In this talk, we will present the use of right-invariant Riemannian metric on the group of diffeomorphisms in image matching.
Our presentation will be mainly theoretical although motivated by applications. First, we will review metric completeness properties on the group of sobolev diffeomorphims as well as geodesic completeness. In the second part, we will propose higher-order variational problems on the group of diffeomorphisms and we will present higher-order reduction method to derive the associated Euler-Lagrange equation. We will focus on the Cauchy problem associated with this Euler-Lagrange equation and the existence of minimizers for the boundary value problem. We will conclude with some open questions.