Jacobi fields in groups of diffeomorphisms and applications

This paper presents a series of applications of the Jacobi evolution equations along geodesics in groups of diffeomorphisms. We describe, in particular, how they can be used to perform implementable gradient descent algorithms for image matching, in several situations, and illustrate this with 2D and 3D experiments. We also discuss parallel translation in the group, and its projection on shape manifolds, and focus in particular on an implementation of these equations using iterated Jacobi fields.