5 - Beyond Riemannian geometry: The affine connection setting for transformation groups

Abstract

In computational anatomy, the statistics from the object space (images, surfaces, etc.) are often lifted to the group of deformation acting on their embedding space. Statistics on transformation groups have been considered in the previous chapters by providing the Lie group with a left- or right-invariant metric, which may (or may not) be consistent with the group action on our original objects. In this chapter we take the point of view of statistics on abstract transformations, independently of their action. In this case it is reasonable to ask that our statistical methods respect the geometric structure of the transformation group. For instance, we would like to have a mean that is stable by the group operations (left and right compositions, inversion). Such a property is ensured for Fréchet means in Lie groups endowed with a biinvariant Riemannian metric, like compact Lie groups (e.g. rotations). Unfortunately, biinvariant Riemannian metrics do not exist for most noncompact and noncommutative Lie groups, including rigid-body transformations in any dimension greater than one. Thus there is a need for the development of a more general non-Riemannian statistical framework for general Lie groups.