PhD defense of Thomas PIERRON

Thomas PIERRON

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

Abstract

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 regidstration, 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.

Key Words

sub-Remannian geometry in infinite dimensions, optimal control, group of diffeomorphisms, Lie groups, shape analysis, computational anatomy

Supervision

• Alain TROUVE

jury 

• Alice Barbora TUMPACH, Directrice de recherche, WPI & Université de Lille (en détachement à l’Institut CNRS Pauli, Vienne, Autriche), Rapporteur
• Klas MODIN, Professor, Chalmers University of Technology and University of Gothenburg, Rapporteur
• Boris KHESIN, Professor, University of Toronto, Examinateur
• Sylvain ARGUILLÈRE, Chargé de recherche, Université de Lille, Examinateur
• Alice LE BRIGANT, Maîtresse de conférences, Université Paris 1 Panthéon Sorbonne, Examinateur
• Barbara GRIS, Chargée de recherche, Sorbonne Université Examinateur
• Ugo BOSCAIN, Directeur de recherche, Sorbonne Université, Examinateur

Publications