Parameter Selection in a Mumford–Shah Geometrical Model for the Detection of Thin Structures

Abstract

We present a variational model to perform the segmentation of thin structures in MRI images (namely codimension 1 objects). It is based on the classical Mumford-Shah functional and we have added geometrical priors as constraints. We precisely describe the structure model (that we call tubes) and write the problem as a bilevel problem. We focus on the lower level optimization problem and give existence, uniqueness and regularity results for the solution. The keypoint is the fact that 2D/3D problems are equivalent to 1D ones. This gives hints to perform an automatic parameter tuning for numerical purpose.

An anisotropic Mumford–Shah model

“…Nevertheless, this result provides a continuous model (1.2) of the discrete energy which may be used as a mirror to understand the proper geometry of the problem. Then, an automatic determination of the (numerous) parameters of the algorithms is possible if we have an apriori on the image we want to get (see [21,2]).…”

A Nonlocal Laplacian-Based Model for Bituminous Surfacing Crack Recovery and its MPI Implementation

“…In [6,20], Aubert and Drogoul introduce a topological gradient-based method for the detection of fine structures in 2D, the major difference with our model lying in the introduction of an auxiliary variable encoding the crack-type singularities. Still in a variational setting, Bergounioux and Vicente ( [9]) propose a model derived from the Mumford-Shah functional and constrained by geometrical priors to perform the segmentation of tube-like structures with small diameter. A limitation of this method rests on its inability to handle junctions of tubes.…”