In shape analysis and computational anatomy, the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework uses strong right-invariant sub-Riemmannian metrics on diffeomorphisms groups to define shape spaces and characterize shape variations through diffeomorphic deformations. In this setting, the diffeomorphisms have only finite regularity, hence the deformation group is in general not a Lie group but a half-Lie group. In this talk, we will discuss a recent work with Alain Trouvé on a natural extension of this setting to more general half-Lie groups to capture new motions. Most results about those metrics will be recovered, and we will derive the associated sub-Riemannian Euler-Arnold equation. We will give the particular example of multi-scale registration.