Convergence of level sets in total variation denoising through variational curvatures in unbounded domains

TL;DR

This paper investigates how the level sets of solutions to total variation denoising converge in unbounded domains, using variational mean curvatures without requiring regularity assumptions on the data.

Contribution

It introduces a geometric convergence analysis of level sets in total variation denoising using variational curvatures, applicable even when level set regularity is lacking.

Findings

Level sets converge geometrically as regularization vanishes.

Explicit variational mean curvatures facilitate convergence analysis.

Results apply to both noisy and noiseless data scenarios.

Abstract

We present some results of geometric convergence of level sets for solutions of total variation denoising as the regularization parameter tends to zero. The common feature among them is that they make use of explicit constructions of variational mean curvatures for general sets of finite perimeter. Consequently, no additional regularity of the level sets of the ideal data is assumed, and in particular the subgradient of the total variation at it could be empty. In exchange, other restrictions on the data or on the noise are required. We consider two cases: characteristic functions with a parameter choice depending on the noise level, and noiseless generic data.