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

Resum

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 $\overline{\Omega}$ of the image is Jordan domain, a rectangle, for instance, and the image $u\in C(\overline{\Omega}) \cap \mathit{WBV}(\Omega)$ (being constant near $\partial \Omega$), we prove that for almost all levels $\lambda$ of $u$, the classical connected components of positive measure of $[u\geq \lambda]$ coincide with the $M$-components of $[u\geq\lambda]$. Thus the notion of $M$-component can be seen as a relaxation of the classical notion of connected component when going from $C(\overline{\Omega})$ to $\mathit{WBV}(\Omega)$.