An anisotropic Mumford–Shah model

Abstract

A variational 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 with a surfacic term perturbed by a Finsler metric. The formulation in the special space of functions with bounded variations is given and, in order to get an energy more adapted for numerics, a result of Γconvergence is proved.

Phase-field approximation of functionals defined on piecewise-rigid maps

“…Then, for these partitions, a recovery sequence matching asymptotically the sharp lower bound can be contructed by creating a layer of order ε around the jump set of the target function u, in which the transition is one-dimensional and is obtained by a suitable scaling of the optimal profile. As for the Ambrosio-Tortorelli approximation of the Mumford-Shah functionals (see also, e.g., [33,7,8,9,58]) also in our case the regularised bulk and surface energy in (1.6) separately converge to their sharp counterparts. Namely, in this case the bulk term in (1.6) vanishes in the limit due to the presence of the diverging parameter k ε , that is, equivalently, limit deformations have (approximate) gradients in SO(n) a.e.…”

“…Remark 3.4 (Approximation of inhomogeneous anisotropic perimeter functionals). Arguing as in the proof of [58] (see also [33,Theorem 3.1]), in view of Proposition 3.1 one can establish a Γ-convergence result for functionals of the form…”

“…It is well-known that the relationship between variational models for brittle fracture and (gradient) damage models can be made rigorous building upon the seminal approximation result of Ambrosio and Tortorelli [5,6] as shown in [33,18,7,8,9,58,43,19], just to mention a few examples. Furthermore, damage models á la Ambrosio-Tortorelli can be also used to approximate fracture models of cohesive-type as shown, e.g., in [1,2,24,29,35,42,16,26].…”

Phase-Field Approximation of Functionals Defined on Piecewise-Rigid Maps

“…Arguing as in the proof of Vicente (2017) (see also Focardi 2001, Theorem 3.1), in view of Proposition 3.2 one can establish a -convergence result for functionals of the form…”