On the regularity of the total variation minimizers

TL;DR

This paper establishes regularity properties of total variation minimizers used in image processing, showing that the solution inherits the Lipschitz regularity of the source term in any dimension and domain.

Contribution

It extends previous regularity results for total variation minimizers to all dimensions and general domains, removing previous restrictions.

Findings

Solutions are Lipschitz continuous if the source term is Lipschitz.

Regularity results hold in any dimension and for any regular domain.

The paper generalizes earlier results limited to low dimensions and convex domains.

Abstract

We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by L. Rudin, S. Osher and E. Fatemi. In particular we show that if the source term $f$ is locally, respectively globally, Lipschitz, then the solution has the same regularity with local, respectively global, Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by V. Caselles, A. Chambolle and M. Novaga for dimension $N\leq 7$ and (in case of the global regularity) for convex domains.