"The matching equation, or EPDiff, can be analyzed as an expression of the evolution of the generalized momentum of diffeomorphic flow of least energy, in Eulerian and Lagrangian coordinates. In particular, the momentum map for singular solutions of the EPDiff yields a canonical Hamiltonian formulation, which in turn provides a complete parametrization of the landmarks by their canonical positions and initial momenta. In the case of set points, a numerical method based on geometric integration and the implementation of a trilinear interpolation approach has been recently presented.
In the following examples the evolution of points (representing surfaces) under the action of EPDiff is shown, given different initial momenta. The colors represent the norm of the velocity, or the momentum of each particle."