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Connected components of sets of finite perimeter and applications to image processing. (English) Zbl 0981.49024

Authors’ abstract: “This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in \(\mathbb{R}^N\), 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 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 mathematical basis to a large class of denoising filters acting on connectedness components of level sets. We introduce a natural domain for these filters, the space \(\text{WBV}(\Omega)\) of functions of weakly bounded variation in \(\Omega\), and show that these filters are also well-behaved in the classical Sobolev and BV spaces”.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
68U10 Computing methodologies for image processing
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