This thesis is dedicated to the development of a new deformation model to study shapes. Deformations, and diffeormophisms in particular, have played a tremendous role in the field of statistical shape analysis, as a proxy to measure and interpret differences between similar objects but with different shapes.

Diffeomorphisms usually result from the integration of a flow of regular velocity fields, whose parameters have not enabled so far a full control of the local behaviour of the deformation.

We propose a new model in which vector fields, and then diffeomorphisms, are built on the combination of a few local and interpretable vector fields. These vector fields are generated thanks to a structure which we named deformation module. This notion of deformation module allows to develop a coherent mathematical framework to generate (modular) large deformations from generators (deformation modules). The deformation module used to build the deformations determines the deformation patterns that are allowed. It can be chosen by the user independently of data shapes and this choice corresponds to the choice of a vocabulary that will be used to study the variability amongst a population of shapes.

We present a simple method to build complex deformation modules through a combination of base-modules in a hierarchical framework. We equip spaces of shapes with a sub-Riemannian metric which takes into account the deformation constraints, linking optimal modular large deformations with a notion of distance between shapes. Estimating an optimal modular large deformation transporting one data shape onto a second one amounts to an interesting new optimal control problem.