Bi-invariant means on Lie groups with Cartan-Schouten connections

Xavier Pennec 1
1 ASCLEPIOS - Analysis and Simulation of Biomedical Images
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : The statistical Riemannian framework was pretty well developped for finite-dimensional manifolds. For Lie groups, left or right invariant metric provide a nice setting as the Lie group becomes a geodesically complete Riemannian manifold, thus also metrically complete. However, this Riemannian approach is fully consistent with the group operations only if a bi-invariant metric exists. Unfortunately, bi-invariant Riemannian metrics do not exist on most non compact and non-commutative Lie groups. In particular, such metrics do not exist in any dimension for {\it rigid-body transformations}, which form the most simple Lie group involved in biomedical image registration. The log-Euclidean framework, initially developed for symmetric positive definite matrices, was proposed as an alternative for affine transformations based on the log of matrices and for (some) diffeomorphisms based on Stationary Velocity Fields (SVFs). The idea is to rely on one-parameter subgroups, for which efficient algorithms exists to compute the deformation from the initial tangent vector (e.g. scaling and squaring). Previously, we showed that this framework allows to define bi-invariant means on Lie groups provided that the square-root (thus the log) of the transformations do exist. The goal of this note is to summarize the mathematical roots of these algorithms and to set the bases for comparing their properties with the left and right invariant metrics. The basis of our developments is the structure of affine connection instead of Riemannian metric. The connection defines the parallel transport, and thus a notion of geodesics (auto-parallel curves). Many local properties of Riemannian manifolds remains valid with affine connection spaces. In particular, there is still a local diffeomorphisms between the manifold and the tangent space using the exp and log maps. We explore invariant connections and show that there is a unique bi-invariant torsion-free Cartan-Schouten connection for which the geodesics are left and right translations of one-parameter subgroups. These group geodesics correspond to the ones of a left-invariant metric for the normal elements of the Lie algebra only. When a bi-invariant metric exists (we show that this is not always the case), then all elements are normal and Riemannian and group geodesics coincide. Finally we summarize the properties of the bi-invariant mean defined as the exponential barycenters of the canonical Cartan connection.
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Communication dans un congrès
Nielsen, Frank and Barbaresco, F. Geometric Science of Information (GSI 2013), Aug 2013, Paris, France. Springer, 8085, pp.59-67, 2013, Lecture Notes in Computer Science - LNCS. 〈10.1007/978-3-642-40020-9_5〉
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Xavier Pennec. Bi-invariant means on Lie groups with Cartan-Schouten connections. Nielsen, Frank and Barbaresco, F. Geometric Science of Information (GSI 2013), Aug 2013, Paris, France. Springer, 8085, pp.59-67, 2013, Lecture Notes in Computer Science - LNCS. 〈10.1007/978-3-642-40020-9_5〉. 〈hal-00846373〉

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