Beyond Riemannian geometry: The affine connection setting for transformation groups

Description

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.In this chapter we partially extend the theory of geometric statistics developed in the previous chapters to affine connection spaces. More particularly, we consider connected Lie groups endowed with the canonical Cartan–Schouten connection (a generally nonmetric connection). We show that this connection provides group geodesics that are completely consistent with the composition and inversion. With such a nonmetric structure, the mean cannot be defined by minimizing the variance as in Riemannian manifolds. However, the characterization of the mean as an exponential barycenter gives us an implicit definition of the mean using a general barycentric equation. Thanks to the properties of the canonical Cartan connection, this mean is naturally biinvariant. In finite dimension this provides strong theoretical bases for the use of one-parameter subgroups. The generalization to infinite dimensions is at the basis of the SVF-framework. From the practical point of view, we show that it leads to efficient and plausible models of atrophy of the brain in Alzheimer's disease.

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