The Piecewise Constant Mumford and Shah Model: Mathematical Analysis
In this chapter, we start with the mathematical proofs and show that the minimizing of the “simplest” segmentation energy discussed in the first chapters, the Mumford and Shah piecewise constant model, yields properties sought by most segmentation devices. Indeed, the most primitive segmentation tool, the “merging”, applied to this variational model, is enough to ensure a small (compact) set of possible segmentations, with no small regions and no thin regions. Uniform a priori estimates for the size and number of the regions can be given for all segmentations obtained by exhaustive “merging”.
In Section 5.1, we state the main existence theorem. In Section 5.2, we give some notation and define the “2-normal segmentations” as a formalization of the segmentations obtained by exhaustive “merging”. We list some useful topological properties of these segmentations. In Section 5.3, we prove several compactness, existence and regularity results for 2-normal and optimal segmentations and obtain a proof of Theorem 5.1. Section 5.4 is devoted to the definition of an algorithm computing 2-normal segmentations and to some numerical experiments on real images.
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