The metric spaces, Euler equations, and normal geodesic image motions of computational anatomy

Over the past several years our group has been studying biological shape in the emerging new discipline of computational anatomy (CA). CA consists of several components: (i) the construction of coordinatized anatomical manifolds, (ii) comparison of anatomical manifolds, and (iii) inference of morphometric change on anatomical manifolds. In this paper we focus on (ii) the comparison of anatomical shapes and structures in imagery via metric mapping. The purpose of this paper is to examine the generation of the geodesics associated with the metric from several points of view, the first the Euler equation describing the geodesic diffeomorphic flow, and the second the variational formulation of the geodesic in terms of the minimizing flow of vector fields which generate them.