发表机构
KU Leuven; Univ. Bordeaux; Inria; University of Catania; Sapienza Universita di Roma（荷鲁芬大学; 波尔多大学; 法国国家数字与电子研究所; 卡塔尼亚大学; 罗马第一大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出两种基于Fokker-Planck方程的粒子方法（移动最小二乘离散与蒙特卡洛求解器）用于有界域内点集配准，分别通过鬼点反射边界和局部对数梯度近似处理障碍与提取渗透路径，实验验证了其性能与可扩展性。
AI 中文摘要
我们提出了两种基于Fokker-Planck方程的有界域内点集配准的粒子方法。第一种方法依赖于S-公式的移动最小二乘离散化，其中移动网格点（粒子）由漂移项随目标分布平流，扩散在动态演化的粒子云上求解。这种设置自然导致粒子云的强烈压缩和扩张。通过一种新颖的鬼点方法施加反射边界条件来处理障碍。第二种方法是针对相关的Langevin随机微分方程的蒙特卡洛求解器。它依赖于演化粒子密度的对数梯度的局部近似，从单个随机轨迹中提取宏观渗透路径。由于其固有的并行性，该方法表现出优异的可扩展性，非常适合高维配准问题。我们通过广泛的数值实验展示了两种方法的主要特征和性能。
英文摘要
We present two particle methods for point set registration in bounded domains based on the Fokker-Planck equation. The first method relies on a moving least squares discretization of the S--formulation, in which moving grid points (particles) are advected by the drift with the target distribution, diffusion is resolved on a dynamically evolving particle cloud. This setting naturally leads to strong compression and expansion of the particle cloud. Obstacles are handled by enforcing reflective boundary conditions through a novel ghost point method. The second method is a Monte Carlo solver for the associated Langevin stochastic differential equation. It relies on a local approximation of the logarithmic gradient of the evolving particle density to extract macroscopic osmotic paths from individual stochastic trajectories. Owing to its inherent parallelism, this method exhibits excellent scalability and is well suited for high-dimensional registration problems. We illustrate the main features and performance of both approaches through extensive numerical experiments.
