Variational problems in SBV and image segmentation

This paper shows how it is possible to prove the existence of solutions of the Mumford-Shah image segmentation functional F(u,K) and deals with the regularity of minimizers and the approximation of F by elliptic functionals defined on Sobolev spaces.

Abstract

We show how it is possible to prove the existence of solutions of the Mumford-Shah image segmentation functional F(u,K) = ∫Ω\K [⊂∇u⊂2 + β(u − g)2]dx + αℋ n − 1(K), u ∈ W 1,2(Ω\K), K ⊄ Ω closed in Ω. We use a weak formulation of the minimum problem in a special class SBV(Ω) of functions of bounded variation. Moreover, we also deal with the regularity of minimizers and the approximation of F by elliptic functionals defined on Sobolev spaces. In this paper, we have collected the main results of Ambrosio and others.