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)


49J20 Existence theories for optimal control problems involving partial differential equations


Zbl 0691.49036
Full Text: DOI EuDML


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