Generalized Sampling Expansion for the Quaternion Linear Canonical Transform

Generalized Sampling Expansion for the Quaternion Linear Canonical Transform | 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 Generalized Sampling Expansion for the Quaternion Linear Canonical Transform Saima Siddiqui, Bingzhao Li, Muhammad Adnan Samad This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3879933/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 13 Apr, 2024 Read the published version in Signal, Image and Video Processing → Version 1 posted 7 You are reading this latest preprint version Abstract The theory of quaternions has gained firm ground in recent times and is being widely explored, with the field ofsignal and image processing being no exception. However, many important aspects of quaternionic signals are yet tobe explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion linear canonical transform (QLCT). To facilitate the intent, we construct a set of quaternion-valued filter functions which are used to construct a system of equations determining the synthesis functions for the process of reconstruction. Besides, as a special case, another sampling formula involving the derivatives of the quaternionic signal is also obtained in the sequel. Since derivatives contain information about the edges and curves appearing in images, therefore, by invoking the higher degrees of freedom pertaining to QLCT, the obtained sampling formula can be used to develop new image scaling algorithms with state-of-the-art flexibility. As an endorsement of the obtained results, an example with simulations demonstrating the signal reconstruction is presented at the end. quaternion algebra quaternionic signals quaternion fourier transform sampling expansion Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 13 Apr, 2024 Read the published version in Signal, Image and Video Processing → Version 1 posted Editorial decision: Revision requested 27 Jan, 2024 Reviews received at journal 27 Jan, 2024 Reviewers agreed at journal 27 Jan, 2024 Reviewers invited by journal 26 Jan, 2024 Submission checks completed at journal 25 Jan, 2024 Editor assigned by journal 25 Jan, 2024 First submitted to journal 19 Jan, 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. 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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-3879933","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":269630005,"identity":"dae29526-66fb-4786-8e81-ebc25dbbfdd0","order_by":0,"name":"Saima Siddiqui","email":"","orcid":"","institution":"Beijing Institute of Technology","correspondingAuthor":false,"prefix":"","firstName":"Saima","middleName":"","lastName":"Siddiqui","suffix":""},{"id":269630006,"identity":"3804b94b-8bc9-40b5-8461-1e4207a9cd79","order_by":1,"name":"Bingzhao Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5klEQVRIiWNgGAWjYBACPmYGNoYEIMOAgYHxMYhkAHPxADYkLczGDAYGRGgBI4gWNmkwRVALOwPbg4c7au3N2Q8fqy4o+MPAz55jwPBzB16HsRsknjnObNmTlnZ7BtBhkj1vDBh7z+D3i0Ri2zE2gwM5Zrd5gFoMbuQYMDO2EdbCY3D+jVkxSIs9kVpqJICGmzGDbZEgrAXol7YDBgY3niVLzzAw5pE486zgYC8eLfz8B9ge/myrszc4n3zwc8EfOTn+9uSND37i0QLU9AFIHIZzeUDEAXwaoKCOCDWjYBSMglEwYgEApRxDsM2M90oAAAAASUVORK5CYII=","orcid":"","institution":"Beijing Institute of Technology","correspondingAuthor":true,"prefix":"","firstName":"Bingzhao","middleName":"","lastName":"Li","suffix":""},{"id":269630007,"identity":"1c6e670f-2262-4b4f-91c6-0fa9ff3b2c76","order_by":2,"name":"Muhammad Adnan Samad","email":"","orcid":"","institution":"Ferghana Polytechnical Institute","correspondingAuthor":false,"prefix":"","firstName":"Muhammad","middleName":"Adnan","lastName":"Samad","suffix":""}],"badges":[],"createdAt":"2024-01-19 21:44:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3879933/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3879933/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11760-024-03157-6","type":"published","date":"2024-04-13T15:01:38+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":54712675,"identity":"1cd63102-46c8-452e-bc6f-cd5ac227aa67","added_by":"auto","created_at":"2024-04-15 15:12:06","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":448722,"visible":true,"origin":"","legend":"","description":"","filename":"GeneralizedSamplingExpansionfortheQuaternionLinearCanonicalTransform.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3879933/v1_covered_c3d5075d-c2b5-40ca-8547-43b4541aa84a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Generalized Sampling Expansion for the Quaternion Linear Canonical Transform","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":"signal-image-and-video-processing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"sivp","sideBox":"Learn more about [Signal, Image and Video Processing](http://link.springer.com/journal/11760)","snPcode":"11760","submissionUrl":"https://submission.nature.com/new-submission/11760/3","title":"Signal, Image and Video Processing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"quaternion algebra, quaternionic signals, quaternion fourier transform, sampling expansion","lastPublishedDoi":"10.21203/rs.3.rs-3879933/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3879933/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The theory of quaternions has gained firm ground in recent times and is being widely explored, with the field ofsignal and image processing being no exception. 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