The M-components of level sets of continuous functions in WBV
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...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2001 |
| País: | España |
| Institución: | Universitat Pompeu Fabra |
| Repositorio: | Repositorio Digital de la UPF |
| OAI Identifier: | oai:repositori.upf.edu:10230/46948 |
| Acceso en línea: | http://hdl.handle.net/10230/46948 http://dx.doi.org/10.5565/PUBLMAT_45201_10 |
| Access Level: | acceso abierto |
| Palabra clave: | Mathematical morphology Level sets Connected components Morse theory Functions of bounded variation Sets of finite perimeter |
| Sumario: | 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). |
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