with D. Smets, On explicit solutions for the problem of Mumford and Shah, Comm. Contemp. Math., 1(2), 1999. 201-212.
We look for explicit image segmentations in the framework of the variational model proposed by Mumford and Shah. We first treat the symmetric case when the "screen" is a disk D and the image is a concentric disk C ⊂ D. We prove the optimal segmentation is either the given disk D or the solution of the associated Neumann problem, depending on both the difference of intensity between the background and the disk, and the distance separating Bdry D and Bdry C. Both segmentations are optimal in some critical cases which we characterize. Our main result is a first step towards a generalization of this behaviour. In case D and C are convex, we prove the following for an optimal segmentation (u,K) such that K ⊂ C : K tends to Bdry C (in the Hausdorff distance) when the difference of intensity between D and C goes to infinity.