Split Bregman Isotropic and Anisotropic Image Deblurring with Kronecker Product Sum Approximations using Single Precision Enlarged-GKB or RSVD Algorithms to provide low rank truncated SVDs 

Split Bregman Isotropic and Anisotropic Image Deblurring with Kronecker Product Sum Approximations using Single Precision Enlarged-GKB or RSVD Algorithms to provide low rank truncated SVDs | 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 Split Bregman Isotropic and Anisotropic Image Deblurring with Kronecker Product Sum Approximations using Single Precision Enlarged-GKB or RSVD Algorithms to provide low rank truncated SVDs Abdulmajeed Alsubhi, Rosemary Renaut This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5390994/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 May, 2025 Read the published version in Numerical Algorithms → Version 1 posted 9 You are reading this latest preprint version Abstract We consider the solution of the ℓ 1 regularized image deblurring problem using isotropic and anisotropic regularization implemented with the split Bregman algorithm. We replace the system matrix A using a Kronecker product approximation obtained via an approximate truncated singular value decomposition for a reordering of the matrix A . To obtain the approximate decomposition for the reordered matrix, we propose the enlarged Golub Kahan Bidiagonalization algorithm that proceeds by enlarging the Krylov subspace beyond either a given rank for the desired approximation, or uses an automatic stopping test that provides a suitable rank for the approximation. The resultant expansion is contrasted with the use of the truncated and the randomized singular value decompositions with the same number of terms. To further extend the scale of problem that can be considered we implement the determination of the approximation using single precision, while performing all steps for the regularization in double precision. The reported numerical tests demonstrate the effectiveness of applying the approximate single precision Kronecker product expansion for A , combined with either isotropic or anisotropic regularization implemented using the split Bregman algorithm, for the solution of image deblurring problems. As the size of the problem increases, our results demonstrate that the major costs are associated with determining the Kronecker product approximation, rather than with the cost of the regularization algorithm. Moreover, the enlarged Golub Kahan Bidiagonalization algorithm competes favorably with the randomized singular value decomposition for estimating the approximate singular value decomposition. MSC Classification: 65F22 , 65F10 , 68W40 ℓ1 regularization Split Bregman Anisotropic and Isotropic Kronecker Product Single and double precision Golub Kahan Bidiagonalization Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 06 May, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 25 Feb, 2025 Reviews received at journal 13 Feb, 2025 Reviews received at journal 09 Feb, 2025 Reviewers agreed at journal 20 Nov, 2024 Reviewers agreed at journal 18 Nov, 2024 Reviewers invited by journal 17 Nov, 2024 Editor assigned by journal 14 Nov, 2024 Submission checks completed at journal 13 Nov, 2024 First submitted to journal 04 Nov, 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-5390994","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":380464874,"identity":"ffd6fd4f-0c5b-4dd9-b679-df38adb8a934","order_by":0,"name":"Abdulmajeed Alsubhi","email":"","orcid":"","institution":"Arizona State University","correspondingAuthor":false,"prefix":"","firstName":"Abdulmajeed","middleName":"","lastName":"Alsubhi","suffix":""},{"id":380464875,"identity":"0e10bba0-a60a-4479-9fbb-ee4be7d75845","order_by":1,"name":"Rosemary Renaut","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAApUlEQVRIiWNgGAWjYPACGwjFQ7yOhDTStRwmQYt5e+/Bz7w/zsvLz0hgfPC2jQgtMmfOJUvzJNw23HAjgdlwLjFaJCRyDEBaGDdIJLBJ8xKlRf6N8W+ehHP282cksP8mTosEjxnQlgOJDTcS2JiJ08KTY2Y5Jy05ecOZh82Sc84Ro4X9jPGNNzZ2tvPbkw9+eFNGhBYkwNhAmvpRMApGwSgYBbgBAJ+qMOEhxUCqAAAAAElFTkSuQmCC","orcid":"","institution":"Arizona State University","correspondingAuthor":true,"prefix":"","firstName":"Rosemary","middleName":"","lastName":"Renaut","suffix":""}],"badges":[],"createdAt":"2024-11-04 23:38:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5390994/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5390994/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-025-02086-w","type":"published","date":"2025-05-06T15:57:12+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":82538170,"identity":"4e8eb3c9-e645-4a43-a66a-3816d4348339","added_by":"auto","created_at":"2025-05-12 16:10:38","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":791527,"visible":true,"origin":"","legend":"","description":"","filename":"NumericalAlgorithm.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5390994/v1_covered_431b5169-ebcd-4c39-a023-cf266e16f092.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Split Bregman Isotropic and Anisotropic Image Deblurring with Kronecker Product Sum Approximations using Single Precision Enlarged-GKB or RSVD Algorithms to provide low rank truncated SVDs ","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":"ℓ1 regularization, Split Bregman, Anisotropic and Isotropic, Kronecker Product, Single and double precision, Golub Kahan Bidiagonalization","lastPublishedDoi":"10.21203/rs.3.rs-5390994/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5390994/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe consider the solution of the \u003cem\u003eℓ\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e regularized \u0026nbsp;image deblurring problem \u0026nbsp;using isotropic and anisotropic regularization implemented with \u0026nbsp;the split Bregman algorithm. We replace the system matrix \u003cem\u003eA\u003c/em\u003e using a Kronecker product approximation obtained via an approximate truncated singular value decomposition for a reordering \u0026nbsp; \u0026nbsp;of the matrix \u003cem\u003eA\u003c/em\u003e. To obtain the approximate decomposition for the reordered matrix, we propose the enlarged Golub Kahan Bidiagonalization algorithm that proceeds by enlarging the Krylov subspace beyond either a given rank for the desired approximation, or uses an automatic stopping test that provides a suitable rank for the approximation. The resultant expansion is contrasted with the use of the truncated and the randomized singular value decompositions with the same number of terms. To further extend the scale of problem that can be considered we implement the determination of the approximation using single precision, while performing all steps for the regularization in double precision. The reported numerical tests demonstrate the effectiveness of applying the approximate single precision Kronecker product expansion for \u003cem\u003eA\u003c/em\u003e, combined with either isotropic or anisotropic regularization implemented using the split Bregman algorithm, for the solution of image deblurring \u0026nbsp;problems. \u0026nbsp;As the size of the problem increases, our results demonstrate that the major costs are associated with determining the Kronecker product approximation, rather than with the cost of the regularization algorithm. 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