The statistical Riemannian framework was pretty well developped for finite-dimensional manifolds [28,20,22,6,7,8,23]. 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 rigid-body transformations, which form the most simple Lie group involved in biomedical image registration.
展开▼
