  The Euler-Poincar\'e (EP) equations describe the geodesic motion on the
diffeomorphism group. For template matching (template deformation), the
Euler-Lagrangian equation, arising from minimizing an energy function, falls
into the Euler-Poincar\'e theory and can be recast into the EP equations. By
casting the EP equations in the Lagrangian (or characteristics) form, we
formulate the equations as a finite dimensional particle system. The evolution
of this particle system describes the geodesic motion of landmark points on a
Riemann manifold. In this paper we present a class of novel algorithms that
take advantage of the structure of the particle system to achieve a fast
matching process between the reference and the target templates. The strong
suit of the proposed algorithms includes (1) the efficient feedback control
iteration, which allows one to find the initial velocity field for driving the
deformation from the reference template to the target one, (2) the use of the
conical kernel in the particle system, which limits the interaction between
particles and thus accelerates the convergence, and (3) the availability of the
implementation of fast-multipole method for solving the particle system, which
could reduce the computational cost from $O(N^2)$ to $O(N\log N)$, where $N$ is
the number of particles. The convergence properties of the proposed algorithms
are analyzed. Finally, we present several examples for both exact and inexact
matchings, and numerically analyze the iterative process to illustrate the
efficiency and the robustness of the proposed algorithms.
