2.6"><mo id="S3.E14.m1.2.6.1" stretchy="false">[</mo><mo id="S3.E14.m1.2.6.2" lspace="0em" rspace="0.167em" stretchy="true">∥</mo><mi id="S3.E14.m1.2.6.3">ε</mi><mmultiscripts id="S3.E14.m1.2.6.4"><mo id="S3.E14.m1.2.6.4.2">−</mo><mprescripts id="S3.E14.m1.2.6.4a"></mprescripts><mi id="S3.E14.m1.2.6.4.3">𝐭</mi><mrow id="S3.E14.m1.2.6.4b"></mrow></mmultiscripts><msub id="S3.E14.m1.2.6.5"><mi id="S3.E14.m1.2.6.5.2">ε</mi><mi id="S3.E14.m1.2.6.5.3">θ</mi></msub><mrow id="S3.E14.m1.2.6.6"><mo id="S3.E14.m1.2.6.6.1" stretchy="false">(</mo><mi id="S3.E14.m1.2.6.6.2">𝐱</mi><mmultiscripts id="S3.E14.m1.2.6.6.3"><mo id="S3.E14.m1.2.6.6.3.2">,</mo><mprescripts id="S3.E14.m1.2.6.6.3a"></mprescripts><mi id="S3.E14.m1.2.6.6.3.3">𝐭</mi><mrow id="S3.E14.m1.2.6.6.3b"></mrow></mmultiscripts><mi id="S3.E14.m1.2.6.6.4">𝐬</mi><mo id="S3.E14.m1.2.6.6.5">,</mo><mi id="S3.E14.m1.2.6.6.6">𝐭</mi><mo id="S3.E14.m1.2.6.6.7" stretchy="false">)</mo></mrow><msup id="S3.E14.m1.2.6.7"><mo id="S3.E14.m1.2.6.7.2" lspace="0em" rspace="0.167em" stretchy="true">∥</mo><mn id="S3.E14.m1.2.6.7.3">2</mn></msup><mo id="S3.E14.m1.2.6.8" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.E14.m1.2c">{\cal L}_{1}={{\rm{E}}_{\bf{t},{x_{0}},{\varepsilon\bf{{}_{t}}}}}[{\left\|{% \varepsilon\bf{{}_{t}}-{\varepsilon_{\theta}}({x\bf{{}_{t}}},s,\bf{t})}\right% \|^{2}}]</annotation><annotation encoding="application/x-llamapun" id="S3.E14.m1.2d">caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = roman_E start_POSTSUBSCRIPT bold_t , bold_x start_POSTSUBSCRIPT bold_0 end_POSTSUBSCRIPT , italic_ε start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT end_POSTSUBSCRIPT [ ∥ italic_ε start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT - italic_ε start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( bold_x start_FLOATSUBSCRIPT bold_t end_FLOATSUBSCRIPT , bold_s , bold_t ) ∥ start_POSTSUPERSCRIPT 2 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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.1">Given the instability of the reverse diffusion process in the proposed network, a combination of mean absolute error (MAE) loss, Structural Similarity Index Measure (SSIM) loss <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib29" title="">29</a>]</cite>, and learned perceptual image patch similarity (LPIPS) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib30" title="">30</a>]</cite> is utilized to constrain the reverse diffusion process, which is also the network reconstruction enhancement process, as formulated below:</p> </div> <div class="ltx_para" id="S3.SS3.p3"> <table class="ltx_equation ltx_eqn_table" id="S3.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="{\cal L}_{2}=\,{\lambda_{1}}{\left\|{\hat{\bf{x}}-\bf{x}}\right\|_{1}}+{% \lambda_{2}}{\rm{SSIM}(\hat{\bf{x}},\bf{x})}+{\lambda_{3}}\left\{\left\|{{f_{2% }}(\hat{\bf{x}})-{f_{2}}(\bf{x})}\right\|^{2}+\left\|{{f_{4}}(\hat{\bf{x}})-{f% _{4}}(\bf{x})}\right\|^{2}\right\}" class="ltx_Math" display="block" id="S3.E15.m1.8"><semantics id="S3.E15.m1.8a"><mrow id="S3.E15.m1.8.8" xref="S3.E15.m1.8.8.cmml"><msub id="S3.E15.m1.8.8.4" xref="S3.E15.m1.8.8.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E15.m1.8.8.4.2" xref="S3.E15.m1.8.8.4.2.cmml">ℒ</mi><mn id="S3.E15.m1.8.8.4.3" xref="S3.E15.m1.8.8.4.3.cmml">2</mn></msub><mo id="S3.E15.m1.8.8.3" rspace="0.448em" xref="S3.E15.m1.8.8.3.cmml">=</mo><mrow id="S3.E15.m1.8.8.2" xref="S3.E15.m1.8.8.2.cmml"><mrow id="S3.E15.m1.7.7.1.1" xref="S3.E15.m1.7.7.1.1.cmml"><msub id="S3.E15.m1.7.7.1.1.3" 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start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∥ over^ start_ARG bold_x end_ARG - bold_x ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_SSIM ( over^ start_ARG bold_x end_ARG , bold_x ) + italic_λ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT { ∥ italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( over^ start_ARG bold_x end_ARG ) - italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( bold_x ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∥ italic_f start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( over^ start_ARG bold_x end_ARG ) - italic_f start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( bold_x ) ∥ start_POSTSUPERSCRIPT 2 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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p3.5">where <math alttext="\hat{\bf{x}}" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><mover accent="true" id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml"><mi id="S3.SS3.p3.1.m1.1.1.2" xref="S3.SS3.p3.1.m1.1.1.2.cmml">𝐱</mi><mo id="S3.SS3.p3.1.m1.1.1.1" xref="S3.SS3.p3.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1"><ci id="S3.SS3.p3.1.m1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1.1">^</ci><ci id="S3.SS3.p3.1.m1.1.1.2.cmml" xref="S3.SS3.p3.1.m1.1.1.2">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">\hat{\bf{x}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">over^ start_ARG bold_x end_ARG</annotation></semantics></math> and <math alttext="\bf{x}" class="ltx_Math" display="inline" id="S3.SS3.p3.2.m2.1"><semantics id="S3.SS3.p3.2.m2.1a"><mi id="S3.SS3.p3.2.m2.1.1" xref="S3.SS3.p3.2.m2.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.2.m2.1b"><ci id="S3.SS3.p3.2.m2.1.1.cmml" xref="S3.SS3.p3.2.m2.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.2.m2.1c">\bf{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.2.m2.1d">bold_x</annotation></semantics></math> denote the reconstructed enhanced image and the ground truth respectively, <math alttext="f_{2}" class="ltx_Math" display="inline" id="S3.SS3.p3.3.m3.1"><semantics id="S3.SS3.p3.3.m3.1a"><msub id="S3.SS3.p3.3.m3.1.1" xref="S3.SS3.p3.3.m3.1.1.cmml"><mi id="S3.SS3.p3.3.m3.1.1.2" xref="S3.SS3.p3.3.m3.1.1.2.cmml">f</mi><mn id="S3.SS3.p3.3.m3.1.1.3" xref="S3.SS3.p3.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.3.m3.1b"><apply id="S3.SS3.p3.3.m3.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.3.m3.1.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.p3.3.m3.1.1.2.cmml" xref="S3.SS3.p3.3.m3.1.1.2">𝑓</ci><cn id="S3.SS3.p3.3.m3.1.1.3.cmml" type="integer" xref="S3.SS3.p3.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.3.m3.1c">f_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.3.m3.1d">italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{4}" class="ltx_Math" display="inline" id="S3.SS3.p3.4.m4.1"><semantics id="S3.SS3.p3.4.m4.1a"><msub id="S3.SS3.p3.4.m4.1.1" xref="S3.SS3.p3.4.m4.1.1.cmml"><mi id="S3.SS3.p3.4.m4.1.1.2" xref="S3.SS3.p3.4.m4.1.1.2.cmml">f</mi><mn id="S3.SS3.p3.4.m4.1.1.3" xref="S3.SS3.p3.4.m4.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.4.m4.1b"><apply id="S3.SS3.p3.4.m4.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.4.m4.1.1.1.cmml" 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xref="S3.SS3.p3.5.m5.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.p3.5.m5.3.3.3.4" xref="S3.SS3.p3.5.m5.3.3.4.cmml">,</mo><msub id="S3.SS3.p3.5.m5.2.2.2.2" xref="S3.SS3.p3.5.m5.2.2.2.2.cmml"><mi id="S3.SS3.p3.5.m5.2.2.2.2.2" xref="S3.SS3.p3.5.m5.2.2.2.2.2.cmml">λ</mi><mn id="S3.SS3.p3.5.m5.2.2.2.2.3" xref="S3.SS3.p3.5.m5.2.2.2.2.3.cmml">2</mn></msub><mo id="S3.SS3.p3.5.m5.3.3.3.5" xref="S3.SS3.p3.5.m5.3.3.4.cmml">,</mo><msub id="S3.SS3.p3.5.m5.3.3.3.3" xref="S3.SS3.p3.5.m5.3.3.3.3.cmml"><mi id="S3.SS3.p3.5.m5.3.3.3.3.2" xref="S3.SS3.p3.5.m5.3.3.3.3.2.cmml">λ</mi><mn id="S3.SS3.p3.5.m5.3.3.3.3.3" xref="S3.SS3.p3.5.m5.3.3.3.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.5.m5.3b"><list id="S3.SS3.p3.5.m5.3.3.4.cmml" xref="S3.SS3.p3.5.m5.3.3.3"><apply id="S3.SS3.p3.5.m5.1.1.1.1.cmml" xref="S3.SS3.p3.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.5.m5.1.1.1.1.1.cmml" xref="S3.SS3.p3.5.m5.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p3.5.m5.1.1.1.1.2.cmml" xref="S3.SS3.p3.5.m5.1.1.1.1.2">𝜆</ci><cn id="S3.SS3.p3.5.m5.1.1.1.1.3.cmml" type="integer" xref="S3.SS3.p3.5.m5.1.1.1.1.3">1</cn></apply><apply id="S3.SS3.p3.5.m5.2.2.2.2.cmml" xref="S3.SS3.p3.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS3.p3.5.m5.2.2.2.2.1.cmml" xref="S3.SS3.p3.5.m5.2.2.2.2">subscript</csymbol><ci id="S3.SS3.p3.5.m5.2.2.2.2.2.cmml" xref="S3.SS3.p3.5.m5.2.2.2.2.2">𝜆</ci><cn id="S3.SS3.p3.5.m5.2.2.2.2.3.cmml" type="integer" xref="S3.SS3.p3.5.m5.2.2.2.2.3">2</cn></apply><apply id="S3.SS3.p3.5.m5.3.3.3.3.cmml" xref="S3.SS3.p3.5.m5.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS3.p3.5.m5.3.3.3.3.1.cmml" xref="S3.SS3.p3.5.m5.3.3.3.3">subscript</csymbol><ci id="S3.SS3.p3.5.m5.3.3.3.3.2.cmml" xref="S3.SS3.p3.5.m5.3.3.3.3.2">𝜆</ci><cn id="S3.SS3.p3.5.m5.3.3.3.3.3.cmml" type="integer" xref="S3.SS3.p3.5.m5.3.3.3.3.3">3</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.5.m5.3c">{\lambda_{1}},{\lambda_{2}},{\lambda_{3}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.5.m5.3d">italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> represents the weight of each loss term.</p> </div> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.4">Furthermore, a combination of MSE loss and Total Variation (TV) loss is employed to constrain the reconstruction of high-frequency information in the image, as shown in the equation below:</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="{\cal L}_{3}={\lambda_{4}}{\left\|\hat{\rm{HF}}-{\rm{HF}}\right\|^{2}}+{% \lambda_{5}}{\rm{TV}}(\hat{\rm{HF}},{\rm{HF}})" class="ltx_Math" display="block" id="S3.E16.m1.3"><semantics id="S3.E16.m1.3a"><mrow id="S3.E16.m1.3.3" xref="S3.E16.m1.3.3.cmml"><msub id="S3.E16.m1.3.3.3" xref="S3.E16.m1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E16.m1.3.3.3.2" xref="S3.E16.m1.3.3.3.2.cmml">ℒ</mi><mn id="S3.E16.m1.3.3.3.3" xref="S3.E16.m1.3.3.3.3.cmml">3</mn></msub><mo id="S3.E16.m1.3.3.2" xref="S3.E16.m1.3.3.2.cmml">=</mo><mrow id="S3.E16.m1.3.3.1" xref="S3.E16.m1.3.3.1.cmml"><mrow id="S3.E16.m1.3.3.1.1" xref="S3.E16.m1.3.3.1.1.cmml"><msub id="S3.E16.m1.3.3.1.1.3" xref="S3.E16.m1.3.3.1.1.3.cmml"><mi id="S3.E16.m1.3.3.1.1.3.2" xref="S3.E16.m1.3.3.1.1.3.2.cmml">λ</mi><mn id="S3.E16.m1.3.3.1.1.3.3" xref="S3.E16.m1.3.3.1.1.3.3.cmml">4</mn></msub><mo id="S3.E16.m1.3.3.1.1.2" xref="S3.E16.m1.3.3.1.1.2.cmml">⁢</mo><msup id="S3.E16.m1.3.3.1.1.1" xref="S3.E16.m1.3.3.1.1.1.cmml"><mrow id="S3.E16.m1.3.3.1.1.1.1.1" xref="S3.E16.m1.3.3.1.1.1.1.2.cmml"><mo id="S3.E16.m1.3.3.1.1.1.1.1.2" 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L}_{3}={\lambda_{4}}{\left\|\hat{\rm{HF}}-{\rm{HF}}\right\|^{2}}+{% \lambda_{5}}{\rm{TV}}(\hat{\rm{HF}},{\rm{HF}})</annotation><annotation encoding="application/x-llamapun" id="S3.E16.m1.3d">caligraphic_L start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ∥ over^ start_ARG roman_HF end_ARG - roman_HF ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_λ start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT roman_TV ( over^ start_ARG roman_HF end_ARG , roman_HF )</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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p4.3">where <math alttext="\hat{\rm{HF}}" class="ltx_Math" display="inline" id="S3.SS3.p4.1.m1.1"><semantics id="S3.SS3.p4.1.m1.1a"><mover accent="true" id="S3.SS3.p4.1.m1.1.1" xref="S3.SS3.p4.1.m1.1.1.cmml"><mi id="S3.SS3.p4.1.m1.1.1.2" xref="S3.SS3.p4.1.m1.1.1.2.cmml">HF</mi><mo id="S3.SS3.p4.1.m1.1.1.1" xref="S3.SS3.p4.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.1.m1.1b"><apply id="S3.SS3.p4.1.m1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1"><ci id="S3.SS3.p4.1.m1.1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1.1">^</ci><ci id="S3.SS3.p4.1.m1.1.1.2.cmml" xref="S3.SS3.p4.1.m1.1.1.2">HF</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.1.m1.1c">\hat{\rm{HF}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.1.m1.1d">over^ start_ARG roman_HF end_ARG</annotation></semantics></math> and <math alttext="\rm{HF}" class="ltx_Math" display="inline" id="S3.SS3.p4.2.m2.1"><semantics id="S3.SS3.p4.2.m2.1a"><mi id="S3.SS3.p4.2.m2.1.1" xref="S3.SS3.p4.2.m2.1.1.cmml">HF</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.2.m2.1b"><ci id="S3.SS3.p4.2.m2.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1">HF</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.2.m2.1c">\rm{HF}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.2.m2.1d">roman_HF</annotation></semantics></math> represent reconstruction of the enhanced high-frequency component and ground truth of high-frequency component, and <math alttext="{\lambda_{4}},{\lambda_{5}}" class="ltx_Math" display="inline" id="S3.SS3.p4.3.m3.2"><semantics id="S3.SS3.p4.3.m3.2a"><mrow id="S3.SS3.p4.3.m3.2.2.2" xref="S3.SS3.p4.3.m3.2.2.3.cmml"><msub id="S3.SS3.p4.3.m3.1.1.1.1" xref="S3.SS3.p4.3.m3.1.1.1.1.cmml"><mi id="S3.SS3.p4.3.m3.1.1.1.1.2" xref="S3.SS3.p4.3.m3.1.1.1.1.2.cmml">λ</mi><mn id="S3.SS3.p4.3.m3.1.1.1.1.3" xref="S3.SS3.p4.3.m3.1.1.1.1.3.cmml">4</mn></msub><mo id="S3.SS3.p4.3.m3.2.2.2.3" xref="S3.SS3.p4.3.m3.2.2.3.cmml">,</mo><msub id="S3.SS3.p4.3.m3.2.2.2.2" xref="S3.SS3.p4.3.m3.2.2.2.2.cmml"><mi id="S3.SS3.p4.3.m3.2.2.2.2.2" xref="S3.SS3.p4.3.m3.2.2.2.2.2.cmml">λ</mi><mn id="S3.SS3.p4.3.m3.2.2.2.2.3" xref="S3.SS3.p4.3.m3.2.2.2.2.3.cmml">5</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.3.m3.2b"><list id="S3.SS3.p4.3.m3.2.2.3.cmml" xref="S3.SS3.p4.3.m3.2.2.2"><apply id="S3.SS3.p4.3.m3.1.1.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p4.3.m3.1.1.1.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p4.3.m3.1.1.1.1.2.cmml" xref="S3.SS3.p4.3.m3.1.1.1.1.2">𝜆</ci><cn id="S3.SS3.p4.3.m3.1.1.1.1.3.cmml" type="integer" xref="S3.SS3.p4.3.m3.1.1.1.1.3">4</cn></apply><apply id="S3.SS3.p4.3.m3.2.2.2.2.cmml" xref="S3.SS3.p4.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS3.p4.3.m3.2.2.2.2.1.cmml" xref="S3.SS3.p4.3.m3.2.2.2.2">subscript</csymbol><ci id="S3.SS3.p4.3.m3.2.2.2.2.2.cmml" xref="S3.SS3.p4.3.m3.2.2.2.2.2">𝜆</ci><cn id="S3.SS3.p4.3.m3.2.2.2.2.3.cmml" type="integer" xref="S3.SS3.p4.3.m3.2.2.2.2.3">5</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.3.m3.2c">{\lambda_{4}},{\lambda_{5}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.3.m3.2d">italic_λ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT</annotation></semantics></math> represents the weight of each loss term.</p> </div> <div class="ltx_para" id="S3.SS3.p5"> <p class="ltx_p" id="S3.SS3.p5.3">In summary, <math alttext="{{\cal L}_{1}}" class="ltx_Math" display="inline" id="S3.SS3.p5.1.m1.1"><semantics id="S3.SS3.p5.1.m1.1a"><msub id="S3.SS3.p5.1.m1.1.1" xref="S3.SS3.p5.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p5.1.m1.1.1.2" xref="S3.SS3.p5.1.m1.1.1.2.cmml">ℒ</mi><mn id="S3.SS3.p5.1.m1.1.1.3" xref="S3.SS3.p5.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.1.m1.1b"><apply id="S3.SS3.p5.1.m1.1.1.cmml" 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xref="S3.SS3.p5.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.2.m2.1b"><apply id="S3.SS3.p5.2.m2.1.1.cmml" xref="S3.SS3.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.2.m2.1.1.1.cmml" xref="S3.SS3.p5.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.p5.2.m2.1.1.2.cmml" xref="S3.SS3.p5.2.m2.1.1.2">ℒ</ci><cn id="S3.SS3.p5.2.m2.1.1.3.cmml" type="integer" xref="S3.SS3.p5.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.2.m2.1c">{{\cal L}_{2}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.2.m2.1d">caligraphic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> facilitates high-quality image generation during the reverse diffusion process. Finally, <math alttext="{{\cal L}_{3}}" class="ltx_Math" display="inline" id="S3.SS3.p5.3.m3.1"><semantics id="S3.SS3.p5.3.m3.1a"><msub id="S3.SS3.p5.3.m3.1.1" xref="S3.SS3.p5.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p5.3.m3.1.1.2" xref="S3.SS3.p5.3.m3.1.1.2.cmml">ℒ</mi><mn id="S3.SS3.p5.3.m3.1.1.3" xref="S3.SS3.p5.3.m3.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.3.m3.1b"><apply id="S3.SS3.p5.3.m3.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.3.m3.1.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.p5.3.m3.1.1.2.cmml" xref="S3.SS3.p5.3.m3.1.1.2">ℒ</ci><cn id="S3.SS3.p5.3.m3.1.1.3.cmml" type="integer" xref="S3.SS3.p5.3.m3.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.3.m3.1c">{{\cal L}_{3}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.3.m3.1d">caligraphic_L start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> emphasizes the reconstruction of the enhanced high-frequency components. As marked in Fig <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.F2" title="Figure 2 ‣ 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">2</span></a>, the total loss function of the proposed network is:</p> <table class="ltx_equation ltx_eqn_table" id="S3.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="{{\cal L}_{{\rm{all}}}}={{\cal L}_{\rm{1}}}+{{\cal L}_{\rm{2}}}+{{\cal L}_{\rm% {3}}}" class="ltx_Math" display="block" id="S3.E17.m1.1"><semantics id="S3.E17.m1.1a"><mrow id="S3.E17.m1.1.1" xref="S3.E17.m1.1.1.cmml"><msub id="S3.E17.m1.1.1.2" xref="S3.E17.m1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E17.m1.1.1.2.2" xref="S3.E17.m1.1.1.2.2.cmml">ℒ</mi><mi id="S3.E17.m1.1.1.2.3" xref="S3.E17.m1.1.1.2.3.cmml">all</mi></msub><mo id="S3.E17.m1.1.1.1" xref="S3.E17.m1.1.1.1.cmml">=</mo><mrow id="S3.E17.m1.1.1.3" xref="S3.E17.m1.1.1.3.cmml"><msub id="S3.E17.m1.1.1.3.2" xref="S3.E17.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E17.m1.1.1.3.2.2" xref="S3.E17.m1.1.1.3.2.2.cmml">ℒ</mi><mn id="S3.E17.m1.1.1.3.2.3" xref="S3.E17.m1.1.1.3.2.3.cmml">1</mn></msub><mo id="S3.E17.m1.1.1.3.1" xref="S3.E17.m1.1.1.3.1.cmml">+</mo><msub id="S3.E17.m1.1.1.3.3" xref="S3.E17.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E17.m1.1.1.3.3.2" xref="S3.E17.m1.1.1.3.3.2.cmml">ℒ</mi><mn id="S3.E17.m1.1.1.3.3.3" xref="S3.E17.m1.1.1.3.3.3.cmml">2</mn></msub><mo id="S3.E17.m1.1.1.3.1a" xref="S3.E17.m1.1.1.3.1.cmml">+</mo><msub id="S3.E17.m1.1.1.3.4" xref="S3.E17.m1.1.1.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E17.m1.1.1.3.4.2" xref="S3.E17.m1.1.1.3.4.2.cmml">ℒ</mi><mn id="S3.E17.m1.1.1.3.4.3" xref="S3.E17.m1.1.1.3.4.3.cmml">3</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E17.m1.1b"><apply id="S3.E17.m1.1.1.cmml" xref="S3.E17.m1.1.1"><eq id="S3.E17.m1.1.1.1.cmml" xref="S3.E17.m1.1.1.1"></eq><apply id="S3.E17.m1.1.1.2.cmml" xref="S3.E17.m1.1.1.2"><csymbol cd="ambiguous" id="S3.E17.m1.1.1.2.1.cmml" xref="S3.E17.m1.1.1.2">subscript</csymbol><ci id="S3.E17.m1.1.1.2.2.cmml" xref="S3.E17.m1.1.1.2.2">ℒ</ci><ci id="S3.E17.m1.1.1.2.3.cmml" xref="S3.E17.m1.1.1.2.3">all</ci></apply><apply id="S3.E17.m1.1.1.3.cmml" xref="S3.E17.m1.1.1.3"><plus id="S3.E17.m1.1.1.3.1.cmml" xref="S3.E17.m1.1.1.3.1"></plus><apply id="S3.E17.m1.1.1.3.2.cmml" xref="S3.E17.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.E17.m1.1.1.3.2.1.cmml" xref="S3.E17.m1.1.1.3.2">subscript</csymbol><ci id="S3.E17.m1.1.1.3.2.2.cmml" xref="S3.E17.m1.1.1.3.2.2">ℒ</ci><cn id="S3.E17.m1.1.1.3.2.3.cmml" type="integer" xref="S3.E17.m1.1.1.3.2.3">1</cn></apply><apply id="S3.E17.m1.1.1.3.3.cmml" xref="S3.E17.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.E17.m1.1.1.3.3.1.cmml" xref="S3.E17.m1.1.1.3.3">subscript</csymbol><ci id="S3.E17.m1.1.1.3.3.2.cmml" xref="S3.E17.m1.1.1.3.3.2">ℒ</ci><cn id="S3.E17.m1.1.1.3.3.3.cmml" type="integer" xref="S3.E17.m1.1.1.3.3.3">2</cn></apply><apply id="S3.E17.m1.1.1.3.4.cmml" xref="S3.E17.m1.1.1.3.4"><csymbol cd="ambiguous" id="S3.E17.m1.1.1.3.4.1.cmml" xref="S3.E17.m1.1.1.3.4">subscript</csymbol><ci id="S3.E17.m1.1.1.3.4.2.cmml" xref="S3.E17.m1.1.1.3.4.2">ℒ</ci><cn id="S3.E17.m1.1.1.3.4.3.cmml" type="integer" xref="S3.E17.m1.1.1.3.4.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E17.m1.1c">{{\cal L}_{{\rm{all}}}}={{\cal L}_{\rm{1}}}+{{\cal L}_{\rm{2}}}+{{\cal L}_{\rm% {3}}}</annotation><annotation encoding="application/x-llamapun" id="S3.E17.m1.1d">caligraphic_L start_POSTSUBSCRIPT roman_all end_POSTSUBSCRIPT = caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + caligraphic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + caligraphic_L start_POSTSUBSCRIPT 3 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"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Experiment and Results</h2> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Dataset</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">Due to the lack of publicly available low-light lensless imaging datasets, we simulated measurements using an established lensless imaging model and actual measured PSF. We used the LOLv2 dataset<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib31" title="">31</a>]</cite>, selecting 1000 pairs of synthetic low-light and normal-light images. These were processed through our lensless imaging model to create a low-light lensless dataset, with 900 pairs for training and 100 for testing.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.1">To validate our method in real-world scenarios, we developed a lensless camera. We projected images onto an LCD screen and captured measurements by adjusting the camera’s acquisition time and exposure via a Raspberry Pi. This approach aligns with actual lensless camera imaging and facilitates labeled dataset collection. We used the "Synthetic" subset of LOLv2, processing the normal-light images for projection and pairing them with captured low-light lensless measurements.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Impletmentation Details</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">The prototype of the lensless camera used in this experiment employs a camera equipped with a IMX219 CMOS sensor, featuring a pixel size of 1.12µm. The dimensions of all ground truth images are adjusted to 384×384, equivalent to the calibrated camera’s field of view, ensuring consistency in size between the input images and ground truth images for the network. We directly utilize Bayer measurements, divided into four channels (R, Gr, B, Gb), as the input for raw imaging, utilizing the full size of 2028×1520×4.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.2">The implementation of our experiments is accomplished within the PyTorch framework. The <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_λ</annotation></semantics></math> parameter in the Wiener filter controls noise suppression, initially set to 50000(as in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib18" title="">18</a>]</cite>) and increased to 80000 for noisier scenes. However, noise reduction is mainly handled by the second-stage diffusion model, which has a greater impact on the final image quality. After the first stage, the region of interest is cropped to 384×384×3. For training, images are randomly cropped into 256×256×3 patches. During testing, the reconstructed image is kept at 384×384×3 without cropping. The Adam optimizer is utilized to train the network for 500 epochs with an initial learning rate of <math alttext="10{{}^{-4}}" class="ltx_math_unparsed" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mrow id="S4.SS2.p2.2.m2.1b"><mn id="S4.SS2.p2.2.m2.1.1">10</mn><msup id="S4.SS2.p2.2.m2.1.2"><mi id="S4.SS2.p2.2.m2.1.2a"></mi><mrow id="S4.SS2.p2.2.m2.1.2.1"><mo id="S4.SS2.p2.2.m2.1.2.1a">−</mo><mn id="S4.SS2.p2.2.m2.1.2.1.2">4</mn></mrow></msup></mrow><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">10{{}^{-4}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">10 start_FLOATSUPERSCRIPT - 4 end_FLOATSUPERSCRIPT</annotation></semantics></math>, decayed by 0.8 every 100 epochs. No weight decay is applied. Exponential Moving Average (EMA) is implemented on model parameters at a rate of 0.9999 to ensure a more stable training process. The dropout value for the resnet blocks within the model is set to 0.3. During the training phase, the diffusion step size T is set to 200, the implicit sampling step size is set to 10, and the batch size is 22. The entire experimental process is executed on a Windows system equipped with 32GB of RAM and two NVIDIA RTX 3090 GPUs.</p> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Quantitative Metrics</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">In addition to qualitative evaluations based on human visual perception, this paper also selects a range of quantitative metrics to effectively assess the experimental results. Apart from the classic MSE to measure the degree of image quality loss, Peak Signal-to-Noise Ratio (PSNR) to reflect the fidelity of image signals, and SSIM to evaluate the similarity of image structures, we have additionally incorporated LPIPS , an index that aligns more closely with human visual perception, as a metric. Unlike traditional error-based evaluation metrics, LPIPS is an image quality assessment metric based on a trained neural network model. It aims to capture differences in human perception by comparing the local perceptual features of two images. These features are obtained by training a deep convolutional neural network on a large dataset of image pairs, where the network learns to map image content into a low dimensional space where images that are perceptually similar to humans have smaller distances. LPIPS considers not only pixel-level differences but also perceptual differences, making it better at predicting human subjective perception. This comprehensive approach ensures an objective and accurate evaluation of the experimental outcomes, while better capturing the nuances of human visual experience.</p> </div> </section> <section class="ltx_subsection" id="S4.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.4 </span>Simulated Reconstruction </h3> <div class="ltx_para" id="S4.SS4.p1"> <p class="ltx_p" id="S4.SS4.p1.1">In the simulation experiments, a point light source was placed 320 mm in front of the random binary mask and 150 mm in height, and the PSF was acquired using the lensless camera constructed in this paper. The random binary mask is 10 mm away from the CMOS sensor. The output resolution of our sensor is 2028 × 1520 with a pixel pitch of 0.014 mm. Based on the forward imaging model of the lensless camera in Section <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S3.SS1" title="3.1 Fisrt Stage ‣ 3 Proposed Method ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">3.1</span></a>, a simulated dataset was obtained using the captured PSF and existing low-light images.</p> </div> <div class="ltx_para" id="S4.SS4.p2"> <p class="ltx_p" id="S4.SS4.p2.1">First, the proposed reconstruction enhancement method is trained and evaluated using the simulated training and test sets. To comprehensively demonstrate the effectiveness of the low-light lens-free reconstruction enhancement method introduced in this paper, we have deliberately selected several well-established methods that perform well under normal lighting conditions for comparison. These methods include ADMM <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib32" title="">32</a>]</cite> with 100 iterations, the purely data-driven U-Net <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib17" title="">17</a>]</cite>, FlatNet <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib18" title="">18</a>]</cite>, which combines generative adversarial networks and perceptual losses, MWDN <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib21" title="">21</a>]</cite> with multi-scale deconvolution, and DeepLIR <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib33" title="">33</a>]</cite>, a two-stage network integrated with an attention mechanism. Unlike previous studies, however, this experiment applies these methods to low-light conditions to assess their actual performance.</p> </div> <figure class="ltx_figure" id="S4.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="1040" id="S4.F3.g1" src="x1.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>The test set results for the simulated dataset, from top to bottom, are the Measurements, ADMM, U-Net, FlatNet, MWDN, DeepLIR, Ours, and Ground Truth.</figcaption> </figure> <div class="ltx_para" id="S4.SS4.p3"> <p class="ltx_p" id="S4.SS4.p3.1">Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.F3" title="Figure 3 ‣ 4.4 Simulated Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">3</span></a> shows the reconstructed images under low-light conditions using different methods, along with the original input and ground truth images. Visual comparison reveals that although these classical methods perform well under normal lighting, their reconstruction results are significantly degraded under low-light conditions, exhibiting noticeable blur, distortion, and color shift. In contrast, the method proposed in this paper maintains high reconstruction quality even under low-light conditions, with clear image details and accurate color restoration, demonstrating its unique advantages in low-light, lens-free reconstruction and enhancement.</p> </div> <div class="ltx_para" id="S4.SS4.p4"> <p class="ltx_p" id="S4.SS4.p4.1">Specifically, compared to traditional optimization methods, generic data-driven networks, physics-driven networks, and data-driven two-stage networks, the images reconstructed by the proposed model exhibit superior visual quality across all samples. While ADMM can recover the basic contours from the raw measurements, it fails to effectively enhance the image brightness, resulting in overall dark reconstruction with hidden details. As a purely data-driven method, U-Net is unable to generate accurate scene images, indicating its limited capability when working with small datasets. FlatNet, which combines generative adversarial networks and perceptual losses, improves reconstruction quality but still struggles with restoring fine details and color accuracy. MWDN achieves better results but still falls short in recovering precise details and brightness. Similar to FlatNet, DeepLIR suffers from significant color distortion. This highlights the increased complexity of data characteristics under low-light conditions. The significant brightness disparity creates a need for brightness enhancement, which causes the original denoising network to lose focus on accurate color and detail restoration. In contrast, the model proposed in this paper shows clear advantages under low-light conditions, with reconstructed images that are closer to the ground truth and richer in both color and detail, thanks to the model’s careful consideration of the unique characteristics of low-light data and its targeted optimization strategies during training.</p> </div> <figure class="ltx_table" id="S4.T1"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 1: </span><span class="ltx_text ltx_font_bold" id="S4.T1.2.1">The average MSE, LPIPS, PSNR and SSIM of the proposed method and several other methods on the simulation test set.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T1.3"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T1.3.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.1.1.1">Method</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.1.1.2">MSE</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.1.1.3">LPIPS</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.1.1.4">PSNR(in dB)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.1.1.5">SSIM</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T1.3.2.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.3.2.1.1">ADMM</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.3.2.1.2">0.1009</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.3.2.1.3">0.3666</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.3.2.1.4">11.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.3.2.1.5">0.3283</td> </tr> <tr class="ltx_tr" id="S4.T1.3.3.2"> <td class="ltx_td ltx_align_center" id="S4.T1.3.3.2.1">U-Net</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.3.2.2">-</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.3.2.3">-</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.3.2.4">-</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.3.2.5">-</td> </tr> <tr class="ltx_tr" id="S4.T1.3.4.3"> <td class="ltx_td ltx_align_center" id="S4.T1.3.4.3.1">FlatNet</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.4.3.2">0.0259</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.4.3.3">0.2099</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.4.3.4">17.05</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.4.3.5">0.4647</td> </tr> <tr class="ltx_tr" id="S4.T1.3.5.4"> <td class="ltx_td ltx_align_center" id="S4.T1.3.5.4.1">MWDN</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.5.4.2">0.0190</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.5.4.3">0.2646</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.5.4.4">17.7218</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.5.4.5">0.6115</td> </tr> <tr class="ltx_tr" id="S4.T1.3.6.5"> <td class="ltx_td ltx_align_center" id="S4.T1.3.6.5.1">DeepLIR</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.6.5.2">0.0636</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.6.5.3">0.2720</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.6.5.4">13.6968</td> <td class="ltx_td ltx_align_center" id="S4.T1.3.6.5.5">0.4463</td> </tr> <tr class="ltx_tr" id="S4.T1.3.7.6"> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.3.7.6.1">Ours</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.3.7.6.2">0.0166</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.3.7.6.3">0.1605</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.3.7.6.4">18.83</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.3.7.6.5">0.5719</td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S4.SS4.p5"> <p class="ltx_p" id="S4.SS4.p5.1">To further quantify the analysis, Table <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.T1" title="Table 1 ‣ 4.4 Simulated Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">1</span></a> presents the average MSE, LPIPS, PSNR, and SSIM of each algorithm on the simulated test dataset. The traditional ADMM method shows poor performance across all metrics due to the high noise and low brightness in the reconstructed images. FlatNet and DeepLIR, as two-stage networks, are able to perform image reconstruction but still struggle with color and detail restoration, leading to suboptimal performance in all metrics. MWDN, by performing reconstruction in a multi-scale space, achieves relatively better results, particularly in SSIM. However, the proposed method combines the physical model of lens-free reconstruction with low-light priors, allowing it to outperform the others across all metrics, demonstrating superior reconstruction quality.</p> </div> <div class="ltx_para" id="S4.SS4.p6"> <p class="ltx_p" id="S4.SS4.p6.1">These results not only validate the effectiveness of the proposed method but also highlight the limitations of existing imaging techniques under low-light conditions, further emphasizing the need for specific optimizations and designs for low-light environments.</p> </div> <div class="ltx_para" id="S4.SS4.p7"> <p class="ltx_p" id="S4.SS4.p7.1">The lensless camera noise model in Section <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S2" title="2 Problem analysis ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">2</span></a> allows for a more accurate simulation of the complex noise characteristics generated during actual CMOS imaging. The model contains a full set of noise components, mainly read noise, Poisson noise and quantisation noise. In order to further validate the robustness and effectiveness of the proposed method, we inject noise into the original measurements in the simulated dataset according to the above noise model. This approach ensures that the features of the simulated dataset are very similar to those of the real-world measurements, which improves the reliability and credibility of the experimental results. Table <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.T2" title="Table 2 ‣ 4.4 Simulated Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">2</span></a> details the values of the key parameters involved in the implementation.</p> </div> <figure class="ltx_table" id="S4.T2"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 2: </span><span class="ltx_text ltx_font_bold" id="S4.T2.2.1">The simulation parameter values of the camera noise added to the simulation data set.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T2.3"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T2.3.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S4.T2.3.1.1.1">Parameters</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T2.3.1.1.2">Values</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T2.3.2.1"> <td class="ltx_td ltx_align_left ltx_border_t" id="S4.T2.3.2.1.1">The maximum light intensity of the camera</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.3.2.1.2">1000</td> </tr> <tr class="ltx_tr" id="S4.T2.3.3.2"> <td class="ltx_td ltx_align_left" id="S4.T2.3.3.2.1">The quantum efficiency of the camera</td> <td class="ltx_td ltx_align_center" id="S4.T2.3.3.2.2">0.7</td> </tr> <tr class="ltx_tr" id="S4.T2.3.4.3"> <td class="ltx_td ltx_align_left" id="S4.T2.3.4.3.1">The standard deviation of read noise</td> <td class="ltx_td ltx_align_center" id="S4.T2.3.4.3.2">2.63</td> </tr> <tr class="ltx_tr" id="S4.T2.3.5.4"> <td class="ltx_td ltx_align_left" id="S4.T2.3.5.4.1">The Analog to Digital Unit (ADU) of the camera</td> <td class="ltx_td ltx_align_center" id="S4.T2.3.5.4.2">0.23</td> </tr> <tr class="ltx_tr" id="S4.T2.3.6.5"> <td class="ltx_td ltx_align_left" id="S4.T2.3.6.5.1">The baseline ADU of the camera</td> <td class="ltx_td ltx_align_center" id="S4.T2.3.6.5.2">4.48</td> </tr> <tr class="ltx_tr" id="S4.T2.3.7.6"> <td class="ltx_td ltx_align_left ltx_border_b" id="S4.T2.3.7.6.1">The number of bits of the camera</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T2.3.7.6.2">8</td> </tr> </tbody> </table> </figure> <figure class="ltx_figure" id="S4.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="903" id="S4.F4.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Reconstruct and enhance results in a simulated dataset with camera noise, from top to bottom, are the Measurements, ADMM, FlatNet, MWDN, DeepLIR, Ours, and Ground Truth.</figcaption> </figure> <div class="ltx_para" id="S4.SS4.p8"> <p class="ltx_p" id="S4.SS4.p8.1">Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.F4" title="Figure 4 ‣ 4.4 Simulated Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">4</span></a> presents the reconstruction examples of various methods on the simulated test dataset with added camera noise. As shown in the figure, ADMM successfully recovers most of the image structure but is heavily contaminated by complex noise, which obscures fine details and does not improve image brightness. While FlatNet and DeepLIR are effective at removing most of the noise and enhancing brightness, they suffer from significant loss of detail and color information. MWDN achieves basic reconstruction and ensures color recovery, but still falls short in terms of noise suppression and fine detail restoration. In contrast, the proposed method not only reduces noise effectively but also significantly enhances image brightness, resulting in visually acceptable reconstruction and enhancement. This demonstrates that our method is highly robust to noise.</p> </div> <figure class="ltx_table" id="S4.T3"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 3: </span><span class="ltx_text ltx_font_bold" id="S4.T3.2.1">The average MSE, LPIPS, PSNR and SSIM of the proposed method and several other methods on the simulation test set.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T3.3"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T3.3.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T3.3.1.1.1">Method</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T3.3.1.1.2">MSE</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T3.3.1.1.3">LPIPS</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T3.3.1.1.4">PSNR(in dB)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T3.3.1.1.5">SSIM</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T3.3.2.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.3.2.1.1">ADMM</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.3.2.1.2">0.1048</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.3.2.1.3">0.4675</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.3.2.1.4">10.70</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T3.3.2.1.5">0.2990</td> </tr> <tr class="ltx_tr" id="S4.T3.3.3.2"> <td class="ltx_td ltx_align_center" id="S4.T3.3.3.2.1">FlatNet</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.3.2.2">0.0482</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.3.2.3">0.2321</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.3.2.4">14.02</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.3.2.5">0.3350</td> </tr> <tr class="ltx_tr" id="S4.T3.3.4.3"> <td class="ltx_td ltx_align_center" id="S4.T3.3.4.3.1">MWDN</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.4.3.2">0.0249</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.4.3.3">0.3003</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.4.3.4">16.4291</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.4.3.5">0.5275</td> </tr> <tr class="ltx_tr" id="S4.T3.3.5.4"> <td class="ltx_td ltx_align_center" id="S4.T3.3.5.4.1">DeepLIR</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.5.4.2">0.0579</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.5.4.3">0.3145</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.5.4.4">13.84</td> <td class="ltx_td ltx_align_center" id="S4.T3.3.5.4.5">0.3595</td> </tr> <tr class="ltx_tr" id="S4.T3.3.6.5"> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.3.6.5.1">Ours</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.3.6.5.2">0.0211</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.3.6.5.3">0.2084</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.3.6.5.4">17.59</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T3.3.6.5.5">0.4951</td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S4.SS4.p9"> <p class="ltx_p" id="S4.SS4.p9.1">To further quantify the analysis, Table <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.T3" title="Table 3 ‣ 4.4 Simulated Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">3</span></a> presents the average MSE, LPIPS, PSNR, and SSIM scores of each algorithm on the simulated test dataset with added camera noise. As shown in the table, with the introduction of camera noise, traditional methods like ADMM show significant deterioration across all metrics, resulting in a noticeable drop in image quality. Although FlatNet and DeepLIR make some improvements in denoising, they still fail to effectively restore details and color, leading to a decline in performance. MWDN demonstrates relatively stable performance in noise handling, but still struggles with fine detail recovery and image brightness enhancement. In contrast, the proposed method shows minimal degradation compared to the noise-free case, with particularly strong results in PSNR and LPIPS.</p> </div> <div class="ltx_para" id="S4.SS4.p10"> <p class="ltx_p" id="S4.SS4.p10.1">These results not only confirm the effectiveness of the proposed method in handling camera noise under low-light conditions, but also highlight the limitations of previous methods under the same conditions, further emphasizing the unique advantages of the proposed approach in solving image reconstruction under low-light environments.</p> </div> </section> <section class="ltx_subsection" id="S4.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.5 </span>Measured Reconstruction</h3> <div class="ltx_para" id="S4.SS5.p1"> <p class="ltx_p" id="S4.SS5.p1.1">This section validates the proposed method through measured experiments. As shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.F5" title="Figure 5 ‣ 4.5 Measured Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">5</span></a>, we placed a self-designed random binary mask in front of the CMOS sensor, with a distance of 10mm between the mask and the CMOS sensor, considering the thickness of the glass covering the CMOS surface and the mask. Then, an LCD display screen used to display the captured target images was positioned 300mm in front of the CMOS sensor. The distance between the screen and the sensor is optimal for our imaging device. The system has a field of view (FOV) of about 26.6°. The target scene is placed at a distance that matches the adopted PSF, ensuring optimal imaging. If the scene is positioned outside this range, image quality degrades due to a mismatch between the assumed and actual system response. The raw resolution collected by the CMOS sensor is 4056×3040, encompassing Bayer measurements with four original channels (R, Gr, B, Gb), and the Bayer array image was converted to an RGB image with the dimensions of 2028 × 1520 × 3.</p> </div> <figure class="ltx_figure" id="S4.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="253" id="S4.F5.g1" src="extracted/6115207/figure/5.png" width="393"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Our self-built lensless imaging system.</figcaption> </figure> <div class="ltx_para" id="S4.SS5.p2"> <p class="ltx_p" id="S4.SS5.p2.1">The specific acquisition process for the measured data involves configuring the exposure time and exposure of the CMOS sensor using a Raspberry Pi to 0.7s and 100, respectively. The CMOS sensor is then set to collect data every 10 seconds, and the collected raw measurement data is saved to a computer. The captured target images are switched on the LCD display screen every 10 seconds until all target images in the original dataset have been traversed, resulting in the final dataset for the measured experiments.</p> </div> <figure class="ltx_figure" id="S4.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="805" id="S4.F6.g1" src="x3.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span> The test set results of the self-built measured dataset, from top to bottom, are the ADMM, FlatNet, MWDN, DeepLIR, Ours, and Ground Truth.</figcaption> </figure> <figure class="ltx_table" id="S4.T4"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 4: </span><span class="ltx_text ltx_font_bold" id="S4.T4.2.1">The average MSE, LPIPS, PSNR and SSIM of the proposed method and several other methods on the simulation test set.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T4.3"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T4.3.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T4.3.1.1.1">Method</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T4.3.1.1.2">MSE</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T4.3.1.1.3">LPIPS</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T4.3.1.1.4">PSNR(in dB)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T4.3.1.1.5">SSIM</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T4.3.2.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T4.3.2.1.1">ADMM</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T4.3.2.1.2">0.1371</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T4.3.2.1.3">0.5710</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T4.3.2.1.4">8.76</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T4.3.2.1.5">0.1952</td> </tr> <tr class="ltx_tr" id="S4.T4.3.3.2"> <td class="ltx_td ltx_align_center" id="S4.T4.3.3.2.1">FlatNet</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.3.2.2">0.0180</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.3.2.3">0.1646</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.3.2.4">18.35</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.3.2.5">0.4952</td> </tr> <tr class="ltx_tr" id="S4.T4.3.4.3"> <td class="ltx_td ltx_align_center" id="S4.T4.3.4.3.1">MWDN</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.4.3.2">0.0118</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.4.3.3">0.1965</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.4.3.4">19.56</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.4.3.5">0.5630</td> </tr> <tr class="ltx_tr" id="S4.T4.3.5.4"> <td class="ltx_td ltx_align_center" id="S4.T4.3.5.4.1">DeepLIR</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.5.4.2">0.0126</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.5.4.3">0.1885</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.5.4.4">19.21</td> <td class="ltx_td ltx_align_center" id="S4.T4.3.5.4.5">0.5166</td> </tr> <tr class="ltx_tr" id="S4.T4.3.6.5"> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T4.3.6.5.1">Ours</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T4.3.6.5.2">0.0071</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T4.3.6.5.3">0.1325</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T4.3.6.5.4">22.02</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T4.3.6.5.5">0.6392</td> </tr> </tbody> </table> </figure> <figure class="ltx_figure" id="S4.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="528" id="S4.F7.g1" src="x4.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span>Reconstruct and enhance results for measured datasets acquired at varying exposure, from top to bottom, are the results for the 0.3s exposure, the 0.5s exposure, the 0.7s exposure, and the ground truth.</figcaption> </figure> <div class="ltx_para" id="S4.SS5.p3"> <p class="ltx_p" id="S4.SS5.p3.1">To evaluate the performance of the proposed reconstruction enhancement algorithm, experiments were conducted on the real-world test set, using ADMM with 100 iterations, FlatNet, MWDN, and DeepLIR as baseline methods. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.F6" title="Figure 6 ‣ 4.5 Measured Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">6</span></a> presents some sample reconstruction results from the real-world validation dataset, along with visual comparisons to the original input and ground truth images.</p> </div> <div class="ltx_para" id="S4.SS5.p4"> <p class="ltx_p" id="S4.SS5.p4.1">As shown in this figure, ADMM produces poor reconstruction quality, only recovering the basic contours of the target, with the image almost entirely overwhelmed by noise. FlatNet and DeepLIR, while effectively removing noticeable noise, suffer from significant loss of color and detail information, resulting in subpar reconstructions. MWDN performs relatively well in preserving color information and recovering the image to some extent, but still struggles with fine details. In contrast, the proposed algorithm significantly improves reconstruction quality, virtually eliminating noise, and effectively preserving both color and detail information, yielding superior visual results compared to the other methods.</p> </div> <div class="ltx_para" id="S4.SS5.p5"> <p class="ltx_p" id="S4.SS5.p5.1">To further quantify the analysis, Table <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.T4" title="Table 4 ‣ 4.5 Measured Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">4</span></a> presents the average MSE, LPIPS, PSNR, and SSIM scores for each algorithm on the real-world test dataset. As shown in the table, ADMM performs the worst across all metrics, with very low scores. In comparison, the two-stage networks FlatNet and DeepLIR show significant improvements over ADMM, but still slightly lag behind MWDN. MWDN achieves the best performance among the baseline methods, particularly showing a notable improvement in SSIM. In contrast, the proposed method outperforms all other methods across all metrics, consistent with the visual results in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.F6" title="Figure 6 ‣ 4.5 Measured Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">6</span></a>, demonstrating its clear advantage in reconstruction quality.</p> </div> <div class="ltx_para" id="S4.SS5.p6"> <p class="ltx_p" id="S4.SS5.p6.1">The results on the measured test set verify the excellent effectiveness of the proposed method in low-light conditions is also verified, and the limitations and shortcomings of previous imaging methods under the same conditions are profoundly revealed, and the unique advantages of the proposed method in solving the problem of image reconstruction in low-light conditions are further highlighted through the comparative analyses.</p> </div> <div class="ltx_para" id="S4.SS5.p7"> <p class="ltx_p" id="S4.SS5.p7.1">To evaluate the robustness of the proposed reconstruction enhancement algorithm under varying low-light conditions, three sets of measured data were collected with exposure times of 0.7s, 0.5s, and 0.3s, respectively. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.F7" title="Figure 7 ‣ 4.5 Measured Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">7</span></a> displays sample reconstruction results from these datasets. Visually, the reconstructed images exhibit no significant differences in detail or color information across the three exposure conditions. All results achieve satisfactory reconstruction and enhancement, indicating that the algorithm maintains high robustness under different low-light scenarios.</p> </div> <div class="ltx_para" id="S4.SS5.p8"> <p class="ltx_p" id="S4.SS5.p8.1">For a more detailed evaluation, the average MSE, LPIPS, PSNR, and SSIM metrics of the reconstructed images under each exposure condition were calculated and are presented in Table <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.T5" title="Table 5 ‣ 4.5 Measured Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">5</span></a>. As the exposure time decreases from 0.7s to 0.3s, these metrics show only minor declines, with no significant performance degradation. This is consistent with the visual results in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S4.F7" title="Figure 7 ‣ 4.5 Measured Reconstruction ‣ 4 Experiment and Results ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">7</span></a>, confirming that the proposed algorithm effectively preserves image quality even under reduced exposure conditions. These findings underscore the excellent performance and stability of the method when applied to varying low-light environments.</p> </div> <figure class="ltx_table" id="S4.T5"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 5: </span><span class="ltx_text ltx_font_bold" id="S4.T5.2.1">The average MSE, LPIPS, PSNR and SSIM of the proposed method on the simulation test set under different low light conditions.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T5.3"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T5.3.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T5.3.1.1.1">exposure times(s)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T5.3.1.1.2">MSE</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T5.3.1.1.3">LPIPS</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T5.3.1.1.4">PSNR(in dB)</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T5.3.1.1.5">SSIM</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T5.3.2.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T5.3.2.1.1">0.3</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T5.3.2.1.2">0.0078</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T5.3.2.1.3">0.1417</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T5.3.2.1.4">21.55</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T5.3.2.1.5">0.6271</td> </tr> <tr class="ltx_tr" id="S4.T5.3.3.2"> <td class="ltx_td ltx_align_center" id="S4.T5.3.3.2.1">0.5</td> <td class="ltx_td ltx_align_center" id="S4.T5.3.3.2.2">0.0073</td> <td class="ltx_td ltx_align_center" id="S4.T5.3.3.2.3">0.1373</td> <td class="ltx_td ltx_align_center" id="S4.T5.3.3.2.4">21.85</td> <td class="ltx_td ltx_align_center" id="S4.T5.3.3.2.5">0.6365</td> </tr> <tr class="ltx_tr" id="S4.T5.3.4.3"> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T5.3.4.3.1">0.7</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T5.3.4.3.2">0.0071</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T5.3.4.3.3">0.1325</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T5.3.4.3.4">22.02</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T5.3.4.3.5">0.6392</td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S4.SS5.p9"> <p class="ltx_p" id="S4.SS5.p9.1">On the dataset used in this study, the proposed method takes approximately 0.4s to reconstruct a single target image, with a memory usage of around 4GB. This demonstrates that the method strikes a balance between performance and computational efficiency, making it suitable for practical applications. Furthermore, the efficiency of the method during the diffusion process further underscores its applicability in large-scale data processing scenarios.</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Discussion</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this work, we focus on addressing the challenges of lensless imaging under low-light conditions, with an emphasis on improving image reconstruction methods. Current lensless imaging techniques predominantly rely on coded-aperture light modulation, which can be broadly categorized into amplitude masks and phase masks. Phase masks, which modulate the phase of incident light instead of its amplitude, generally offer higher light throughput. This characteristic makes them more suitable for low-light scenarios compared to amplitude masks. However, despite their advantages, phase-mask-based systems often fail to match the reconstruction quality of traditional lens cameras, necessitating further advancements in reconstruction algorithms.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">To evaluate the proposed method, we conducted experiments using a self-built amplitude-mask-based lensless camera. Additionally, to assess its performance on phase-mask systems, we utilized the publicly available DiffuserCam dataset <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib17" title="">17</a>]</cite>. While the original dataset was captured under normal lighting, we simulated low-light conditions using a camera noise model described in Section <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S2" title="2 Problem analysis ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">2</span></a>. Experimental results, illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S5.F8" title="Figure 8 ‣ 5 Discussion ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">8</span></a>, compare our method with traditional Wiener deconvolution. Under low-light conditions, phase-mask systems exhibit significant noise and insufficient brightness. In contrast, our method improves image brightness, effectively suppresses noise, and produces visually realistic reconstructions, as demonstrated in the second row of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#S5.F8" title="Figure 8 ‣ 5 Discussion ‣ A generative approach for lensless imaging in low-light conditions"><span class="ltx_text ltx_ref_tag">8</span></a>.</p> </div> <figure class="ltx_figure" id="S5.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="261" id="S5.F8.g1" src="x5.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8: </span>Reconstruction results on the diffusercam dataset with added camera noise in low-light conditions.</figcaption> </figure> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">Low-light imaging presents challenges that extend beyond the mere issue of insufficient illumination. In such scenarios, the interaction between faint target light and varying environmental light conditions can introduce additional complexities, such as uneven illumination, color distortion, and interference. Most lensless imaging systems, including ours, are typically tested in controlled indoor environments, where light conditions can be precisely managed. These experiments often involve re-photographing scenes displayed on monitors to minimize environmental light interference.</p> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">A recent study addressed this issue <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2501.03511v1#bib.bib24" title="">24</a>]</cite>, sharing a similar conceptual framework with us, by employing a diffusion model conditioned on imaging results influenced by outdoor environmental lighting. In addition, they augmented their system with an array of metalenses to gather additional information, enabling promising results under real-world broadband illumination. Their work also underscores the importance of multiplexed measurements, integrating hardware enhancements, such as custom-designed nanophotonic arrays in their way, to modulate supplementary information for computational imaging.</p> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1">Future low-light applications of lensless imaging outside laboratory settings face dual challenges: insufficient target light and interference from environmental light. These issues suggest that multiplexed measurements could play a critical role in overcoming these limitations. However, implementing such measurements in physical systems remains an open problem, particularly given the spatial constraints inherent in low-light applications. Addressing these challenges will require innovative approaches to optimize both hardware design and computational algorithms, paving the way for robust lensless imaging systems suitable for real-world environments.</p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Conclusion</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">In summary, this paper proposed an innovative two-stage, model-driven generative reconstruction framework for lensless high-quality reconstruction under low-light conditions. In the first stage, a learnable Wiener filter-based module generates an initial, noisy reconstruction. The result is then transformed into the wavelet domain using a 2D discrete wavelet transform, producing lower-dimensional subbands for efficient processing. In the second stage, a noise-robust conditional diffusion generative model is applied to progressively refine the reconstruction, incorporating forward diffusion and backward denoising during training to ensure stable outputs. The experimental results show that the proposed method provides a substantial improvement in image brightness, noise reduction and overall sharpness in low-light conditions. It also reveals the limitations in previous reconstruction approaches, and demonstrates the unique advantages of the proposed method in solving the image reconstruction problem in low-light conditions.</p> </div> <div class="ltx_para" id="S6.p2"> <span class="ltx_ERROR undefined" id="S6.p2.1">\bmsection</span> <p class="ltx_p" id="S6.p2.2">FundingThis work was supported in part by the National Natural Science Foundation of China under Grants 62471113, 62305049 and 62371104, and in part by Sichuan Science and Technology Program 2024NS-FSC0479 and 2024NS-FSC1439.</p> </div> <div class="ltx_para" id="S6.p3"> <span class="ltx_ERROR undefined" id="S6.p3.1">\bmsection</span> <p class="ltx_p" id="S6.p3.2">DisclosuresThe authors declare that there are no conflicts of interest.</p> </div> <div class="ltx_para" id="S6.p4"> <span class="ltx_ERROR undefined" id="S6.p4.1">\bmsection</span> <p class="ltx_p" id="S6.p4.2">Data availabilityData underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> F. Heide, M. Rouf, M. B. Hullin, <em class="ltx_emph ltx_font_italic" id="bib.bib1.1.1">et al.</em>, “High-quality computational imaging through simple lenses,” <span class="ltx_ERROR undefined" id="bib.bib1.2.2">\JournalTitle</span>ACM Transactions on Graphics (TOG) <span class="ltx_text ltx_font_bold" id="bib.bib1.3.3">32</span>, 1–14 (2013). </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> S. Li, Y. Gao, J. Wu, <em class="ltx_emph ltx_font_italic" id="bib.bib2.1.1">et al.</em>, “Lensless camera: Unraveling the breakthroughs and prospects,” <span class="ltx_ERROR undefined" id="bib.bib2.2.2">\JournalTitle</span>Fundamental Research (2024). </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_tag_bibitem">[3]</span> <span class="ltx_bibblock"> Y. Zhang, X. Liu, and E. Y. Lam, “Single-shot inline holography using a physics-aware diffusion model,” <span class="ltx_ERROR undefined" id="bib.bib3.1.1">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib3.2.2">32</span>, 10444–10460 (2024). </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_tag_bibitem">[4]</span> <span class="ltx_bibblock"> K. Wang, L. Song, C. Wang, <em class="ltx_emph ltx_font_italic" id="bib.bib4.1.1">et al.</em>, “On the use of deep learning for phase recovery,” <span class="ltx_ERROR undefined" id="bib.bib4.2.2">\JournalTitle</span>Light: Science & Applications <span class="ltx_text ltx_font_bold" id="bib.bib4.3.3">13</span>, 4 (2024). </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_tag_bibitem">[5]</span> <span class="ltx_bibblock"> L. Song and E. Y. Lam, “Iterative phase retrieval with a sensor mask,” <span class="ltx_ERROR undefined" id="bib.bib5.1.1">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib5.2.2">30</span>, 25788–25802 (2022). </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_tag_bibitem">[6]</span> <span class="ltx_bibblock"> B. Huang, J. Li, B. Yao, <em class="ltx_emph ltx_font_italic" id="bib.bib6.1.1">et al.</em>, “Enhancing image resolution of confocal fluorescence microscopy with deep learning,” <span class="ltx_ERROR undefined" id="bib.bib6.2.2">\JournalTitle</span>PhotoniX <span class="ltx_text ltx_font_bold" id="bib.bib6.3.3">4</span>, 2 (2023). </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_tag_bibitem">[7]</span> <span class="ltx_bibblock"> Z. Ge, H. Wei, F. Xu, <em class="ltx_emph ltx_font_italic" id="bib.bib7.1.1">et al.</em>, “Millisecond autofocusing microscopy using neuromorphic event sensing,” <span class="ltx_ERROR undefined" id="bib.bib7.2.2">\JournalTitle</span>Optics and Lasers in Engineering <span class="ltx_text ltx_font_bold" id="bib.bib7.3.3">160</span>, 107247 (2023). </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_tag_bibitem">[8]</span> <span class="ltx_bibblock"> T. T. N. Mai, E. Y. Lam, and C. Lee, “Deep unrolled low-rank tensor completion for high dynamic range imaging,” <span class="ltx_ERROR undefined" id="bib.bib8.1.1">\JournalTitle</span>IEEE Transactions on Image Processing <span class="ltx_text ltx_font_bold" id="bib.bib8.2.2">31</span>, 5774–5787 (2022). </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_tag_bibitem">[9]</span> <span class="ltx_bibblock"> Y. Zhu, T. Zeng, K. Liu, <em class="ltx_emph ltx_font_italic" id="bib.bib9.1.1">et al.</em>, “Full scene underwater imaging with polarization and an untrained network,” <span class="ltx_ERROR undefined" id="bib.bib9.2.2">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib9.3.3">29</span>, 41865–41881 (2021). </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_tag_bibitem">[10]</span> <span class="ltx_bibblock"> M. S. Asif, A. Ayremlou, A. Sankaranarayanan, <em class="ltx_emph ltx_font_italic" id="bib.bib10.1.1">et al.</em>, “Flatcam: Thin, lensless cameras using coded aperture and computation,” <span class="ltx_ERROR undefined" id="bib.bib10.2.2">\JournalTitle</span>IEEE Transactions on Computational Imaging <span class="ltx_text ltx_font_bold" id="bib.bib10.3.3">3</span>, 384–397 (2016). </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_tag_bibitem">[11]</span> <span class="ltx_bibblock"> V. Boominathan, J. K. Adams, J. T. Robinson, and A. Veeraraghavan, “Phlatcam: Designed phase-mask based thin lensless camera,” <span class="ltx_ERROR undefined" id="bib.bib11.1.1">\JournalTitle</span>IEEE transactions on pattern analysis and machine intelligence <span class="ltx_text ltx_font_bold" id="bib.bib11.2.2">42</span>, 1618–1629 (2020). </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_tag_bibitem">[12]</span> <span class="ltx_bibblock"> V. Boominathan, J. T. Robinson, L. Waller, and A. Veeraraghavan, “Recent advances in lensless imaging,” <span class="ltx_ERROR undefined" id="bib.bib12.1.1">\JournalTitle</span>Optica <span class="ltx_text ltx_font_bold" id="bib.bib12.2.2">9</span>, 1–16 (2021). </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_tag_bibitem">[13]</span> <span class="ltx_bibblock"> F. Liu, J. Wu, and L. Cao, “Autofocusing of fresnel zone aperture lensless imaging for qr code recognition,” <span class="ltx_ERROR undefined" id="bib.bib13.1.1">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib13.2.2">31</span>, 15889–15903 (2023). </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_tag_bibitem">[14]</span> <span class="ltx_bibblock"> S. Goswami, P. Wani, G. Gupta, and B. Javidi, “Assessment of lateral resolution of single random phase encoded lensless imaging systems,” <span class="ltx_ERROR undefined" id="bib.bib14.1.1">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib14.2.2">31</span>, 11213–11226 (2023). </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_tag_bibitem">[15]</span> <span class="ltx_bibblock"> Y. Zhang, Z. Wu, Y. Xu, and J. Huangfu, “Dual-branch fusion model for lensless imaging,” <span class="ltx_ERROR undefined" id="bib.bib15.1.1">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib15.2.2">31</span>, 19463–19477 (2023). </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_tag_bibitem">[16]</span> <span class="ltx_bibblock"> X. Pan, X. Chen, T. Nakamura, and M. Yamaguchi, “Incoherent reconstruction-free object recognition with mask-based lensless optics and the transformer,” <span class="ltx_ERROR undefined" id="bib.bib16.1.1">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib16.2.2">29</span>, 37962–37978 (2021). </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_tag_bibitem">[17]</span> <span class="ltx_bibblock"> K. Monakhova, J. Yurtsever, G. Kuo, <em class="ltx_emph ltx_font_italic" id="bib.bib17.1.1">et al.</em>, “Learned reconstructions for practical mask-based lensless imaging,” <span class="ltx_ERROR undefined" id="bib.bib17.2.2">\JournalTitle</span>Optics express <span class="ltx_text ltx_font_bold" id="bib.bib17.3.3">27</span>, 28075–28090 (2019). </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_tag_bibitem">[18]</span> <span class="ltx_bibblock"> S. S. Khan, V. Sundar, V. Boominathan, <em class="ltx_emph ltx_font_italic" id="bib.bib18.1.1">et al.</em>, “Flatnet: Towards photorealistic scene reconstruction from lensless measurements,” <span class="ltx_ERROR undefined" id="bib.bib18.2.2">\JournalTitle</span>IEEE Transactions on Pattern Analysis and Machine Intelligence <span class="ltx_text ltx_font_bold" id="bib.bib18.3.3">44</span>, 1934–1948 (2020). </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_tag_bibitem">[19]</span> <span class="ltx_bibblock"> T. Zeng and E. Y. Lam, “Robust reconstruction with deep learning to handle model mismatch in lensless imaging,” <span class="ltx_ERROR undefined" id="bib.bib19.1.1">\JournalTitle</span>IEEE Transactions on Computational Imaging <span class="ltx_text ltx_font_bold" id="bib.bib19.2.2">7</span>, 1080–1092 (2021). </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_tag_bibitem">[20]</span> <span class="ltx_bibblock"> O. Kingshott, N. Antipa, E. Bostan, and K. Akşit, “Unrolled primal-dual networks for lensless cameras,” <span class="ltx_ERROR undefined" id="bib.bib20.1.1">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib20.2.2">30</span>, 46324–46335 (2022). </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_tag_bibitem">[21]</span> <span class="ltx_bibblock"> Y. Li, Z. Li, K. Chen, <em class="ltx_emph ltx_font_italic" id="bib.bib21.1.1">et al.</em>, “Mwdns: reconstruction in multi-scale feature spaces for lensless imaging,” <span class="ltx_ERROR undefined" id="bib.bib21.2.2">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib21.3.3">31</span>, 39088–39101 (2023). </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_tag_bibitem">[22]</span> <span class="ltx_bibblock"> H. Qian, H. Ling, and X. Lu, “Robust unrolled network for lensless imaging with enhanced resistance to model mismatch and noise,” <span class="ltx_ERROR undefined" id="bib.bib22.1.1">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib22.2.2">32</span>, 30267–30283 (2024). </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_tag_bibitem">[23]</span> <span class="ltx_bibblock"> X. Cai, Z. You, H. Zhang, <em class="ltx_emph ltx_font_italic" id="bib.bib23.1.1">et al.</em>, “Phocolens: Photorealistic and consistent reconstruction in lensless imaging,” <span class="ltx_ERROR undefined" id="bib.bib23.2.2">\JournalTitle</span>arXiv preprint arXiv:2409.17996 (2024). </span> </li> <li class="ltx_bibitem" id="bib.bib24"> <span class="ltx_tag ltx_tag_bibitem">[24]</span> <span class="ltx_bibblock"> P. Chakravarthula, J. Sun, X. Li, <em class="ltx_emph ltx_font_italic" id="bib.bib24.1.1">et al.</em>, “Thin on-sensor nanophotonic array cameras,” <span class="ltx_ERROR undefined" id="bib.bib24.2.2">\JournalTitle</span>ACM Transactions on Graphics (TOG) <span class="ltx_text ltx_font_bold" id="bib.bib24.3.3">42</span>, 1–18 (2023). </span> </li> <li class="ltx_bibitem" id="bib.bib25"> <span class="ltx_tag ltx_tag_bibitem">[25]</span> <span class="ltx_bibblock"> E. Bezzam, S. Peters, and M. Vetterli, “Let there be light: Robust lensless imaging under external illumination with deep learning,” <span class="ltx_ERROR undefined" id="bib.bib25.1.1">\JournalTitle</span>arXiv preprint arXiv:2409.16766 (2024). </span> </li> <li class="ltx_bibitem" id="bib.bib26"> <span class="ltx_tag ltx_tag_bibitem">[26]</span> <span class="ltx_bibblock"> S. Goswami, G. Krishnan, and B. Javidi, “Robustness of single random phase encoding lensless imaging with camera noise,” <span class="ltx_ERROR undefined" id="bib.bib26.1.1">\JournalTitle</span>Optics Express <span class="ltx_text ltx_font_bold" id="bib.bib26.2.2">32</span>, 4916–4930 (2024). </span> </li> <li class="ltx_bibitem" id="bib.bib27"> <span class="ltx_tag ltx_tag_bibitem">[27]</span> <span class="ltx_bibblock"> J. Ho, A. Jain, and P. Abbeel, “Denoising diffusion probabilistic models,” <span class="ltx_ERROR undefined" id="bib.bib27.1.1">\JournalTitle</span>Advances in neural information processing systems <span class="ltx_text ltx_font_bold" id="bib.bib27.2.2">33</span>, 6840–6851 (2020). </span> </li> <li class="ltx_bibitem" id="bib.bib28"> <span class="ltx_tag ltx_tag_bibitem">[28]</span> <span class="ltx_bibblock"> H. Jiang, A. Luo, H. Fan, <em class="ltx_emph ltx_font_italic" id="bib.bib28.1.1">et al.</em>, “Low-light image enhancement with wavelet-based diffusion models,” <span class="ltx_ERROR undefined" id="bib.bib28.2.2">\JournalTitle</span>ACM Transactions on Graphics (TOG) <span class="ltx_text ltx_font_bold" id="bib.bib28.3.3">42</span>, 1–14 (2023). </span> </li> <li class="ltx_bibitem" id="bib.bib29"> <span class="ltx_tag ltx_tag_bibitem">[29]</span> <span class="ltx_bibblock"> Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” <span class="ltx_ERROR undefined" id="bib.bib29.1.1">\JournalTitle</span>IEEE transactions on image processing <span class="ltx_text ltx_font_bold" id="bib.bib29.2.2">13</span>, 600–612 (2004). </span> </li> <li class="ltx_bibitem" id="bib.bib30"> <span class="ltx_tag ltx_tag_bibitem">[30]</span> <span class="ltx_bibblock"> R. Zhang, P. Isola, A. A. Efros, <em class="ltx_emph ltx_font_italic" id="bib.bib30.1.1">et al.</em>, “The unreasonable effectiveness of deep features as a perceptual metric,” in <em class="ltx_emph ltx_font_italic" id="bib.bib30.2.2">Proceedings of the IEEE conference on computer vision and pattern recognition,</em> (2018), pp. 586–595. </span> </li> <li class="ltx_bibitem" id="bib.bib31"> <span class="ltx_tag ltx_tag_bibitem">[31]</span> <span class="ltx_bibblock"> W. Yang, W. Wang, H. Huang, <em class="ltx_emph ltx_font_italic" id="bib.bib31.1.1">et al.</em>, “Sparse gradient regularized deep retinex network for robust low-light image enhancement,” <span class="ltx_ERROR undefined" id="bib.bib31.2.2">\JournalTitle</span>IEEE Transactions on Image Processing <span class="ltx_text ltx_font_bold" id="bib.bib31.3.3">30</span>, 2072–2086 (2021). </span> </li> <li class="ltx_bibitem" id="bib.bib32"> <span class="ltx_tag ltx_tag_bibitem">[32]</span> <span class="ltx_bibblock"> E. Bezzam, S. Kashani, M. Vetterli, and M. Simeoni, “Lenslesspicam: A hardware and software platform for lensless computational imaging with a raspberry pi,” <span class="ltx_ERROR undefined" id="bib.bib32.1.1">\JournalTitle</span>arXiv preprint arXiv:2206.01430 (2022). </span> </li> <li class="ltx_bibitem" id="bib.bib33"> <span class="ltx_tag ltx_tag_bibitem">[33]</span> <span class="ltx_bibblock"> A. Poudel and U. Nakarmi, “Deeplir: Attention-based approach for mask-based lensless image reconstruction,” in <em class="ltx_emph ltx_font_italic" id="bib.bib33.1.1">Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision,</em> (2024), pp. 431–439. </span> </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Tue Jan 7 04:09:58 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>