Directional Projection of the Hessian: A Robust Method for Edge Detection

Directional Projection of the Hessian: A Robust Method for Edge Detection | 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 Directional Projection of the Hessian: A Robust Method for Edge Detection Mohamed LAJILI, Fella Hayat Laroui This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9115139/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Edge detection remains a fundamental component in many computer vision and image processing applications, including image segmentation, object recognition, and scene understanding. Despite extensive research, achieving both high precision and robustness in the presence of noise and structural ambiguity remains challenging. This paper introduces a novel edge detection method based on Directional Hessian Projection (DHP), which exploits second-order differential structure by projecting the local Hessian matrix onto multiple orientations. The proposed approach constructs a directional edge map by averaging the absolute directional curvatures obtained from these projections. This mathematically grounded formulation enhances the detection of curvature-rich and fine-scale structures while improving robustness to noise and filtering artifacts. Extensive experiments on synthetic and natural images, including the BSDS500 benchmark and the MultiCue dataset, show that DHP consistently outperforms several classical edge detectors such as Sobel, Canny, Marr–Hildreth, and wavelet-based methods. Quantitatively, DHP achieves lower MSE, higher PSNR and accuracy, and an average precision (AP) of 0.77 on BSDS500. On the MultiCue dataset, it reaches an ODS of 0.901 and an OIS of 0.93, with precision of 0.85 and recall of 0.96. Compared with deep learning approaches, DHP does not require large annotated datasets while narrowing the performance gap with learning-based methods. Applied Mathematics Theoretical Computer Science Numerical Analysis Edge Detection Local Features Second-order derivatives Curvature analysis Directional Moments Noisy Image Directional filtering. Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted 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. 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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-9115139","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":605677933,"identity":"1a168126-1128-438b-a2b4-4b704aff1e5f","order_by":0,"name":"Mohamed LAJILI","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7ElEQVRIiWNgGAWjYFAC5gYGCQMLHn4gUwKIeRjYGRg+4NfCCNIiwSPZANPCzMA4g6AWkGKDAxAtDAS16DYwNn6wKJCQMT5+xvDGxx11MvxALc0VeLSYHWBslgA5zOxMjrHlzDOHeSSbGRgbz+DX0gDRciDHTJq37QCPwWEG9ocNBGz5AdJi3P/GTPpvWx2P/WGgLQS0tIFtMZAA2sLYxsxjwExIy2HGNguQFokbz4ote9sO80gcZmzEr+V48+HbEn9s7Pn7kzfe+NlWZ8/f3nwQrxZgLDAwS6AKMeLVAFFCIH2MglEwCkbBSAcAOOFDuTxRSWwAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0002-9508-3587","institution":"Audensiel","correspondingAuthor":true,"prefix":"","firstName":"Mohamed","middleName":"","lastName":"LAJILI","suffix":""},{"id":605677934,"identity":"5e96a8a4-090d-45a8-8afb-8ce21d946730","order_by":1,"name":"Fella Hayat Laroui","email":"","orcid":"","institution":"Aix Marseille University","correspondingAuthor":false,"prefix":"","firstName":"Fella","middleName":"Hayat","lastName":"Laroui","suffix":""}],"badges":[],"createdAt":"2026-03-13 13:11:18","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9115139/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9115139/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104782258,"identity":"50403f5b-36c2-4b31-927f-b9c8db9e8d32","added_by":"auto","created_at":"2026-03-17 07:57:04","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3262357,"visible":true,"origin":"","legend":"","description":"","filename":"HessianMatrixEdgeDetection8.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9115139/v1_covered_540f4773-b037-46a6-955f-fb58cc1735b3.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eDirectional Projection of the Hessian: A Robust Method for Edge Detection\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Edge Detection, Local Features, Second-order derivatives, Curvature analysis, Directional Moments, Noisy Image, Directional filtering.","lastPublishedDoi":"10.21203/rs.3.rs-9115139/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9115139/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eEdge detection remains a fundamental component in many computer vision and image processing applications, including image segmentation, object recognition, and scene understanding. 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