author = {Rumpf, Martin and Wirth, Benedikt},
  title = {Discrete geodesic calculus in shape space and applications in the
	space of viscous fluidic objects},
  journal = {SIAM J. Imaging Sci.},
  year = {2013},
  volume = {6},
  pages = {2581--2602},
  number = {4},
  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. 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 developed in the context of shape spaces with
	shapes that are described via deformations of a given reference shape,
	and it is applied to a particular 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.},
  eprint = {1210.0822},
  pdf = { 1},
  fjournal = {SIAM Journal on Imaging Sciences},
  owner = {rumpf}