Geometry of diffeomorphism groups and shape matching

Abstract

The large deformation matching (LDM) framework is a method for registration of images and other data structures, used in computational anatomy. We show how to reformulate the large deformation matching framework for registration in a geometric way. The general framework also allows to generalize the large deformation matching framework to include multiple scales by using the iterated semidirect product of groups. An important ingredient in the LDM framework is the choice of a suitable Riemannian metric on the space of diffeomorphisms. Since the space in question is infinite-dimensional, not every choice of the metric is suitable. In particular the geodesic distance, which is defined as the infimum over the length of all paths connecting two points, may vanish. For the family of Sobolev-type Hs-metrics on the diffeomorphism groups of R and S1 we establish that the geodesic distance vanishes for metrics of order 0 ≤ s ≤ 1 2 . The geodesic distance also vanishes for the L2-metric on the Virasoro-Bott group, which is a central extension of the diffeomorphism group of the circle. Vanishing of geodesic distance implies that the length-functional, which assigns to each curve in the manifold its length, has no global minima, when restricted to paths with fixed endpoints. We show that for the L2-metric on the diffeomorphism group of R and the Virasoro-Bott group doesn’t have any local minima either. The large deformation matching framework is not the only approach to the registration and shape comparison. For curves and surfaces it is possible to define a Riemannian metric directly on the space of curves or surfaces and use geodesics with respect to this metric to measure differences in shape. We use the family of Sobolev-type metrics on surfaces from [7]. We show how to discretize the geodesic equations and solve the boundary value problem via a shooting method on the initial velocity. The discrete equations are implemented via the finite element method.