Transport of Relational Structures in Groups of Diffeomorphisms

This paper focuses on the issue of translating the relative variation of one shape with respect to another in a template centered representation. The context is the theory of Diffeomorphic Pattern Matching which provides a representation of the space of shapes of objects, including images and point sets, as an infinite dimensional Riemannian manifold which is acted upon by groups of diffeomorphisms. We discuss two main options for achieving our goal; the first one is the parallel translation, based on the Riemannian metric; the second one, based on the group action, is the coadjoint transport. These methods are illustrated with 3D experiments.