A Class of Fast Geodesic Shooting Algorithms for Template Deformation and Its Applications via the N-particle System of the Euler-poincaré Equations

The Euler-Poincaré di↵erential (EPDi↵) equations are of general interest as evolution equations on Riemannian manifolds endowed with Sobolev metrics. The EPDi↵ equations describe geodesic motion on the di↵eomorphism group. The deformation between the reference and the target templates, following the path of minimum energy, satisfies the EPDi↵ equations. The EPDi↵ equations can be written as a finite-dimensional particle system. The evolution of this particle system describes the geodesic landmark evolution in template deformation. 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. The strong suit of the proposed algorithms includes (1) the e cient feedback control iteration that allows one to find the initial velocity field for driving the deformation from reference to target, (2) the use of the conical kernel in the particle system that limits the influence between particles to accelerate the convergence, and (3) the availability of the implementation of fast-multipole method for solving the particle system. 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 e ciency and the robustness of the proposed algorithms. keywords: Euler-Poincaré equations, di↵eomorphism group, particle system, geodesic landmark evolution, template deformation.