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arXiv · 2404.03125

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

Martin Alkämper·Stephan Hilb·Andreas Langer
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Abstract

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.

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Martin Alkämper, Stephan Hilb, Andreas Langer. 2025-02-18. A primal-dual adaptive finite element method for total variation minimization. https://arxiv.org/abs/2404.03125

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