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).
@article{urn:eudml:doc:41437,
title = {The M-components of level sets of continuous functions in WBV.},
journal = {Publicacions Matem\`atiques},
volume = {45},
year = {2001},
pages = {477-527},
mrnumber = {MR1876918},
zbl = {0991.54013},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41437}
}
Ballester, Coloma; Caselles, Vicent. The M-components of level sets of continuous functions in WBV.. Publicacions Matemàtiques, Tome 45 (2001) pp. 477-527. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41437/