Point-set registration in bounded domains via the Fokker-Planck equation

TL;DR

The paper proposes a point-set registration approach in bounded domains by evolving references and targets through the Fokker-Planck equation, with densities $ ho_0$ and $ ho_ ext{∞}$ estimated by Gaussian mixtures. A stabilized finite-element discretization of FP is used in space with Crank-Nicolson time stepping, and two particle transport schemes are explored: an explicit ODE-based method and a gradient-flow–based method that solves a linearized optimal transport problem; boundary proximity is controlled by a regularized potential term. Numerical experiments in two-dimensional geometries demonstrate rapid, monotone convergence to the target density, with the gradient-flow transport delivering more accurate deformations than the ODE approach, especially in complex cylinder domains. The framework is applicable to model-order reduction and data assimilation tasks, and future work includes adaptive time stepping and adaptive mesh strategies along with developing specialized FP solvers to reduce computational cost.

Abstract

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.

Figures (8)

• Figure 1: transport of Gaussian distributions across a cylinder. (a) initial distribution. (b) target distribution.
• Figure 2: transport of Gaussian distributions across a cylinder. (a) behavior of the $L^2$ error $\| \rho(\cdot, t) - \rho_\infty \|_{L^1(\Omega)}$ for several choices of the regularization parameter $\epsilon$. (b) deformed point cloud at final time for $\epsilon=0$ and target point cloud. (c) behavior of select trajectories for $\epsilon=0$. (d) behavior of one trajectory for several values of $\epsilon=0$.
• Figure 3: point-set registration across a cylinder. (a) reference (red) and target point (blue) clouds. (b) initial condition $\rho_0$. (c) target density $\rho_\infty$.
• Figure 4: point-set registration across a cylinder. (a) behavior of the $L^1$ error $\| \rho(\cdot, t) - \rho_\infty \|_{L^1(\Omega)}$ for both coarse and fine meshes. (b) coarse mesh ($N_{\rm hf}=13623$). (c) fine mesh ($N_{\rm hf}= 121425$).
• Figure 5: point-set registration across a cylinder; density profiles for three time instants.

Theorems & Definitions (6)

• Theorem 2.1
• Theorem 2.2
• proof
• proof
• proof
• proof