Anisotropic Mumford-Shah Model and its approximation with Gamma-convergence

A new model is introduced for the segmentation problem of thin structures, like tubes or thin plates, in an image. The energy is based on the Mumford-Shah model and it introduces as a new variable a continuous and anisotropic perturbation of the Hausdorff measure. A relaxed formulation in the special space of functions with bounded variations is given and the existence of a solution is established. In order to get an energy more adapted for numerics, an approximation with Γ-convergence and its complete proof are given. Introduction This work is motivated by the problem of segmentation of sets strongly elongated in some directions as, for example, tubes or thin plates in an image of dimension n ∈ {2; 3}. Let Ω ⊂ R be an open bounded domain and g ∈ L∞(Ω). We denote by Hn−1 the (n−1)-dimensional Hausdorff measure. The model we introduce in this paper consists in minimizing E(u,K,M) = ∫