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A variational method in image segmentation: Existence and approximation results. (English) Zbl 0772.49006
The paper deals with a variational approach to the image segmentation problem, recently proposed by D. Mumford and J. Shah [Common. Pure Appl. Math. 42, No. 5, 577-684 (1989; Zbl 0691.49036)]. Given a bounded domain $$\Omega\subset{\mathbf R}^ 2$$ and a function $$g\in L^ \infty(\Omega)$$, the problem consists in finding a pair $$(\bar u,\overline K)$$ minimizing the functional $J(u,K)=\int_{\Omega\backslash K} \nabla u^ 2 dx+\int_{\Omega\backslash K} u-g^ 2 dx+{\mathcal H}^ 1(K\cap\Omega)$ among all pairs $$(u,K)$$ with $$K\subset{\mathbf R}^ 2$$ closed and $$u\in C^ 1(\Omega\backslash K)$$. Existence of a weak solution of this problem (with $$K$$ replaced by $$S_ u$$, the jump set of $$u$$) can be obtained in the so-called class $$\text{SBV}(\Omega)$$ of special functions of bounded variation in $$\Omega$$. The aim of the paper is to prove that any minimizer of the relaxed formulation of the problem actually corresponds to a minimizer of $$J$$. This is obtained by showing the existence of $$\gamma>0$$ such that ${{\mathcal H}^ 1(S_ u\cap B_ \rho(x))\over\rho}<\gamma\Longrightarrow u\in C^ 1(B_{\rho/2}(x))$ for any minimizer $$u\in\text{SBV}(\Omega)$$ and any ball $$B_ \rho(x)\subset\Omega$$.
Reviewer: L.Ambrosio (Roma)

##### MSC:
 49J20 Existence theories for optimal control problems involving partial differential equations
##### Keywords:
image segmentation
Full Text:
##### References:
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