{"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/the-l1-potts-functional-for-robust-jump","title":"The L1-Potts functional for robust jump-sparse reconstruction","arxiv_id":"1207.4642","date":"2012-07-19","proceeding":null,"authors":["Andreas Weinmann","Martin Storath","Laurent Demaret"],"abstract":"We investigate the non-smooth and non-convex $L^1$-Potts functional in discrete and continuous time. We show $\\Gamma$-convergence of discrete $L^1$-Potts functionals towards their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete $L^1$-Potts problem, we introduce an $O(n^2)$ time and $O(n)$ space algorithm to compute an exact minimizer. We apply $L^1$-Potts minimization to the problem of recovering piecewise constant signals from noisy measurements $f.$ It turns out that the $L^1$-Potts functional has a quite interesting blind deconvolution property. In fact, we show that mildly blurred jump-sparse signals are reconstructed by minimizing the $L^1$-Potts functional. Furthermore, for strongly blurred signals and known blurring operator, we derive an iterative reconstruction algorithm.","url_abs":"https://arxiv.org/abs/1207.4642v3","url_pdf":"https://arxiv.org/pdf/1207.4642v3.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-l1-potts-functional-for-robust-jump","repo_url":"https://github.com/mstorath/Pottslab","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}