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

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).