Sub-Riemannian Methods in Shape Analysis

Because they reduce the search domains for geodesic paths in manifolds, sub-Riemannian methods can be used both as computational and modeling tools in designing distances between complex objects. This chapter provides a review of the methods that have been recently introduced in this context to study shapes, with a special focus on shape spaces defined as homogeneous spaces under the action of diffeomorphisms. It describes sub-Riemannian methods that are based on control points, possibly enhanced with geometric information, and their generalization to deformation modules. It also discusses the introduction of implicit constraints on geodesic evolution, and the associated computational challenges. Several examples and numerical results are provided as illustrations.