Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields

Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields Dhruv Meduri, Mohit Sharma, Vijay Natarajan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4334752/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 04 Jun, 2024 Read the published version in The Visual Computer → Version 1 posted 4 You are reading this latest preprint version Abstract The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. It captures the regions of topological changes in the bivariate field. The Jacobi set is a bivariate analog of critical points, and may correspond to features of interest. In the specific case of time-varying fields and when one of the scalar fields is time, the Jacobi set corresponds to temporal tracks of critical points, and serves as a feature-tracking graph. The Jacobi set of a bivariate field or a time-varying scalar field is complex, resulting in cluttered visualizations that are difficult to analyze. This paper addresses the problem of Jacobi set simplification. Specifically, we use the time-varying scalar field scenario to introduce a method that computes a reduced Jacobi set. The method is based on a stability measure called robustness that was originally developed for vector fields and helps capture the structural stability of critical points. We also present a mathematical analysis for the method, and describe an implementation for 2D time-varying scalar fields. Applications to both synthetic and real-world datasets demonstrate the effectiveness of the method for tracking features. Time-varying fields Visualization Jacobi set Critical points Topological simplification Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 04 Jun, 2024 Read the published version in The Visual Computer → Version 1 posted Editorial decision: Revision requested 09 May, 2024 Editor assigned by journal 29 Apr, 2024 Submission checks completed at journal 29 Apr, 2024 First submitted to journal 27 Apr, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4334752","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":298966088,"identity":"4f90409a-b6ab-4a82-8f51-b5f0af03bd74","order_by":0,"name":"Dhruv Meduri","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAElEQVRIie3QsUoDMRjA8YRAu5xmzQ36DJ8ciNDbfJHvEO4Gm1IQipud7HLg2scoHNxqJKvW9dzyABXOLYOCSbuJjR0d8ocEPsiPhBASi/3HGKksQn7H/YBTvyduDQJkSIww0xLTuSfgwV+EU5OaXiMoPx1CQDMEBD3JuuuiN0CKx+GLIv1MB8gAEaG6Oe9kI9zDijqZIF2uQyRBhTCibSdXO0LGwI7uQ4R/zBEYbZaysVvCN8C+goRcuY+6LFZCtrtbhLuFBkiqSelImYnn9/YCQWR1t4Gnel3tJcevqqT2Mz/lC9m82dv8ZPEwPjN2NtpLfia2uzr4fCwWi8V+7Rt4oVhlxgyY5gAAAABJRU5ErkJggg==","orcid":"","institution":"University of Utah","correspondingAuthor":true,"prefix":"","firstName":"Dhruv","middleName":"","lastName":"Meduri","suffix":""},{"id":298966089,"identity":"bb947a47-41ec-463b-9251-fe3dfa55ffa5","order_by":1,"name":"Mohit Sharma","email":"","orcid":"","institution":"Indian Institute of Science Bangalore","correspondingAuthor":false,"prefix":"","firstName":"Mohit","middleName":"","lastName":"Sharma","suffix":""},{"id":298966090,"identity":"3ea87487-0697-4777-ac70-9a519f8293f8","order_by":2,"name":"Vijay Natarajan","email":"","orcid":"","institution":"Indian Institute of Science Bangalore","correspondingAuthor":false,"prefix":"","firstName":"Vijay","middleName":"","lastName":"Natarajan","suffix":""}],"badges":[],"createdAt":"2024-04-27 15:36:28","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4334752/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4334752/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00371-024-03484-2","type":"published","date":"2024-06-05T00:50:28+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":57806063,"identity":"5d55ecc2-84da-4ef4-a5c1-801d7b9391b2","added_by":"auto","created_at":"2024-06-06 00:50:37","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2956725,"visible":true,"origin":"","legend":"","description":"","filename":"cgi2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4334752/v1_covered_73947dee-4ee8-4bbe-b716-1eccea1f00eb.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"the-visual-computer","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"tvcj","sideBox":"Learn more about [The Visual Computer](http://link.springer.com/journal/371)","snPcode":"371","submissionUrl":"https://submission.nature.com/new-submission/371/3","title":"The Visual Computer","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Time-varying fields, Visualization, Jacobi set, Critical points, Topological simplification","lastPublishedDoi":"10.21203/rs.3.rs-4334752/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4334752/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. 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