  The geometric approach to diffeomorphic image registration known as "large
deformation by diffeomorphic metric mapping" (LDDMM) is based on a left action
of diffeomorphisms on images, and a right-invariant metric on a diffeomorphism
group, usually defined using a reproducing kernel. We explore the use of
left-invariant metrics on diffeomorphism groups, based on reproducing kernels
defined in the body coordinates of a source image. This perspective, which we
call Left-LDM, allows us to consider non-isotropic spatially-varying kernels,
which can be interpreted as describing variable deformability of the source
image. We also show a simple relationship between LDDMM and the new approach,
implying that spatially-varying kernels are interpretable in the same way in
LDDMM. We conclude with a discussion of a class of kernels that enforce a soft
mirror-symmetry constraint, which we validate in numerical experiments on a
model of a lesioned brain.
