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

TL;DR

This paper develops an adaptive finite element method for total variation minimization, enhancing image processing tasks like inpainting and optical flow estimation with improved efficiency and accuracy.

Contribution

It introduces a primal-dual semi-smooth Newton method with adaptive grid refinement based on a-posteriori error estimates for TV minimization.

Findings

Significant speed-up in optical flow computation.

Effective adaptive grid generation for image inpainting.

Improved accuracy in total variation minimization tasks.

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.