author = {Rumpf, Martin and Wirth, Benedikt},
  title = {A nonlinear elastic shape averaging approach},
  journal = {SIAM J. Imaging Sci.},
  year = {2009},
  volume = {2},
  pages = {800--833},
  number = {3},
  abstract = {A physically motivated approach is presented to compute a shape average
	of a given number of shapes. An elastic deformation is assigned to
	each shape. The shape average is then described as the common image
	under all elastic deformations of the given shapes, which minimizes
	the total elastic energy stored in these deformations. The underlying
	nonlinear elastic energy measures the local change of length, area,
	and volume. It is invariant under rigid body motions, and isometries
	are local minimizers. The model is relaxed involving a further energy
	which measures how well the elastic deformation image of a particular
	shape matches the average shape, and a suitable shape prior can be
	considered for the shape average. Shapes are represented via their
	edge sets, which also allows for an application to averaging image
	morphologies described via ensembles of edge sets. To make the approach
	computationally tractable, sharp edges are approximated via phase
	fields, and a corresponding variational phase field model is derived.
	Finite elements are applied for the spatial discretization, and a
	multi-scale alternating minimization approach allows the efficient
	computation of shape averages in 2D and 3D. Various applications,
	e. g. averaging the shape of feet or human organs, underline the
	qualitative properties of the presented approach.},
  doi = {10.1137/080738337},
  pdf = { 1},
  fjournal = {SIAM Journal on Imaging Sciences}