  This paper addresses the morphing of manifold-valued images based on the time
discrete geodesic paths model of Berkels, Effland and Rumpf 2015. Although for
our manifold-valued setting such an interpretation of the energy functional is
not available so far, the model is interesting on its own. We prove the
existence of a minimizing sequence within the set of $L^2(\Omega,\mathcal{H})$
images having values in a finite dimensional Hadamard manifold $\mathcal{H}$
together with a minimizing sequence of admissible diffeomorphisms. To this end,
we show that the continuous manifold-valued functions are dense in
$L^2(\Omega,\mathcal{H})$. We propose a space discrete model based on a finite
difference approach on staggered grids, where we focus on the linearized
elastic potential in the regularizing term. The numerical minimization
alternates between i) the computation of a deformation sequence between given
images via the parallel solution of certain registration problems for
manifold-valued images, and ii) the computation of an image sequence with fixed
first (template) and last (reference) frame based on a given sequence of
deformations via the solution of a system of equations arising from the
corresponding Euler-Lagrange equation. Numerical examples give a proof of the
concept of our ideas.
