Numerical minimization of the Mumford--Shah functional
In this paper the authors are interested in the numerical implementation of the Mumford-Shah functional, a functional used in the calculus of variations to approach the segmentation problems; the approach is taken by using the finite elements method (the minima of the discretized functional \(\Gamma\)-converge in the strong \(L^2\) topology to the minima of a functional, similar to the Mumford-Shah functional, but taking into account also the geometry of the triangulation used in the finite elements method). To find the minimum of the discrete functional, the authors use a quasi-Newton algorithm (taking into account that the Hessian of the discrete functional is not in general symmetric). The authors also give a one-dimensional model problem, pointing out the fact that there are local mimima, both for the discrete as well as for the continuous functional. To this aim, the authors propose two algorithms, both based on a multiscaled technique, which give the convergence to the global minimizer.
scientific article; zbMATH DE number 1898327