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

We consider a variational model for image segmentation proposed in Sandberg et al. (2010) [12]. In such a model the image domain is partitioned into a finite collection of subsets denoted as 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 (Mumford and Shah, 1989 [1]), but modifying the geometric part of the Mumford-Shah energy with the introduction of a suitable scale term. The results of computer experiments discussed in [12] show that the resulting variational model has several properties which are relevant for applications. In this 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.