We present a point set registration method in bounded domains based on the solution to the Fokker Planck equation. Our approach leverages (i) density estimation based on Gaussian mixture models; (ii) a stabilized finite element discretization of the Fokker Planck equation; (iii) a specialized method for the integration of the particles. We review relevant properties of the Fokker Planck equation that provide the foundations for the numerical method. We discuss two strategies for the integration of the particles and we propose a regularization technique to control the distance of the particles from the boundary of the domain. We perform extensive numerical experiments for two two-dimensional model problems to illustrate the many features of the method.

翻译：本文提出了一种基于Fokker-Planck方程求解的有界域点集配准方法。该方法整合了三个核心要素：(i) 基于高斯混合模型的密度估计；(ii) Fokker-Planck方程的稳定化有限元离散格式；(iii) 针对粒子积分的专门化处理技术。我们系统回顾了Fokker-Planck方程的相关数学性质，这些性质为数值方法的构建提供了理论基础。文中详细探讨了两种粒子积分策略，并提出了一种正则化技术以控制粒子与区域边界的距离。通过对两个二维模型问题进行大量数值实验，全面展示了该方法的多方面特性。