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

Luigi Ambrosio; Vicent Caselles; Simon Masnou; Jean-Michel Morel

Journal of the European Mathematical Society (2001)

- Volume: 003, Issue: 1, page 39-92
- ISSN: 1435-9855

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topAmbrosio, Luigi, et al. "Connected components of sets of finite perimeter and applications to image processing." Journal of the European Mathematical Society 003.1 (2001): 39-92. <http://eudml.org/doc/277749>.

@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 -

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