This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in R^N, introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the so-called M-connected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set of finite perimeter from the family of the boundaries of its components. In the two dimensional case we show that this notion of connectedness is comparable with the topological one, modulo the choice of a suitable representative in the equivalence class. Our strong motivation for this study is a mathematical justification of all those operations in image processing that involve connectedness and boundaries. As an application, we use this weak notion of connectedness to provide a rigorous mathemati...

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We propose a variational framework for determining global minimizers of rough energy functionals used in image segmentation. Segmentation is achieved by minimizing an energy model, which is comprised of two parts: the first part is the interaction between the observed data and the model, the second is a regularity term. The optimal boundaries are the set of curves that globally minimize the energy functional.