Stable regular critical points of the Mumford-Shah functional are local minimizers

TL;DR

This paper proves that regular critical points of the Mumford-Shah functional with positive definite second variation are local minimizers in the L^1-topology, establishing a link between criticality and minimality.

Contribution

It demonstrates that regular critical points with positive definite second variation are necessarily local minimizers, clarifying the stability properties of solutions to the Mumford-Shah problem.

Findings

Regular critical points with positive definite second variation are local minimizers.

Such minimizers are isolated in the L^1-topology.

The result applies to the Mumford-Shah functional in image segmentation.

Abstract

In this paper it is shown that any regular critical point of the Mumford-Shah functional, with positive definite second variation, is an isolated local minimizer with respect to competitors which are sufficiently close in the L^1-topology.