  In some implementations of the Large Deformation Diffeomorphic Metric Mapping
formulation for image registration we consider the motion of particles which
locally translate image data. We then lift the motion of the particles to
obtain a motion on the entire image. However, it is certainly possible to
consider particles which do more than translate, and this is what will be
described in this paper. As the unreduced Lagrangian associated to EPDiff
possesses $\Diff(M)$ symmetry, it must also exhibit $G \subset \Diff(M)$
symmetry, for any Lie subgroup. In this paper we will describe a tower of Lie
groups $G^{(0)} \subseteq G^{(1)} \subseteq G^{(2)} \subseteq...$ which
correspond to preserving $k$-th order jet-data. The reduced configuration
spaces $Q^{(k)} := \Diff(M) / G^{(k)}$ will be finite-dimensional (in
particular, $Q^{(0)}$ is the configuration manifold for $N$ particles in $M$).
We will observe that $G^{(k)}$ is a normal subgroup of $G^{(0)}$ and so the
quotient $G^{(0)} / G^{(k)}$ is itself a (finite dimensional) Lie group which
acts on $Q^{(k)}$. This makes $Q^{(k)}$ a principle bundle over $Q^{(0)}$ and
the reduced geodesic equations on $Q^{(k)}$ will possess $G^{(0)} /
G^{(k)}$-symmetry. Noether's theorem implies the existence of conserved momenta
for the reduced system on $T^{\ast}Q^{(k)}$.

Dynamical Systems;Computer Vision and Pattern Recognition