Finite-differences discretizations of the mumford-shah functional

Abstract

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