Density Properties and Existence Theory for the Mumford-Shah Minimizers
The Essentiality Property: Around every point x of K one has that ℋ1(K ∩ D(R)) > 0, where D(R) denotes the disk B(x,R). In other terms, no point of K is “isolated”.
The Uniform Density Property: There is a universal constant β such that at every point x ∈ K, ℋ1(K∩D(R)) > βR. This clearly is a stronger nonsparseness property for K.
The Concentration Property: For every ε > 0, there is α > 0 such that every disk D(R) centered on the edge set K contains a subdisk D′ with radius larger than αR inside which K is “concentrated”, that is ℋ1(K ∩ D′) ≥ (1 - ε)diam(D′). This property clearly implies the Uniform Density Property.
The First Projection Property: There is a universal constant β1 such that ∀x ∈ K, ℋ1(p1(K ∩ D(R))) + ℋ1(p2(K ∩ D(R))) ≥ βR, where p1 and p2 denote the projectors in two orthogonal directions.
From the Uniform Projection Property we deduce, by using the results of Chapter 12 about projections of 1-sets, that the Mumford-Shah minima and quasiminima (in the sense of Property (M)) must be rectifiable.
From the Uniform Projection Property, we deduce that any minimal or quasiminimal (in the sense of property (M)) rectifiable segmentation can be approximated by a finite set of curves.
From both the Projection Property and the Uniform Concentration Property, we deduce, by using the results of Chapter 10, that the set of segmentations satisfying (M) is compact and that the minimum of the Mumford-Shah energy is attained for some uniformly rectifiable segmentation.
We end the chapter by showing that the minimal segmentation may be nonunique, which somehow matches the computer scientist’s intuition tha t more than one segmentation can be “good” for a given image.
KeywordsConcentration Property Universal Constant Uniform Density Projection Property Minimality Property
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