Connected components of sets of finite perimeter and applications to image processing

Abstract

Abstract. This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in IR 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 mathematical basis to a large class of denoising filters acting on connected components of level sets. We introduce a natural domain for these filters, the space WBV¢ ¤ £ ¦ ¥ of functions of weakly bounded variation in £ , and show that these filters are also well behaved in the classical Sobolev and BV spaces.

Local simplicity, topology, and sets of finite perimeter

“…We also show that, for locally simple sets, our decomposition agrees exactly with that of [8,Theorem 1]. Our result therefore allows us to make use of theorems and constructions from [8] Finally, as an example application we consider the method of L 1 TV-minimization, a variational method for image reconstruction related to the popular ROF model of L. Rudin, S. Osher, and E. Fatemi [35]. L 1 TV-minimization is a well-known method, first considered in a discrete setting by S. Alliney and M. Nikolova (see, for example, [4] and [34]), and later in a continuous setting for n 2 by T. Chan and S. Esedoglu [20], W. Allard ([1], [2], [3]), and others.…”

“…We also establish a decomposition theorem (Theorem 6), expressing measure-theoretic interiors and exteriors of locally simple sets uniquely as countable unions of disjoint, open connected sets, and we show that the perimeter of the set is the sum of the perimeters of the components. We also show that, for locally simple sets, our decomposition agrees exactly with that of [8,Theorem 1]. Our result therefore allows us to make use of theorems and constructions from [8] Finally, as an example application we consider the method of L 1 TV-minimization, a variational method for image reconstruction related to the popular ROF model of L. Rudin, S. Osher, and E. Fatemi [35].…”

“…Indeed, the importance of connectedness in image processing was a strong motivating factor for L. Ambrosio, V. Caselles, S. Masnou, and J.-M. Morel. In their paper [8], they specialized H. Federer's notion of indecomposable integral currents ( [25, 4.2.25])-the measure-theoretic analogues of connected sets-to the context of sets of finite perimeter. In that setting they established results analogous to the usual topological theorems concerning connectedness, so that topology-like connectedness arguments could be imitated in the finite perimeter setting, and so that operations in image processing that used connectedness in that setting could be rigorously studied and justified.…”

Crack Initiation in Brittle Materials

“…In the case Γ is sufficiently regular, the Euler-Lagrange equation satisfied by u Γ is 5) where n denotes the normal vector to ∂ N Ω ∪ ∂Γ . In the sequel, we will write…”

“…Let us divide E in the union of its indecomposable components according to [5,Theorem 1], that is, let (E i ) i∈N be a family of sets with finite perimeter in…”