[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-608368-105":59,"doc-detail-608368-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","corrected-laplacians-closer-cuts-and-segmentation-with-shape-priors","Corrected Laplacians - Closer Cuts and Segmentation with Shape Priors","","We optimize over corrected laplacians (CL) tied to a weighted graph to improve the average case normalized cut (NCut). Unlike edge-relaxation SDPs, CL optimization exploits matrix sparsity by updating only diagonal entries. This helps image segmentation where vertex counts scale with pixel counts. The method integrates statistical shape models as priors, yielding foreground regions consistent with a parameterized shape family. Experiments on MRI validate quantitative gains versus spectral relaxation for graph partitioning.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/corrected-laplacians-closer-cuts-and-segmentation-with-shape-priors/608368/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/corrected-laplacians-closer-cuts-and-segmentation-with-shape-priors/608368.png","ImageObject",300,407,{"name":92,"@type":93},"lilac","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-10-17",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the goal of optimizing corrected laplacians (CL) in the paper?","Question",{"text":112,"@type":113},"The paper optimizes CL associated with a weighted graph to improve the average case normalized cut (NCut) used for graph-based image segmentation.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does CL optimization differ from edge-relaxation SDPs?",{"text":117,"@type":113},"CL optimization naturally exploits matrix sparsity by operating only on the diagonal, avoiding the broader edge-relaxation structure of SDP approaches.",{"name":119,"@type":110,"acceptedAnswer":120},"How are statistical shape models used to guide segmentation?",{"text":121,"@type":113},"The authors incorporate prior information through statistical shape models, producing foreground regions consistent with a parameterized family of shapes and enabling domain-specific shape constraints.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},608368,1792657969,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":34,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144},3985747861865,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Corrected Laplacians: Closer Cuts and Segmentation with Shape Priors  \nDavid Tolliver Gary L. Miller Robert T. Collins  \nSchool of Computer Science  \nCarnegie Mellon University  \nPittsburgh, PA 15213  \nAbstract  \nWe optimize over the set of corrected laplacians (CL) associated with a weighted graph to improve the average case normalized cut (NCut) of a graph. Unlike edge-relaxation SDPs, optimizing over the set CL naturally exploits the matrix sparsity by operating solely on the diagonal. This structure is critical to image segmentation applications because the number of vertices is generally proportional to the number of pixels in the image. CL optimization provides a guiding principle for improving the combinatorial solution over the spectral relaxation, which is important because small improvements in the cut cost often result in significant improvements in the perceptual relevance of the segmentation. We develop an optimization procedure to accommodate prior information in the form of statistical shape models, resulting in a segmentation method that produces foreground regions which are consistent with a parameterized family of shapes. We validate our technique with ground truth on MRI medical images, providing a quantitative comparison against results produced by current spectral relaxation approaches to graph partitioning.  \n1 Introduction  \nThe normalized cut (NCut), proposed by Shi and Malik [9], provides a rigorous computational foundation for image segmentation problems. A graph is constructed from the image such that the pixels constitute the vertex set, connected by weighted edges representing similarity between nearby pixels. This formulation allows the image segmentation problem to be computed via graph partitioning. The partition measure associated with the NCut seeks to simultaneously minimize the cross segment connectivity and maximize the within segment association. Yet directly computing the optimal partition under the NCut is an NP-hard combinatorial problem.  \nThe authors of [9] develop a spectral relaxation of the normalized cut criterion to approximate their NP-hard objective function. This approximate solution, obtained from  \nFigure 1 . The eigenvector derived from the standard spectral relaxation to the normalized cut is shown on the top row. The eigenvector derived from the shape based Corrected Laplacian is shown on the bottom row. We see that the result is a superior segmentation, both visually and w.r.t. the cut criterion.  \na standard eigencalculation, is optimal on the continuous definition. Relaxations of this form were further extended by Yu and Shi in [12, 13] to allow constraints to be included in the cut function. They also derive multiway cuts. Ideally, the spectral relaxation affords efficient computation of nearly optimal combinatorial solutions. However, the bound associated with this relaxation is loose. Moreover, the approximation suffers from spurious structure introduced by the constraints on the target eigenvectors (known as the Fiedler space1 ) . As a result the spectral algorithm may generate poor solutions for apparently simple segmentation problems (see Figure 1) .  \n1The Fiedler space of a graph is the minimal p−eigenspace of a generalized graph laplacian. These vectors possess powerful properties, preserved by the NCut. For example, Fiedler vectors with minimal support partition the graph into connected subgraphs [5] . In image processing this result assures us that for most topologies, graph partitioning will result in connected components in the image plane.  \nWe propose an additional optimization over the particular matrix encoding of the graph. This optimization is performed over a restricted class of generalized laplacians encoding the weighted graph known as corrected laplacians (CL) . The result is an efficient means of improving the cut value aimed at image segmentation. By maximizing the Feidler value (λ2) a tighter bound on the true optimal value of the cut is achieved,","cbCaivxHAXpfxDrp","https://ap.wps.com/l/cbCaivxHAXpfxDrp","pdf",478144,"English","# Introduction\n## Graph Based Image Segmentation","[{\"question\":\"What is the goal of optimizing corrected laplacians (CL) in the paper?\",\"answer\":\"The paper optimizes CL associated with a weighted graph to improve the average case normalized cut (NCut) used for graph-based image segmentation.\"},{\"question\":\"How does CL optimization differ from edge-relaxation SDPs?\",\"answer\":\"CL optimization naturally exploits matrix sparsity by operating only on the diagonal, avoiding the broader edge-relaxation structure of SDP approaches.\"},{\"question\":\"How are statistical shape models used to guide segmentation?\",\"answer\":\"The authors incorporate prior information through statistical shape models, producing foreground regions consistent with a parameterized family of shapes and enabling domain-specific shape constraints.\"}]","Corrected Laplacians - Closer Cuts and Segmentation with Shape Priors | PDF",1792246246,18]