Morphing of Manifold-Valued Images Inspired by Discrete Geodesics in Image Spaces

This paper addresses the morphing of manifold-valued images based on the time discrete geodesic paths model of Berkels, Effland, and Rumpf [SIAM J. Imaging Sci., 8 (2015), pp. 1457--1488]. 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 $L2(\Omega,\mathcal{H})$ 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 $L2(\Omega,\mathcal{H})$. 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.