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Abstract. This paper is devoted to piecewise-constant segmentation of images using a curve evolution approach in a variational formulation. The problem to be solved is also called the minimal partition problem, as formulated by Mumford and Shah [20]. The proposed new models are extensions of the techniques previously introduced in [9], [10], [27]. We represent here the set of boundaries of the segmentation implicitly, by a multilayer of level-lines of a continuous function. In the standard approach of front propagation, only one level line is used to represent the boundary. The multilayer idea is inspired from previous work on island dynamics for epitaxial growth [14], [4]. Using a multilayer level set approach, the computational cost is smaller and in some applications, a nested structure of the level lines can be useful.

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This chapter presents an overview of the Mumford and Shah model for image segmentation. It discusses its various formulations, some of its properties, the mathematical framework, and several approximations. It also presents numerical algorithms and segmentation results using the Ambrosio-Tortorelli phase-field approximations on one hand and level set formulations on the other hand. Several applications of the Mumford-Shah problem to image restoration are also presented.