| Abstract: |
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 [omega] of the image is Jordan domain, a rectangle, for instance, and the image u [member of] C([omega]) [intersection] WBV([omega]) (being constant near [delta omega]), we prove that for almost all levels [lambda] of u, the classical connected components of positive measure of[u [greater than or equal] [lambda]] coincide with the M-components of [u [greater than or equal] [lambda]]. Thus the notion of M-component can be seen as a relaxation of the classical notion of connected component when going from C([omega]) to WBV([omega]). |
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