Mumford-Shah Problem Research Articles

Stationarity of the crack-front for the Mumford–Shah problem in 3D

In this paper we exhibit a family of stationary solutions of the Mumford–Shah functional in R3, arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet [4]. We also give a local version in a finite cylinder and prove an energy estimate for minimizers. Numerical illustrations indicate the stationary solutions are unlikely minimizers and show how the dependence on axial variable impacts the geometry of the discontinuity set. A self-contained proof of the stationarity of the cracktip function for the Mumford–Shah problem in 2D is presented.

Existence of minimizers for the 2d stationary Griffith fracture model

We consider the variational formulation of the Griffith fracture model in two spatial dimensions and prove the existence of strong minimizers, that is deformation fields that are continuously differentiable outside a closed jump set and that minimize the relevant energy. To this aim, we show that minimizers of the weak formulation of the problem, set in the function space GSBD2 and whose existence is well known, are actually strong minimizers following the approach developed by De Giorgi, Carriero, and Leaci in the corresponding scalar setting of the Mumford–Shah problem.

Joint image reconstruction and segmentation using the Potts model

We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford–Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation. We focus on Radon data, where we in particular consider limited data situations. For instance, our method is able to recover all segments of the Shepp–Logan phantom from seven angular views only. We illustrate the practical applicability on a real positron emission tomography dataset. As further applications, we consider spherical Radon data as well as blurred data.

Color Image Segmentation by the Vector-Valued Allen–Cahn Phase-Field Model: A Multigrid Solution

We present an efficient numerical solution of a PDE-driven model for color image segmentation and give numerical examples of the results. The method combines the vector-valued Allen-Cahn phase field equation with initial data fitting terms with prescribed interface width and fidelity constants. Efficient numerical solution is achieved using a multigrid splitting of a finite element space, thereby producing an efficient and robust method for the segmentation of large images. We also present the use of adaptive mesh refinement to further speed up the segmentation process.

Monotonicity and separation for the Mumford–Shah problem

We prove some monotonicity properties of the global Mumford–Shah minimizers as defined by Bonnet in [4]. The main consequence is that the only solutions for which the complement in the plane of the singular set is not connected correspond to lines and propellers. We also get a boundary version of the Mumford–Shah conjecture.

Implementation of an adaptive finite-element approximation of the Mumford-Shah functional

We present and detail a method for the numerical solving of the Mumford-Shah problem, based on a finite element method and on adaptive meshes. We start with the formulation introduced in [13], detail its numerical implementation and then propose a variant which is proved to converge to the Mumford-Shah problem. A few experiments are illustrated.

Flatness and finiteness in the Mumford-Shah problem

We extend A. Bonnet's results about the solutions of the global Mumford-Shah problem (see Bonnet (1996)) by replacing his connected assumption by more natural flatness assumptions. Nous étendons les résultats d'A. Bonnet sur les solutions du problème de Mumford-Shah global (voir Bonnet (1996)) en remplaçant son hypothèse de connexité par des hypothèses de platitude plus naturelles.