Image Segmentation by Variational Methods: Mumford and Shah Functional and the Discrete Approximations

In this paper we discuss the links between Mumford and Shah’s variational problem for (signal and) image segmentation, based on an energy functional of a continuous grey-level function, and the numerical algorithms proposed to solve it. These numerical approaches are based on a discrete functional. We recall that, in one dimension, this discrete functional is asymptotically equivalent to the continuous functional. This can be summarized in a Γ-convergence result. We show that the same result holds in dimension two, provided that the continuous energy is adapted to the anisotropy of the discrete approaches. We display a few experimental results in dimensions one and two.