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The Euler-Poincar\'e (EP) equations describe the geodesic motion on thediffeomorphism group. For template matching (template deformation), theEuler-Lagrangian equation, arising from minimizing an energy function, fallsinto the Euler-Poincar\'e theory and can be recast into the EP equations. Bycasting the EP equations in the Lagrangian (or characteristics) form, weformulate the equations as a finite dimensional particle system. The evolutionof this particle system describes the geodesic motion of landmark points on aRiemann manifold. In this paper we present a class of novel algorithms thattake advantage of the structure of the particle system to achieve a fastmatching process between the reference and the target templates. The strongsuit of the proposed algorithms includes (1) the efficient feedback controliteration, which allows one to find the initial velocity field for driving thedeformation from the reference template to the target one, (2) the use of theconical kernel in the particle system, which limits the interaction betweenparticles and thus accelerates the convergence, and (3) the availability of theimplementation of fast-multipole method for solving the particle system, whichcould reduce the computational cost from $O(N^2)$ to $O(N\log N)$, where $N$ isthe number of particles. The convergence properties of the proposed algorithmsare analyzed. Finally, we present several examples for both exact and inexactmatchings, and numerically analyze the iterative process to illustrate theefficiency and the robustness of the proposed algorithms.

A class of fast geodesic shooting algorithms for template matching and  its applications via the $N$-particle system of the Euler-Poincar\'e  equations