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

Abstract

top
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

top

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 -

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.