# Density Properties and Existence Theory for the Mumford-Shah Minimizers

• Jean Michel Morel
• Sergio Solimini
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 14)

## Abstract

In this chapter, we shall find a first application of the elimination techniques which we have introduced in the preceding chapter. We shall prove more 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, let us say that we shall prove (in this order):
• 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.

We shall also prove a Second Projection Property, which is in the same relation to the first as the Concentration Property is to the Uniform Density Property. We could have directly stated and proved the two last density properties, which are stronger, but this would have made the exposition tedious and technical. As a matter of fact, each one of the above-mentioned properties will be used in the proof of existence of minimizers for the Mumford-Shah energy. To be more precise, we shall make in the second part of this chapter the following deductions.
• 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.

## Keywords

Concentration Property Universal Constant Uniform Density Projection Property Minimality Property
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