Variational time discretization of geodesic calculus

We analyse a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps and discrete parallel transport, and we prove convergence to their continuous counterparts. The presented analysis is based on the direct method in the calculus of variation, on Γ-convergence and on weighted finite element error estimation. The convergence results of the discrete geodesic calculus are experimentally confirmed for a basic model on a two-dimensional Riemannian manifold. This provides a theoretical basis for the application to shape spaces in computer vision, for which we present one specific example.