A primal-dual adaptive finite element method for total variation minimization A primal-dual adaptive finite

• Published in: Advances in computational mathematics Vol. 51; no. 5
• Main Authors: Alkämper, Martin, Hilb, Stephan, Langer, Andreas
• Format: Journal Article
• Language: English
• Published: New York Springer US 21.08.2025
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Visualization; Inpainting; Total variation; data-fidelity; 90C25; 94A08; A-posteriori error estimate; Optical flow estimation; 65N50; 49M29; Non-smooth optimisation; Adaptive finite element discretization; Combined; 65N30
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-025-10254-8

Based on previous work, we extend a primal-dual semi-smooth Newton method for minimizing a general L 1 - L 2 - 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.