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

• Autores: Coloma Ballester Nicolau, Vicent Caselles Costa
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 45, Nº 2, 2001, págs. 477-527
• Idioma: inglés
• Títulos paralelos:
  • Los M-componentes de conjuntos de nivel de funciones continuas en WBV
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• Resumen

  • 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 O' of the image is Jordan domain, a rectangle, for instance, and the image u Î C(O') n WBV(O) (being constant near ?O), we prove that for almost all levels ? of u, the classical connected components of positive measure of [u = ?] coincide with the M-components of[ u = ?]. Thus the notion of M-component can be seen as a relaxation ofthe classical notion of connected component when going from C(O') to WBV(O).