We prove a homogenization result for Mumford-Shah-type energies associated to a brittle composite material with weak inclusions distributed periodically at a scale ${\varepsilon}>0$. The matrix and the inclusions in the material have the same elastic moduli but very different toughness moduli, with the ratio of the toughness modulus in the matrix and in the inclusions being $1/β_{\varepsilon}$, with $β_{\varepsilon}>0$ small. We show that the high-contrast behaviour of the composite leads to the emergence of interesting effects in the limit: The volume and surface energy densities interact by $Γ$-convergence, and the limit volume energy is not a quadratic form in the critical scaling $β_{\varepsilon} = {\varepsilon}$, unlike the ${\varepsilon}$-energies, and unlike the extremal limit cases.
We propose a $Γ$-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.
We establish a new monotonicity formula for minimizers of the Mumford-Shah functional in planar domains. Our formula follows the spirit of Bucur-Luckhaus, but works with the David-Léger entropy instead of the energy. Interestingly, this allows for a sharp truncation threshold. In particular, our monotonicity formula is able to discriminate between points at a $C^1$ interface and any other type of singularity in terms of a finite gap in the entropy. As a corollary, we prove an optimal lower bound on the energy density around any nonsmooth point for minimizers of the Mumford-Shah functional.
We give an alternative proof of the regularity, up to the loose end, of minimizers, resp. critical points of the Mumford-Shah functional when they are sufficiently close to the cracktip, resp. they consist of a single arc terminating at an interior point.
We prove that if $(u,Γ)$ is a minimizer of the functional $$ J(u,Γ)=\int_{B_1(0)\setminus Γ}|\nabla u|^2dx +\H^1(Γ) $$ and $Γ$ connects $\partial B_1(0)$ to a point in the interior, then $Γ$ satisfies a point-wise $C^{2,α}$-estimate at the crack-tip. This means that the Mumford-Shah functional satisfies an additional, and previously unknown, Euler-Lagrange condition. ******* The previous version of the paper contained some mistakes, which has been fixed. More explanations/details has been added in Section 6.
We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford-Shah functionals converge to a minimizer of a continuum Mumford-Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate …
We show that functions in $GSBV^p$ in three-dimensional space with small variation in $2$ of $3$ directions are close to a function of one variable outside an exceptional set. Bounds on the volume and the perimeter in these two directions of the exceptional sets are provided. As a key tool we prove an approximation result for such functions by functions in $W^{1,p}$. For this we present a two-dimensional countable ball construction that allows to carefully remove the jumps of the function. As a direct application, we show $Γ$-convergence of an anisotropic three-dimensional Mumford-Shah model t…
We consider in 2D the following special case of the Mumford-Shah functional $$ J(u, Γ)=\int_{B_1\backslashΓ} |\nabla u|^2 dx + λ^2 \fracπ{2} \mathcal{H}^1(Γ). $$ It is known that if the minimizer has a crack-tip in the ball $B_1$ (assume at the origin), then $u\approx λ\Im \sqrt{z}$ at this point. We calculate higher order terms in the asymptotic expansion, where the homogeneity orders of those terms appear to be solutions to a certain trigonometric relation.
In electrical impedance tomography, we aim to solve the conductivity within a target body through electrical measurements made on the surface of the target. This inverse conductivity problem is severely ill-posed, especially in real applications with only partial boundary data available. Thus regularization has to be introduced. Conventionally regularization promoting smooth features is used, however, the Mumford--Shah regularizer familiar for image segmentation is more appropriate for targets consisting of several distinct objects or materials. It is, however, numerically challenging. We show…
In this paper we introduce a new phase field approximation of the Mumford-Shah functional similar to the well-known one from Ambrosio and Tortorelli. However, in our setting the phase field is allowed to be a function of bounded variation, instead of an $H^1$-function. In the context of image segmentation, we also show how this new approximation can be used for numerical computations, which contains a total variation minimization of the phase field variable, as it appears in many problems of image processing. A comparison to the classical Ambrosio-Tortorelli approximation, where the phase fiel…
