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

TL;DR

This paper introduces a class of fast geodesic shooting algorithms for template matching based on the Euler-Poincaré equations, utilizing particle systems and fast-multipole methods to significantly improve computational efficiency.

Contribution

The paper develops novel algorithms leveraging particle system structures and fast-multipole techniques for rapid and robust template matching within the Euler-Poincaré framework.

Findings

Algorithms achieve $O(N ext{log}N)$ complexity with fast-multipole methods.

Numerical examples demonstrate high efficiency and robustness.

Convergence properties are rigorously analyzed.

Abstract

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.