Andreas Prohl
ETH Zurich, Switzerland
Automatic denoising and segmentation are principle tasks in image processing. Mathematically, the goal is to detect/preserve essential geometric structures (edges, domains,...) during the filtering process to extract relevant image information. The first part of the talk addresses analysis of the total variation flow for initial data $u_0 \in L^2(\Omega)$ by relating it to the prescribed mean curvature flow; then, a numerical analysis for a fully discrete realization is presented which leads to proper scaling laws to balance regularization and discretization effects. The second part of the talk addresses analysis and numerics of the gradient flow for a $\Gamma$-convergent approximation by Ambrosio and Tortorelli of the Mumford-Shah functional. I present a numerical scheme which guarantees convergence of the whole sequence $\{ (\, u_h, \varphi_h\, )\}_h$ towards weak solutions of the limiting problem; in particular, this analysis is based on explicit characterization of a (possible) singularity set. Computational experiments for both scenarios are presented to illustrate solution's behavior of the discussed problems. --- This is joint work with X.~Feng (Univ. of Tennessee).