Abstract

Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure a time discrete geodesic calculus is developed and applications to shape space are explored. Thereby, the dissimilarity measure is derived from a spring type energy whose Hessian reproduces the underlying Riemannian metric. Using this measure to define length and energy on discrete paths on the manifold a discrete analog of classical geodesic calculus can be developed. The notion of discrete geodesics defined as energy minimizing paths gives rise to discrete logarithmic and exponential maps and enables to introduce a time discrete parallel transport as well. The relation of this time discrete to the actual, time continuous Riemannian calculus is explored and the new concept is applied to a shape space in which shapes are considered as boundary contours of physical objects consisting of viscous material. The flexibility and computational efficiency of the approach is demonstrated for topology preserving morphing, the interplay of paths in shape space and local shape variations as associated generators, the extrapolation of paths, and the transfer of geometric features.