Titlebook: Variational Methods in Image Segmentation; with seven image pro Jean Michel Morel,Sergio Solimini Book 1995 Birkhäuser Boston 1995 Mathemat

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The Piecewise Constant Mumford and Shah Model: Mathematical Analysisormalization 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

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Semicontinuity Properties of the Hausdorff Measureve that the set of closed subsets of a compact set is compact. It would be extremely convenient to have the following continuity property: If a sequence of sets A. converges for the Hausdorff distance towards a set A, then the Hausdorff measures of the A. also converge to the Hausdorff measure of A.

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Properties of Regular and Rectifiable Setsrties of unrectifiable sets. It is first proved that there is a universal constant c(α) such that at almost every point of a fully unrectifiable set, the lower conic density is larger that c(α). This is a local and clear cut computational criterion for distinguishing rectifiable points from unrectif

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Small Oscillation Coverings and the Excision Methodnimum, we shall conclude that K is not too “ragged”, whatever the disk D is, and regularity properties for K will follow. Let us now see roughly which kind of raggedness condition we shall consider. We shall assume that the edge set can be covered by a family of sets D. on whose boundary, ∂D., the f

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Density Properties and Existence Theory for the Mumford-Shah Minimizerse and more precise density properties for minimal segmentations and, more generally, for segmentations satisfying the minimality property (M). Recall that this property essentially states that no excision of a disk can decrease the Mumford-Shah energy of the segmentation. To make a long story short,

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