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Zbl 0757.49027
Richardson, Thomas J.
Limit theorems for a variational problem arising in computer vision. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 19, No.1, 1-49 (1992). ISSN 0391-173X

We consider minimizers of the functional, $E(f,\Gamma)=\beta\int\sb \Omega(f-g)\sp 2dx+\int\sb{\Omega\backslash\Gamma}\vert\nabla f\vert\sp 2dx+{\cal H}\sp 1(\Gamma)$ where $\Omega$ is an open rectangle in ${\germ R}\sp 2$, $\Gamma\subset\Omega$ is a relatively closed set ${\cal H}\sp 1$ is one dimensional Hausdorff measure, $g\in L\sp \infty(\Omega)$, $f\in W\sp{1,2}(\Omega\backslash\Gamma)$, and $\beta$ is a scalar. This is a ``free discontinuity'' problem with applications in computer vision. It is proved that if $g$ is piecewise Lipschitz with a discontinuity set $S\sb g$ satisfying ${\cal H}\sp 1(S\sb g)<\infty$, except for some corruption depending on $\beta$, then optimal $\Gamma$ will approximate $S\sb g$ arbitrarily well with respect to the Hausdorff metric for $\beta$ sufficiently large.
[T.J.Richardson (Murray Hill)]

MSC 2000:: *49N99 Miscellaneous problems of optimal control

Keywords: free discontinuity problem; computer vision

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