Published Paper
(2001)
Authors:
Luigi Ambrosio - Vicent Caselles - Simon Masnou - Jean-Michel Morel
Journal: Journal of EMS. [MathSciNet]
Number: 3
Pages: 213-266
Abstract:
This paper contains a systematic analysis of a natural
measure theoretic notion of connectedness for sets of finite
perimeter in \Rn, 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(U) of functions of weakly bounded variation in U and
show that these filters are also well behaved in the
classical Sobolev and BV spaces. ![[PS]](/style/ps.png)
![[PDF]](/style/pdf.png)
[BibTeX Entry]
Available Files:
abstract.tex (abstract.ps, abstract.pdf)
paper.ps (paper.pdf)
