Non-local approximation of the Mumford-Shah functional
The paper deals with the approximation of the Mumford-Shah functional \[ F(u):=\int_\Omega |\nabla u|^2 dx+\beta{\mathcal H}^{n-1}(\Omega\cap S_u) \] defined in the class SBV\((\Omega)\) of special functions with bounded variation (here \(S_u\) is the Lebesgue discontinuity set of \(u\)). It is easy to see that \(F\) can't be approximated (in the sense of \(\Gamma\)-convergence) by integral functionals defined on Sobolev spaces and depending only on one variable. The main result of the paper is the proof that an approximation can be achieved with non local functionals, whose lack of locality tends to \(0\). Specifically, the approximating functionals are given by \[ F_\varepsilon(u):={1\over\varepsilon}\int_\Omega f\biggl(\varepsilon \hbox{\vrule height3.5pt depth-2.8pt width6pt}\mkern-18mu\int\nolimits_{B_\varepsilon(x)\cap\Omega} |\nabla u|^2(y) dy\biggr) dx,\qquad\qquad \varepsilon\downarrow 0, \] where \(f(t)\) is any continuous increasing function such that \(f'_+(0)=1\) and \(f(\infty)=\beta/2\). Allowing \(f\) to depend on \(\varepsilon\), suitable convergence theorems of minimum points to minimum points for the Mumford-Shah image segmentation problem \[ \min\biggl\{F(u)+\alpha\int_\Omega (u-g)^2 dx:\;u\in \text{SBV}(\Omega)\biggr\} \] are estabilished.
special functions with bounded variation
scientific article; zbMATH DE number 1019672