{"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/generalized-minkowski-sets-for-the","title":"Generalized Minkowski sets for the regularization of inverse problems","arxiv_id":"1903.03942","date":"2019-03-10","proceeding":null,"authors":[],"abstract":"Many works on inverse problems in the imaging sciences consider\nregularization via one or more penalty functions or constraint sets. When the\nmodels/images are not easily described using one or a few penalty\nfunctions/constraints, additive model descriptions for regularization lead to\nbetter imaging results. These include cartoon-texture decomposition,\nmorphological component analysis, and robust principal component analysis;\nmethods that typically rely on penalty functions. We propose a regularization\nframework, based on the Minkowski set, that merges the strengths of additive\nmodels and constrained formulations. We generalize the Minkowski set, such that\nthe model parameters are the sum of two components, each of which is\nconstrained to an intersection of sets. Furthermore, the sum of the components\nis also an element of another intersection of sets. These generalizations allow\nus to include multiple pieces of prior knowledge on each of the components, as\nwell as on the sum of components, which is necessary to ensure physical\nfeasibility of partial-differential-equation based parameters estimation\nproblems. We derive the projection operation onto the generalized Minkowski\nsets and construct an algorithm based on the alternating direction method of\nmultipliers. We illustrate how we benefit from using more prior knowledge in\nthe form of the generalized Minkowski set using seismic waveform inversion and\nvideo background-anomaly separation.","url_abs":"http://arxiv.org/abs/1903.03942v1","url_pdf":"http://arxiv.org/pdf/1903.03942v1.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":"generalized-minkowski-sets-for-the","repo_url":"https://github.com/slimgroup/SetIntersectionProjection.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"additive-models","task_name":"Additive models"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}