APPROXIMATION OF FUNCTIONALS DEPENDING ON JUMPS BY ELLIPTIC FUNCTIONALS VIA GAMMA-CONVERGENCE

We show how it is possible to approximate the Mumford-Shah (see[29]) image segmentation functional [GRAPHICS] by elliptic functionals defined on Sobolev spaces. The heuristic idea is to consider functionals G(h)(u, z) with z ranging between 0 and 1 and related to the set K. The minimizing z(h) are near to 1 in a neighborhood of the set K, and far from the neighborhood they are very small. The neighborhood shrinks as h --> + infinity. For a similar approach to the problem compare Kulkarni; see [25]. The approximation of G(h) to G takes place in a variational sense, the De Giorgi-GAMMA-convergence.