# The M-components of level sets of continuous functions in WBV.

• Volume: 45, Issue: 2, page 477-527
• ISSN: 0214-1493

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## Abstract

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We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV. Both function spaces are frequently used as models for images. Thus, if the domain Ω' of the image is Jordan domain, a rectangle, for instance, and the image u ∈ C(Ω') ∩ WBV(Ω) (being constant near ∂Ω), we prove that for almost all levels λ of u, the classical connected components of positive measure of [u ≥ λ] coincide with the M-components of[ u ≥ λ]. Thus the notion of M-component can be seen as a relaxation ofthe classical notion of connected component when going from C(Ω') to WBV(Ω).

## How to cite

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Ballester, Coloma, and Caselles, Vicent. "The M-components of level sets of continuous functions in WBV.." Publicacions Matemàtiques 45.2 (2001): 477-527. <http://eudml.org/doc/41437>.

@article{Ballester2001,
abstract = {We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV. Both function spaces are frequently used as models for images. Thus, if the domain Ω' of the image is Jordan domain, a rectangle, for instance, and the image u ∈ C(Ω') ∩ WBV(Ω) (being constant near ∂Ω), we prove that for almost all levels λ of u, the classical connected components of positive measure of [u ≥ λ] coincide with the M-components of[ u ≥ λ]. Thus the notion of M-component can be seen as a relaxation ofthe classical notion of connected component when going from C(Ω') to WBV(Ω).},
author = {Ballester, Coloma, Caselles, Vicent},
journal = {Publicacions Matemàtiques},
keywords = {Teoría de la medida; Funciones de variación acotada; Funciones de variable real; Funciones continuas; Mapa topográfico; Teoría de Morse; level sets; connected components; Morse theory; functions of bounded variation; sets of finite perimeter},
language = {eng},
number = {2},
pages = {477-527},
title = {The M-components of level sets of continuous functions in WBV.},
url = {http://eudml.org/doc/41437},
volume = {45},
year = {2001},
}

TY - JOUR
AU - Ballester, Coloma
AU - Caselles, Vicent
TI - The M-components of level sets of continuous functions in WBV.
JO - Publicacions Matemàtiques
PY - 2001
VL - 45
IS - 2
SP - 477
EP - 527
AB - We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV. Both function spaces are frequently used as models for images. Thus, if the domain Ω' of the image is Jordan domain, a rectangle, for instance, and the image u ∈ C(Ω') ∩ WBV(Ω) (being constant near ∂Ω), we prove that for almost all levels λ of u, the classical connected components of positive measure of [u ≥ λ] coincide with the M-components of[ u ≥ λ]. Thus the notion of M-component can be seen as a relaxation ofthe classical notion of connected component when going from C(Ω') to WBV(Ω).
LA - eng
KW - Teoría de la medida; Funciones de variación acotada; Funciones de variable real; Funciones continuas; Mapa topográfico; Teoría de Morse; level sets; connected components; Morse theory; functions of bounded variation; sets of finite perimeter
UR - http://eudml.org/doc/41437
ER -

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