Calculus of Variations and Geometric Measure Theory

S. H. Kang - R. March

Existence and regularity of minimizers of a functional for unsupervised multiphase segmentation

created by march on 29 Aug 2012
modified on 02 Dec 2013


Published Paper

Inserted: 29 aug 2012
Last Updated: 2 dec 2013

Journal: Nonlinear Analysis - Theory, Methods and Applications
Volume: 76
Pages: 181-201
Year: 2013


We consider a variational model for image segmentation proposed by Sandberg, Kang and Chan. In such a model the image domain is partitioned into a finite collection of subsets denoted phases. The segmentation is unsupervised, i.e., the model finds automatically an optimal number of phases, which are not required to be connected subsets. Unsupervised segmentation is obtained by minimizing a functional of the Mumford-Shah type, but modifying the geometric part of the Mumford-Shah energy with the introduction of a suitable scale term. The results of computer experiments discussed by Sandberg, Kang and Chan show that the resulting variational model has several properties which are relevant for applications. In the present paper we investigate the theoretical properties of the model. We study the existence of minimizers of the corresponding functional, first looking for a weak solution in a class of phases constituted by sets of finite perimeter. Then we find various regularity properties of such minimizers, particularly we study the structure of triple junctions by determining their optimal angles.