A note on convergence of solutions of total variation regularized linear inverse problems

TL;DR

This paper extends convergence results of total variation regularized solutions from denoising to general linear inverse problems, demonstrating the applicability across various domains and illustrating behavior with numerical examples.

Contribution

It generalizes the convergence of total variation solutions under source conditions to a broader class of inverse problems beyond denoising.

Findings

Convergence of level-sets in general inverse problems under source conditions

Applicability to denoising, deblurring, and Radon transform inversion

Numerical illustrations of convergence behavior

Abstract

In a recent paper by A. Chambolle et al. [Geometric properties of solutions to the total variation denoising problem. Inverse Problems 33, 2017] it was proven that if the subgradient of the total variation at the noise free data is not empty, the level-sets of the total variation denoised solutions converge to the level-sets of the noise free data with respect to the Hausdorff distance. The condition on the subgradient corresponds to the source condition introduced by Burger and Osher [Convergence rates of convex variational regularization. Inverse Problems 20, 2004], who proved convergence rates results with respect to the Bregman distance under this condition. We generalize the result of Chambolle et al. to total variation regularization of general linear inverse problems under such a source condition. As particular applications we present denoising in bounded and unbounded, convex and non convex domains, deblurring and inversion of the circular Radon transform. In all these examples the convergence result applies. Moreover, we illustrate the convergence behavior through numerical examples.