A generalisation of flat morphology, II: main properties, duality and hybrid operators

A generalisation of flat morphology, II: main properties, duality and hybrid operators | 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 A generalisation of flat morphology, II: main properties, duality and hybrid operators Christian Ronse This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2657672/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 28 Mar, 2023 Read the published version in Journal of Mathematical Imaging and Vision → Version 1 posted 4 You are reading this latest preprint version Abstract In a previous paper, we gave a new general theory of the construction of flat operators on grey-level or multivalued images from operators on binary images. While the traditional approach was based on threshold superposition, we rely instead on threshold summation, and this allows a correct formulation for non-increasing flat operators, and also for operators with non-binary outputs. We obtained then some basic properties of flat operators, valid for both increasing and non-increasing operators. Here we pursue this work by investigating further properties of flat operators, which differ in the increasing and non-increasing cases, in particular the composition, join and meet of operators, and the commutation with contrast mappings. We study duality under inversion and characterise discrete linear convolution operators as flat operators. This allows to integrate various hybrid morphological operators into our framework. MSC Classification: 68U10 , 06B23 , 06E30 , 06F20 , 26A45 generalised flat morphological operator threshold summation duality linearity Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 28 Mar, 2023 Read the published version in Journal of Mathematical Imaging and Vision → Version 1 posted Editorial decision: Accepted 07 Mar, 2023 Editor assigned by journal 07 Mar, 2023 Submission checks completed at journal 06 Mar, 2023 First submitted to journal 05 Mar, 2023 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-2657672","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":181265812,"identity":"68e74759-5b4d-466e-a9b5-08048641fcfd","order_by":0,"name":"Christian Ronse","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwUlEQVRIiWNgGAWjYPACGwYGZgiLn1gtaSAtjA1AlmQDkVoOgwgitfDPPnzs4Y+K84nb2dmfP66oYZAwJ6RH4lxaujHPmduJO5t5DBvPHGOQkDlAQIsBD4+ZNGPb7cQNh3kYGxvYGOokCDkMpEXy579zQC3sDxsb/jFIEKVFgrfhAFALg2FjYxsRWiTOsKVJ8xxLNgY6zHBmY58EYS38PczHJH/U2MluOH/8wceGbzaEtWDYSqqGUTAKRsEoGAXYAABhEDlfhZkReAAAAABJRU5ErkJggg==","orcid":"","institution":"Université de Strasbourg","correspondingAuthor":true,"prefix":"","firstName":"Christian","middleName":"","lastName":"Ronse","suffix":""}],"badges":[],"createdAt":"2023-03-05 14:44:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2657672/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2657672/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10851-023-01145-y","type":"published","date":"2023-03-28T20:15:02+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":44723859,"identity":"466212aa-066e-4af5-b877-4fc9d2d7eedf","added_by":"auto","created_at":"2023-10-16 20:22:31","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":729032,"visible":true,"origin":"","legend":"","description":"","filename":"SUB.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2657672/v1_covered_f0c80ffb-4ff7-40f6-98bc-42e1f94ab9da.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A generalisation of flat morphology, II: main properties, duality and hybrid operators","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":"journal-of-mathematical-imaging-and-vision","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jmiv","sideBox":"Learn more about [Journal of Mathematical Imaging and Vision](http://link.springer.com/journal/10851)","snPcode":"10851","submissionUrl":"https://submission.nature.com/new-submission/10851/3","title":"Journal of Mathematical Imaging and Vision","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"generalised flat morphological operator, threshold summation, duality, linearity","lastPublishedDoi":"10.21203/rs.3.rs-2657672/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2657672/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn a previous paper, we gave a new general theory of the construction of flat operators on grey-level or multivalued images from operators on binary images. 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