BézierFormer: Affine-Invariant Shape Classification via Control Point Attention

BézierFormer: Affine-Invariant Shape Classification via Control Point Attention | 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 BézierFormer: Affine-Invariant Shape Classification via Control Point Attention Xiao Liu, Jean-Michel Morel, Roy Y. He This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8755906/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 9 You are reading this latest preprint version Abstract We address the resolution paradox in modern deep learning, where networks receive far more spatial information than they demonstrably utilize for shape classification. We show that it is possible to train networks directly on spatially sparse and structurally compressed shape representations rather than dense pixel grids. Specifically, we extract vector graphic representations of shapes from raster images, and train on control points of curves, which naturally encode the sparse, localized features (corners, curvature extrema) that both human vision and interpretability studies identify as critical for recognition. To effectively process these sparse geometric primitives, we propose an attention-based architecture, BézierFormer, that processes each parametric curve independently through shared-weight transformations, then synthesizes global shape understanding through tailored attention mechanisms. This combination of sparse vector graphic training data and segment-wise processing with attention-based synthesis achieves computational efficiency while maintaining high discriminative power, demonstrating that classification can be performed with dramatically fewer geometric primitives than pixels in conventional approaches. Shape classification geometric representation vector graphics affine-shortening flow Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 12 May, 2026 Reviews received at journal 29 Apr, 2026 Reviews received at journal 08 Mar, 2026 Reviewers agreed at journal 09 Feb, 2026 Reviewers agreed at journal 09 Feb, 2026 Reviewers invited by journal 09 Feb, 2026 Editor assigned by journal 04 Feb, 2026 Submission checks completed at journal 04 Feb, 2026 First submitted to journal 01 Feb, 2026 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-8755906","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":588616950,"identity":"d1ae0155-bb30-43a4-9897-611da2c1e773","order_by":0,"name":"Xiao Liu","email":"","orcid":"","institution":"City University of Hong Kong","correspondingAuthor":false,"prefix":"","firstName":"Xiao","middleName":"","lastName":"Liu","suffix":""},{"id":588616951,"identity":"992fa533-c1cf-4ccc-a7f3-0ae81101a012","order_by":1,"name":"Jean-Michel Morel","email":"","orcid":"","institution":"Lingnan University","correspondingAuthor":false,"prefix":"","firstName":"Jean-Michel","middleName":"","lastName":"Morel","suffix":""},{"id":588616952,"identity":"d095dac8-a873-4cac-bf4c-47a401d809ab","order_by":2,"name":"Roy Y. He","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVRIiWNgGAWjYHACNgaGChsQCQLMDAwSbMRoOZPGwMZGkhbGtsMMDERrMW/vMXvwge28PJ98j5kEQ4V1YoN0WwJeLTJnzpgbzuC5bdjGxgPUciY9sUHm2AG8WiQkcsykeSRuJ7CBtABdmNggkd5AWMsfg3NQLf+I1cKQcACqpQGkJY2Aw3iOlUn2HEgG+iWt2CLhWLpxm0RaAn4t7M3bJH7+s5OXbz688caHGmvZfok0A7xaGBg4DBAMkPGEI5KB/QE6YxSMglEwCkYBKgAA0oo72Ae/CicAAAAASUVORK5CYII=","orcid":"","institution":"City University of Hong Kong","correspondingAuthor":true,"prefix":"","firstName":"Roy","middleName":"Y.","lastName":"He","suffix":""}],"badges":[],"createdAt":"2026-02-01 11:53:28","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8755906/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8755906/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102745682,"identity":"fceb753b-4fd6-499d-b18f-92fbc9ea8daa","added_by":"auto","created_at":"2026-02-16 08:53:18","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1800650,"visible":true,"origin":"","legend":"","description":"","filename":"New.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8755906/v1_covered_32fb8df3-1b46-4a7f-9cbd-b04b016fcdf5.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"BézierFormer: Affine-Invariant Shape Classification via Control Point Attention","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-mathematical-imaging-and-vision","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jmiv","sideBox":"Learn more about [Journal of Mathematical Imaging and Vision](http://link.springer.com/journal/10851)","snPcode":"10851","submissionUrl":"https://submission.nature.com/new-submission/10851/3","title":"Journal of Mathematical Imaging and Vision","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Shape classification, geometric representation, vector graphics, affine-shortening flow","lastPublishedDoi":"10.21203/rs.3.rs-8755906/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8755906/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We address the resolution paradox in modern deep learning, where networks receive far more spatial information than they demonstrably utilize for shape classification. We show that it is possible to train networks directly on spatially sparse and structurally compressed shape representations rather than dense pixel grids. Specifically, we extract vector graphic representations of shapes from raster images, and train on control points of curves, which naturally encode the sparse, localized features (corners, curvature extrema) that both human vision and interpretability studies identify as critical for recognition. To effectively process these sparse geometric primitives, we propose an attention-based architecture, BézierFormer, that processes each parametric curve independently through shared-weight transformations, then synthesizes global shape understanding through tailored attention mechanisms. This combination of sparse vector graphic training data and segment-wise processing with attention-based synthesis achieves computational efficiency while maintaining high discriminative power, demonstrating that classification can be performed with dramatically fewer geometric primitives than pixels in conventional approaches.","manuscriptTitle":"BézierFormer: Affine-Invariant Shape Classification via Control Point Attention","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-11 18:33:42","doi":"10.21203/rs.3.rs-8755906/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-05-12T07:34:50+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-29T09:42:42+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-03-08T14:04:08+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"252851578929739433785336015444279572662","date":"2026-02-10T01:01:08+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"76609648670726710832557172873916102525","date":"2026-02-09T14:31:02+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-02-09T07:14:46+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-02-04T13:58:54+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-02-04T05:18:18+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Mathematical Imaging and Vision","date":"2026-02-01T11:44:58+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-mathematical-imaging-and-vision","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jmiv","sideBox":"Learn more about [Journal of Mathematical Imaging and Vision](http://link.springer.com/journal/10851)","snPcode":"10851","submissionUrl":"https://submission.nature.com/new-submission/10851/3","title":"Journal of Mathematical Imaging and Vision","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"819b004b-b892-4ff7-bc3d-17736179685f","owner":[],"postedDate":"February 11th, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Revision requested","date":"2026-05-12T07:34:50+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-29T09:42:42+00:00","index":10,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2026-05-12T07:43:27+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-11 18:33:42","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8755906","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8755906","identity":"rs-8755906","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}