Improved parameterized preconditioner for linear system from multiplicative half-quadratic regularized image restoration

Improved parameterized preconditioner for linear system from multiplicative half-quadratic regularized image restoration | 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 Improved parameterized preconditioner for linear system from multiplicative half-quadratic regularized image restoration Peipei Zhao, Ruijiang Wu, Ruiping Wen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8705355/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 01 May, 2026 Read the published version in Numerical Algorithms → Version 1 posted 7 You are reading this latest preprint version Abstract Image restoration is a fundamental problem in image processing, recovering a clear image from its degraded observation is an ill-posed problem, which can be addressed effectively by regularized techniques. Image restoration can usually be solved by minimizing a cost function consisting of a data-fidelity term and a regularization term. In this paper, we consider the multiplicative half-quadratic regularized image restoration problem and the Newton method is employed to solve the regularized model. At each step of the Newton iteration, a linear system of equations with symmetric positive definite coefficient matrix needs to be solved. Based on the block triangular decomposition of the coefficient matrix, by introducing a parameter, and taking the truncated Taylor expansion form of the Schur complement inverse matrix, we design a parameterized preconditioner of the coefficient matrix and combine it into the conjugate gradient method to solve the linear system efficiently. The spectral properties of the preconditioned matrix are further analyzed. Numerical experiments demonstrate that the proposed preconditioner for solving the multiplicative half-quadratic regularization image restoration problem is more robust and effective, in reducing both the number of iterations and the overall computation time compared with the existing preconditioners. Image restoration Multiplicative HQ regularization Truncated Taylor expansion Parameterized preconditioner Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 01 May, 2026 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 28 Mar, 2026 Reviews received at journal 28 Mar, 2026 Reviewers agreed at journal 06 Feb, 2026 Reviewers invited by journal 02 Feb, 2026 Editor assigned by journal 31 Jan, 2026 Submission checks completed at journal 30 Jan, 2026 First submitted to journal 26 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-8705355","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":584919796,"identity":"63d37696-cd91-4887-83c4-68526c4d91c2","order_by":0,"name":"Peipei Zhao","email":"","orcid":"","institution":"Taiyuan Normal University","correspondingAuthor":false,"prefix":"","firstName":"Peipei","middleName":"","lastName":"Zhao","suffix":""},{"id":584919797,"identity":"b8df0f36-a4ef-45a3-bd71-ac8b623682b7","order_by":1,"name":"Ruijiang Wu","email":"","orcid":"","institution":"Taiyuan Normal University","correspondingAuthor":false,"prefix":"","firstName":"Ruijiang","middleName":"","lastName":"Wu","suffix":""},{"id":584919804,"identity":"ae7c4b42-45f2-431e-a90c-0aedbc81d0ce","order_by":2,"name":"Ruiping Wen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAm0lEQVRIiWNgGAWjYNCCCgk5eRK1nLEwNmwgSQdjW0UiwwFiVRsc7z32uXCeRAJjA/PDRzeI0nLmXPLsmdsk8tgZ2IyNc4jRYnYjx5iZd5tEMWMDD5s0cVruvwFqmSOR2HCAaC03eIBaGkjRYn8G6LAZxySMDZuJ9Ytk+xlj5oKaOjl59uaHj4nSAgLMSCRJWkbBKBgFo2AU4AIApGAqnEKLO8MAAAAASUVORK5CYII=","orcid":"","institution":"Taiyuan Normal University","correspondingAuthor":true,"prefix":"","firstName":"Ruiping","middleName":"","lastName":"Wen","suffix":""}],"badges":[],"createdAt":"2026-01-27 03:38:22","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8705355/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8705355/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-026-02380-1","type":"published","date":"2026-05-01T15:58:05+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":108437681,"identity":"f9d814e2-3d0c-46c2-a942-6acd104c0501","added_by":"auto","created_at":"2026-05-04 16:02:10","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3012472,"visible":true,"origin":"","legend":"","description":"","filename":"Manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8705355/v1_covered_f8035ccf-e292-4696-96a1-8c785ff99f0e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Improved parameterized preconditioner for linear system from multiplicative half-quadratic regularized image restoration","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":"Image restoration, Multiplicative HQ regularization, Truncated Taylor expansion, Parameterized preconditioner","lastPublishedDoi":"10.21203/rs.3.rs-8705355/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8705355/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Image restoration is a fundamental problem in image processing, recovering a clear image from its degraded observation is an ill-posed problem, which can be addressed effectively by regularized techniques. 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