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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation</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> <meta content=" Medical Image Processing, Ultrasound Imaging, Semantic Segmentation, Probabilistic Model " lang="en" name="keywords"/> <base href="/html/2411.14250v1/"/></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/2411.14250v1#S1" title="In CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S2" title="In CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">Method</span></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/2411.14250v1#S2.SS1" title="In II Method ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-A</span> </span><span class="ltx_text ltx_font_italic">Multi-group channel shifted downsampling</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S2.SS2" title="In II Method ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-B</span> </span><span class="ltx_text ltx_font_italic">Contour Probabilistic Modeling</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S2.SS3" title="In II Method ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-C</span> </span><span class="ltx_text ltx_font_italic">Gating-based feature filtering module</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S3" title="In CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Experiments</span></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/2411.14250v1#S3.SS1" title="In III Experiments ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-A</span> </span><span class="ltx_text ltx_font_italic">Datasets and Experimental Settings </span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S3.SS2" title="In III Experiments ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-B</span> </span><span class="ltx_text ltx_font_italic">Performance Comparison</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S3.SS3" title="In III Experiments ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-C</span> </span><span class="ltx_text ltx_font_italic">Ablation Studies</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S4" title="In CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Conclusion</span></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Ruiguo Yu<sup class="ltx_sup" id="id12.12.id1"><span class="ltx_text ltx_font_italic" id="id12.12.id1.1">1,2,3,4</span></sup>, Yiyang Zhang<sup class="ltx_sup" id="id13.13.id2"><span class="ltx_text ltx_font_italic" id="id13.13.id2.1">2,3,4</span></sup>, Yuan Tian<sup class="ltx_sup" id="id14.14.id3"><span class="ltx_text ltx_font_italic" id="id14.14.id3.1">1,2,3</span></sup>, Zhiqiang Liu<sup class="ltx_sup" id="id15.15.id4"><span class="ltx_text ltx_font_italic" id="id15.15.id4.1">1,2,3</span></sup>, Xuewei Li<sup class="ltx_sup" id="id16.16.id5"><span class="ltx_text ltx_font_italic" id="id16.16.id5.1">1,2,3,4</span></sup>, Jie Gao<sup class="ltx_sup" id="id17.17.id6"><span class="ltx_text ltx_font_italic" id="id17.17.id6.1">1,2,3,∗</span></sup> </span><span class="ltx_author_notes"><sup class="ltx_sup" id="id18.18.id1"><span class="ltx_text ltx_font_italic" id="id18.18.id1.1">∗</span></sup> Corresponding author. <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id19.19.id1">1</sup> College of Intelligence and Computing, Tianjin University, Tianjin, 300350, China. </span> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id20.20.id1">2</sup> Tianjin Key Laboratory of Cognitive Computing and Application, Tianjin, 300350, China. </span> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id21.21.id1">3</sup> Tianjin Key Laboratory of Advanced Networking, Tianjin, 300350, China. </span> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id22.22.id1">4</sup> School of Future Technology, Tianjin University, Tianjin, 300350, China. </span> <span class="ltx_contact ltx_role_affiliation"> {rgyu, zyy_0203, tiany, tjubeisong, lixuewei, gaojie}@tju.edu.cn </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id23.id1">Deep learning-based segmentation methods are widely utilized for detecting lesions in ultrasound images. Throughout the imaging procedure, the attenuation and scattering of ultrasound waves cause contour blurring and the formation of artifacts, limiting the clarity of the acquired ultrasound images. To overcome this challenge, we propose a contour-based probabilistic segmentation model CP-UNet, which guides the segmentation network to enhance its focus on contour during decoding. We design a novel down-sampling module to enable the contour probability distribution modeling and encoding stages to acquire global-local features. Furthermore, the Gaussian Mixture Model utilizes optimized features to model the contour distribution, capturing the uncertainty of lesion boundaries. Extensive experiments with several state-of-the-art deep learning segmentation methods on three ultrasound image datasets show that our method performs better on breast and thyroid lesions segmentation.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> Medical Image Processing, Ultrasound Imaging, Semantic Segmentation, Probabilistic Model </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Ultrasound imaging is widely used for various disease diagnoses due to its low cost, simple operation, and non-invasiveness. With the development of computer-aided diagnosis (CAD) technology, deep learning-based segmentation methods have been applied to lesion segmentation in medical ultrasound images. However, inhomogeneous lesion region distribution, speckle noise, and imaging artifacts in ultrasound images increase the difficulty of the segmentation task. Structural boundary line ambiguity, such as the unclear transition between lesions and normal tissues as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S1.F1" title="Figure 1 ‣ I Introduction ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">1</span></a>(b)(c)(d), and contour heterogeneity, including irregular variations in lesion shapes and textures illustrated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S1.F1" title="Figure 1 ‣ I Introduction ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">1</span></a>(e)(f), both contribute to contour blurring. These factors collectively affect the accuracy of segmentation.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Existing approaches primarily focus on optimizing convolutional neural networks to enhance the accuracy of medical image segmentation.UNet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib1" title="">1</a>]</cite> and its variants<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib2" title="">2</a>]</cite><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib3" title="">3</a>]</cite><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib4" title="">4</a>]</cite>effectively integrate features at different scales through an encoder-decoder architecture, which enables the network to extract fine-grained local features from the image. However, A limited receptive field limits the perception of detailed features by convolutional operations and can only process local information within the size of the convolutional kernel. To alleviate the limitations of convolutional operations, ReAgU-Net <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib5" title="">5</a>]</cite> and UNeXt <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib6" title="">6</a>]</cite> propose different strategies to expand the receptive field. While enhancing the contextual connection of the deep encoding stage, the shallow encoding stage still does not introduce the global connection of detailed features, which affects the accurate recognition of blurred contours. Other methods based on the attention mechanism, such as ConTrans<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib7" title="">7</a>]</cite>, ACC-Unet <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib8" title="">8</a>]</cite>, Segnetr <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib9" title="">9</a>]</cite>, utilize the improved self-attention mechanism to establish connections between detail features and enhance the global local connections in the full-stage encoding. However, they need to develop interconnections between all features, which is unnecessary for the sparsity of contours in ultrasound images. We utilize the fusion strategy in the attention mechanism to establish global links that fuse sparse features while preserving detailed features.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="335" id="S1.F1.g1" src="x1.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Enlarged images of nodules in six ultrasound images: (a) Clear contours; (b)-(d) Blurred contour edges; (e)-(f) Irregular shapes. The green contour line is physician-labeled</figcaption> </figure> <figure class="ltx_figure" id="S1.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="259" id="S1.F2.g1" src="x2.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Framework of CP-UNet. CMP, Contour Probabilistic Modeling. MgCSD, Multi-group channel shifted downsampling. GF, Gating-based feature filtering module</figcaption> </figure> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">There are many methods to create constraints specifically for the contour features to guide the network in enhancing its focus on the contours to address the domain of ambiguous contours in ultrasound images. For example, GGNet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib10" title="">10</a>]</cite> and Wavelet U-Net++<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib11" title="">11</a>]</cite> both provide constraints on the extent of the lesion region for the encoded intermediate features. AGMT<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib12" title="">12</a>]</cite> investigates a transformer-based breast ultrasound loss function that enhances the perception of the lesion’s shape. However, the loss function cannot fully express the contour features to learn the specific features of the contour. When the local edge line of the contour is blurred, it needs to be complemented by the global edge line characteristics, such as pixels around the edge line, contour shape, and range characteristics. We use the Gaussian Mixture Model to simulate the distributional features of the synthetic contours to break the limitation of a single loss constraint and establish generalized features to guide the contour representation in the segmentation network.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">To obtain generalized contour features to guide the segmentation process, we propose CP-UNet, in which the Multi-group Channel Shift Downsampling (MgCSD) utilizes a fusion strategy in the attention mechanism to give global connections to preserve detailed features and provide more comprehensive features. Then, the Contour Probabilistic Modeling Module (CPM) in the network breaks the limitation of a single loss constraint on the contour to build generalized contour features. Finally, the gating mechanism-based feature filtering module (GF) provides effective fusion strategies for contour features.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">The contributions of this paper are summarized as follows:</p> </div> <div class="ltx_para" id="S1.p6"> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><math alttext="\bullet" class="ltx_Math" display="inline" id="S1.I1.ix1.1.1.m1.1"><semantics id="S1.I1.ix1.1.1.m1.1b"><mo id="S1.I1.ix1.1.1.m1.1.1" xref="S1.I1.ix1.1.1.m1.1.1.cmml">∙</mo><annotation-xml encoding="MathML-Content" id="S1.I1.ix1.1.1.m1.1c"><ci id="S1.I1.ix1.1.1.m1.1.1.cmml" xref="S1.I1.ix1.1.1.m1.1.1">∙</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix1.1.1.m1.1d">\bullet</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix1.1.1.m1.1e">∙</annotation></semantics></math></span> <div class="ltx_para" id="S1.I1.ix1.p1"> <p class="ltx_p" id="S1.I1.ix1.p1.1">We propose a contour probabilistic modeling-guided ultrasound image segmentation network, which fits the contour features with probability distributions and guides the enhancement of the contour representation of the features at different stages to improve the segmentation results.</p> </div> </li> <li class="ltx_item" id="S1.I1.ix2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><math alttext="\bullet" class="ltx_Math" display="inline" id="S1.I1.ix2.1.1.m1.1"><semantics id="S1.I1.ix2.1.1.m1.1b"><mo id="S1.I1.ix2.1.1.m1.1.1" xref="S1.I1.ix2.1.1.m1.1.1.cmml">∙</mo><annotation-xml encoding="MathML-Content" id="S1.I1.ix2.1.1.m1.1c"><ci id="S1.I1.ix2.1.1.m1.1.1.cmml" xref="S1.I1.ix2.1.1.m1.1.1">∙</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix2.1.1.m1.1d">\bullet</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix2.1.1.m1.1e">∙</annotation></semantics></math></span> <div class="ltx_para" id="S1.I1.ix2.p1"> <p class="ltx_p" id="S1.I1.ix2.p1.1">MgCSD is proposed instead of traditional convolutional downsampling to create global-local links for full-stage coding features. GF can fuse contour features into the decoding stage, bridging the semantic gap between MgCSD and CPM and downsampling results.</p> </div> </li> <li class="ltx_item" id="S1.I1.ix3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><math alttext="\bullet" class="ltx_Math" display="inline" id="S1.I1.ix3.1.1.m1.1"><semantics id="S1.I1.ix3.1.1.m1.1b"><mo id="S1.I1.ix3.1.1.m1.1.1" xref="S1.I1.ix3.1.1.m1.1.1.cmml">∙</mo><annotation-xml encoding="MathML-Content" id="S1.I1.ix3.1.1.m1.1c"><ci id="S1.I1.ix3.1.1.m1.1.1.cmml" xref="S1.I1.ix3.1.1.m1.1.1">∙</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix3.1.1.m1.1d">\bullet</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix3.1.1.m1.1e">∙</annotation></semantics></math></span> <div class="ltx_para" id="S1.I1.ix3.p1"> <p class="ltx_p" id="S1.I1.ix3.p1.1">Extensive experiments on BUSI <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib13" title="">13</a>]</cite>, DDTI <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib14" title="">14</a>]</cite>, and one private thyroid ultrasound images dataset demonstrate that our approach consistently improves the segmentation accuracy of breast nodules and thyroid nodules, outperforming the strong baseline and state-of-the-art medical image segmentation methods.</p> </div> </li> </ul> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">Method</span> </h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">The CP-UNet architecture proposed is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S1.F2" title="Figure 2 ‣ I Introduction ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">2</span></a>(a). Firstly, the image is downsampled using multiple multi-group channel shift downsampling (MgCSD) modules. Secondly, the results of MgCSD are fed into the contour probabilistic modeling (CPM) module to fit the contour features of the original image using the mixed Gaussian distribution to fit the contour features. Finally, the gating-based feature filtering module (GF) fuses the up-sampling result, the down-sampling consequence, and the CPM sampling result of the same dimension at the corresponding stage. Finally, the nodule’s segmentation result is obtained. The following subsections will introduce the MgCSD, CPM, and GF modules.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS1.5.1.1">II-A</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS1.6.2">Multi-group channel shifted downsampling</span> </h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1">Shallow convolutional layers can effectively capture localized edge line features but often lack a grasp of contour structure. Deeper convolutional layers, expanding the receptive field and capturing the contextual relationships of contour features, may lose detailed representations like edge lines. We propose a downsampling method based on spatial shift weighting of multiple groups of channels.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.1">The shift operation is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S1.F2" title="Figure 2 ‣ I Introduction ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">2</span></a>(b), where the features are divided into groups according to the channel dimensions. The first group remains fixed, and the rest are shifted across the width and the height according to cyclic upward and cyclic rightward shifts, leaving the relative positions of the elements in the regions. The non-feature areas are filled with zeros after the shifts. 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encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">F_{supply}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">italic_F start_POSTSUBSCRIPT italic_s italic_u italic_p italic_p italic_l italic_y end_POSTSUBSCRIPT</annotation></semantics></math> in terms of the channel dimensions by utilizing the result of global average pooling, which is fused to obtain the globally enhanced weights <math alttext="F_{L}" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.1"><semantics id="S2.SS1.p4.3.m3.1a"><msub id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml"><mi id="S2.SS1.p4.3.m3.1.1.2" xref="S2.SS1.p4.3.m3.1.1.2.cmml">F</mi><mi id="S2.SS1.p4.3.m3.1.1.3" xref="S2.SS1.p4.3.m3.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.3.m3.1b"><apply id="S2.SS1.p4.3.m3.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.3.m3.1.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p4.3.m3.1.1.2.cmml" 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xref="S2.E1.m1.2.2.4.3.cmml">L</mi></msub><mo id="S2.E1.m1.2.2.3" xref="S2.E1.m1.2.2.3.cmml">=</mo><mrow id="S2.E1.m1.2.2.2" xref="S2.E1.m1.2.2.2.cmml"><mrow id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.3.cmml">G</mi><mo id="S2.E1.m1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S2.E1.m1.1.1.1.1.4" xref="S2.E1.m1.1.1.1.1.4.cmml">A</mi><mo id="S2.E1.m1.1.1.1.1.2a" xref="S2.E1.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S2.E1.m1.1.1.1.1.5" xref="S2.E1.m1.1.1.1.1.5.cmml">P</mi><mo id="S2.E1.m1.1.1.1.1.2b" xref="S2.E1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.1.cmml"><mo id="S2.E1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.1.cmml"><msub id="S2.E1.m1.1.1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.E1.m1.1.1.1.1.1.1.1.2.2" xref="S2.E1.m1.1.1.1.1.1.1.1.2.2.cmml">F</mi><mrow 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xref="S2.E1.m1.2.2.2.2.1"><csymbol cd="latexml" id="S2.E1.m1.2.2.2.2.1.1.1.cmml" xref="S2.E1.m1.2.2.2.2.1.1.1">direct-sum</csymbol><apply id="S2.E1.m1.2.2.2.2.1.1.2.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2"><csymbol cd="ambiguous" id="S2.E1.m1.2.2.2.2.1.1.2.1.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2">subscript</csymbol><ci id="S2.E1.m1.2.2.2.2.1.1.2.2.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2.2">𝐹</ci><apply id="S2.E1.m1.2.2.2.2.1.1.2.3.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2.3"><times id="S2.E1.m1.2.2.2.2.1.1.2.3.1.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2.3.1"></times><ci id="S2.E1.m1.2.2.2.2.1.1.2.3.2.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2.3.2">𝑠</ci><ci id="S2.E1.m1.2.2.2.2.1.1.2.3.3.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2.3.3">ℎ</ci><ci id="S2.E1.m1.2.2.2.2.1.1.2.3.4.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2.3.4">𝑖</ci><ci id="S2.E1.m1.2.2.2.2.1.1.2.3.5.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2.3.5">𝑓</ci><ci id="S2.E1.m1.2.2.2.2.1.1.2.3.6.cmml" xref="S2.E1.m1.2.2.2.2.1.1.2.3.6">𝑡</ci></apply></apply><apply id="S2.E1.m1.2.2.2.2.1.1.3.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S2.E1.m1.2.2.2.2.1.1.3.1.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3">subscript</csymbol><ci id="S2.E1.m1.2.2.2.2.1.1.3.2.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3.2">𝐹</ci><apply id="S2.E1.m1.2.2.2.2.1.1.3.3.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3.3"><times id="S2.E1.m1.2.2.2.2.1.1.3.3.1.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3.3.1"></times><ci id="S2.E1.m1.2.2.2.2.1.1.3.3.2.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3.3.2">𝑠</ci><ci id="S2.E1.m1.2.2.2.2.1.1.3.3.3.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3.3.3">𝑢</ci><ci id="S2.E1.m1.2.2.2.2.1.1.3.3.4.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3.3.4">𝑝</ci><ci id="S2.E1.m1.2.2.2.2.1.1.3.3.5.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3.3.5">𝑝</ci><ci id="S2.E1.m1.2.2.2.2.1.1.3.3.6.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3.3.6">𝑙</ci><ci id="S2.E1.m1.2.2.2.2.1.1.3.3.7.cmml" xref="S2.E1.m1.2.2.2.2.1.1.3.3.7">𝑦</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.2c">F_{L}=GAP(F_{shift}\oplus F_{supply})\otimes(F_{shift}\oplus F_{supply})</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.2d">italic_F start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = italic_G italic_A italic_P ( italic_F start_POSTSUBSCRIPT italic_s italic_h italic_i italic_f italic_t end_POSTSUBSCRIPT ⊕ italic_F start_POSTSUBSCRIPT italic_s italic_u italic_p italic_p italic_l italic_y end_POSTSUBSCRIPT ) ⊗ ( italic_F start_POSTSUBSCRIPT italic_s italic_h italic_i italic_f italic_t end_POSTSUBSCRIPT ⊕ italic_F start_POSTSUBSCRIPT italic_s italic_u italic_p italic_p italic_l italic_y 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">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p4.5">where <math alttext="\otimes" class="ltx_Math" display="inline" id="S2.SS1.p4.4.m1.1"><semantics id="S2.SS1.p4.4.m1.1a"><mo id="S2.SS1.p4.4.m1.1.1" xref="S2.SS1.p4.4.m1.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.4.m1.1b"><csymbol cd="latexml" id="S2.SS1.p4.4.m1.1.1.cmml" xref="S2.SS1.p4.4.m1.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.4.m1.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.4.m1.1d">⊗</annotation></semantics></math> and <math alttext="\oplus" class="ltx_Math" display="inline" id="S2.SS1.p4.5.m2.1"><semantics id="S2.SS1.p4.5.m2.1a"><mo id="S2.SS1.p4.5.m2.1.1" xref="S2.SS1.p4.5.m2.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.5.m2.1b"><csymbol cd="latexml" id="S2.SS1.p4.5.m2.1.1.cmml" xref="S2.SS1.p4.5.m2.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.5.m2.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.5.m2.1d">⊕</annotation></semantics></math> are element-wise operation</p> </div> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.6">Complete local feature preservation is provided by convolutional downsampling, denoted as <math alttext="F_{R}" class="ltx_Math" display="inline" id="S2.SS1.p5.1.m1.1"><semantics id="S2.SS1.p5.1.m1.1a"><msub id="S2.SS1.p5.1.m1.1.1" xref="S2.SS1.p5.1.m1.1.1.cmml"><mi id="S2.SS1.p5.1.m1.1.1.2" xref="S2.SS1.p5.1.m1.1.1.2.cmml">F</mi><mi id="S2.SS1.p5.1.m1.1.1.3" xref="S2.SS1.p5.1.m1.1.1.3.cmml">R</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.1.m1.1b"><apply id="S2.SS1.p5.1.m1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.1.m1.1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p5.1.m1.1.1.2.cmml" xref="S2.SS1.p5.1.m1.1.1.2">𝐹</ci><ci id="S2.SS1.p5.1.m1.1.1.3.cmml" xref="S2.SS1.p5.1.m1.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.1.m1.1c">F_{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.1.m1.1d">italic_F start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math>.The result of adding <math alttext="F_{L}" class="ltx_Math" display="inline" id="S2.SS1.p5.2.m2.1"><semantics id="S2.SS1.p5.2.m2.1a"><msub id="S2.SS1.p5.2.m2.1.1" xref="S2.SS1.p5.2.m2.1.1.cmml"><mi id="S2.SS1.p5.2.m2.1.1.2" xref="S2.SS1.p5.2.m2.1.1.2.cmml">F</mi><mi id="S2.SS1.p5.2.m2.1.1.3" xref="S2.SS1.p5.2.m2.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.2.m2.1b"><apply id="S2.SS1.p5.2.m2.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.1.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p5.2.m2.1.1.2.cmml" xref="S2.SS1.p5.2.m2.1.1.2">𝐹</ci><ci id="S2.SS1.p5.2.m2.1.1.3.cmml" xref="S2.SS1.p5.2.m2.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m2.1c">F_{L}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.2.m2.1d">italic_F start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="F_{R}" class="ltx_Math" display="inline" id="S2.SS1.p5.3.m3.1"><semantics id="S2.SS1.p5.3.m3.1a"><msub id="S2.SS1.p5.3.m3.1.1" xref="S2.SS1.p5.3.m3.1.1.cmml"><mi id="S2.SS1.p5.3.m3.1.1.2" xref="S2.SS1.p5.3.m3.1.1.2.cmml">F</mi><mi id="S2.SS1.p5.3.m3.1.1.3" xref="S2.SS1.p5.3.m3.1.1.3.cmml">R</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.3.m3.1b"><apply id="S2.SS1.p5.3.m3.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.3.m3.1.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p5.3.m3.1.1.2.cmml" xref="S2.SS1.p5.3.m3.1.1.2">𝐹</ci><ci id="S2.SS1.p5.3.m3.1.1.3.cmml" xref="S2.SS1.p5.3.m3.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.3.m3.1c">F_{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.3.m3.1d">italic_F start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math> will be mapped to obtain an interrelationship matrix <math alttext="O_{mixer}" class="ltx_Math" display="inline" id="S2.SS1.p5.4.m4.1"><semantics id="S2.SS1.p5.4.m4.1a"><msub id="S2.SS1.p5.4.m4.1.1" xref="S2.SS1.p5.4.m4.1.1.cmml"><mi id="S2.SS1.p5.4.m4.1.1.2" xref="S2.SS1.p5.4.m4.1.1.2.cmml">O</mi><mrow id="S2.SS1.p5.4.m4.1.1.3" xref="S2.SS1.p5.4.m4.1.1.3.cmml"><mi id="S2.SS1.p5.4.m4.1.1.3.2" xref="S2.SS1.p5.4.m4.1.1.3.2.cmml">m</mi><mo id="S2.SS1.p5.4.m4.1.1.3.1" xref="S2.SS1.p5.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p5.4.m4.1.1.3.3" xref="S2.SS1.p5.4.m4.1.1.3.3.cmml">i</mi><mo id="S2.SS1.p5.4.m4.1.1.3.1a" xref="S2.SS1.p5.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p5.4.m4.1.1.3.4" xref="S2.SS1.p5.4.m4.1.1.3.4.cmml">x</mi><mo id="S2.SS1.p5.4.m4.1.1.3.1b" xref="S2.SS1.p5.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p5.4.m4.1.1.3.5" xref="S2.SS1.p5.4.m4.1.1.3.5.cmml">e</mi><mo id="S2.SS1.p5.4.m4.1.1.3.1c" xref="S2.SS1.p5.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p5.4.m4.1.1.3.6" xref="S2.SS1.p5.4.m4.1.1.3.6.cmml">r</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.4.m4.1b"><apply id="S2.SS1.p5.4.m4.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.4.m4.1.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.p5.4.m4.1.1.2.cmml" xref="S2.SS1.p5.4.m4.1.1.2">𝑂</ci><apply id="S2.SS1.p5.4.m4.1.1.3.cmml" xref="S2.SS1.p5.4.m4.1.1.3"><times id="S2.SS1.p5.4.m4.1.1.3.1.cmml" xref="S2.SS1.p5.4.m4.1.1.3.1"></times><ci id="S2.SS1.p5.4.m4.1.1.3.2.cmml" xref="S2.SS1.p5.4.m4.1.1.3.2">𝑚</ci><ci id="S2.SS1.p5.4.m4.1.1.3.3.cmml" xref="S2.SS1.p5.4.m4.1.1.3.3">𝑖</ci><ci id="S2.SS1.p5.4.m4.1.1.3.4.cmml" xref="S2.SS1.p5.4.m4.1.1.3.4">𝑥</ci><ci id="S2.SS1.p5.4.m4.1.1.3.5.cmml" xref="S2.SS1.p5.4.m4.1.1.3.5">𝑒</ci><ci id="S2.SS1.p5.4.m4.1.1.3.6.cmml" xref="S2.SS1.p5.4.m4.1.1.3.6">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.4.m4.1c">O_{mixer}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.4.m4.1d">italic_O start_POSTSUBSCRIPT italic_m italic_i italic_x italic_e italic_r end_POSTSUBSCRIPT</annotation></semantics></math>. For the <math alttext="F_{R}" class="ltx_Math" display="inline" id="S2.SS1.p5.5.m5.1"><semantics id="S2.SS1.p5.5.m5.1a"><msub id="S2.SS1.p5.5.m5.1.1" xref="S2.SS1.p5.5.m5.1.1.cmml"><mi id="S2.SS1.p5.5.m5.1.1.2" xref="S2.SS1.p5.5.m5.1.1.2.cmml">F</mi><mi id="S2.SS1.p5.5.m5.1.1.3" xref="S2.SS1.p5.5.m5.1.1.3.cmml">R</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.5.m5.1b"><apply id="S2.SS1.p5.5.m5.1.1.cmml" xref="S2.SS1.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.5.m5.1.1.1.cmml" xref="S2.SS1.p5.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p5.5.m5.1.1.2.cmml" xref="S2.SS1.p5.5.m5.1.1.2">𝐹</ci><ci id="S2.SS1.p5.5.m5.1.1.3.cmml" xref="S2.SS1.p5.5.m5.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.5.m5.1c">F_{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.5.m5.1d">italic_F start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="O_{mixer}" class="ltx_Math" display="inline" id="S2.SS1.p5.6.m6.1"><semantics id="S2.SS1.p5.6.m6.1a"><msub id="S2.SS1.p5.6.m6.1.1" xref="S2.SS1.p5.6.m6.1.1.cmml"><mi id="S2.SS1.p5.6.m6.1.1.2" xref="S2.SS1.p5.6.m6.1.1.2.cmml">O</mi><mrow id="S2.SS1.p5.6.m6.1.1.3" xref="S2.SS1.p5.6.m6.1.1.3.cmml"><mi id="S2.SS1.p5.6.m6.1.1.3.2" xref="S2.SS1.p5.6.m6.1.1.3.2.cmml">m</mi><mo id="S2.SS1.p5.6.m6.1.1.3.1" xref="S2.SS1.p5.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p5.6.m6.1.1.3.3" xref="S2.SS1.p5.6.m6.1.1.3.3.cmml">i</mi><mo id="S2.SS1.p5.6.m6.1.1.3.1a" xref="S2.SS1.p5.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p5.6.m6.1.1.3.4" xref="S2.SS1.p5.6.m6.1.1.3.4.cmml">x</mi><mo id="S2.SS1.p5.6.m6.1.1.3.1b" xref="S2.SS1.p5.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p5.6.m6.1.1.3.5" xref="S2.SS1.p5.6.m6.1.1.3.5.cmml">e</mi><mo id="S2.SS1.p5.6.m6.1.1.3.1c" xref="S2.SS1.p5.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p5.6.m6.1.1.3.6" xref="S2.SS1.p5.6.m6.1.1.3.6.cmml">r</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.6.m6.1b"><apply id="S2.SS1.p5.6.m6.1.1.cmml" xref="S2.SS1.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.6.m6.1.1.1.cmml" xref="S2.SS1.p5.6.m6.1.1">subscript</csymbol><ci id="S2.SS1.p5.6.m6.1.1.2.cmml" xref="S2.SS1.p5.6.m6.1.1.2">𝑂</ci><apply id="S2.SS1.p5.6.m6.1.1.3.cmml" xref="S2.SS1.p5.6.m6.1.1.3"><times id="S2.SS1.p5.6.m6.1.1.3.1.cmml" xref="S2.SS1.p5.6.m6.1.1.3.1"></times><ci id="S2.SS1.p5.6.m6.1.1.3.2.cmml" xref="S2.SS1.p5.6.m6.1.1.3.2">𝑚</ci><ci id="S2.SS1.p5.6.m6.1.1.3.3.cmml" xref="S2.SS1.p5.6.m6.1.1.3.3">𝑖</ci><ci id="S2.SS1.p5.6.m6.1.1.3.4.cmml" xref="S2.SS1.p5.6.m6.1.1.3.4">𝑥</ci><ci id="S2.SS1.p5.6.m6.1.1.3.5.cmml" xref="S2.SS1.p5.6.m6.1.1.3.5">𝑒</ci><ci id="S2.SS1.p5.6.m6.1.1.3.6.cmml" xref="S2.SS1.p5.6.m6.1.1.3.6">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.6.m6.1c">O_{mixer}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.6.m6.1d">italic_O start_POSTSUBSCRIPT italic_m italic_i italic_x italic_e italic_r end_POSTSUBSCRIPT</annotation></semantics></math> preserves the global features with the spatial features of the right branch. The weighting is the same as that of the spatial self-attention module in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib15" title="">15</a>]</cite>, and the fusion process is represented as follows:</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="F=O_{mixer}\otimes F_{L}\oplus O_{mixer}\otimes F_{R}" class="ltx_Math" display="block" id="S2.E2.m1.1"><semantics id="S2.E2.m1.1a"><mrow id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml"><mi id="S2.E2.m1.1.1.2" xref="S2.E2.m1.1.1.2.cmml">F</mi><mo id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.cmml">=</mo><mrow id="S2.E2.m1.1.1.3" xref="S2.E2.m1.1.1.3.cmml"><mrow id="S2.E2.m1.1.1.3.2" xref="S2.E2.m1.1.1.3.2.cmml"><msub id="S2.E2.m1.1.1.3.2.2" xref="S2.E2.m1.1.1.3.2.2.cmml"><mi id="S2.E2.m1.1.1.3.2.2.2" xref="S2.E2.m1.1.1.3.2.2.2.cmml">O</mi><mrow id="S2.E2.m1.1.1.3.2.2.3" xref="S2.E2.m1.1.1.3.2.2.3.cmml"><mi id="S2.E2.m1.1.1.3.2.2.3.2" xref="S2.E2.m1.1.1.3.2.2.3.2.cmml">m</mi><mo id="S2.E2.m1.1.1.3.2.2.3.1" xref="S2.E2.m1.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.3.2.2.3.3" xref="S2.E2.m1.1.1.3.2.2.3.3.cmml">i</mi><mo id="S2.E2.m1.1.1.3.2.2.3.1a" xref="S2.E2.m1.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.3.2.2.3.4" xref="S2.E2.m1.1.1.3.2.2.3.4.cmml">x</mi><mo id="S2.E2.m1.1.1.3.2.2.3.1b" xref="S2.E2.m1.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.3.2.2.3.5" xref="S2.E2.m1.1.1.3.2.2.3.5.cmml">e</mi><mo id="S2.E2.m1.1.1.3.2.2.3.1c" xref="S2.E2.m1.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.3.2.2.3.6" xref="S2.E2.m1.1.1.3.2.2.3.6.cmml">r</mi></mrow></msub><mo id="S2.E2.m1.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.E2.m1.1.1.3.2.1.cmml">⊗</mo><msub id="S2.E2.m1.1.1.3.2.3" xref="S2.E2.m1.1.1.3.2.3.cmml"><mi id="S2.E2.m1.1.1.3.2.3.2" xref="S2.E2.m1.1.1.3.2.3.2.cmml">F</mi><mi id="S2.E2.m1.1.1.3.2.3.3" xref="S2.E2.m1.1.1.3.2.3.3.cmml">L</mi></msub></mrow><mo id="S2.E2.m1.1.1.3.1" xref="S2.E2.m1.1.1.3.1.cmml">⊕</mo><mrow id="S2.E2.m1.1.1.3.3" xref="S2.E2.m1.1.1.3.3.cmml"><msub id="S2.E2.m1.1.1.3.3.2" xref="S2.E2.m1.1.1.3.3.2.cmml"><mi id="S2.E2.m1.1.1.3.3.2.2" xref="S2.E2.m1.1.1.3.3.2.2.cmml">O</mi><mrow id="S2.E2.m1.1.1.3.3.2.3" xref="S2.E2.m1.1.1.3.3.2.3.cmml"><mi id="S2.E2.m1.1.1.3.3.2.3.2" xref="S2.E2.m1.1.1.3.3.2.3.2.cmml">m</mi><mo id="S2.E2.m1.1.1.3.3.2.3.1" xref="S2.E2.m1.1.1.3.3.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.3.3.2.3.3" xref="S2.E2.m1.1.1.3.3.2.3.3.cmml">i</mi><mo id="S2.E2.m1.1.1.3.3.2.3.1a" xref="S2.E2.m1.1.1.3.3.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.3.3.2.3.4" xref="S2.E2.m1.1.1.3.3.2.3.4.cmml">x</mi><mo id="S2.E2.m1.1.1.3.3.2.3.1b" xref="S2.E2.m1.1.1.3.3.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.3.3.2.3.5" xref="S2.E2.m1.1.1.3.3.2.3.5.cmml">e</mi><mo id="S2.E2.m1.1.1.3.3.2.3.1c" xref="S2.E2.m1.1.1.3.3.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.3.3.2.3.6" xref="S2.E2.m1.1.1.3.3.2.3.6.cmml">r</mi></mrow></msub><mo id="S2.E2.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.E2.m1.1.1.3.3.1.cmml">⊗</mo><msub id="S2.E2.m1.1.1.3.3.3" xref="S2.E2.m1.1.1.3.3.3.cmml"><mi id="S2.E2.m1.1.1.3.3.3.2" xref="S2.E2.m1.1.1.3.3.3.2.cmml">F</mi><mi id="S2.E2.m1.1.1.3.3.3.3" xref="S2.E2.m1.1.1.3.3.3.3.cmml">R</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.m1.1b"><apply id="S2.E2.m1.1.1.cmml" xref="S2.E2.m1.1.1"><eq id="S2.E2.m1.1.1.1.cmml" xref="S2.E2.m1.1.1.1"></eq><ci id="S2.E2.m1.1.1.2.cmml" xref="S2.E2.m1.1.1.2">𝐹</ci><apply id="S2.E2.m1.1.1.3.cmml" xref="S2.E2.m1.1.1.3"><csymbol cd="latexml" id="S2.E2.m1.1.1.3.1.cmml" xref="S2.E2.m1.1.1.3.1">direct-sum</csymbol><apply id="S2.E2.m1.1.1.3.2.cmml" xref="S2.E2.m1.1.1.3.2"><csymbol cd="latexml" id="S2.E2.m1.1.1.3.2.1.cmml" xref="S2.E2.m1.1.1.3.2.1">tensor-product</csymbol><apply 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id="S2.E2.m1.1.1.3.2.3.3.cmml" xref="S2.E2.m1.1.1.3.2.3.3">𝐿</ci></apply></apply><apply id="S2.E2.m1.1.1.3.3.cmml" xref="S2.E2.m1.1.1.3.3"><csymbol cd="latexml" id="S2.E2.m1.1.1.3.3.1.cmml" xref="S2.E2.m1.1.1.3.3.1">tensor-product</csymbol><apply id="S2.E2.m1.1.1.3.3.2.cmml" xref="S2.E2.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.E2.m1.1.1.3.3.2.1.cmml" xref="S2.E2.m1.1.1.3.3.2">subscript</csymbol><ci id="S2.E2.m1.1.1.3.3.2.2.cmml" xref="S2.E2.m1.1.1.3.3.2.2">𝑂</ci><apply id="S2.E2.m1.1.1.3.3.2.3.cmml" xref="S2.E2.m1.1.1.3.3.2.3"><times id="S2.E2.m1.1.1.3.3.2.3.1.cmml" xref="S2.E2.m1.1.1.3.3.2.3.1"></times><ci id="S2.E2.m1.1.1.3.3.2.3.2.cmml" xref="S2.E2.m1.1.1.3.3.2.3.2">𝑚</ci><ci id="S2.E2.m1.1.1.3.3.2.3.3.cmml" xref="S2.E2.m1.1.1.3.3.2.3.3">𝑖</ci><ci id="S2.E2.m1.1.1.3.3.2.3.4.cmml" xref="S2.E2.m1.1.1.3.3.2.3.4">𝑥</ci><ci id="S2.E2.m1.1.1.3.3.2.3.5.cmml" xref="S2.E2.m1.1.1.3.3.2.3.5">𝑒</ci><ci id="S2.E2.m1.1.1.3.3.2.3.6.cmml" xref="S2.E2.m1.1.1.3.3.2.3.6">𝑟</ci></apply></apply><apply id="S2.E2.m1.1.1.3.3.3.cmml" xref="S2.E2.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.E2.m1.1.1.3.3.3.1.cmml" xref="S2.E2.m1.1.1.3.3.3">subscript</csymbol><ci id="S2.E2.m1.1.1.3.3.3.2.cmml" xref="S2.E2.m1.1.1.3.3.3.2">𝐹</ci><ci id="S2.E2.m1.1.1.3.3.3.3.cmml" xref="S2.E2.m1.1.1.3.3.3.3">𝑅</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.1c">F=O_{mixer}\otimes F_{L}\oplus O_{mixer}\otimes F_{R}</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.1d">italic_F = italic_O start_POSTSUBSCRIPT italic_m italic_i italic_x italic_e italic_r end_POSTSUBSCRIPT ⊗ italic_F start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ⊕ italic_O start_POSTSUBSCRIPT italic_m italic_i italic_x italic_e italic_r end_POSTSUBSCRIPT ⊗ italic_F 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">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p5.9">where <math alttext="O_{mixer},F_{L},F_{R}\in{\mathbb{R}}^{C\times\dfrac{h}{2}\times\dfrac{w}{2}}" class="ltx_Math" display="inline" id="S2.SS1.p5.7.m1.3"><semantics id="S2.SS1.p5.7.m1.3a"><mrow id="S2.SS1.p5.7.m1.3.3" xref="S2.SS1.p5.7.m1.3.3.cmml"><mrow id="S2.SS1.p5.7.m1.3.3.3.3" xref="S2.SS1.p5.7.m1.3.3.3.4.cmml"><msub id="S2.SS1.p5.7.m1.1.1.1.1.1" xref="S2.SS1.p5.7.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p5.7.m1.1.1.1.1.1.2" xref="S2.SS1.p5.7.m1.1.1.1.1.1.2.cmml">O</mi><mrow id="S2.SS1.p5.7.m1.1.1.1.1.1.3" xref="S2.SS1.p5.7.m1.1.1.1.1.1.3.cmml"><mi id="S2.SS1.p5.7.m1.1.1.1.1.1.3.2" xref="S2.SS1.p5.7.m1.1.1.1.1.1.3.2.cmml">m</mi><mo id="S2.SS1.p5.7.m1.1.1.1.1.1.3.1" xref="S2.SS1.p5.7.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p5.7.m1.1.1.1.1.1.3.3" 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italic_x italic_e italic_r end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_C × divide start_ARG italic_h end_ARG start_ARG 2 end_ARG × divide start_ARG italic_w end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\otimes" class="ltx_Math" display="inline" id="S2.SS1.p5.8.m2.1"><semantics id="S2.SS1.p5.8.m2.1a"><mo id="S2.SS1.p5.8.m2.1.1" xref="S2.SS1.p5.8.m2.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.8.m2.1b"><csymbol cd="latexml" id="S2.SS1.p5.8.m2.1.1.cmml" xref="S2.SS1.p5.8.m2.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.8.m2.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.8.m2.1d">⊗</annotation></semantics></math> and <math alttext="\oplus" class="ltx_Math" display="inline" id="S2.SS1.p5.9.m3.1"><semantics id="S2.SS1.p5.9.m3.1a"><mo id="S2.SS1.p5.9.m3.1.1" xref="S2.SS1.p5.9.m3.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.9.m3.1b"><csymbol cd="latexml" id="S2.SS1.p5.9.m3.1.1.cmml" xref="S2.SS1.p5.9.m3.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.9.m3.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.9.m3.1d">⊕</annotation></semantics></math> are element-wise operation</p> </div> <div class="ltx_para" id="S2.SS1.p6"> <p class="ltx_p" id="S2.SS1.p6.1">MgCSD replaces and enhances the traditional downsampling module and can be used as a direct replacement in the encoding phase. In particular, note that since the global weighted branch targets feature rather than the original image, the simple convolution of the first layer is retained when used to map the original image into the feature space.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS2.5.1.1">II-B</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS2.6.2">Contour Probabilistic Modeling</span> </h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">Due to the individual variability in the presentation of a lesion’s contour, when the contour’s local edge lines are unclear, downsampling in relation to the global can be supplemented with contour information from other regions. When the shape of the contour is irregular, supplementing local edge lines requires consideration of global contour variations. We use probabilistic modeling to fit the generalized contour features, representing the shape of the contour, the edge line, the degree of blurring, and other information. In contrast, the Gaussian mixture distribution can be fitted to any distribution. So, the probability density function of the hybrid Gaussian distribution can be expressed as several Gaussian distributions of the linear weighting of the weights and the sum of the values of one.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.1">Taking advantage of the Gaussian mixture distribution, we assume that the K sets of Gaussian distribution features of the contour are learned from the original image X, denoted as <math alttext="\mu_{A}(\textit{X};\theta),\sigma_{A}(\textit{X};\theta)" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.6"><semantics id="S2.SS2.p2.1.m1.6a"><mrow id="S2.SS2.p2.1.m1.6.6.2" xref="S2.SS2.p2.1.m1.6.6.3.cmml"><mrow id="S2.SS2.p2.1.m1.5.5.1.1" xref="S2.SS2.p2.1.m1.5.5.1.1.cmml"><msub id="S2.SS2.p2.1.m1.5.5.1.1.2" xref="S2.SS2.p2.1.m1.5.5.1.1.2.cmml"><mi id="S2.SS2.p2.1.m1.5.5.1.1.2.2" xref="S2.SS2.p2.1.m1.5.5.1.1.2.2.cmml">μ</mi><mi id="S2.SS2.p2.1.m1.5.5.1.1.2.3" xref="S2.SS2.p2.1.m1.5.5.1.1.2.3.cmml">A</mi></msub><mo id="S2.SS2.p2.1.m1.5.5.1.1.1" xref="S2.SS2.p2.1.m1.5.5.1.1.1.cmml">⁢</mo><mrow id="S2.SS2.p2.1.m1.5.5.1.1.3.2" xref="S2.SS2.p2.1.m1.5.5.1.1.3.1.cmml"><mo id="S2.SS2.p2.1.m1.5.5.1.1.3.2.1" stretchy="false" xref="S2.SS2.p2.1.m1.5.5.1.1.3.1.cmml">(</mo><mtext class="ltx_mathvariant_italic" id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1a.cmml">X</mtext><mo id="S2.SS2.p2.1.m1.5.5.1.1.3.2.2" xref="S2.SS2.p2.1.m1.5.5.1.1.3.1.cmml">;</mo><mi id="S2.SS2.p2.1.m1.2.2" xref="S2.SS2.p2.1.m1.2.2.cmml">θ</mi><mo id="S2.SS2.p2.1.m1.5.5.1.1.3.2.3" stretchy="false" xref="S2.SS2.p2.1.m1.5.5.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.1.m1.6.6.2.3" xref="S2.SS2.p2.1.m1.6.6.3.cmml">,</mo><mrow id="S2.SS2.p2.1.m1.6.6.2.2" xref="S2.SS2.p2.1.m1.6.6.2.2.cmml"><msub id="S2.SS2.p2.1.m1.6.6.2.2.2" xref="S2.SS2.p2.1.m1.6.6.2.2.2.cmml"><mi id="S2.SS2.p2.1.m1.6.6.2.2.2.2" xref="S2.SS2.p2.1.m1.6.6.2.2.2.2.cmml">σ</mi><mi id="S2.SS2.p2.1.m1.6.6.2.2.2.3" xref="S2.SS2.p2.1.m1.6.6.2.2.2.3.cmml">A</mi></msub><mo id="S2.SS2.p2.1.m1.6.6.2.2.1" xref="S2.SS2.p2.1.m1.6.6.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.1.m1.6.6.2.2.3.2" xref="S2.SS2.p2.1.m1.6.6.2.2.3.1.cmml"><mo id="S2.SS2.p2.1.m1.6.6.2.2.3.2.1" stretchy="false" xref="S2.SS2.p2.1.m1.6.6.2.2.3.1.cmml">(</mo><mtext class="ltx_mathvariant_italic" id="S2.SS2.p2.1.m1.3.3" xref="S2.SS2.p2.1.m1.3.3a.cmml">X</mtext><mo id="S2.SS2.p2.1.m1.6.6.2.2.3.2.2" xref="S2.SS2.p2.1.m1.6.6.2.2.3.1.cmml">;</mo><mi id="S2.SS2.p2.1.m1.4.4" xref="S2.SS2.p2.1.m1.4.4.cmml">θ</mi><mo id="S2.SS2.p2.1.m1.6.6.2.2.3.2.3" stretchy="false" xref="S2.SS2.p2.1.m1.6.6.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.6b"><list id="S2.SS2.p2.1.m1.6.6.3.cmml" xref="S2.SS2.p2.1.m1.6.6.2"><apply id="S2.SS2.p2.1.m1.5.5.1.1.cmml" xref="S2.SS2.p2.1.m1.5.5.1.1"><times id="S2.SS2.p2.1.m1.5.5.1.1.1.cmml" xref="S2.SS2.p2.1.m1.5.5.1.1.1"></times><apply id="S2.SS2.p2.1.m1.5.5.1.1.2.cmml" xref="S2.SS2.p2.1.m1.5.5.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.5.5.1.1.2.1.cmml" xref="S2.SS2.p2.1.m1.5.5.1.1.2">subscript</csymbol><ci id="S2.SS2.p2.1.m1.5.5.1.1.2.2.cmml" xref="S2.SS2.p2.1.m1.5.5.1.1.2.2">𝜇</ci><ci id="S2.SS2.p2.1.m1.5.5.1.1.2.3.cmml" xref="S2.SS2.p2.1.m1.5.5.1.1.2.3">𝐴</ci></apply><list id="S2.SS2.p2.1.m1.5.5.1.1.3.1.cmml" xref="S2.SS2.p2.1.m1.5.5.1.1.3.2"><ci id="S2.SS2.p2.1.m1.1.1a.cmml" xref="S2.SS2.p2.1.m1.1.1"><mtext class="ltx_mathvariant_italic" id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">X</mtext></ci><ci id="S2.SS2.p2.1.m1.2.2.cmml" xref="S2.SS2.p2.1.m1.2.2">𝜃</ci></list></apply><apply id="S2.SS2.p2.1.m1.6.6.2.2.cmml" xref="S2.SS2.p2.1.m1.6.6.2.2"><times id="S2.SS2.p2.1.m1.6.6.2.2.1.cmml" xref="S2.SS2.p2.1.m1.6.6.2.2.1"></times><apply id="S2.SS2.p2.1.m1.6.6.2.2.2.cmml" xref="S2.SS2.p2.1.m1.6.6.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.6.6.2.2.2.1.cmml" xref="S2.SS2.p2.1.m1.6.6.2.2.2">subscript</csymbol><ci id="S2.SS2.p2.1.m1.6.6.2.2.2.2.cmml" xref="S2.SS2.p2.1.m1.6.6.2.2.2.2">𝜎</ci><ci id="S2.SS2.p2.1.m1.6.6.2.2.2.3.cmml" xref="S2.SS2.p2.1.m1.6.6.2.2.2.3">𝐴</ci></apply><list id="S2.SS2.p2.1.m1.6.6.2.2.3.1.cmml" xref="S2.SS2.p2.1.m1.6.6.2.2.3.2"><ci id="S2.SS2.p2.1.m1.3.3a.cmml" xref="S2.SS2.p2.1.m1.3.3"><mtext class="ltx_mathvariant_italic" id="S2.SS2.p2.1.m1.3.3.cmml" xref="S2.SS2.p2.1.m1.3.3">X</mtext></ci><ci id="S2.SS2.p2.1.m1.4.4.cmml" xref="S2.SS2.p2.1.m1.4.4">𝜃</ci></list></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.6c">\mu_{A}(\textit{X};\theta),\sigma_{A}(\textit{X};\theta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.6d">italic_μ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( X ; 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italic_σ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( X ; italic_θ ) = { italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( X ; italic_θ ) , italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( X ; italic_θ ) } start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.1">To fuse the obtained contour features into the decoding stage, we use the reparameterization trick <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib16" title="">16</a>]</cite> to sample the obtained K sets of Gaussian distributions. As shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S1.F2" title="Figure 2 ‣ I Introduction ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">2</span></a>(a), the sampling result is weighted and summed with the adaptive learnable weights <math alttext="\Omega" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><mi id="S2.SS2.p4.1.m1.1.1" mathvariant="normal" xref="S2.SS2.p4.1.m1.1.1.cmml">Ω</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><ci id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1">Ω</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">\Omega</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">roman_Ω</annotation></semantics></math> to achieve the purpose of sampling from the mixed Gaussian distribution. The whole process is called adaptive homoscedastic resampling and is expressed as follows:</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="G=\Omega(z\cdot\sigma_{A}+\mu_{A})" class="ltx_Math" display="block" id="S2.E3.m1.1"><semantics id="S2.E3.m1.1a"><mrow id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml"><mi id="S2.E3.m1.1.1.3" xref="S2.E3.m1.1.1.3.cmml">G</mi><mo id="S2.E3.m1.1.1.2" xref="S2.E3.m1.1.1.2.cmml">=</mo><mrow id="S2.E3.m1.1.1.1" xref="S2.E3.m1.1.1.1.cmml"><mi id="S2.E3.m1.1.1.1.3" mathvariant="normal" xref="S2.E3.m1.1.1.1.3.cmml">Ω</mi><mo id="S2.E3.m1.1.1.1.2" xref="S2.E3.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.1.1.cmml"><mo id="S2.E3.m1.1.1.1.1.1.2" stretchy="false" xref="S2.E3.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E3.m1.1.1.1.1.1.1" 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id="S2.E3.m1.1b"><apply id="S2.E3.m1.1.1.cmml" xref="S2.E3.m1.1.1"><eq id="S2.E3.m1.1.1.2.cmml" xref="S2.E3.m1.1.1.2"></eq><ci id="S2.E3.m1.1.1.3.cmml" xref="S2.E3.m1.1.1.3">𝐺</ci><apply id="S2.E3.m1.1.1.1.cmml" xref="S2.E3.m1.1.1.1"><times id="S2.E3.m1.1.1.1.2.cmml" xref="S2.E3.m1.1.1.1.2"></times><ci id="S2.E3.m1.1.1.1.3.cmml" xref="S2.E3.m1.1.1.1.3">Ω</ci><apply id="S2.E3.m1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.1.1.1.1.1"><plus id="S2.E3.m1.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.1.1.1.1.1.1.1"></plus><apply id="S2.E3.m1.1.1.1.1.1.1.2.cmml" xref="S2.E3.m1.1.1.1.1.1.1.2"><ci id="S2.E3.m1.1.1.1.1.1.1.2.1.cmml" xref="S2.E3.m1.1.1.1.1.1.1.2.1">⋅</ci><ci id="S2.E3.m1.1.1.1.1.1.1.2.2.cmml" xref="S2.E3.m1.1.1.1.1.1.1.2.2">𝑧</ci><apply id="S2.E3.m1.1.1.1.1.1.1.2.3.cmml" xref="S2.E3.m1.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.1.1.1.2.3.1.cmml" xref="S2.E3.m1.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="S2.E3.m1.1.1.1.1.1.1.2.3.2.cmml" xref="S2.E3.m1.1.1.1.1.1.1.2.3.2">𝜎</ci><ci id="S2.E3.m1.1.1.1.1.1.1.2.3.3.cmml" xref="S2.E3.m1.1.1.1.1.1.1.2.3.3">𝐴</ci></apply></apply><apply id="S2.E3.m1.1.1.1.1.1.1.3.cmml" xref="S2.E3.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.1.1.1.3.1.cmml" xref="S2.E3.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.E3.m1.1.1.1.1.1.1.3.2.cmml" xref="S2.E3.m1.1.1.1.1.1.1.3.2">𝜇</ci><ci id="S2.E3.m1.1.1.1.1.1.1.3.3.cmml" xref="S2.E3.m1.1.1.1.1.1.1.3.3">𝐴</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.1c">G=\Omega(z\cdot\sigma_{A}+\mu_{A})</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.1d">italic_G = roman_Ω ( italic_z ⋅ italic_σ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT + italic_μ start_POSTSUBSCRIPT italic_A 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.SS2.p4.4">where <math alttext="\Omega\in{\mathcal{R}}^{T\times K}" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m1.1"><semantics id="S2.SS2.p4.2.m1.1a"><mrow id="S2.SS2.p4.2.m1.1.1" xref="S2.SS2.p4.2.m1.1.1.cmml"><mi id="S2.SS2.p4.2.m1.1.1.2" mathvariant="normal" xref="S2.SS2.p4.2.m1.1.1.2.cmml">Ω</mi><mo id="S2.SS2.p4.2.m1.1.1.1" xref="S2.SS2.p4.2.m1.1.1.1.cmml">∈</mo><msup id="S2.SS2.p4.2.m1.1.1.3" xref="S2.SS2.p4.2.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.2.m1.1.1.3.2" xref="S2.SS2.p4.2.m1.1.1.3.2.cmml">ℛ</mi><mrow id="S2.SS2.p4.2.m1.1.1.3.3" xref="S2.SS2.p4.2.m1.1.1.3.3.cmml"><mi id="S2.SS2.p4.2.m1.1.1.3.3.2" xref="S2.SS2.p4.2.m1.1.1.3.3.2.cmml">T</mi><mo id="S2.SS2.p4.2.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p4.2.m1.1.1.3.3.1.cmml">×</mo><mi id="S2.SS2.p4.2.m1.1.1.3.3.3" xref="S2.SS2.p4.2.m1.1.1.3.3.3.cmml">K</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m1.1b"><apply id="S2.SS2.p4.2.m1.1.1.cmml" xref="S2.SS2.p4.2.m1.1.1"><in id="S2.SS2.p4.2.m1.1.1.1.cmml" xref="S2.SS2.p4.2.m1.1.1.1"></in><ci id="S2.SS2.p4.2.m1.1.1.2.cmml" xref="S2.SS2.p4.2.m1.1.1.2">Ω</ci><apply id="S2.SS2.p4.2.m1.1.1.3.cmml" xref="S2.SS2.p4.2.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.2.m1.1.1.3.1.cmml" xref="S2.SS2.p4.2.m1.1.1.3">superscript</csymbol><ci id="S2.SS2.p4.2.m1.1.1.3.2.cmml" xref="S2.SS2.p4.2.m1.1.1.3.2">ℛ</ci><apply id="S2.SS2.p4.2.m1.1.1.3.3.cmml" xref="S2.SS2.p4.2.m1.1.1.3.3"><times id="S2.SS2.p4.2.m1.1.1.3.3.1.cmml" xref="S2.SS2.p4.2.m1.1.1.3.3.1"></times><ci id="S2.SS2.p4.2.m1.1.1.3.3.2.cmml" xref="S2.SS2.p4.2.m1.1.1.3.3.2">𝑇</ci><ci id="S2.SS2.p4.2.m1.1.1.3.3.3.cmml" xref="S2.SS2.p4.2.m1.1.1.3.3.3">𝐾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m1.1c">\Omega\in{\mathcal{R}}^{T\times K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m1.1d">roman_Ω ∈ caligraphic_R start_POSTSUPERSCRIPT italic_T × italic_K end_POSTSUPERSCRIPT</annotation></semantics></math>,<math alttext="T" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m2.1"><semantics id="S2.SS2.p4.3.m2.1a"><mi id="S2.SS2.p4.3.m2.1.1" xref="S2.SS2.p4.3.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m2.1b"><ci id="S2.SS2.p4.3.m2.1.1.cmml" xref="S2.SS2.p4.3.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m2.1d">italic_T</annotation></semantics></math> corresponds to the number of eigen channels in the corresponding stage. z is sampled from the standard normal distribution <math alttext="\mathcal{N}(0,1)" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m3.2"><semantics id="S2.SS2.p4.4.m3.2a"><mrow id="S2.SS2.p4.4.m3.2.3" xref="S2.SS2.p4.4.m3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.4.m3.2.3.2" xref="S2.SS2.p4.4.m3.2.3.2.cmml">𝒩</mi><mo id="S2.SS2.p4.4.m3.2.3.1" xref="S2.SS2.p4.4.m3.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p4.4.m3.2.3.3.2" xref="S2.SS2.p4.4.m3.2.3.3.1.cmml"><mo id="S2.SS2.p4.4.m3.2.3.3.2.1" stretchy="false" xref="S2.SS2.p4.4.m3.2.3.3.1.cmml">(</mo><mn id="S2.SS2.p4.4.m3.1.1" xref="S2.SS2.p4.4.m3.1.1.cmml">0</mn><mo id="S2.SS2.p4.4.m3.2.3.3.2.2" xref="S2.SS2.p4.4.m3.2.3.3.1.cmml">,</mo><mn id="S2.SS2.p4.4.m3.2.2" xref="S2.SS2.p4.4.m3.2.2.cmml">1</mn><mo id="S2.SS2.p4.4.m3.2.3.3.2.3" stretchy="false" xref="S2.SS2.p4.4.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m3.2b"><apply id="S2.SS2.p4.4.m3.2.3.cmml" xref="S2.SS2.p4.4.m3.2.3"><times id="S2.SS2.p4.4.m3.2.3.1.cmml" xref="S2.SS2.p4.4.m3.2.3.1"></times><ci id="S2.SS2.p4.4.m3.2.3.2.cmml" xref="S2.SS2.p4.4.m3.2.3.2">𝒩</ci><interval closure="open" id="S2.SS2.p4.4.m3.2.3.3.1.cmml" xref="S2.SS2.p4.4.m3.2.3.3.2"><cn id="S2.SS2.p4.4.m3.1.1.cmml" type="integer" xref="S2.SS2.p4.4.m3.1.1">0</cn><cn id="S2.SS2.p4.4.m3.2.2.cmml" type="integer" xref="S2.SS2.p4.4.m3.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m3.2c">\mathcal{N}(0,1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m3.2d">caligraphic_N ( 0 , 1 )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.1">Since the image contour is not only a closed geometric image, the pixel features around the contour line are also crucial for constructing the contour distribution, so the result from the mask-processed image Y mapped by the same feature extractor as CPM can be used, which is expressed as <math alttext="\mu_{B}(\textit{Y};\theta^{\prime}),\ sigma_{B}(\textit{Y};\theta^{\prime})={% \left\{\mu_{i}(\textit{Y};\theta^{\prime}),\sigma_{i}(\textit{Y};\theta^{% \prime})\right\}}_{i=0}^{K}" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.8"><semantics id="S2.SS2.p5.1.m1.8a"><mrow id="S2.SS2.p5.1.m1.8.8" xref="S2.SS2.p5.1.m1.8.8.cmml"><mrow id="S2.SS2.p5.1.m1.6.6.2.2" xref="S2.SS2.p5.1.m1.6.6.2.3.cmml"><mrow id="S2.SS2.p5.1.m1.5.5.1.1.1" xref="S2.SS2.p5.1.m1.5.5.1.1.1.cmml"><msub id="S2.SS2.p5.1.m1.5.5.1.1.1.3" xref="S2.SS2.p5.1.m1.5.5.1.1.1.3.cmml"><mi id="S2.SS2.p5.1.m1.5.5.1.1.1.3.2" xref="S2.SS2.p5.1.m1.5.5.1.1.1.3.2.cmml">μ</mi><mi id="S2.SS2.p5.1.m1.5.5.1.1.1.3.3" xref="S2.SS2.p5.1.m1.5.5.1.1.1.3.3.cmml">B</mi></msub><mo id="S2.SS2.p5.1.m1.5.5.1.1.1.2" xref="S2.SS2.p5.1.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p5.1.m1.5.5.1.1.1.1.1" xref="S2.SS2.p5.1.m1.5.5.1.1.1.1.2.cmml"><mo id="S2.SS2.p5.1.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p5.1.m1.5.5.1.1.1.1.2.cmml">(</mo><mtext class="ltx_mathvariant_italic" 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end_POSTSUPERSCRIPT ) , italic_s italic_i italic_g italic_m italic_a start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( Y ; italic_θ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = { italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( Y ; italic_θ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( Y ; italic_θ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) } start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.p6"> <p class="ltx_p" id="S2.SS2.p6.2">To constrain the contour distribution to fit the actual distribution of the contour and focus on the critical area <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib17" title="">17</a>]</cite>, the KL dispersion will align <math alttext="\mu_{A},\sigma_{A}" class="ltx_Math" display="inline" id="S2.SS2.p6.1.m1.2"><semantics id="S2.SS2.p6.1.m1.2a"><mrow id="S2.SS2.p6.1.m1.2.2.2" xref="S2.SS2.p6.1.m1.2.2.3.cmml"><msub id="S2.SS2.p6.1.m1.1.1.1.1" xref="S2.SS2.p6.1.m1.1.1.1.1.cmml"><mi id="S2.SS2.p6.1.m1.1.1.1.1.2" xref="S2.SS2.p6.1.m1.1.1.1.1.2.cmml">μ</mi><mi id="S2.SS2.p6.1.m1.1.1.1.1.3" xref="S2.SS2.p6.1.m1.1.1.1.1.3.cmml">A</mi></msub><mo id="S2.SS2.p6.1.m1.2.2.2.3" xref="S2.SS2.p6.1.m1.2.2.3.cmml">,</mo><msub id="S2.SS2.p6.1.m1.2.2.2.2" xref="S2.SS2.p6.1.m1.2.2.2.2.cmml"><mi id="S2.SS2.p6.1.m1.2.2.2.2.2" xref="S2.SS2.p6.1.m1.2.2.2.2.2.cmml">σ</mi><mi id="S2.SS2.p6.1.m1.2.2.2.2.3" xref="S2.SS2.p6.1.m1.2.2.2.2.3.cmml">A</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.1.m1.2b"><list id="S2.SS2.p6.1.m1.2.2.3.cmml" xref="S2.SS2.p6.1.m1.2.2.2"><apply id="S2.SS2.p6.1.m1.1.1.1.1.cmml" xref="S2.SS2.p6.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p6.1.m1.1.1.1.1.1.cmml" xref="S2.SS2.p6.1.m1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p6.1.m1.1.1.1.1.2.cmml" xref="S2.SS2.p6.1.m1.1.1.1.1.2">𝜇</ci><ci id="S2.SS2.p6.1.m1.1.1.1.1.3.cmml" xref="S2.SS2.p6.1.m1.1.1.1.1.3">𝐴</ci></apply><apply id="S2.SS2.p6.1.m1.2.2.2.2.cmml" xref="S2.SS2.p6.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p6.1.m1.2.2.2.2.1.cmml" xref="S2.SS2.p6.1.m1.2.2.2.2">subscript</csymbol><ci id="S2.SS2.p6.1.m1.2.2.2.2.2.cmml" xref="S2.SS2.p6.1.m1.2.2.2.2.2">𝜎</ci><ci id="S2.SS2.p6.1.m1.2.2.2.2.3.cmml" xref="S2.SS2.p6.1.m1.2.2.2.2.3">𝐴</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.1.m1.2c">\mu_{A},\sigma_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.1.m1.2d">italic_μ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> toward <math alttext="\mu_{B},\sigma_{B}" class="ltx_Math" display="inline" id="S2.SS2.p6.2.m2.2"><semantics id="S2.SS2.p6.2.m2.2a"><mrow id="S2.SS2.p6.2.m2.2.2.2" xref="S2.SS2.p6.2.m2.2.2.3.cmml"><msub id="S2.SS2.p6.2.m2.1.1.1.1" xref="S2.SS2.p6.2.m2.1.1.1.1.cmml"><mi id="S2.SS2.p6.2.m2.1.1.1.1.2" xref="S2.SS2.p6.2.m2.1.1.1.1.2.cmml">μ</mi><mi id="S2.SS2.p6.2.m2.1.1.1.1.3" xref="S2.SS2.p6.2.m2.1.1.1.1.3.cmml">B</mi></msub><mo id="S2.SS2.p6.2.m2.2.2.2.3" xref="S2.SS2.p6.2.m2.2.2.3.cmml">,</mo><msub id="S2.SS2.p6.2.m2.2.2.2.2" xref="S2.SS2.p6.2.m2.2.2.2.2.cmml"><mi id="S2.SS2.p6.2.m2.2.2.2.2.2" xref="S2.SS2.p6.2.m2.2.2.2.2.2.cmml">σ</mi><mi id="S2.SS2.p6.2.m2.2.2.2.2.3" xref="S2.SS2.p6.2.m2.2.2.2.2.3.cmml">B</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.2.m2.2b"><list id="S2.SS2.p6.2.m2.2.2.3.cmml" xref="S2.SS2.p6.2.m2.2.2.2"><apply id="S2.SS2.p6.2.m2.1.1.1.1.cmml" xref="S2.SS2.p6.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p6.2.m2.1.1.1.1.1.cmml" xref="S2.SS2.p6.2.m2.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p6.2.m2.1.1.1.1.2.cmml" xref="S2.SS2.p6.2.m2.1.1.1.1.2">𝜇</ci><ci id="S2.SS2.p6.2.m2.1.1.1.1.3.cmml" xref="S2.SS2.p6.2.m2.1.1.1.1.3">𝐵</ci></apply><apply id="S2.SS2.p6.2.m2.2.2.2.2.cmml" xref="S2.SS2.p6.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p6.2.m2.2.2.2.2.1.cmml" xref="S2.SS2.p6.2.m2.2.2.2.2">subscript</csymbol><ci id="S2.SS2.p6.2.m2.2.2.2.2.2.cmml" xref="S2.SS2.p6.2.m2.2.2.2.2.2">𝜎</ci><ci id="S2.SS2.p6.2.m2.2.2.2.2.3.cmml" xref="S2.SS2.p6.2.m2.2.2.2.2.3">𝐵</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.2.m2.2c">\mu_{B},\sigma_{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.2.m2.2d">italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math>, which is denoted as:</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 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xref="S2.E4.m1.1.2.3.3"><divide id="S2.E4.m1.1.2.3.3.1.cmml" xref="S2.E4.m1.1.2.3.3"></divide><cn id="S2.E4.m1.1.2.3.3.2.cmml" type="integer" xref="S2.E4.m1.1.2.3.3.2">1</cn><cn id="S2.E4.m1.1.2.3.3.3.cmml" type="integer" xref="S2.E4.m1.1.2.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.1c">KL_{AB}=\log{\dfrac{\sigma_{B}}{\sigma_{A}}}+\dfrac{\sigma_{A}^{2}+(\mu_{A}-% \mu_{B})^{2}}{2\sigma_{B}^{2}}-\dfrac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.1d">italic_K italic_L start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = roman_log divide start_ARG italic_σ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_ARG start_ARG italic_σ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_σ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_μ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT - italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p6.4">where <math alttext="\mu_{A},\sigma_{A}" class="ltx_Math" display="inline" id="S2.SS2.p6.3.m1.2"><semantics id="S2.SS2.p6.3.m1.2a"><mrow id="S2.SS2.p6.3.m1.2.2.2" xref="S2.SS2.p6.3.m1.2.2.3.cmml"><msub id="S2.SS2.p6.3.m1.1.1.1.1" xref="S2.SS2.p6.3.m1.1.1.1.1.cmml"><mi id="S2.SS2.p6.3.m1.1.1.1.1.2" xref="S2.SS2.p6.3.m1.1.1.1.1.2.cmml">μ</mi><mi id="S2.SS2.p6.3.m1.1.1.1.1.3" xref="S2.SS2.p6.3.m1.1.1.1.1.3.cmml">A</mi></msub><mo 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xref="S2.SS2.p6.3.m1.2.2.2.2.2">𝜎</ci><ci id="S2.SS2.p6.3.m1.2.2.2.2.3.cmml" xref="S2.SS2.p6.3.m1.2.2.2.2.3">𝐴</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.3.m1.2c">\mu_{A},\sigma_{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.3.m1.2d">italic_μ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mu_{B},\sigma_{B}" class="ltx_Math" display="inline" id="S2.SS2.p6.4.m2.2"><semantics id="S2.SS2.p6.4.m2.2a"><mrow id="S2.SS2.p6.4.m2.2.2.2" xref="S2.SS2.p6.4.m2.2.2.3.cmml"><msub id="S2.SS2.p6.4.m2.1.1.1.1" xref="S2.SS2.p6.4.m2.1.1.1.1.cmml"><mi id="S2.SS2.p6.4.m2.1.1.1.1.2" xref="S2.SS2.p6.4.m2.1.1.1.1.2.cmml">μ</mi><mi id="S2.SS2.p6.4.m2.1.1.1.1.3" xref="S2.SS2.p6.4.m2.1.1.1.1.3.cmml">B</mi></msub><mo id="S2.SS2.p6.4.m2.2.2.2.3" xref="S2.SS2.p6.4.m2.2.2.3.cmml">,</mo><msub id="S2.SS2.p6.4.m2.2.2.2.2" 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xref="S2.SS2.p6.4.m2.2.2.2.2.3">𝐵</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.4.m2.2c">\mu_{B},\sigma_{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.4.m2.2d">italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math>, are both composed of K sets of Gaussian distributions, each of which is a pair of combinations of mean-variance values.</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS3.5.1.1">II-C</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS3.6.2">Gating-based feature filtering module</span> </h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">As shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S1.F2" title="Figure 2 ‣ I Introduction ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">2</span></a>(c), to fuse the multiple data inputs sampled on each stage, a gating mechanism filters the input features <math alttext="F\in{\mathbb{R}}^{n\times h\times w}," class="ltx_Math" display="inline" id="S2.SS3.p1.1.m1.1"><semantics id="S2.SS3.p1.1.m1.1a"><mrow id="S2.SS3.p1.1.m1.1.1.1" xref="S2.SS3.p1.1.m1.1.1.1.1.cmml"><mrow id="S2.SS3.p1.1.m1.1.1.1.1" xref="S2.SS3.p1.1.m1.1.1.1.1.cmml"><mi id="S2.SS3.p1.1.m1.1.1.1.1.2" xref="S2.SS3.p1.1.m1.1.1.1.1.2.cmml">F</mi><mo id="S2.SS3.p1.1.m1.1.1.1.1.1" xref="S2.SS3.p1.1.m1.1.1.1.1.1.cmml">∈</mo><msup id="S2.SS3.p1.1.m1.1.1.1.1.3" xref="S2.SS3.p1.1.m1.1.1.1.1.3.cmml"><mi id="S2.SS3.p1.1.m1.1.1.1.1.3.2" xref="S2.SS3.p1.1.m1.1.1.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS3.p1.1.m1.1.1.1.1.3.3" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.cmml"><mi id="S2.SS3.p1.1.m1.1.1.1.1.3.3.2" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.2.cmml">n</mi><mo id="S2.SS3.p1.1.m1.1.1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.1.cmml">×</mo><mi id="S2.SS3.p1.1.m1.1.1.1.1.3.3.3" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.3.cmml">h</mi><mo id="S2.SS3.p1.1.m1.1.1.1.1.3.3.1a" lspace="0.222em" rspace="0.222em" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.1.cmml">×</mo><mi id="S2.SS3.p1.1.m1.1.1.1.1.3.3.4" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.4.cmml">w</mi></mrow></msup></mrow><mo id="S2.SS3.p1.1.m1.1.1.1.2" xref="S2.SS3.p1.1.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.1.m1.1b"><apply id="S2.SS3.p1.1.m1.1.1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1.1"><in id="S2.SS3.p1.1.m1.1.1.1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.1"></in><ci id="S2.SS3.p1.1.m1.1.1.1.1.2.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.2">𝐹</ci><apply id="S2.SS3.p1.1.m1.1.1.1.1.3.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p1.1.m1.1.1.1.1.3.1.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.3">superscript</csymbol><ci id="S2.SS3.p1.1.m1.1.1.1.1.3.2.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.3.2">ℝ</ci><apply id="S2.SS3.p1.1.m1.1.1.1.1.3.3.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3"><times id="S2.SS3.p1.1.m1.1.1.1.1.3.3.1.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.1"></times><ci id="S2.SS3.p1.1.m1.1.1.1.1.3.3.2.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.2">𝑛</ci><ci id="S2.SS3.p1.1.m1.1.1.1.1.3.3.3.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.3">ℎ</ci><ci id="S2.SS3.p1.1.m1.1.1.1.1.3.3.4.cmml" xref="S2.SS3.p1.1.m1.1.1.1.1.3.3.4">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.1.m1.1c">F\in{\mathbb{R}}^{n\times h\times w},</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.1.m1.1d">italic_F ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_h × italic_w end_POSTSUPERSCRIPT ,</annotation></semantics></math> such that important features are retained and irrelevant or noisy features are suppressed. F is denoted as follows:</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="F=F_{up}\oplus F_{skip}\oplus F_{contour}" class="ltx_Math" display="block" id="S2.E5.m1.1"><semantics id="S2.E5.m1.1a"><mrow id="S2.E5.m1.1.1" xref="S2.E5.m1.1.1.cmml"><mi id="S2.E5.m1.1.1.2" xref="S2.E5.m1.1.1.2.cmml">F</mi><mo id="S2.E5.m1.1.1.1" xref="S2.E5.m1.1.1.1.cmml">=</mo><mrow id="S2.E5.m1.1.1.3" xref="S2.E5.m1.1.1.3.cmml"><msub id="S2.E5.m1.1.1.3.2" xref="S2.E5.m1.1.1.3.2.cmml"><mi id="S2.E5.m1.1.1.3.2.2" xref="S2.E5.m1.1.1.3.2.2.cmml">F</mi><mrow id="S2.E5.m1.1.1.3.2.3" xref="S2.E5.m1.1.1.3.2.3.cmml"><mi id="S2.E5.m1.1.1.3.2.3.2" xref="S2.E5.m1.1.1.3.2.3.2.cmml">u</mi><mo id="S2.E5.m1.1.1.3.2.3.1" xref="S2.E5.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2.E5.m1.1.1.3.2.3.3" 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xref="S2.E5.m1.1.1.3.3">subscript</csymbol><ci id="S2.E5.m1.1.1.3.3.2.cmml" xref="S2.E5.m1.1.1.3.3.2">𝐹</ci><apply id="S2.E5.m1.1.1.3.3.3.cmml" xref="S2.E5.m1.1.1.3.3.3"><times id="S2.E5.m1.1.1.3.3.3.1.cmml" xref="S2.E5.m1.1.1.3.3.3.1"></times><ci id="S2.E5.m1.1.1.3.3.3.2.cmml" xref="S2.E5.m1.1.1.3.3.3.2">𝑠</ci><ci id="S2.E5.m1.1.1.3.3.3.3.cmml" xref="S2.E5.m1.1.1.3.3.3.3">𝑘</ci><ci id="S2.E5.m1.1.1.3.3.3.4.cmml" xref="S2.E5.m1.1.1.3.3.3.4">𝑖</ci><ci id="S2.E5.m1.1.1.3.3.3.5.cmml" xref="S2.E5.m1.1.1.3.3.3.5">𝑝</ci></apply></apply><apply id="S2.E5.m1.1.1.3.4.cmml" xref="S2.E5.m1.1.1.3.4"><csymbol cd="ambiguous" id="S2.E5.m1.1.1.3.4.1.cmml" xref="S2.E5.m1.1.1.3.4">subscript</csymbol><ci id="S2.E5.m1.1.1.3.4.2.cmml" xref="S2.E5.m1.1.1.3.4.2">𝐹</ci><apply id="S2.E5.m1.1.1.3.4.3.cmml" xref="S2.E5.m1.1.1.3.4.3"><times id="S2.E5.m1.1.1.3.4.3.1.cmml" xref="S2.E5.m1.1.1.3.4.3.1"></times><ci id="S2.E5.m1.1.1.3.4.3.2.cmml" xref="S2.E5.m1.1.1.3.4.3.2">𝑐</ci><ci id="S2.E5.m1.1.1.3.4.3.3.cmml" xref="S2.E5.m1.1.1.3.4.3.3">𝑜</ci><ci id="S2.E5.m1.1.1.3.4.3.4.cmml" xref="S2.E5.m1.1.1.3.4.3.4">𝑛</ci><ci id="S2.E5.m1.1.1.3.4.3.5.cmml" xref="S2.E5.m1.1.1.3.4.3.5">𝑡</ci><ci id="S2.E5.m1.1.1.3.4.3.6.cmml" xref="S2.E5.m1.1.1.3.4.3.6">𝑜</ci><ci id="S2.E5.m1.1.1.3.4.3.7.cmml" xref="S2.E5.m1.1.1.3.4.3.7">𝑢</ci><ci id="S2.E5.m1.1.1.3.4.3.8.cmml" xref="S2.E5.m1.1.1.3.4.3.8">𝑟</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.1c">F=F_{up}\oplus F_{skip}\oplus F_{contour}</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.1d">italic_F = italic_F start_POSTSUBSCRIPT italic_u italic_p end_POSTSUBSCRIPT ⊕ italic_F start_POSTSUBSCRIPT italic_s italic_k italic_i italic_p end_POSTSUBSCRIPT ⊕ italic_F start_POSTSUBSCRIPT italic_c italic_o italic_n italic_t italic_o italic_u 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">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p1.5">where <math alttext="F_{up}\in{\mathbb{R}}^{c\times h\times w}" class="ltx_Math" display="inline" id="S2.SS3.p1.2.m1.1"><semantics id="S2.SS3.p1.2.m1.1a"><mrow id="S2.SS3.p1.2.m1.1.1" xref="S2.SS3.p1.2.m1.1.1.cmml"><msub id="S2.SS3.p1.2.m1.1.1.2" xref="S2.SS3.p1.2.m1.1.1.2.cmml"><mi id="S2.SS3.p1.2.m1.1.1.2.2" xref="S2.SS3.p1.2.m1.1.1.2.2.cmml">F</mi><mrow id="S2.SS3.p1.2.m1.1.1.2.3" xref="S2.SS3.p1.2.m1.1.1.2.3.cmml"><mi id="S2.SS3.p1.2.m1.1.1.2.3.2" xref="S2.SS3.p1.2.m1.1.1.2.3.2.cmml">u</mi><mo id="S2.SS3.p1.2.m1.1.1.2.3.1" xref="S2.SS3.p1.2.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.2.m1.1.1.2.3.3" xref="S2.SS3.p1.2.m1.1.1.2.3.3.cmml">p</mi></mrow></msub><mo id="S2.SS3.p1.2.m1.1.1.1" xref="S2.SS3.p1.2.m1.1.1.1.cmml">∈</mo><msup id="S2.SS3.p1.2.m1.1.1.3" xref="S2.SS3.p1.2.m1.1.1.3.cmml"><mi id="S2.SS3.p1.2.m1.1.1.3.2" xref="S2.SS3.p1.2.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS3.p1.2.m1.1.1.3.3" xref="S2.SS3.p1.2.m1.1.1.3.3.cmml"><mi id="S2.SS3.p1.2.m1.1.1.3.3.2" xref="S2.SS3.p1.2.m1.1.1.3.3.2.cmml">c</mi><mo id="S2.SS3.p1.2.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS3.p1.2.m1.1.1.3.3.1.cmml">×</mo><mi id="S2.SS3.p1.2.m1.1.1.3.3.3" xref="S2.SS3.p1.2.m1.1.1.3.3.3.cmml">h</mi><mo id="S2.SS3.p1.2.m1.1.1.3.3.1a" lspace="0.222em" rspace="0.222em" xref="S2.SS3.p1.2.m1.1.1.3.3.1.cmml">×</mo><mi id="S2.SS3.p1.2.m1.1.1.3.3.4" xref="S2.SS3.p1.2.m1.1.1.3.3.4.cmml">w</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.2.m1.1b"><apply id="S2.SS3.p1.2.m1.1.1.cmml" xref="S2.SS3.p1.2.m1.1.1"><in id="S2.SS3.p1.2.m1.1.1.1.cmml" xref="S2.SS3.p1.2.m1.1.1.1"></in><apply id="S2.SS3.p1.2.m1.1.1.2.cmml" xref="S2.SS3.p1.2.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p1.2.m1.1.1.2.1.cmml" xref="S2.SS3.p1.2.m1.1.1.2">subscript</csymbol><ci id="S2.SS3.p1.2.m1.1.1.2.2.cmml" xref="S2.SS3.p1.2.m1.1.1.2.2">𝐹</ci><apply id="S2.SS3.p1.2.m1.1.1.2.3.cmml" xref="S2.SS3.p1.2.m1.1.1.2.3"><times id="S2.SS3.p1.2.m1.1.1.2.3.1.cmml" xref="S2.SS3.p1.2.m1.1.1.2.3.1"></times><ci id="S2.SS3.p1.2.m1.1.1.2.3.2.cmml" xref="S2.SS3.p1.2.m1.1.1.2.3.2">𝑢</ci><ci id="S2.SS3.p1.2.m1.1.1.2.3.3.cmml" xref="S2.SS3.p1.2.m1.1.1.2.3.3">𝑝</ci></apply></apply><apply id="S2.SS3.p1.2.m1.1.1.3.cmml" xref="S2.SS3.p1.2.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p1.2.m1.1.1.3.1.cmml" xref="S2.SS3.p1.2.m1.1.1.3">superscript</csymbol><ci id="S2.SS3.p1.2.m1.1.1.3.2.cmml" xref="S2.SS3.p1.2.m1.1.1.3.2">ℝ</ci><apply id="S2.SS3.p1.2.m1.1.1.3.3.cmml" xref="S2.SS3.p1.2.m1.1.1.3.3"><times id="S2.SS3.p1.2.m1.1.1.3.3.1.cmml" xref="S2.SS3.p1.2.m1.1.1.3.3.1"></times><ci id="S2.SS3.p1.2.m1.1.1.3.3.2.cmml" xref="S2.SS3.p1.2.m1.1.1.3.3.2">𝑐</ci><ci id="S2.SS3.p1.2.m1.1.1.3.3.3.cmml" xref="S2.SS3.p1.2.m1.1.1.3.3.3">ℎ</ci><ci id="S2.SS3.p1.2.m1.1.1.3.3.4.cmml" xref="S2.SS3.p1.2.m1.1.1.3.3.4">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.2.m1.1c">F_{up}\in{\mathbb{R}}^{c\times h\times w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.2.m1.1d">italic_F start_POSTSUBSCRIPT italic_u italic_p end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_c × italic_h × italic_w end_POSTSUPERSCRIPT</annotation></semantics></math> is the result of the sampling on the previous layer of decoder result, <math alttext="F_{skip}" class="ltx_Math" display="inline" id="S2.SS3.p1.3.m2.1"><semantics id="S2.SS3.p1.3.m2.1a"><msub id="S2.SS3.p1.3.m2.1.1" xref="S2.SS3.p1.3.m2.1.1.cmml"><mi id="S2.SS3.p1.3.m2.1.1.2" xref="S2.SS3.p1.3.m2.1.1.2.cmml">F</mi><mrow id="S2.SS3.p1.3.m2.1.1.3" xref="S2.SS3.p1.3.m2.1.1.3.cmml"><mi id="S2.SS3.p1.3.m2.1.1.3.2" xref="S2.SS3.p1.3.m2.1.1.3.2.cmml">s</mi><mo id="S2.SS3.p1.3.m2.1.1.3.1" xref="S2.SS3.p1.3.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.3.m2.1.1.3.3" xref="S2.SS3.p1.3.m2.1.1.3.3.cmml">k</mi><mo id="S2.SS3.p1.3.m2.1.1.3.1a" xref="S2.SS3.p1.3.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.3.m2.1.1.3.4" xref="S2.SS3.p1.3.m2.1.1.3.4.cmml">i</mi><mo id="S2.SS3.p1.3.m2.1.1.3.1b" xref="S2.SS3.p1.3.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.3.m2.1.1.3.5" xref="S2.SS3.p1.3.m2.1.1.3.5.cmml">p</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.3.m2.1b"><apply id="S2.SS3.p1.3.m2.1.1.cmml" xref="S2.SS3.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.3.m2.1.1.1.cmml" xref="S2.SS3.p1.3.m2.1.1">subscript</csymbol><ci id="S2.SS3.p1.3.m2.1.1.2.cmml" xref="S2.SS3.p1.3.m2.1.1.2">𝐹</ci><apply id="S2.SS3.p1.3.m2.1.1.3.cmml" xref="S2.SS3.p1.3.m2.1.1.3"><times id="S2.SS3.p1.3.m2.1.1.3.1.cmml" xref="S2.SS3.p1.3.m2.1.1.3.1"></times><ci id="S2.SS3.p1.3.m2.1.1.3.2.cmml" xref="S2.SS3.p1.3.m2.1.1.3.2">𝑠</ci><ci id="S2.SS3.p1.3.m2.1.1.3.3.cmml" xref="S2.SS3.p1.3.m2.1.1.3.3">𝑘</ci><ci id="S2.SS3.p1.3.m2.1.1.3.4.cmml" xref="S2.SS3.p1.3.m2.1.1.3.4">𝑖</ci><ci id="S2.SS3.p1.3.m2.1.1.3.5.cmml" xref="S2.SS3.p1.3.m2.1.1.3.5">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.3.m2.1c">F_{skip}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.3.m2.1d">italic_F start_POSTSUBSCRIPT italic_s italic_k italic_i italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="F_{contour}" class="ltx_Math" display="inline" id="S2.SS3.p1.4.m3.1"><semantics id="S2.SS3.p1.4.m3.1a"><msub id="S2.SS3.p1.4.m3.1.1" xref="S2.SS3.p1.4.m3.1.1.cmml"><mi id="S2.SS3.p1.4.m3.1.1.2" xref="S2.SS3.p1.4.m3.1.1.2.cmml">F</mi><mrow id="S2.SS3.p1.4.m3.1.1.3" xref="S2.SS3.p1.4.m3.1.1.3.cmml"><mi id="S2.SS3.p1.4.m3.1.1.3.2" xref="S2.SS3.p1.4.m3.1.1.3.2.cmml">c</mi><mo id="S2.SS3.p1.4.m3.1.1.3.1" xref="S2.SS3.p1.4.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.4.m3.1.1.3.3" xref="S2.SS3.p1.4.m3.1.1.3.3.cmml">o</mi><mo id="S2.SS3.p1.4.m3.1.1.3.1a" xref="S2.SS3.p1.4.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.4.m3.1.1.3.4" xref="S2.SS3.p1.4.m3.1.1.3.4.cmml">n</mi><mo id="S2.SS3.p1.4.m3.1.1.3.1b" xref="S2.SS3.p1.4.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.4.m3.1.1.3.5" xref="S2.SS3.p1.4.m3.1.1.3.5.cmml">t</mi><mo id="S2.SS3.p1.4.m3.1.1.3.1c" xref="S2.SS3.p1.4.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.4.m3.1.1.3.6" xref="S2.SS3.p1.4.m3.1.1.3.6.cmml">o</mi><mo id="S2.SS3.p1.4.m3.1.1.3.1d" xref="S2.SS3.p1.4.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.4.m3.1.1.3.7" xref="S2.SS3.p1.4.m3.1.1.3.7.cmml">u</mi><mo id="S2.SS3.p1.4.m3.1.1.3.1e" xref="S2.SS3.p1.4.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.4.m3.1.1.3.8" xref="S2.SS3.p1.4.m3.1.1.3.8.cmml">r</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.4.m3.1b"><apply id="S2.SS3.p1.4.m3.1.1.cmml" xref="S2.SS3.p1.4.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.4.m3.1.1.1.cmml" xref="S2.SS3.p1.4.m3.1.1">subscript</csymbol><ci id="S2.SS3.p1.4.m3.1.1.2.cmml" xref="S2.SS3.p1.4.m3.1.1.2">𝐹</ci><apply id="S2.SS3.p1.4.m3.1.1.3.cmml" xref="S2.SS3.p1.4.m3.1.1.3"><times id="S2.SS3.p1.4.m3.1.1.3.1.cmml" xref="S2.SS3.p1.4.m3.1.1.3.1"></times><ci id="S2.SS3.p1.4.m3.1.1.3.2.cmml" xref="S2.SS3.p1.4.m3.1.1.3.2">𝑐</ci><ci id="S2.SS3.p1.4.m3.1.1.3.3.cmml" xref="S2.SS3.p1.4.m3.1.1.3.3">𝑜</ci><ci id="S2.SS3.p1.4.m3.1.1.3.4.cmml" xref="S2.SS3.p1.4.m3.1.1.3.4">𝑛</ci><ci id="S2.SS3.p1.4.m3.1.1.3.5.cmml" xref="S2.SS3.p1.4.m3.1.1.3.5">𝑡</ci><ci id="S2.SS3.p1.4.m3.1.1.3.6.cmml" xref="S2.SS3.p1.4.m3.1.1.3.6">𝑜</ci><ci id="S2.SS3.p1.4.m3.1.1.3.7.cmml" xref="S2.SS3.p1.4.m3.1.1.3.7">𝑢</ci><ci id="S2.SS3.p1.4.m3.1.1.3.8.cmml" xref="S2.SS3.p1.4.m3.1.1.3.8">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.4.m3.1c">F_{contour}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.4.m3.1d">italic_F start_POSTSUBSCRIPT italic_c italic_o italic_n italic_t italic_o italic_u italic_r end_POSTSUBSCRIPT</annotation></semantics></math> are the coded features of the corresponding stage and the contour features derived from the adaptive resampling of the same source, respectively, with the same size as <math alttext="F_{up}" class="ltx_Math" display="inline" id="S2.SS3.p1.5.m4.1"><semantics id="S2.SS3.p1.5.m4.1a"><msub id="S2.SS3.p1.5.m4.1.1" xref="S2.SS3.p1.5.m4.1.1.cmml"><mi id="S2.SS3.p1.5.m4.1.1.2" xref="S2.SS3.p1.5.m4.1.1.2.cmml">F</mi><mrow id="S2.SS3.p1.5.m4.1.1.3" xref="S2.SS3.p1.5.m4.1.1.3.cmml"><mi id="S2.SS3.p1.5.m4.1.1.3.2" xref="S2.SS3.p1.5.m4.1.1.3.2.cmml">u</mi><mo id="S2.SS3.p1.5.m4.1.1.3.1" xref="S2.SS3.p1.5.m4.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p1.5.m4.1.1.3.3" xref="S2.SS3.p1.5.m4.1.1.3.3.cmml">p</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.5.m4.1b"><apply id="S2.SS3.p1.5.m4.1.1.cmml" xref="S2.SS3.p1.5.m4.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.5.m4.1.1.1.cmml" xref="S2.SS3.p1.5.m4.1.1">subscript</csymbol><ci id="S2.SS3.p1.5.m4.1.1.2.cmml" xref="S2.SS3.p1.5.m4.1.1.2">𝐹</ci><apply id="S2.SS3.p1.5.m4.1.1.3.cmml" xref="S2.SS3.p1.5.m4.1.1.3"><times id="S2.SS3.p1.5.m4.1.1.3.1.cmml" xref="S2.SS3.p1.5.m4.1.1.3.1"></times><ci id="S2.SS3.p1.5.m4.1.1.3.2.cmml" xref="S2.SS3.p1.5.m4.1.1.3.2">𝑢</ci><ci id="S2.SS3.p1.5.m4.1.1.3.3.cmml" xref="S2.SS3.p1.5.m4.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.5.m4.1c">F_{up}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.5.m4.1d">italic_F start_POSTSUBSCRIPT italic_u italic_p end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.2">The GF module establishes channel-based gating branches to create dynamically selected intermediate representations for high-frequency information <math alttext="F" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mi id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><ci id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">italic_F</annotation></semantics></math>. The gating branch is established using the linear mapping Linear, the scale-invariant convolution, and the activation function GELU, and the gating signals <math alttext="G" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.1"><semantics id="S2.SS3.p2.2.m2.1a"><mi id="S2.SS3.p2.2.m2.1.1" xref="S2.SS3.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.1b"><ci id="S2.SS3.p2.2.m2.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.1d">italic_G</annotation></semantics></math> obtained all contribute to the hidden units from the feature information of different channels at the exact spatial location. Finally, the gating signal weights are weighted to each embedded channel feature using the scale operation, which can be expressed as:</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="F_{gate}=G^{T}\otimes Linear(F)" class="ltx_Math" display="block" id="S2.E6.m1.1"><semantics id="S2.E6.m1.1a"><mrow id="S2.E6.m1.1.2" xref="S2.E6.m1.1.2.cmml"><msub id="S2.E6.m1.1.2.2" xref="S2.E6.m1.1.2.2.cmml"><mi id="S2.E6.m1.1.2.2.2" xref="S2.E6.m1.1.2.2.2.cmml">F</mi><mrow id="S2.E6.m1.1.2.2.3" xref="S2.E6.m1.1.2.2.3.cmml"><mi id="S2.E6.m1.1.2.2.3.2" xref="S2.E6.m1.1.2.2.3.2.cmml">g</mi><mo id="S2.E6.m1.1.2.2.3.1" xref="S2.E6.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.E6.m1.1.2.2.3.3" xref="S2.E6.m1.1.2.2.3.3.cmml">a</mi><mo id="S2.E6.m1.1.2.2.3.1a" xref="S2.E6.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.E6.m1.1.2.2.3.4" xref="S2.E6.m1.1.2.2.3.4.cmml">t</mi><mo id="S2.E6.m1.1.2.2.3.1b" xref="S2.E6.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.E6.m1.1.2.2.3.5" xref="S2.E6.m1.1.2.2.3.5.cmml">e</mi></mrow></msub><mo id="S2.E6.m1.1.2.1" xref="S2.E6.m1.1.2.1.cmml">=</mo><mrow id="S2.E6.m1.1.2.3" xref="S2.E6.m1.1.2.3.cmml"><mrow id="S2.E6.m1.1.2.3.2" xref="S2.E6.m1.1.2.3.2.cmml"><msup id="S2.E6.m1.1.2.3.2.2" xref="S2.E6.m1.1.2.3.2.2.cmml"><mi id="S2.E6.m1.1.2.3.2.2.2" xref="S2.E6.m1.1.2.3.2.2.2.cmml">G</mi><mi id="S2.E6.m1.1.2.3.2.2.3" xref="S2.E6.m1.1.2.3.2.2.3.cmml">T</mi></msup><mo id="S2.E6.m1.1.2.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.E6.m1.1.2.3.2.1.cmml">⊗</mo><mi id="S2.E6.m1.1.2.3.2.3" xref="S2.E6.m1.1.2.3.2.3.cmml">L</mi></mrow><mo id="S2.E6.m1.1.2.3.1" xref="S2.E6.m1.1.2.3.1.cmml">⁢</mo><mi id="S2.E6.m1.1.2.3.3" xref="S2.E6.m1.1.2.3.3.cmml">i</mi><mo id="S2.E6.m1.1.2.3.1a" xref="S2.E6.m1.1.2.3.1.cmml">⁢</mo><mi id="S2.E6.m1.1.2.3.4" xref="S2.E6.m1.1.2.3.4.cmml">n</mi><mo id="S2.E6.m1.1.2.3.1b" 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xref="S2.E6.m1.1.2.3.2.2.2">𝐺</ci><ci id="S2.E6.m1.1.2.3.2.2.3.cmml" xref="S2.E6.m1.1.2.3.2.2.3">𝑇</ci></apply><ci id="S2.E6.m1.1.2.3.2.3.cmml" xref="S2.E6.m1.1.2.3.2.3">𝐿</ci></apply><ci id="S2.E6.m1.1.2.3.3.cmml" xref="S2.E6.m1.1.2.3.3">𝑖</ci><ci id="S2.E6.m1.1.2.3.4.cmml" xref="S2.E6.m1.1.2.3.4">𝑛</ci><ci id="S2.E6.m1.1.2.3.5.cmml" xref="S2.E6.m1.1.2.3.5">𝑒</ci><ci id="S2.E6.m1.1.2.3.6.cmml" xref="S2.E6.m1.1.2.3.6">𝑎</ci><ci id="S2.E6.m1.1.2.3.7.cmml" xref="S2.E6.m1.1.2.3.7">𝑟</ci><ci id="S2.E6.m1.1.1.cmml" xref="S2.E6.m1.1.1">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.1c">F_{gate}=G^{T}\otimes Linear(F)</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.1d">italic_F start_POSTSUBSCRIPT italic_g italic_a italic_t italic_e end_POSTSUBSCRIPT = italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ⊗ italic_L italic_i italic_n italic_e italic_a italic_r ( italic_F )</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="S2.SS3.p2.11">where <math alttext="G\in{\mathbb{R}}^{c\times n},n=h\times w" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m1.2"><semantics id="S2.SS3.p2.3.m1.2a"><mrow id="S2.SS3.p2.3.m1.2.2.2" xref="S2.SS3.p2.3.m1.2.2.3.cmml"><mrow id="S2.SS3.p2.3.m1.1.1.1.1" xref="S2.SS3.p2.3.m1.1.1.1.1.cmml"><mi id="S2.SS3.p2.3.m1.1.1.1.1.2" xref="S2.SS3.p2.3.m1.1.1.1.1.2.cmml">G</mi><mo id="S2.SS3.p2.3.m1.1.1.1.1.1" xref="S2.SS3.p2.3.m1.1.1.1.1.1.cmml">∈</mo><msup id="S2.SS3.p2.3.m1.1.1.1.1.3" xref="S2.SS3.p2.3.m1.1.1.1.1.3.cmml"><mi id="S2.SS3.p2.3.m1.1.1.1.1.3.2" xref="S2.SS3.p2.3.m1.1.1.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS3.p2.3.m1.1.1.1.1.3.3" xref="S2.SS3.p2.3.m1.1.1.1.1.3.3.cmml"><mi id="S2.SS3.p2.3.m1.1.1.1.1.3.3.2" 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id="S2.SS3.p2.7.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.7.m5.1d">italic_n</annotation></semantics></math> is consistent with the embedding dimension of the <math alttext="Linear(F)" class="ltx_Math" display="inline" id="S2.SS3.p2.8.m6.1"><semantics id="S2.SS3.p2.8.m6.1a"><mrow id="S2.SS3.p2.8.m6.1.2" xref="S2.SS3.p2.8.m6.1.2.cmml"><mi id="S2.SS3.p2.8.m6.1.2.2" xref="S2.SS3.p2.8.m6.1.2.2.cmml">L</mi><mo id="S2.SS3.p2.8.m6.1.2.1" xref="S2.SS3.p2.8.m6.1.2.1.cmml">⁢</mo><mi id="S2.SS3.p2.8.m6.1.2.3" xref="S2.SS3.p2.8.m6.1.2.3.cmml">i</mi><mo id="S2.SS3.p2.8.m6.1.2.1a" xref="S2.SS3.p2.8.m6.1.2.1.cmml">⁢</mo><mi id="S2.SS3.p2.8.m6.1.2.4" xref="S2.SS3.p2.8.m6.1.2.4.cmml">n</mi><mo id="S2.SS3.p2.8.m6.1.2.1b" xref="S2.SS3.p2.8.m6.1.2.1.cmml">⁢</mo><mi id="S2.SS3.p2.8.m6.1.2.5" xref="S2.SS3.p2.8.m6.1.2.5.cmml">e</mi><mo id="S2.SS3.p2.8.m6.1.2.1c" xref="S2.SS3.p2.8.m6.1.2.1.cmml">⁢</mo><mi id="S2.SS3.p2.8.m6.1.2.6" xref="S2.SS3.p2.8.m6.1.2.6.cmml">a</mi><mo id="S2.SS3.p2.8.m6.1.2.1d" xref="S2.SS3.p2.8.m6.1.2.1.cmml">⁢</mo><mi id="S2.SS3.p2.8.m6.1.2.7" xref="S2.SS3.p2.8.m6.1.2.7.cmml">r</mi><mo id="S2.SS3.p2.8.m6.1.2.1e" xref="S2.SS3.p2.8.m6.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p2.8.m6.1.2.8.2" xref="S2.SS3.p2.8.m6.1.2.cmml"><mo id="S2.SS3.p2.8.m6.1.2.8.2.1" stretchy="false" xref="S2.SS3.p2.8.m6.1.2.cmml">(</mo><mi id="S2.SS3.p2.8.m6.1.1" xref="S2.SS3.p2.8.m6.1.1.cmml">F</mi><mo id="S2.SS3.p2.8.m6.1.2.8.2.2" stretchy="false" xref="S2.SS3.p2.8.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.8.m6.1b"><apply id="S2.SS3.p2.8.m6.1.2.cmml" xref="S2.SS3.p2.8.m6.1.2"><times id="S2.SS3.p2.8.m6.1.2.1.cmml" xref="S2.SS3.p2.8.m6.1.2.1"></times><ci id="S2.SS3.p2.8.m6.1.2.2.cmml" xref="S2.SS3.p2.8.m6.1.2.2">𝐿</ci><ci id="S2.SS3.p2.8.m6.1.2.3.cmml" xref="S2.SS3.p2.8.m6.1.2.3">𝑖</ci><ci id="S2.SS3.p2.8.m6.1.2.4.cmml" xref="S2.SS3.p2.8.m6.1.2.4">𝑛</ci><ci id="S2.SS3.p2.8.m6.1.2.5.cmml" xref="S2.SS3.p2.8.m6.1.2.5">𝑒</ci><ci id="S2.SS3.p2.8.m6.1.2.6.cmml" xref="S2.SS3.p2.8.m6.1.2.6">𝑎</ci><ci id="S2.SS3.p2.8.m6.1.2.7.cmml" xref="S2.SS3.p2.8.m6.1.2.7">𝑟</ci><ci id="S2.SS3.p2.8.m6.1.1.cmml" xref="S2.SS3.p2.8.m6.1.1">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.8.m6.1c">Linear(F)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.8.m6.1d">italic_L italic_i italic_n italic_e italic_a italic_r ( italic_F )</annotation></semantics></math> operation, and <math alttext="G^{T}" class="ltx_Math" display="inline" id="S2.SS3.p2.9.m7.1"><semantics id="S2.SS3.p2.9.m7.1a"><msup id="S2.SS3.p2.9.m7.1.1" xref="S2.SS3.p2.9.m7.1.1.cmml"><mi id="S2.SS3.p2.9.m7.1.1.2" xref="S2.SS3.p2.9.m7.1.1.2.cmml">G</mi><mi id="S2.SS3.p2.9.m7.1.1.3" xref="S2.SS3.p2.9.m7.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.9.m7.1b"><apply id="S2.SS3.p2.9.m7.1.1.cmml" 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encoding="application/x-llamapun" id="S2.SS3.p2.10.m8.1d">italic_G</annotation></semantics></math>.<math alttext="F_{gate}" class="ltx_Math" display="inline" id="S2.SS3.p2.11.m9.1"><semantics id="S2.SS3.p2.11.m9.1a"><msub id="S2.SS3.p2.11.m9.1.1" xref="S2.SS3.p2.11.m9.1.1.cmml"><mi id="S2.SS3.p2.11.m9.1.1.2" xref="S2.SS3.p2.11.m9.1.1.2.cmml">F</mi><mrow id="S2.SS3.p2.11.m9.1.1.3" xref="S2.SS3.p2.11.m9.1.1.3.cmml"><mi id="S2.SS3.p2.11.m9.1.1.3.2" xref="S2.SS3.p2.11.m9.1.1.3.2.cmml">g</mi><mo id="S2.SS3.p2.11.m9.1.1.3.1" xref="S2.SS3.p2.11.m9.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p2.11.m9.1.1.3.3" xref="S2.SS3.p2.11.m9.1.1.3.3.cmml">a</mi><mo id="S2.SS3.p2.11.m9.1.1.3.1a" xref="S2.SS3.p2.11.m9.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p2.11.m9.1.1.3.4" xref="S2.SS3.p2.11.m9.1.1.3.4.cmml">t</mi><mo id="S2.SS3.p2.11.m9.1.1.3.1b" xref="S2.SS3.p2.11.m9.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.p2.11.m9.1.1.3.5" xref="S2.SS3.p2.11.m9.1.1.3.5.cmml">e</mi></mrow></msub><annotation-xml encoding="MathML-Content" 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end_POSTSUBSCRIPT</annotation></semantics></math>will serve as the input for the next stage of upsampling.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">Experiments</span> </h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS1.5.1.1">III-A</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS1.6.2">Datasets and Experimental Settings </span> </h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.1">This paper validates the model in this paper on two types of ultrasound datasets, breast and thyroid. The breast ultrasound dataset BUSI<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib13" title="">13</a>]</cite> removed 133 images of standard cases without breast lesions, using 665 images containing nodules. The thyroid ultrasound dataset DDTI<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib14" title="">14</a>]</cite> removes labeled corrupted images and expands images with multiple nodules, using 872 images. The experiments are supplemented with the private thyroid ultrasound dataset TUI, which contains 15,233 images, to validate the results thoroughly.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.3">The entire network is optimized using the SGD algorithm with a momentum of 0.9, weight decay of 0.01, batch size of 64, and epoch 200. This paper’s initial learning rate is set to 1e-3 and is scaled down with the training process using the cosine annealing algorithm. The network is trained on a single RTX-3090 GPU. We use VGG16BN as a feature extractor in CPM. We use a combination of binary cross entropy (BCE), dice loss, and KL dispersion loss to train CP-UNet, expressed as:</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="\mathscr{L}=BCE(\hat{y},y)+Dice(\hat{y},y)+KL_{AB}" class="ltx_Math" display="block" id="S3.E7.m1.4"><semantics id="S3.E7.m1.4a"><mrow id="S3.E7.m1.4.5" xref="S3.E7.m1.4.5.cmml"><mi class="ltx_font_mathscript" id="S3.E7.m1.4.5.2" xref="S3.E7.m1.4.5.2.cmml">ℒ</mi><mo id="S3.E7.m1.4.5.1" xref="S3.E7.m1.4.5.1.cmml">=</mo><mrow id="S3.E7.m1.4.5.3" xref="S3.E7.m1.4.5.3.cmml"><mrow id="S3.E7.m1.4.5.3.2" xref="S3.E7.m1.4.5.3.2.cmml"><mi id="S3.E7.m1.4.5.3.2.2" xref="S3.E7.m1.4.5.3.2.2.cmml">B</mi><mo id="S3.E7.m1.4.5.3.2.1" xref="S3.E7.m1.4.5.3.2.1.cmml">⁢</mo><mi id="S3.E7.m1.4.5.3.2.3" xref="S3.E7.m1.4.5.3.2.3.cmml">C</mi><mo 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id="S3.E7.m1.4.5.3.4.3.3.2.cmml" xref="S3.E7.m1.4.5.3.4.3.3.2">𝐴</ci><ci id="S3.E7.m1.4.5.3.4.3.3.3.cmml" xref="S3.E7.m1.4.5.3.4.3.3.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m1.4c">\mathscr{L}=BCE(\hat{y},y)+Dice(\hat{y},y)+KL_{AB}</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m1.4d">script_L = italic_B italic_C italic_E ( over^ start_ARG italic_y end_ARG , italic_y ) + italic_D italic_i italic_c italic_e ( over^ start_ARG italic_y end_ARG , italic_y ) + italic_K italic_L start_POSTSUBSCRIPT italic_A italic_B 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">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p2.2">where y is the predicted outcome, <math alttext="\hat{y}" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mover accent="true" id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml"><mi id="S3.SS1.p2.1.m1.1.1.2" xref="S3.SS1.p2.1.m1.1.1.2.cmml">y</mi><mo id="S3.SS1.p2.1.m1.1.1.1" xref="S3.SS1.p2.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><apply id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1"><ci id="S3.SS1.p2.1.m1.1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1.1">^</ci><ci id="S3.SS1.p2.1.m1.1.1.2.cmml" xref="S3.SS1.p2.1.m1.1.1.2">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">\hat{y}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">over^ start_ARG italic_y end_ARG</annotation></semantics></math> is the target outcome, and <math alttext="KL_{AB}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2" xref="S3.SS1.p2.2.m2.1.1.2.cmml">K</mi><mo id="S3.SS1.p2.2.m2.1.1.1" xref="S3.SS1.p2.2.m2.1.1.1.cmml">⁢</mo><msub id="S3.SS1.p2.2.m2.1.1.3" xref="S3.SS1.p2.2.m2.1.1.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.2" xref="S3.SS1.p2.2.m2.1.1.3.2.cmml">L</mi><mrow id="S3.SS1.p2.2.m2.1.1.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.3.2" xref="S3.SS1.p2.2.m2.1.1.3.3.2.cmml">A</mi><mo id="S3.SS1.p2.2.m2.1.1.3.3.1" xref="S3.SS1.p2.2.m2.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS1.p2.2.m2.1.1.3.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.3.cmml">B</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><times id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1"></times><ci id="S3.SS1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2">𝐾</ci><apply id="S3.SS1.p2.2.m2.1.1.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.2">𝐿</ci><apply id="S3.SS1.p2.2.m2.1.1.3.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3"><times id="S3.SS1.p2.2.m2.1.1.3.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.1"></times><ci id="S3.SS1.p2.2.m2.1.1.3.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.2">𝐴</ci><ci id="S3.SS1.p2.2.m2.1.1.3.3.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">KL_{AB}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_K italic_L start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT</annotation></semantics></math> is the result of (<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S2.E4" title="In II-B Contour Probabilistic Modeling ‣ II Method ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">4</span></a>).</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS2.5.1.1">III-B</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS2.6.2">Performance Comparison</span> </h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">In this paper, we compare the performance of CP-UNet with widely used medical image segmentation methods, which include convolutional baselines such as UNet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib1" title="">1</a>]</cite>, UNet++<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib2" title="">2</a>]</cite>, ResUNet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib3" title="">3</a>]</cite>; SETR<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib18" title="">18</a>]</cite>, TransUNet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib19" title="">19</a>]</cite>, an improved segmentation network based on transformer, UNETR<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib20" title="">20</a>]</cite>, MedT<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib21" title="">21</a>]</cite>; and Attention-based medical image methods<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib21" title="">21</a>]</cite>. To make a fair comparison, this paper downloads the code of their public implementations and retrains them on all datasets.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.1">Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S3.F3" title="Figure 3 ‣ III-B Performance Comparison ‣ III Experiments ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">3</span></a> shows the results of the segmentation visualization on the three datasets, and it can be seen that our method exhibits a relatively average level of regional delineation of the nodes, but for the trend direction of the contours, the performance is excellent and close to that of the truth map.</p> </div> <figure class="ltx_table" id="S3.T1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE I: </span>Performance Comparison with Other Method</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S3.T1.1" style="width:433.6pt;height:228.4pt;vertical-align:-0.8pt;"><span class="ltx_transformed_inner" style="transform:translate(-40.4pt,21.2pt) scale(0.84302496761495,0.84302496761495) ;"> <table class="ltx_tabular ltx_align_middle" id="S3.T1.1.1"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T1.1.1.1.1"> <td class="ltx_td ltx_border_tt" id="S3.T1.1.1.1.1.1"></td> <td class="ltx_td ltx_align_center ltx_border_tt" colspan="2" id="S3.T1.1.1.1.1.2">BUSI</td> <td class="ltx_td ltx_align_center ltx_border_tt" colspan="2" id="S3.T1.1.1.1.1.3">DDTI</td> <td class="ltx_td ltx_align_center ltx_border_tt" colspan="2" id="S3.T1.1.1.1.1.4">TUI</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.2.2"> <td class="ltx_td" id="S3.T1.1.1.2.2.1"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.2.2.2">IoU(%)</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.2.2.3">Dice(%)</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.2.2.4">IoU(%)</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.2.2.5">Dice(%)</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.2.2.6">IoU(%)</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.2.2.7">Dice(%)</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.3.3"> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.3.3.1">UNet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib1" title="">1</a>]</cite> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.3.3.2">0.5070</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.3.3.3">0.6729</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.3.3.4">0.6692</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.3.3.5">0.7999</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.3.3.6">0.7965</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.3.3.7">0.8866</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.4.4"> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.4.4.1">UNet++<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib2" title="">2</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.4.4.2">0.5935</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.4.4.3">0.7444</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.4.4.4">0.7299</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.4.4.5">0.8429</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.4.4.6">0.8034</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.4.4.7">0.8901</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.5.5"> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.5.5.1">ResUNet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib3" title="">3</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.5.5.2">0.5517</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.5.5.3">0.7069</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.5.5.4">0.7121</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.5.5.5">0.8294</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.5.5.6">0.7631</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.5.5.7">0.8631</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.6.6"> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.6.6.1">SETR<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib18" title="">18</a>]</cite> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.6.6.2">0.2293</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.6.6.3">0.3720</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.6.6.4">0.2864</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.6.6.5">0.4434</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.6.6.6">0.2629</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.6.6.7">0.4141</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.7.7"> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.7.7.1">MedT<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib21" title="">21</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.7.7.2">0.5155</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.7.7.3">0.6790</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.7.7.4">0.5873</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.7.7.5">0.7366</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.7.7.6">0.6382</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.7.7.7">0.7752</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.8.8"> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.8.8.1">Swin-Unet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib22" title="">22</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.8.8.2">0.4850</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.8.8.3">0.6503</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.8.8.4">0.4982</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.8.8.5">0.6602</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.8.8.6">0.3978</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.8.8.7">0.5636</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.9.9"> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.9.9.1">TransUNet(pretrained)<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib19" title="">19</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.9.9.2">0.6454</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.9.9.3">0.7812</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.9.9.4">0.8147</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.9.9.5">0.8963</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.9.9.6">0.8481</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.9.9.7">0.9166</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.10.10"> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.10.10.1">TransUNet(w/o pretrained)<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib19" title="">19</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.10.10.2">0.5966</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.10.10.3">0.7448</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.10.10.4">0.7612</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.10.10.5">0.8625</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.10.10.6">0.8066</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.10.10.7">0.8909</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.11.11"> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.11.11.1">UNeXt<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib6" title="">6</a>]</cite> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.11.11.2">0.5919</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.11.11.3">0.7428</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.11.11.4">0.7746</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.11.11.5">0.8722</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.11.11.6">0.8358</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.11.11.7">0.9104</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.12.12"> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.12.12.1">Acc-Unet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib8" title="">8</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.12.12.2">0.5989</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.12.12.3">0.7463</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.12.12.4">0.7483</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.12.12.5">0.8541</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.12.12.6">0.8255</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.12.12.7">0.8994</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.13.13"> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.13.13.1">GGNet<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib10" title="">10</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.13.13.2">0.5410</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.13.13.3">0.7020</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.13.13.4">0.5594</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.13.13.5">0.7154</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.13.13.6">0.6744</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.13.13.7">0.8039</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.14.14"> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.14.14.1">AAU-net<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#bib.bib15" title="">15</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.14.14.2">0.6286</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.14.14.3">0.7714</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.14.14.4">0.7731</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.14.14.5">0.8704</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.14.14.6">0.7802</td> <td class="ltx_td ltx_align_center" id="S3.T1.1.1.14.14.7">0.8754</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1.15.15"> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T1.1.1.15.15.1">ours</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T1.1.1.15.15.2"><span class="ltx_text ltx_font_bold" id="S3.T1.1.1.15.15.2.1">0.6445</span></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T1.1.1.15.15.3"><span class="ltx_text ltx_font_bold" id="S3.T1.1.1.15.15.3.1">0.7827</span></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T1.1.1.15.15.4"><span class="ltx_text ltx_font_bold" id="S3.T1.1.1.15.15.4.1">0.8136</span></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T1.1.1.15.15.5"><span class="ltx_text ltx_font_bold" id="S3.T1.1.1.15.15.5.1">0.8972</span></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T1.1.1.15.15.6"><span class="ltx_text ltx_font_bold" id="S3.T1.1.1.15.15.6.1">0.8496</span></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T1.1.1.15.15.7"><span class="ltx_text ltx_font_bold" id="S3.T1.1.1.15.15.7.1">0.9183</span></td> </tr> </tbody> </table> </span></div> </figure> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="297" id="S3.F3.g1" src="x3.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Visualization of the results on two ultrasound images. The first three rows show the segmentation results of the breast ultrasound image, and the last row of the thyroid ultrasound</figcaption> </figure> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.1">Table <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S3.T1" title="TABLE I ‣ III-B Performance Comparison ‣ III Experiments ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">I</span></a> reports the experimental results of this paper’s method and all competing methods on two ultrasound datasets, thyroid (DDTI, TUI) and breast (BUSI). On both the BUSI and DDTI datasets, the convolution-based baseline performs poorly compared to the model in this paper, which delivers about ten percent improvement and is comparable to the results of the pre-trained TransUNet on the ImageNet-21K dataset. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S3.F4" title="Figure 4 ‣ III-B Performance Comparison ‣ III Experiments ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">4</span></a> shows that when the segmentation metrics are similar, CP-UNet is better at capturing contour zigzag variations than the pre-trained TransUNet, proving the effectiveness of our contour-specific modeling.</p> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.1">The experimental results in the last two columns of Table <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S3.T1" title="TABLE I ‣ III-B Performance Comparison ‣ III Experiments ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">I</span></a> show that with sufficient data support, the attention-based method obtains a more stable enhancement and performs better than the convolutional baseline. At the same time, the model in this paper can achieve better results than the attention-based baseline, which indicates that the network proposed in this paper can build better support for the features in addition to the sufficient amount of data. Compared to the pure convolutional baseline, the method in this paper has a more significant improvement in BUSI than TUI and a considerable improvement over the attention-based baseline, indicating that the network proposed in this paper can extract more practical information for segmentation enhancement on a smaller dataset.</p> </div> <div class="ltx_para" id="S3.SS2.p5"> <p class="ltx_p" id="S3.SS2.p5.1">Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S3.F4" title="Figure 4 ‣ III-B Performance Comparison ‣ III Experiments ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">4</span></a> presents the segmentation results of ultrasound images obtained using different comparison methods. The first three rows display the segmentation results for breast nodules, while the last row shows the results for thyroid nodules. Except for the first row, the contours in the other images are unclear. Although the segmentation results of the pre-trained TransUNet appear similar to the ground truth, it fails to capture the intricate variations of the contours, rendering them smoother compared to our model. As shown in the figure, our model demonstrates superior performance in contour delineation.</p> </div> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="281" id="S3.F4.g1" src="x4.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Visualization of contour segmentation performance by TransUNet and CP-UNet. The red boxes are poorly outlined areas, and the green line is the output of the model.</figcaption> </figure> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS3.5.1.1">III-C</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS3.6.2">Ablation Studies</span> </h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">To validate the effectiveness of each module, we performed an ablation study on the BUSI dataset. Table <a class="ltx_ref" href="https://arxiv.org/html/2411.14250v1#S3.T2" title="TABLE II ‣ III-C Ablation Studies ‣ III Experiments ‣ CP-UNet: Contour-based Probabilistic Model for Medical Ultrasound Images Segmentation"><span class="ltx_text ltx_ref_tag">II</span></a> shows the comparative results of the different components of the methodology in this paper. The baseline is constructed after culling all the network’s MgCSD, GF, and CPM modules. Note that when culling the MgCSD, the network is missing encoders; at this point, we use the encoders in the unit to replace the MgCSD modules in each stage.</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.1">Adding all three modules alone improves the experimental effect relative to the baseline. MgCSD improves the model effect by about <math alttext="10\%" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><mrow id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml"><mn id="S3.SS3.p2.1.m1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.2.cmml">10</mn><mo id="S3.SS3.p2.1.m1.1.1.1" xref="S3.SS3.p2.1.m1.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><apply id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1"><csymbol cd="latexml" id="S3.SS3.p2.1.m1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1.1">percent</csymbol><cn id="S3.SS3.p2.1.m1.1.1.2.cmml" type="integer" xref="S3.SS3.p2.1.m1.1.1.2">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">10\%</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">10 %</annotation></semantics></math>, suggesting that the encoding strategy of mutual global-local feature enhancement is effective. The impact of the CPM module alone has a limited effect on the baseline enhancement in line with our expectations, as the CPM module needs the MgCSD to provide it with better quality features, as well as the GF module for its filtering, as evidenced by the experimental comparison of MgCSD+CPM and GF+CPM versus each module alone.GF+CPM is slightly less effective than GF alone. We do not rule out that it is due to the lack of information in CPM caused by the absence of MgCSD because GF+MgCSD+CPM is higher than GF+MgCSD. Together, the three modules can achieve the best results, suggesting that the GF feature fusion strategy is necessary for decoding the three data sources.</p> </div> <figure class="ltx_table" id="S3.T2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">TABLE II: </span>Performance of Ablation Study.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S3.T2.1"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T2.1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T2.1.1.1.1">MgCSD</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T2.1.1.1.2">GF</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T2.1.1.1.3">CPM</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T2.1.1.1.4">IoU</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T2.1.1.1.5">Dice</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T2.1.2.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.1.2.1.1">-</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.1.2.1.2">-</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.1.2.1.3">-</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.1.2.1.4">0.5439</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T2.1.2.1.5">0.7038</td> </tr> <tr class="ltx_tr" id="S3.T2.1.3.2"> <td class="ltx_td ltx_align_center" id="S3.T2.1.3.2.1">✓</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.3.2.2">-</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.3.2.3">-</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.3.2.4">0.6410</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.3.2.5">0.7785</td> </tr> <tr class="ltx_tr" id="S3.T2.1.4.3"> <td class="ltx_td ltx_align_center" id="S3.T2.1.4.3.1">-</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.4.3.2">✓</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.4.3.3">-</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.4.3.4">0.5752</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.4.3.5">0.7299</td> </tr> <tr class="ltx_tr" id="S3.T2.1.5.4"> <td class="ltx_td ltx_align_center" id="S3.T2.1.5.4.1">-</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.5.4.2">-</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.5.4.3">✓</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.5.4.4">0.5550</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.5.4.5">0.7067</td> </tr> <tr class="ltx_tr" id="S3.T2.1.6.5"> <td class="ltx_td ltx_align_center" id="S3.T2.1.6.5.1">✓</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.6.5.2">✓</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.6.5.3">-</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.6.5.4">0.6323</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.6.5.5">0.7739</td> </tr> <tr class="ltx_tr" id="S3.T2.1.7.6"> <td class="ltx_td ltx_align_center" id="S3.T2.1.7.6.1">-</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.7.6.2">✓</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.7.6.3">✓</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.7.6.4">0.5713</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.7.6.5">0.7208</td> </tr> <tr class="ltx_tr" id="S3.T2.1.8.7"> <td class="ltx_td ltx_align_center" id="S3.T2.1.8.7.1">✓</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.8.7.2">-</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.8.7.3">✓</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.8.7.4">0.6420</td> <td class="ltx_td ltx_align_center" id="S3.T2.1.8.7.5">0.7795</td> </tr> <tr class="ltx_tr" id="S3.T2.1.9.8"> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T2.1.9.8.1">✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T2.1.9.8.2">✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T2.1.9.8.3">✓</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T2.1.9.8.4"><span class="ltx_text ltx_font_bold" id="S3.T2.1.9.8.4.1">0.6445</span></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T2.1.9.8.5"><span class="ltx_text ltx_font_bold" id="S3.T2.1.9.8.5.1">0.7827</span></td> </tr> </tbody> </table> </figure> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Conclusion</span> </h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">For focus blurred lesion contour, we propose a contour-based probabilistic modeling medical ultrasound image segmentation network (CP-UNet) for lesion segmentation in ultrasound images. Adaptive homologous resampling and hybrid Gaussian distribution modeling of global semantic features in the CPM module optimize the predicted contours to help improve segmentation performance. In this paper, the network is evaluated on various ultrasound images on public and private datasets and compared with state-of-the-art methods. In future work, we will explore the potential of contour-based attention mechanisms to improve the diagnostic accuracy for nodules.</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"> O. Ronneberger, P. Fischer, and T. 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