Existence and Regularity of Solutions to a Variational Problem of Mumford and Shah: A Constructive Approach

We study a variational problem arising in the approach of Mumford and Shah to the image segmentation problem of computer vision. Given f epsilon L(infinity) (D) for a domain D in R(2), the simplified Mumford-Shah energy associated to a decomposition D = Omega(1) U ... U Omega(N) is [GRAPHICS] where alpha > 0 is a constant, c(Omega i), is the average of f(x) on Omega(i), and where \textbackslash{}Gamma\textbackslash{} is the length of the boundary of the regions Omega(i) not in partial derivative D. Mumford and Shah showed, using geometric measure theory, that for a continuous f a minimizing Gamma{*} exists that is piecewise C-2. We prove this result constructively, and also extend it to show for general bounded measurable f that a minimizer exists. Furthermore, we prove that every minimizer must be piecewise C-1,C-1. Our approach is to study E(0)[Gamma,alpha] on the class of piecewise linear Gamma.