On the String Shaped Singularities of Linear Combinations of Vector Fields on the Solution of the Poisson Image Equation

On the String Shaped Singularities of Linear Combinations of Vector Fields on the Solution of the Poisson Image Equation | 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 On the String Shaped Singularities of Linear Combinations of Vector Fields on the Solution of the Poisson Image Equation Alexander C. Poltzer, Kiana N. Jones, Dane Sheridan, Nikolay M. Sirakov This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6173833/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract This research is a continuation of work that developed the vector fields (VFs) $\nabla\hat{\phi}$, $\overline{\nabla}\hat{\phi}$, $\nabla\hat{\psi}$, and $\overline{\nabla}\hat{\psi}$ generated from the solution of the Poisson Image Equation. In this study, we develop new VFs generated from linear combinations of the above VFs after proving they are not constant multiples of each other. Further, we classify their singular points (SPs), determine their locations, and formulate mappings between critical points (CPs) and corresponding VF SPs. From the new VFs, we discover and describe a novel feature that we name SP ribbons (SPRs), which are strings of SPs in homogeneous regions. We also prove the conditions for SPR existence and explore SP shape invariance under affine transformations. On the basis of the theory, we developed MATLAB code that embeds the new VFs into images and conducted numerous experiments with public image databases. Further, we showcase convolutional neural network (CNN) experiments that demonstrate that embedding our VFs into a medical image database improves machine learning (ML) image classification accuracy. The paper concludes with a discussion on the advantages of using the new VFs and SPRs and outlines directions for future work. Poisson equation vector fields linear combinations critical points and singular points machine learning classification Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions 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-6173833","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":488286122,"identity":"6a3f2233-622b-4ce4-a36f-95a2e90a1d8e","order_by":0,"name":"Alexander C. 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