  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. The dissimilarity
measure is derived from a deformation energy whose Hessian reproduces the
underlying Riemannian metric, and it is used to define length and energy of
discrete paths in shape space. The notion of discrete geodesics defined as
energy minimizing paths gives rise to a discrete logarithmic map, a variational
definition of a discrete exponential map, and a time discrete parallel
transport. This 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 shape morphing, the representation of
paths in shape space via local shape variations as path generators, shape
extrapolation via discrete geodesic flow, and the transfer of geometric
features.
