On explicit solutions for the problem of Mumford and Shah
(1999) Communications in Contemporary Mathematics — Vol. 1, n° 2, p. 201-212 (1999)
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- Abstract
- 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 Omega and the image is a concentric disk D subset of Omega. 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 partial derivative Omega and partial derivative D. 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 Omega and D are convex, we prove the following for an optimal segmentation (u, K) such that K subset of D: K tends to partial derivative D (in the Hausdorff distance) when the difference of intensity between Omega and D goes to infinity.
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De Pauw, T., & Smets, D. (1999). On explicit solutions for the problem of Mumford and Shah. Communications in Contemporary Mathematics, 1(2), 201-212. https://doi.org/10.1142/S0219199799000092 (Original work published 1999)