  This paper discusses the mathematical framework for designing methods of
large deformation matching (LDM) for image registration in computational
anatomy. After reviewing the geometrical framework of LDM image registration
methods, a theorem is proved showing that these methods may be designed by
using the actions of diffeomorphisms on the image data structure to define
their associated momentum representations as (cotangent lift) momentum maps. To
illustrate its use, the momentum map theorem is shown to recover the known
algorithms for matching landmarks, scalar images and vector fields. After
briefly discussing the use of this approach for Diffusion Tensor (DT) images,
we explain how to use momentum maps in the design of registration algorithms
for more general data structures. For example, we extend our methods to
determine the corresponding momentum map for registration using semidirect
product groups, for the purpose of matching images at two different length
scales. Finally, we discuss the use of momentum maps in the design of image
registration algorithms when the image data is defined on manifolds instead of
vector spaces.
