Roberto Camassa

Roberto Camassa, Dongyang Kuang, Long Lee

The Euler-Poincaré (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-Poincaré 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.

Roberto Camassa, Dongyang Kuang, Long Lee

We study a class of partial differential equations (PDEs) in the family of the so-called Euler-Poincaré differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular attention to the mathematical properties of this system when the associated class of elliptic operators possesses non-smooth kernels. By casting the system in its Lagrangian (or characteristics) form, we first formulate a particles system algorithm in free space with homogeneous Dirichlet boundary conditions for the evolving fields. We next examine the deformation of the system when non-homogeneous "constant stream" boundary conditions are assumed. We show how this simple change at the boundary deeply affects the nature of the evolution, from hyperbolic-like to dispersive with a non-trivial dispersion relation, and examine the potentially regularizing properties of singular kernels offered by this deformation. From the particle algorithm viewpoint, kernel singularities affect the existence and uniqueness of solutions to the corresponding ordinary differential equations systems. We illustrate this with the case when the operator kernel assumes a conical shape over the spatial variables, and examine in detail two-particle dynamics under the resulting lack of Lipschitz-continuity. Curiously, we find that for the conically-shaped kernels the motion of the related two-dimensional waves can become completely integrable under appropriate initial data. This reduction projects the two-dimensional system to the one-dimensional completely integrable Shallow-Water equation [Camassa, R. and Holm, D. D., Phys. Rev. Lett., 71, 1961-1964, 1993], while retaining the full dependence on two spatial dimensions for the single channel solutions.