Symmetries in LDDMM with higher order momentum distributions

In the landmark large deformation diffeomorphic metric mapping (landmark-LDDMM) 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 also apply local rotations, scalings, and sheerings. These locally linear transformations manifest as a result of a non-trivial first derivative of a diffeomorphism. In this article, we seek to understand a structurally augmented particle which applies these local transformations. Moreover, we seek to go beyond locally linear transformations by considering the algebra behind the k-th order Taylor expansions of a diffeomorphism, a.k.a. the k-jet of a diffeomorphism. The particles which result from understanding the algebra of k-jets permit the use of higher-order local deformations. Additionally, these particles have internal symmetries which lead to conserved momenta when solving for geodesics between images. Knowledge of these structures provide both a sanity check and guide for future implementations of the LDDMM formalism.