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

Coloma Ballester; Vicent Caselles

Publicacions Matemàtiques (2001)

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

## Access Full Article

top## Abstract

top## How to cite

topBallester, 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 -

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.