Inferring the Geometry of Convex Shapes from Their Gauss Digitization

Inferring the Geometry of Convex Shapes from Their Gauss Digitization | 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 Inferring the Geometry of Convex Shapes from Their Gauss Digitization Jacques-Olivier Lachaud, David Coeurjolly, Tristan Roussillon This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8742703/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 11 You are reading this latest preprint version Abstract This paper studies how well we can infer the geometry of a (smooth or not) convex shape X from the convex hull Yh of its Gauss digitization with a given gridstep h. Without smoothness constraint on X, we first present results concerning the proximity of facet normal vectors to the shape normal vectors, as well as a relation between the number of lattice points just above a facet and its area. Then, further results can be obtained when X is smooth, that are valid in arbitrary dimension d. More precisely, we show that the boundary of Yh is Hausdorff-close to the boundary of X with distance less than $\sqrt{d}h$, and that the vertices of Yh are even much closer (some $O(h^{\frac{2d}{d+1}})$). Our main result states that the geometric normal vectors to the facets of Yh tend to the smooth shape normals with a speed $O(h^{\frac{1}{2}}), and the bound is tight. Finally we compare experimentally the performances of several normal estimators built upon the normal vectors to the facets of Yh with state-of-the-art estimators. We also perform statistical analyses over the facets of digitized convex hulls, like their area, diameter or width as a function of the digitization gridstep. Both our new theoretical properties and our numerical experiments confirm that the convex hull of a digitized shape provide relevant information on the geometry of the underlying Euclidean convex shape, and can be used to construct fast and accurate geometric estimators. Geometric inference Gauss digitization Convex hull geometry Digital normal estimation Digital geometry Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 15 Apr, 2026 Reviews received at journal 08 Apr, 2026 Reviewers agreed at journal 03 Mar, 2026 Reviews received at journal 20 Feb, 2026 Reviews received at journal 20 Feb, 2026 Reviewers agreed at journal 11 Feb, 2026 Reviewers agreed at journal 09 Feb, 2026 Reviewers invited by journal 08 Feb, 2026 Editor assigned by journal 04 Feb, 2026 Submission checks completed at journal 02 Feb, 2026 First submitted to journal 30 Jan, 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. 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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-8742703","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":589865080,"identity":"3405668a-19cb-4dff-a913-6bd31a4661dd","order_by":0,"name":"Jacques-Olivier Lachaud","email":"data:image/png;base64,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","orcid":"","institution":"Université Savoie Mont Blanc","correspondingAuthor":true,"prefix":"","firstName":"Jacques-Olivier","middleName":"","lastName":"Lachaud","suffix":""},{"id":589865087,"identity":"b5e79310-524e-417c-9b37-3289f83319ff","order_by":1,"name":"David Coeurjolly","email":"","orcid":"","institution":"French National Centre for Scientific Research","correspondingAuthor":false,"prefix":"","firstName":"David","middleName":"","lastName":"Coeurjolly","suffix":""},{"id":589865089,"identity":"9efd6487-273a-4b70-b219-3dd51bda547e","order_by":2,"name":"Tristan Roussillon","email":"","orcid":"","institution":"Institut National des Sciences Appliquées de Lyon","correspondingAuthor":false,"prefix":"","firstName":"Tristan","middleName":"","lastName":"Roussillon","suffix":""}],"badges":[],"createdAt":"2026-01-30 15:23:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8742703/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8742703/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102747588,"identity":"69a22903-4ad2-4b28-a92c-bdb5704cac36","added_by":"auto","created_at":"2026-02-16 09:05:00","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":13129822,"visible":true,"origin":"","legend":"","description":"","filename":"JMIV2026.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8742703/v1_covered_98983f5c-9b86-4b03-b117-dbe398043814.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Inferring the Geometry of Convex Shapes from Their Gauss Digitization","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":"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":"Geometric inference, Gauss digitization, Convex hull geometry, Digital normal estimation, Digital geometry","lastPublishedDoi":"10.21203/rs.3.rs-8742703/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8742703/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper studies how well we can infer the geometry of a (smooth or not) convex shape X from the convex hull Yh of its Gauss digitization with a given gridstep h. 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