Finite-differences discretizations of the mumford-shah functional

About two years ago, Gobbino [21] gave a proof of a De Giorgi's conjecture on the approximation of the Mumford-Shah energy by means of finite-differences based non-local functionals. In this work, we introduce a discretized version of De Giorgi's approximation, that may be seen as a generalization of Blake and Zisserman's “weak membrane” energy (first introduced in the image segmentation framework). A simple adaptation of Gobbino's results allows us to compute the Γ-limit of this discrete functional as the discretization step goes to zero; this generalizes a previous work by the author on the “weak membrane” model [10]. We deduce how to design in a systematic way discrete image segmentation functionals with “less anisotropy” than Blake and Zisserman's original energy, and we show in some numerical experiments how it improves the method.