Right-invariant sub-Riemannian geometry in the Banach setting and applications to large deformations models in shape analysis

This thesis is dedicated to the study of strong right-invariant sub-Riemannian geometry on infinite dimensional groups and shape spaces. In particular, it extends the geometric framework of large deformations, which models shape spaces as Banach manifolds acted upon by groups of diffeomorphisms. We broaden this setting by allowing other groups of deformations to act on shapes, giving rise to new matching problems. We consider the setting of half-Lie groups and we first study theoretical properties of strong right-invariant sub-Riemannian structures on these groups. Under suitable regularity conditions, we establish completeness results for such structures and prove global existence of their associated Hamiltonian flows. We then derive regularity conditions on the action of half-Lie groups on the shape spaces that make it possible to induce corresponding right-invariant metrics. This framework enables to define general variational matching problems for shape analysis using the sub-Riemannian energy. Several applications are presented. In chapter 7, we propose a multiscale approach for performing registration, and we define in particular a setting to couple diffeomorphic deformations with the action of finite dimensional Lie groups, such as isometries or scalings. Chapters 8 and 9 then explore the anisotropy of shapes, which we characterize by metrics on the ambient space. We define various group actions on these metrics and therefore transport the anisotropic features of shapes.