[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-280765-105":3,"detail-sidebar-cat-1-en-105":81,"doc-detail-280765-en":127},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","a-finite-element-method-for-deformable-models","A Finite Element Method for Deformable Models","","Deformable models for elastic structures are presented for model-based image interpretation using an alternative to Euler-Lagrange variational methods. The work introduces a direct minimisation of the underlying energy integral via the Finite Element Method, outlines the procedure, and demonstrates the approach. Physical interpretations are given for the energy terms, including image-intensity influence that favors dark artery-like structures. Computational advantages and flexibility over ELT are discussed for complex geometry and boundary conditions.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/a-finite-element-method-for-deformable-models/280765/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/a-finite-element-method-for-deformable-models/280765.png","ImageObject",442,249,{"name":42,"@type":43},"Sage","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-19","2026-09-16",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",9,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What problem does the paper address in deformable model image interpretation?","Question",{"text":63,"@type":64},"It addresses how to derive an optimal global solution for deformable models by minimizing an energy integral rather than relying on Euler-Lagrange-based formulations.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How does the proposed method differ from Euler-Lagrange Theory (ELT)?",{"text":68,"@type":64},"The method minimises the energy integral directly using the Finite Element Method, avoiding the need to transform the integral into differential equations solved numerically.",{"name":70,"@type":61,"acceptedAnswer":71},"Why is FEM advantageous for this application?",{"text":72,"@type":64},"FEM is more straightforward, offers greater flexibility for energy integrals and boundary conditions, and is better suited to geometrically complex problems or discontinuous loads and properties.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},280765,1791725967,{"code":4,"msg":82,"data":83},"success",[84,88,93,98,103,108,113,118,123],{"id":85,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":86,"slug":87},11,90,"presentations",{"id":89,"doc_module":22,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},12,"Resumes",80,"resumes",{"id":94,"doc_module":22,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},14,"Invoices",70,"invoices",{"id":99,"doc_module":22,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},15,"Posters",60,"posters",{"id":104,"doc_module":22,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},16,"Social Media",50,"social-media",{"id":109,"doc_module":22,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},17,"Forms",40,"forms",{"id":114,"doc_module":22,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},18,"Letters",30,"letters",{"id":119,"doc_module":22,"doc_module_name":25,"category_name":120,"show_sort_weight":121,"slug":122},21,"Paper Templates",5,"papers-templates",{"id":124,"doc_module":22,"doc_module_name":25,"category_name":125,"show_sort_weight":4,"slug":126},158,"General","general-158",{"code":4,"msg":82,"data":128},{"doc_id":79,"user_id":129,"nickname":42,"user_avatar":130,"doc_module":22,"category_id":85,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":136,"language":137,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":138,"faqs":139,"seo_title":140,"seo_description":12,"update_tm":141,"read_time":26},687197207057,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","# A Finite Element Method for Deformable Models\n\nPersephoni Karaolani,G.D.Sullivan,K.D.Baker &M.J.Baines  \nIntelligent Systems Group,Department of Computer ScienceUniversity of Reading,RG62AX,UK,P.Karaolani@reading.ac.uk  \n·The term lvg(s)i²(the norm of the first derivative ofthe position vector)makes the model act like anelastic string.  \nDeformable models of elastic structures have beenproposed for use in image analysis.Previous work hasused a variational approach,based on the Euler-Lagrange theory.In this paper an alternativemathematical treatment is introduced,based on a directminimisation of the underlying energy integral using theFinite Element Method.The method is outlined anddemonstrated,and its principal advantages for model-based image interpretation are explained.  \n·The termkv(sI²(the nom of the secondderivative of the positional vector)makes the modelact like a thin bar.  \n## INTRODUCTION\n\nModels allow the use of high-level knowledge aboutexpected structures to control the visual interpretationprocess.Most work on model-based vision hasconsidered rigid [1]or parameterised [2]models.Recently,deformable models have also attractedattention,e.g.[3,4,5].Deformable models are likely tohave particular relevance in medical imagingapplications,as a means of encoding the shapes ofanatomical organs,which are inherently variable.  \nAs an example,consider an X-ray image of an artery.Context-free image processing may produce goodevidence of an artery in the regions A and B(see Figure.4(a)),but fail to identify its position between A and B.One way to detect the artery in this space is to use amodel of an \"ideal\"artery,which can be deformedwithin known constraints under the influence of thefragmentary image data,to derive an optimal globalsolution.  \nA simple artery model can be specified by minimisingthe energy integral given in expression(1)overfunctions v(s),as proposed by Kass et.al.[3].  \n### (1)\n\nWhere v(s)=(x(s),y(s))is the position vector as afunction of the intrinsic parameter s,the normalizedlength of the line with vertices v₁,V₂··Ym [note:v;(i=1..M)denotes v(s;)].The terms vsand vssdenotefirst and second derivatives of v with respect to s;andα(s),β(s)are weighting functions.  \nExpression(1)contains three terms,the minimisation ofwhich have simple physical interpretations.  \n·The final term,I(v(s)),is used to allow chosenfeatures of the image intensity values to influencethe model.In the present treatment we use theimage grey-values along v(s),so that the term isminimised in dark areas of the image.  \nIn effect,the minimisation of expression(1)expressesthe fact that an artery is a stiff,elastic,dark entity.Theoptimal global solution of the position of an artery isthe choice of v(s)which minimises the total energygiven in(1).In general this is a non-linear problemwhich must be solved using iterative numerical methods,which produce successive deformations of the positionv(s).  \nA method for minimising integral(1)based on the useof the Euler-Lagrange Theory(ELT)has been presentedpreviously [3,4,5].  This paper presents an alternativeapproach based on the use of the Finite Element Method(FEM).In recent years,FEM has been used extensivelyin computational studies of elasticity in structuralmechanics.It offers benefits over the more traditionalELT,which it has now superceded in many areas ofstudy.We report here a modification to the FEM whichallows it to be used for model-based vision,and discussits computational advantages over ELT.  \nENERGY MINIMISATION BY ELT&FEM  \nBoth ELT and FEM may be used to solve variationalproblems,including the minimisation of energy integralssuch as expression(1).The principles of both methodshave been treated extensively in the literature [6,7].Thetwo approaches are illustrated in Figure 1,and theirmain characteristics are outlined briefly below.  \n(i)Euler-Lagrange Theory  \nThe ELT solves a variational problem by reducing it tothe solution of differential equations.The ELT methodused by [3,4,5]de","cbCaidx6LtG0pOHf","https://ap.wps.com/l/cbCaidx6LtG0pOHf","pdf",1967507,6,"English","# Introduction\n## Energy minimisation by ELT & FEM\n## The advantages of FEM over ELT","[{\"question\":\"What problem does the paper address in deformable model image interpretation?\",\"answer\":\"It addresses how to derive an optimal global solution for deformable models by minimizing an energy integral rather than relying on Euler-Lagrange-based formulations.\"},{\"question\":\"How does the proposed method differ from Euler-Lagrange Theory (ELT)?\",\"answer\":\"The method minimises the energy integral directly using the Finite Element Method, avoiding the need to transform the integral into differential equations solved numerically.\"},{\"question\":\"Why is FEM advantageous for this application?\",\"answer\":\"FEM is more straightforward, offers greater flexibility for energy integrals and boundary conditions, and is better suited to geometrically complex problems or discontinuous loads and properties.\"}]","A Finite Element Method for Deformable Models | PDF",1789556508]