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On a variational problem from image processing. (English)
J. Fourier Anal. Appl. Spec. Iss., 161-187 (1995).
Summary: Let $Ω$ be a reasonably smooth domain in $R^2$, and let $g\in L^\infty(Ω)$ be given ($g$ often represents an image). The Mumford-Shah functional is defined by $$J(u,K)= \iint_{Ω\backslash K}|u-g|^2+ \iint_{Ω\backslash K}|\nabla u|^2+ H^1(K),$$ where $K$ is a closed set in $Ω$ with finite one-dimensional Hausdorff measure $H^1(K)$, and $u$ is a $C^1$-function on $Ω\backslash K$. The purpose of this note 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 $J(\cdot)$. 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.