A Parallel Douglas Rachford Algorithm for Restoring Images with Values in Symmetric Hadamard Manifolds

Ronny Bergmann, Johannes Persch, Gabriele Steidl

Published 2015-12-09Version 1

We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a total variation like regularizing term. To solve the convex minimization problem, we extend the Douglas-Rachford algorithm and its parallel version to symmetric Hadamard manifolds. For the convergence proof we investigate the corresponding reflection operators. We prove that the reflections of certain distance functions on the manifold are nonexpansive which is an interesting result on its own. Furthermore, the reflection of the involved indicator function of a special closed convex set is nonexpansive on manifolds with constant curvature. The performance of the generalized Douglas-Rachford algorithm for our model is based on analytic expressions for the proximal mappings. It requires the evaluation of exponential and logarithmic functions which can be done efficiently. Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods as the cyclic proximal point algorithm or half-quadratic minimization. Numerical convergence is also observed for the manifold of symmetric positive definite matrices with the affine invariant metric which does not have a constant curvature.