{"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/enhanced-image-approximation-using-shifted","title":"Enhanced image approximation using shifted rank-1 reconstruction","arxiv_id":"1810.01681","date":"2018-10-03","proceeding":null,"authors":["Florian Boßmann","Jianwei Ma"],"abstract":"Low rank approximation has been extensively studied in the past. It is most suitable to reproduce rectangular like structures in the data. In this work we introduce a generalization using shifted rank-1 matrices to approximate $A\\in\\mathbb{C}^{M\\times N}$. These matrices are of the form $S_{\\lambda}(uv^*)$ where $u\\in\\mathbb{C}^M$, $v\\in\\mathbb{C}^N$ and $\\lambda\\in\\mathbb{Z}^N$.The operator $S_{\\lambda}$ circularly shifts the k-th column of $uv^*$ by $\\lambda_k$. These kind of shifts naturally appear in applications, where an object $u$ is observed in $N$ measurements at different positions indicated by the shift $\\lambda$. The vector $v$ gives the observation intensity. Exemplary, a seismic wave can be recorded at $N$ sensors with different time of arrival $\\lambda$; Or a car moves through a video changing its position in every frame. We present theoretical results as well as an efficient algorithm to calculate a shifted rank-1 approximation in $O(NM \\log M)$. The benefit of the proposed method is demonstrated in numerical experiments. A comparison to other sparse approximation methods is given. Finally, we illustrate the utility of the extracted parameters for direct information extraction in several applications including video processing or non-destructive testing.","url_abs":"http://arxiv.org/abs/1810.01681v1","url_pdf":"http://arxiv.org/pdf/1810.01681v1.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":"enhanced-image-approximation-using-shifted","repo_url":"https://github.com/Vitaly-Protasov/NLA-project","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"enhanced-image-approximation-using-shifted","repo_url":"https://github.com/sevenysw/MathGeo2018","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"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}