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Variational problems in SBV and image segmentation

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Abstract

We show how it is possible to prove the existence of solutions of the Mumford-Shah image segmentation functional

F(u,K) = ∫Ω\K [⊂∇u⊂2 + β(u − g)2]dx + αℋn − 1(K), u ∈ W 1,2(Ω\K), K ⊄ Ω closed in Ω.

We use a weak formulation of the minimum problem in a special class SBV(Ω) of functions of bounded variation. Moreover, we also deal with the regularity of minimizers and the approximation of F by elliptic functionals defined on Sobolev spaces. In this paper, we have collected the main results of Ambrosio and others.

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Authors and Affiliations

  1. Università di Roma, Tor Vergata, Rome, Italy

    Luigi Ambrosio

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  1. Luigi Ambrosio
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Ambrosio, L. Variational problems in SBV and image segmentation. Acta Appl Math 17, 1–40 (1989). https://doi.org/10.1007/BF00052492

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