Variational problems in image segmentation and $Gamma$-convergence methods

Variational models for image segmentation aim to recover a piecewise smooth approximation of a given input image together with a discontinuity set which represents the boundaries of the segmentation. In particular, the variational method introduced by Mumford and Shah includes the length of the discontinuity boundaries in the energy. Because of the presence of such a geometric term, the minimization of the corresponding functional is a difficult numerical problem. We consider a mathematical framework for the Mumford-Shah functional and we discuss the computational issue. We suggest the use of the Γ-convergence theory to approximate the functional by elliptic functionals which are convenient for the purpose of numerical computation. We then discuss the design of an iterative numerical scheme for image segmentation based on the Γ-convergent approximation. The relation between the Mumford-Shah model and the Perona-Malik equation will be also discussed.