This paper introduces the use of unbalanced optimal transport methods as a
similarity measure for diffeomorphic matching of imaging data. The similarity
measure is a key object in diffeomorphic registration methods that, together
with the regularization on the deformation, defines the optimal deformation.
Most often, these similarity measures are local or non local but simple enough
to be computationally fast. We build on recent theoretical and numerical
advances in optimal transport to propose fast and global similarity measures
that can be used on surfaces or volumetric imaging data. This new similarity
measure is computed using a fast generalized Sinkhorn algorithm. We apply this
new metric in the LDDMM framework on synthetic and real data, fibres bundles
and surfaces and show that better matching results are obtained.