  We are interested in restoring images having values in a symmetric Hadamard
manifold by minimizing a functional with a quadratic data term and 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. The core of the Douglas-Rachford algorithm are reflections
of the functions involved in the functional to be minimized. In the Euclidean
setting the reflections of convex lower semicontinuous functions are
nonexpansive. As a consequence, convergence results for Krasnoselski-Mann
iterations 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 expansive
reflections. However, splitting our restoration functional in an appropriate
way, we have only to deal with special functions namely, several distance-like
functions and an indicator functions of a special convex sets. We prove that
the reflections of certain distance-like functions on Hadamard manifolds are
nonexpansive which is an interesting result on its own. Furthermore, the
reflection of the involved indicator function is nonexpansive on Hadamard
manifolds with constant curvature so that the Douglas-Rachford algorithm
converges here.
  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 in our experiments on the Hadamard manifold of symmetric positive
definite matrices with the affine invariant metric which does not have a
constant curvature.
