Bi-invariant Means on Lie Groups with Cartan-Schouten Connections

In computational anatomy, one needs to perform statistics on shapes and transformations, and to transport these statistics from one geometry (e.g. a given subject) to another (e.g. the template). The geometric structure that appeared to be the best suited so far was the Riemannian setting. The statistical Riemannian framework was indeed pretty well developped for finite-dimensional manifolds and some matrix Lie groups [16, 9, 10, 5, 6, 7, 12]. An important generalization of some of the tools was done for infinite dimensional Lie groups of diffeomorphisms with the so called Large Deformation Diffeomorphic Metric Mapping framework developped by Miller, Trouvé and Younes [14, 15]. In both cases, the basic idea for Lie groups is to rely on a left or right invariant metric. As these metrics are geodesically complete Riemannian manifolds, they are metrically complete. This provides a very nice setting for generalizing many usual notions such as the mean or median value, the covariance matrix and the Principle Component Analysis (PCA). However, the geodesics are not always easy to compute and one can question the choice of the metric which determines many of the properties. For instance, there is no closed form for geodesics in groups of diffeomorphisms and they need to be determined through a rather expensive optimization process. On a Lie group, this Riemannian approach is consistent with the group operations if a bi-invariant metric exists, which is for example the case for compact groups such as rotations [11, 8]. In this case, the bi-invariant Fréchet mean has many desirable invariance properties: it is invariant with respect to leftand right-multiplication, as well as inversion. Unfortunately, bi-invariant Riemannian metrics do not exist for most non compact and non-commutative Lie groups. In particular, such metrics do not exist in any dimension for rigid-body transformations, which form the most simple Lie group involved in biomedical image registration. The log-Euclidean framework was proposed as an alternative by Arsigny for