Symmetries in LDDMM with higher order momentum distributions

Abstract

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 $T^{\ast}Q^{(k)}$.