Investigation of parallel and globally convergent iterative schemes for nonlinear variational image smoothing and segmentation

We consider a nonquadratic convex variational segmentation approach and investigate numerical schemes to allow for an efficient computation of the global minimum on a parallel architecture. We focus on iterative schemes for which we can show global convergence to the unique solution irrespective of the starting point. In the context of (semi-)automated image processing tasks, such a feature is of utmost importance. We characterize various approaches that have been proposed in the literature as special cases of a general iterative scheme. Among these approaches are the linearization technique introduced by Geman and Reynolds (1992) and the half-quadratic regularization scheme proposed by Geman and Yang (see IEEE Trans. Image Proc., no.4, p.932-45, 1995). As a result, we can show global convergence to the unique solution under weaker conditions. Efficient Krylov subspace solvers for the resulting linear systems have been implemented on a parallel architecture to assess the performance of these numerical schemes. Experimental results concerning convergence rates and speed-up are reported. Due to the similarity of the segmentation approach considered here with total variation based image restoration methods, our results are relevant for this latter class of methods as well.