Acta Applicandae Mathematica

, Volume 17, Issue 1, pp 1–40 | Cite as

Variational problems in SBV and image segmentation

  • Luigi Ambrosio


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(ug)2]dx + αℋn − 1(K), uW1,2(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.

AMS subject classifications (1980)

48A21 46E30 

Key words

BV function semicontinuity compactness image segmentation 


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Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • Luigi Ambrosio
    • 1
  1. 1.Università di RomaRomeItaly

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