Asymptotic behavior of a Geman and McClure discrete model
The paper deals with the asymptotic behaviour of the discrete functionals \(E_h:{\mathbb{R}}^{N^2}\to{\mathbb{R}}\) defined by \[ E_h(u):=h^2\biggl\{\Psi_h(u)+\nu\sum_{i, j=1}^N(u_{i,j}-g_{i,j})^2\biggr\} \] and proposed by Geman and McClure. Here \(\mu>0\) is a real parameter, \[ \varphi_\mu(t):={t^2\over 1+\mu t^2}, \] \[ \Psi_h(u):=\sum_{i, j=1}^N \varphi_\mu(|u_{i+1,j}-u_{i,j}|/h)+ \varphi_\mu(|u_{i,j+1}-u_{i,j}|/h), \] and \(h=1/N\). Extending previous work by Chambolle, where instead of \(\varphi_\mu\) a truncated parabola was considered, the author shows that \(E_h\) \(\Gamma\)-converge to the anisotropic Mumford--Shah functional \[ \int_\Omega |\nabla u|^2+\nu(u-g)^2 dx+\alpha\Lambda_1(S_u) \] provided \(\mu=\mu_h=h/\alpha\). The anisotropy here comes from the term \(\Lambda_1(S_u)\), which stands for the ``cab driver'' length of \(S_u\).
scientific article; zbMATH DE number 1419727