 
Summary: Riemannian Elasticity: A Statistical
Regularization Framework
for Nonlinear Registration
X. Pennec, R. Stefanescu, V. Arsigny, P. Fillard, and N. Ayache
INRIA Sophia  Projet Epidaure, 2004 Route des Lucioles BP 93,
06902 Sophia Antipolis Cedex, France
Xavier.Pennec@sophia.inria.fr
Abstract. In intersubject registration, one often lacks a good model
of the transformation variability to choose the optimal regularization.
Some works attempt to model the variability in a statistical way, but the
reintroduction in a registration algorithm is not easy. In this paper, we
interpret the elastic energy as the distance of the GreenSt Venant strain
tensor to the identity, which reflects the deviation of the local deforma
tion from a rigid transformation. By changing the Euclidean metric for a
more suitable Riemannian one, we define a consistent statistical frame
work to quantify the amount of deformation. In particular, the mean
and the covariance matrix of the strain tensor can be consistently and
efficiently computed from a population of nonlinear transformations.
These statistics are then used as parameters in a Mahalanobis distance
to measure the statistical deviation from the observed variability, giving
