Some properties of minimizers for the Chan-Esedoglu L1TV functional

TL;DR

This paper characterizes minimizers of the Chan-Esedoglu L1TV functional, revealing geometric bounds on their boundaries and how small perturbations affect minimizers, with connections to recent theoretical results.

Contribution

It provides new geometric characterizations of minimizers for the L1TV functional, including boundary bounds and stability properties, extending understanding of these minimizers.

Findings

Existence of minimizers with boundaries between unions of specific-radius balls.

Small perturbations in the domain allow for controlled modifications of minimizers.

Connections made to recent theoretical results by Allard on L1TV minimizers.

Abstract

We present two results characterizing minimizers of the Chan-Esedoglu L1TV functional $F(u) \equiv \int |\nabla u | dx + \lambda \int |u - f| dx $; $u,f:\Bbb{R}^n \to \Bbb{R}$. If we restrict to $u = \chi_{\Sigma}$ and $f = \chi_{\Omega}$, $\Sigma, \Omega \in \Bbb{R}^n$, the $L^1$TV functional reduces to $E(\Sigma) = \Per(\Sigma) + \lambda |\Sigma\vartriangle \Omega |$. We show that there is a minimizer $\Sigma$ such that its boundary $\partial\Sigma$ lies between the union of all balls of radius $\frac{n}{\lambda}$ contained in $\Omega$ and the corresponding union of $\frac{n}{\lambda}$-balls in $\Omega^c$. We also show that if a ball of radius $\frac{n}{\lambda} + \epsilon$ is almost contained in $\Omega$, a slightly smaller concentric ball can be added to $\Sigma$ to get another minimizer. Finally, we comment on recent results Allard has obtained on $L^1$TV minimizers and how these relate to our results.