An approach to the minimization of the Mumford–Shah functional using $\Gamma$-convergence and topological asymptotic expansion

Abstract

In this paper, we present a method for the numerical minimization of the Mumford–Shah functional that is based on the idea of topological asymptotic expansions. The basic idea is to cover the expected edge set with balls of radius $ϵ>0$ and use the number of balls, multiplied with $2ϵ$, as an estimate for the length of the edge set. We introduce a functional based on this idea and prove that it converges in the sense of  $\Gamma$-limits to the Mumford–Shah functional. Moreover, we show that ideas from topological asymptotic analysis can be used for determining where to position the balls covering the edge set. The results of the proposed method are presented by means of two numerical examples and compared with the results of the classical approximation due to Ambrosio and Tortorelli.