{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-numerical-framework-for-efficient-motion","title":"A Numerical Framework for Efficient Motion Estimation on Evolving Sphere-Like Surfaces based on Brightness and Mass Conservation Laws","arxiv_id":"1805.01006","date":"2018-05-02","proceeding":null,"authors":["Lukas F. Lang"],"abstract":"In this work we consider brightness and mass conservation laws for motion\nestimation on evolving Riemannian 2-manifolds that allow for a radial\nparametrisation from the 2-sphere. While conservation of brightness constitutes\nthe foundation for optical flow methods and has been generalised to said\nscenario, we formulate in this article the principle of mass conservation for\ntime-varying surfaces which are embedded in Euclidean 3-space and derive a\ngeneralised continuity equation. The main motivation for this work is efficient\ncell motion estimation in time-lapse (4D) volumetric fluorescence microscopy\nimages of a living zebrafish embryo. Increasing spatial and temporal resolution\nof modern microscopes require efficient analysis of such data. With this\napplication in mind we address this need and follow an emerging paradigm in\nthis field: dimensional reduction. In light of the ill-posedness of considered\nconservation laws we employ Tikhonov regularisation and propose the use of\nspatially varying regularisation functionals that recover motion only in\nregions with cells. For the efficient numerical solution we devise a Galerkin\nmethod based on compactly supported (tangent) vectorial basis functions.\nFurthermore, for the fast and accurate estimation of the evolving sphere-like\nsurface from scattered data we utilise surface interpolation with\nspatio-temporal regularisation. We present numerical results based on\naforementioned zebrafish microscopy data featuring fluorescently labelled\ncells.","url_abs":"http://arxiv.org/abs/1805.01006v1","url_pdf":"http://arxiv.org/pdf/1805.01006v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"a-numerical-framework-for-efficient-motion","repo_url":"https://github.com/lukaslang/ofcm","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"motion-estimation","task_name":"Motion Estimation"},{"task_slug":"optical-flow-estimation","task_name":"Optical Flow Estimation"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}