Revistes Catalanes amb Accés Obert (RACO)

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

C. Ballester, Vicente Caselles


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 $\mathit{WBV}(\Omega)$.

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