  Diffeomorphic matching (only one of several names for this technique) is a
technique for non-rigid registration of curves and surfaces in which the curve
or surface is embedded in the flow of a time-series of vector fields. One seeks
the flow between two topologically-equivalent curves or surfaces which
minimises some metric defined on the vector fields, \emph{i.e.} the flow
closest to the identity in some sense.
  In this paper, we describe a new particle-mesh discretisation for the
evolution of the geodesic flow and the embedded shape. Particle-mesh algorithms
are very natural for this problem because Lagrangian particles (particles
moving with the flow) can represent the movement of the shape whereas the
vector field is Eulerian and hence best represented on a static mesh. We
explain the derivation of the method, and prove conservation properties: the
discrete method has a set of conserved momenta corresponding to the
particle-relabelling symmetry which converge to conserved quantities in the
continuous problem. We also introduce a new discretisation for the geometric
current matching condition of (Vaillant and Glaunes, 2005). We illustrate the
method and the derived properties with numerical examples.
