Logarithmic norm minimization of quaternion matrix decomposition for color image sparse representation

Logarithmic norm minimization of quaternion matrix decomposition for color image sparse representation | 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 Logarithmic norm minimization of quaternion matrix decomposition for color image sparse representation Xiaomin Cai, Yi-Fen Ke, ChangFeng Ma, Yajun Xie, Ri-Wei Liao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4335898/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 19 Jul, 2024 Read the published version in Numerical Algorithms → Version 1 posted 9 You are reading this latest preprint version Abstract In this paper, incorporating the quaternion matrix framework, the logarithmic norm of quaternion matrices is employed to approximate rank. Unlike conventional sparse representation techniques for matrices, which treat RGB channels separately, quaternion-based methods maintain image structure by representing color images within a pure quaternion matrix. Leveraging the logarithmic norm, factorization and truncation techniques can be applied for proficient image recovery. Optimization of these approaches is facilitated through an alternate minimization framework, supplemented by meticulous mathematical scrutiny ensuring convergence. Finally, some numerical examples are used to demonstrate the effectiveness of the proposed algorithms. Quaternion matrix Matrix decomposition Logarithmic norm Color image reconstruction Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 19 Jul, 2024 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 30 Jun, 2024 Reviews received at journal 27 Jun, 2024 Reviews received at journal 15 May, 2024 Reviewers agreed at journal 15 May, 2024 Reviewers agreed at journal 10 May, 2024 Reviewers invited by journal 10 May, 2024 Editor assigned by journal 10 May, 2024 Submission checks completed at journal 09 May, 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. 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-4335898","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":302737797,"identity":"3e98ce2a-db22-41b7-863a-7256098e60ec","order_by":0,"name":"Xiaomin Cai","email":"","orcid":"","institution":"Fujian Normal University","correspondingAuthor":false,"prefix":"","firstName":"Xiaomin","middleName":"","lastName":"Cai","suffix":""},{"id":302737798,"identity":"3a355cc7-586e-4fae-b31f-61a3ebdd12c3","order_by":1,"name":"Yi-Fen Ke","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVRIie3LPwrCMBTH8VcE6xBwfaWoV2gpiOBl2qVdqjh2cMhUl4CH8QA+CdQldXZwcHJy0AuI8c8c6yaYL4RH4PcBsNl+MNSPCoC+y+nxbzUkCiBi9A0BTRLxFE2ItxAhUXHIhFNXCMU44W5NRuIzFRCp00TALkVQWcLZNDaSPuaBvJZysgY1RKeUCUcWmMngHNDmJjP2JLcGxNcD2nAZvwhvQDyRzogqGQpQ0SiusqhkuZngVq4uNJcDhircX+bj3tJVZgLQeQ+QAGJ92x/2Ovf4ul3+eWuz2Wz/2R23zEwcu3VVPgAAAABJRU5ErkJggg==","orcid":"","institution":"Fujian Normal University","correspondingAuthor":true,"prefix":"","firstName":"Yi-Fen","middleName":"","lastName":"Ke","suffix":""},{"id":302737799,"identity":"cfc4bdb2-9157-4685-9f37-66baf4be35c3","order_by":2,"name":"ChangFeng Ma","email":"","orcid":"","institution":"Fujian Normal University","correspondingAuthor":false,"prefix":"","firstName":"ChangFeng","middleName":"","lastName":"Ma","suffix":""},{"id":302737800,"identity":"23cdac2e-5a7b-4bff-af93-b33efe8ed7d8","order_by":3,"name":"Yajun Xie","email":"","orcid":"","institution":"Fuzhou University of International Studies and Trade","correspondingAuthor":false,"prefix":"","firstName":"Yajun","middleName":"","lastName":"Xie","suffix":""},{"id":302737801,"identity":"3ae7c6a7-c16e-4506-9966-0ccf63f66feb","order_by":4,"name":"Ri-Wei Liao","email":"","orcid":"","institution":"Fujian Normal University","correspondingAuthor":false,"prefix":"","firstName":"Ri-Wei","middleName":"","lastName":"Liao","suffix":""}],"badges":[],"createdAt":"2024-04-28 02:55:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4335898/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4335898/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-024-01887-9","type":"published","date":"2024-07-19T16:04:52+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":61594634,"identity":"760ffd30-927f-4a2f-abf9-55af600e9b0a","added_by":"auto","created_at":"2024-08-01 17:13:23","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4536311,"visible":true,"origin":"","legend":"","description":"","filename":"20240401.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4335898/v1_covered_38c87ed6-9cce-4c14-938c-02916cb9f1b7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Logarithmic norm minimization of quaternion matrix decomposition for color image sparse representation","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":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Quaternion matrix, Matrix decomposition, Logarithmic norm, Color image reconstruction","lastPublishedDoi":"10.21203/rs.3.rs-4335898/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4335898/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this paper, incorporating the quaternion matrix framework, the logarithmic norm of quaternion matrices is employed to approximate rank. 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