A sub-Riemannian modular approach for diffeomorphic deformations.

B. Gris (ENS Cachan). Slides

The structure of a collection of shapes (such as a series of segmented anatomical structures) is commonly studied through diffeomorphisms warping one shape onto another. These diffeomorphisms are obtained as flows of vector fields, and then a metric on the space of vector fields allows to build a metric on the space of shapes. Different choices of vector fields and metrics lead to different metrics on the shape space, and various models have been developed to be adapted to various problems. However these non parametric frameworks do not allow to study the structure of data through a descriptive language for deformations as introduced by Ulf Grenander. In this talk we will present an attempt in that direction. We constrain vector fields to be generated by a small dimension base of interpretable vector fields, which depend on the shape and evolve with it during the integration of the flow. Then, studying deformations transporting one shape onto another amounts to an optimal control problem in finite dimension, and enables to equip the shape space with a sub-Riemannian metric. In order to ensure the coherence of this framework, we introduce the concept of deformation module which is a structure, stable under combination, generating vector fields of particular, user-defined type.