On a Variational Problem from Image Processing

Let Ω be a reasonably smooth domain in R², and let g ∈ L^∞(Ω) be given (g often represents an image). The Mumford-Shah functional is defined by J(u, K) = ʃʃ_Ω\K |u − g|² + ʃʃ_Ω\K |∇u |² + H¹ (K), where K is a closed set in Ω with finite one-dimensional Hausdorffmeasure H¹ (K), and u is a C¹-function on Ω\K.

The purpose of these notes is to give a fairly brief account of various regularity results on the set K when the pair (u, K) is an irreducible minimizer for the functional J (.). In particular we show that K satisfies the “elimination property” and the “concentration property” of Dal Maso, Morel, and Solimini, the “property of projections” of Dibos and Koepfler, and that it is contained in an Ahlfors-regular curve.