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We are interested in restoring images having values in a symmetric Hadamardmanifold by minimizing a functional with a quadratic data term and a totalvariation like regularizing term. To solve the convex minimization problem, weextend the Douglas-Rachford algorithm and its parallel version to symmetricHadamard manifolds. The core of the Douglas-Rachford algorithm are reflectionsof the functions involved in the functional to be minimized. In the Euclideansetting the reflections of convex lower semicontinuous functions arenonexpansive. As a consequence, convergence results for Krasnoselski-Manniterations imply the convergence of the Douglas-Rachford algorithm.Unfortunately, this general results does not carry over to Hadamard manifolds,where proper convex lower semicontinuous functions can have expansivereflections. However, splitting our restoration functional in an appropriateway, we have only to deal with special functions namely, several distance-likefunctions and an indicator functions of a special convex sets. We prove thatthe reflections of certain distance-like functions on Hadamard manifolds arenonexpansive which is an interesting result on its own. Furthermore, thereflection of the involved indicator function is nonexpansive on Hadamardmanifolds with constant curvature so that the Douglas-Rachford algorithmconverges here.  Several numerical examples demonstrate the advantageous performance of thesuggested algorithm compared to other existing methods as the cyclic proximalpoint algorithm or half-quadratic minimization. Numerical convergence is alsoobserved in our experiments on the Hadamard manifold of symmetric positivedefinite matrices with the affine invariant metric which does not have aconstant curvature.

A Parallel Douglas Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds