Registering two medical images consists in computing a mapping between the organs of interest they contain. Although this mapping is dense in space, it can only be accurately estimated based on significant intensity variations in the images, which is a sparse information. Using deformation regularization properties that are physiologically meaningful is then one of the keys to estimate pertinent mappings. In the LDDMM framework these regularization properties are directly related to the right-invariant metric which controls the optimal deformation. In this chapter we then present different methodologies related to this degree of freedom. After briefly introducing the LDDMM framework, we present a simple strategy to regularize the mappings at different scales and a more advanced technique to make it possible to estimate a sliding motion at predefined locations. We then propose to switch the paradigm of right-invariant metrics to left-invariant ones, so that spatially adaptive metrics can be used in LDDMM. In the last part, we review different attempts to optimize these spatially adaptive metrics and propose a new evolution of LDDMM that incorporates spatially adaptive metrics.