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

@article{Ambrosio2001,
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 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 $WBV(\Omega )$ of functions of weakly bounded variation in $\Omega $, and show that these filters are also well behaved in the classical Sobolev and BV spaces.},
author = {Ambrosio, Luigi, Caselles, Vicent, Masnou, Simon, Morel, Jean-Michel},
journal = {Journal of the European Mathematical Society},
keywords = {theory of currents; $M$-connected components; denoising filters; functions of weakly bounded variation; sets of finite perimeter; image processing; weak notion of connectedness; denoising filters},
language = {eng},
number = {1},
pages = {39-92},
publisher = {European Mathematical Society Publishing House},
title = {Connected components of sets of finite perimeter and applications to image processing},
url = {http://eudml.org/doc/277749},
volume = {003},
year = {2001},
}

TY - JOUR
AU - Ambrosio, Luigi
AU - Caselles, Vicent
AU - Masnou, Simon
AU - Morel, Jean-Michel
TI - Connected components of sets of finite perimeter and applications to image processing
JO - Journal of the European Mathematical Society
PY - 2001
PB - European Mathematical Society Publishing House
VL - 003
IS - 1
SP - 39
EP - 92
AB - 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 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 $WBV(\Omega )$ of functions of weakly bounded variation in $\Omega $, and show that these filters are also well behaved in the classical Sobolev and BV spaces.
LA - eng
KW - theory of currents; $M$-connected components; denoising filters; functions of weakly bounded variation; sets of finite perimeter; image processing; weak notion of connectedness; denoising filters
UR - http://eudml.org/doc/277749
ER -