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Left-Invariant Riemannian Elasticity: a distance on shape diffeomorphisms ?

In inter-subject 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 re-introduction in a registration algorithm is not easy. In [1], we interpreted the elastic energy as the distance of the Green-St Venant strain tensor to the identity. By changing the Euclidean metric for a more suitable Riemannian one, we defined a consistent statistical framework to quantify the amount of deformation. In particular, the mean and the covariance matrix of the strain tensor could be efficiently computed from a population of non-linear transformations and introduced as parameters in a Mahalanobis distance to measure the statistical deviation from the observed variability. This statistical Riemannian elasticity was able to handle anisotropic deformations but its isotropic stationary version was locally inverse-consistent. In this paper, we investigate how to modify the Riemannian elasticity to make it globally inverse consistent. This allows to define a left-invariant "distance" between shape diffeomorphisms that we call the left-invariant Riemannian elasticity. Such a closed form energy on diffeomorphisms can optimize it directly without relying on a time and memory consuming numerical optimization of the geodesic path.