We present 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 feasible 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, with its
projections on shape manifolds, and focus in particular on an implementation
of the associated equations using iterated Jacobi fields.