author = {Berkels, Benjamin and Effland, Alexander and Rumpf, Martin},
  title = {Time discrete geodesic paths in the space of images},
  journal = {SIAM J. Imaging Sci.},
  year = {2015},
  volume = {8},
  pages = {1457--1488},
  number = {3},
  abstract = {In this paper the space of images is considered as a Riemannian manifold
	using the metamorphosis approach, where the underlying Riemannian
	metric simultaneously measures the cost of image transport and intensity
	variation. A robust and effective variational time discretization
	of geodesics paths is proposed. This requires to minimize a discrete
	path energy consisting of a sum of consecutive image matching functionals
	over a set of image intensity maps and pairwise matching deformations.
	For square-integrable input images the existence of discrete, connecting
	geodesic paths defined as minimizers of this variational problem
	is shown. Furthermore, $\Gamma$-convergence of the underlying discrete
	path energy to the continuous path energy is proved. This includes
	a diffeomorphism property for the induced transport and the existence
	of a square-integrable weak material derivative in space and time.
	A spatial discretization via finite elements combined with an alternating
	descent scheme in the set of image intensity maps and the set of
	matching deformations is presented to approximate discrete geodesic
	paths numerically. Computational results underline the efficiency
	of the proposed approach and demonstrate important qualitative properties.},
  doi = {10.1137/140970719},
  eprint = {1503.02001},
  pdf = { 1},
  fjournal = {SIAM Journal on Imaging Sciences}