{"title":"A primal-dual adaptive finite element method for total variation minimization","authors":"Martin Alkämper, Stephan Hilb, Andreas Langer","doi":"10.1007/s10444-025-10254-8","DOIUrl":null,"url":null,"abstract":"<div><p>Based on previous work, we extend a primal-dual semi-smooth Newton method for minimizing a general <span>\\(\\varvec{L^1}\\)</span>-<span>\\(\\varvec{L^2}\\)</span>-<span>\\(\\varvec{TV}\\)</span> functional over the space of functions of bounded variations by adaptivity in a finite element setting. For automatically generating an adaptive grid, we introduce indicators based on a-posteriori error estimates. Further, we discuss data interpolation methods on unstructured grids in the context of image processing and present a pixel-based interpolation method. The efficiency of our derived adaptive finite element scheme is demonstrated on image inpainting and the task of computing the optical flow in image sequences. In particular, for optical flow estimation, we derive an adaptive finite element coarse-to-fine scheme which allows resolving large displacements and speeds up the computing time significantly.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 5","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10254-8.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Computational Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10444-025-10254-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Based on previous work, we extend a primal-dual semi-smooth Newton method for minimizing a general \(\varvec{L^1}\)-\(\varvec{L^2}\)-\(\varvec{TV}\) functional over the space of functions of bounded variations by adaptivity in a finite element setting. For automatically generating an adaptive grid, we introduce indicators based on a-posteriori error estimates. Further, we discuss data interpolation methods on unstructured grids in the context of image processing and present a pixel-based interpolation method. The efficiency of our derived adaptive finite element scheme is demonstrated on image inpainting and the task of computing the optical flow in image sequences. In particular, for optical flow estimation, we derive an adaptive finite element coarse-to-fine scheme which allows resolving large displacements and speeds up the computing time significantly.
期刊介绍：
Advances in Computational Mathematics publishes high quality, accessible and original articles at the forefront of computational and applied mathematics, with a clear potential for impact across the sciences. The journal emphasizes three core areas: approximation theory and computational geometry; numerical analysis, modelling and simulation; imaging, signal processing and data analysis.
This journal welcomes papers that are accessible to a broad audience in the mathematical sciences and that show either an advance in computational methodology or a novel scientific application area, or both. Methods papers should rely on rigorous analysis and/or convincing numerical studies.
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