The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature

The study of shapes and their similarities is central in computer vision, in that it allows to recognize and classify objects from their representation. One has the interest of defining a distance function between shapes, which both embodies the meaning of similarity between shapes for the application and task that one has in mind, and is at the same time mathematically sound and treatable. In recent years the use of differential-geometric techniques for the study of shape deformation has become popular in the field of pattern analysis: the central idea is to endow "shape spaces" with the structure of a Riemannian manifold, so that one can talk about length of a path, geodesic distance, Euler-Lagrange equations, et cetera; however, the geometry and in particular the curvature of shape manifolds have remained, until very recently, largely unexplored. This thesis first introduces a class of Riemannian metrics on the shape space of landmark points that arise from fluid flow ideas, illustrating the structure of the geodesic equations that derive from such metrics. The central part of this work consists in the computation of the Riemannian curvature tensor and sectional for the landmarks manifold, which first necessitates solving the highly non-trivial problem of expressing sectional curvature in terms of the partial derivatives of the cometric tensor. The effects of curvature on the qualitative dynamics of landmarks and are then explored, for example verifying the existence of conjugate points in regions of positive curvature and the divergence of geodesics on regions of negative curvature. Finally, the potential application of the results to the statistical analysis of medical images is briefly discussed.