  In the framework of large deformation diffeomorphic metric mapping (LDDMM),
we develop a multi-scale theory for the diffeomorphism group based on previous
works. The purpose of the paper is (1) to develop in details a variational
approach for multi-scale analysis of diffeomorphisms, (2) to generalise to
several scales the semidirect product representation and (3) to illustrate the
resulting diffeomorphic decomposition on synthetic and real images. We also
show that the approaches presented in other papers and the mixture of kernels
are equivalent.
