DISCRETE APPROXIMATION OF A FREE DISCONTINUITY PROBLEM

We approximate by discrete GAMMA-convergence a functional proposed by Mumford-Shah for a variational approach to image segmentation. Such a functional is first relaxed with a sequence of nonconvex functionals, which in turn, are discretized by piecewise linear finite elements. Under a suitable relation between the relaxation parameter epsilon and the meshsize h, the convergence of the discrete functionals and the compactness of any sequence of discrete minimizers are proved. The proof relies on the techniques of GAMMA-convergence and on the properties of the Lagrange interpolation and Clement operators.