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

Ballester, Coloma; Caselles, Vicente

doi:http://dx.doi.org/10.5565/PUBLMAT_45201_10

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

Citation

Ballester C, Caselles V. The M-components of level sets of continuous functions in WBV. Publ Mat. 2001;45(2):477-527. DOI: 10.5565/PUBLMAT_45201_10

Abstract

We prove that the topographic map structure ofupper semicontinuous functions, defined in terms of classical connected components ofits level sets, and offunctions ofbounded variation (or a generalization, the WBV functions), defined in terms of M-connected components ofits 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 ofthe 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 ofpositive 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 ofconnected component when going from C(Ω) to WBV (Ω).