Search

We propose a $\Gamma$-convergent discrete approximation of the Mumford-Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford-Shah functional in any dimension., Comment: 47 pages, reorganized version

In this paper, we present a method for the numerical minimization of the Mumford-Shah functional that is based on the idea of topological asymptotic expansions. The basic idea is to cover the expected edge set with balls of radius \epsilon > 0 and use the number of balls, multiplied with 2\epsilon, as an estimate for the length of the edge set. We introduce a functional based on this idea and prove that it converges in the sense of \Gamma-limits to the Mumford-Shah functional. Moreover, we show that ideas from topological asymptotic analysis can be used for determining where to position the balls covering the edge set. The results of the proposed method are presented by means of two numerical examples and compared with the results of the classical approximation due to Ambrosio and Tortorelli., Comment: 28 pages, 2 figures

We study the asymptotic limit of obstacle problems for Mumford–Shah type functionals with pgrowth in periodically perforated domains via the Γ -convergence of the associated free-discontinuity energies. In the limit a non-trivial penalization term related to the 1-capacity of the reference hole appears if and only if the size of the perforation scales like en/(n−1), e being its periodicity. We give two different formulations of the obstacle problem to include also perforations with Lebesgue measure zero.

Under what conditions can one say something about the geometric structure of the singular set of a function? A famous result of this type states that the singular set (= set of non-Lebesgue points) of a function of bounded variation on R n is (countably) rectifiable. In this paper we shall be concerned with quantitative forms of rectifiability, and we give a quantitative version of this theorem. We also give uniform rectifiability results for the singular sets of minimizers of higher-dimensional versions of the Mumford-Shah functional. Along the way we shall encounter some generalizations of the usual topological notion of a set “separating” points in the complement, including one which is based on the failure of Poincare inequalities on the complement of the given set.

We present a series of results on the regularity of edges in the Mumford-Shah minimization process. We prove that an isolated connected component of the edge set is a finite union of C 1,1 arcs. A fundamental step of the proof is the characterization of the four possible global-minimizers having a connected edge.