The graded group action framework for sub-riemannian orbit models in shape spaces

In the standard orbit model on shape analysis, a group of diffeomorphism on the ambient space equipped with a right invariant sub-riemannian metric acts on a space of shapes and induces a sub-riemannian structure on various spaces. An important example is given by the Large Deformation Diffeomorphic Metric Mapping (LDDMM) theory that has been developed initially in the context of medical imaging and image registration. However, the standard theory does not cover many interesting settings emerging in applications. We provide here an extended setting, the graded group action (GGA) framework, specifying regularity conditions to get most of the well known results on the orbit model for general groups and shape spaces equipped with a smooth structure of Banach manifold with application to multi-scale shape spaces. A specific study of the Euler-Poincar{\'e} equations inside the GCA framework leads to a uniqueness result for the momentum map trajectory lifted from shape spaces with different complexities deciphering possible benefits of over-parametrization in shooting algorithms.