Mumford-Shah Minimization Research Articles

Endpoint regularity for 2d Mumford-Shah minimizers: On a theorem of Andersson and Mikayelyan

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.

A selective review on Mumford–Shah minimizers

This is a review paper on the Mumford-Shah functional, with particular interest in blow-up limits, low dimensions (2 and 3), and connection with fracture theory. It includes in particular a sketch proof of the 3D regularity result near minimal cones from my thesis, and a presentation of a few more recent results that appeared in the last two years.

A rigidity result for global Mumford–Shah minimizers in dimension three

We study global Mumford–Shah minimizers in RN, introduced by Bonnet as blow-up limits of Mumford–Shah minimizers. We prove a new monotonicity formula for the energy of u when the singular set K is contained in an (N−1)-dimensional cone. We then use this monotonicity to prove that for any reduced global minimizer (u,K) in R3, if K is contained in a half-plane and touches its edge, then it is the half-plane itself. This partially answers to a question of Guy David.

Four color theorem and convex relaxation for image segmentation with any number of regions

Image segmentation is an essential problem in imaging science. One of the most successful segmentation models is the piecewise constant Mumford-Shah minimization model. This minimization problem is however difficult to carry out, mainly due to the non-convexity of the energy. Recent advances based on convex relaxation methods are capable of estimating almost perfectly the geometry of the regions to be segmented when the mean intensity and the number of segmented regions are known a priori. The next important challenge is to provide a tight approximation of the optimal geometry, mean intensity and the number of regions simultaneously while keeping the computational time and memory usage reasonable. In this work, we propose a new algorithm that combines convex relaxation methods with the four color theorem to deal with the unsupervised segmentation problem. More precisely, the proposed algorithm can segment any a priori unknown number of regions with only four intensity functions and four indicator (``labeling') functions. The number of regions in our segmentation model is decided by one parameter that controls the regularization strength of the geometry, i.e., the total length of the boundary of all the regions. The segmented image function can take as many constant values as needed.

Regularity of the singular set for Mumford-Shah minimizers in R3 near a minimal cone

We show that if (u;K) is a minimizer of the Mumford-Shah functional in an open set of R 3 , and if x2 K and r > 0 are such that K is close enough to a minimal cone of type P (a plane), Y (three half planes meeting with 120 angles) or T (cone over a regular tetrahedron centered at the origin) in terms of Hausdor distance in B(x;r), then K is C 1; equivalent to the minimal cone in B(x;cr) where c < 1 is an universal constant.

Mumford-Shah minimizers on thin plates

We show that limits of Mumford-Shah minimizers in product domains Ω = Ω′× (0,t), t small, are Mumford-Shah minimizers in one less dimension. The main ingredient of the proof is a symmetry argument from Dal Maso, Morel, and Solimini.

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.

On the regularity of edges in image segmentation

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 C1,1 arcs. A fundamental step of the proof is the characterization of the four possible global-minimizers having a connected edge.