The LDDMM formulation treats images as fluids, and defines the distance between two images as the cost of transforming one image into the other. While this method is sufficient for grayscale images, many images carry other geometric data, such as diffusion tensors. I am currently applying the work being done on advection in complex fluids to LDDMM, by invoking the jet-groupoid . With Stefan Sommer this geometric formulation is being used to achieve higher order spatial accuracy in the LDDMM formulation (see images below) . We have also written a survey article together on the use of Lie group symmetries in image registration .
 H.O. Jacobs. Symmetries in LDDMM with higher order momentum distributions, 12 pages, Mathematical Foundations of Computational Anatomy at MICCAI (2013)
 H.O. Jacobs, S. Sommer, Higher order spatial accuracy in diffeomorphic image registration, to appear in J. Geomemtry, Imaging, and Computing (2015) (arXiv:1412.7504)
 S. Sommer, H.O. Jacobs, Symmetry in image registration and deformation modeling, Symmetry (2015), 7(2), 599-624 (link)
MRI data courtesy of http://www.mindboggle.info/papers/evaluation_NeuroImage2009/data/MGH10.php