Solving Unbalanced Optimal Transport on Point Cloud by Tangent Radial Basis Function Method

TL;DR

This work tackles unbalanced optimal transport on surfaces represented by point clouds, where mesh-based discretizations are costly. It develops a dynamic formulation and an ADMM scheme that reduce the computation to solving a space–time Poisson equation on the surface. A novel meshless Tangent Radial Basis Function (TRBF) method discretizes the elliptic operator on the point cloud by operating in tangent planes, enabling efficient, mesh-free computation. A fast time-direction solver via spectral decomposition further accelerates the PDE solves, and numerical experiments on smooth and general point clouds demonstrate accuracy, scalability, and mass-splitting capability, offering a practical tool for UOT on geometric data.

Abstract

In this paper, we solve unbalanced optimal transport (UOT) problem on surfaces represented by point clouds. Based on alternating direction method of multipliers algorithm, the original UOT problem can be solved by an iteration consists of three steps. The key ingredient is to solve a Poisson equation on point cloud which is solved by tangent radial basis function (TRBF) method. The proposed TRBF method requires only the point cloud and normal vectors to discretize the Poisson equation which simplify the computation significantly. Numerical experiments conducted on point clouds with varying geometry and topology demonstrate the effectiveness of the proposed method.

Figures (9)

• Figure 4.2.1: There is an additional circle around the center point $\boldsymbol{x}_c$, and all neighbor points $\boldsymbol{x}\in\Lambda(\boldsymbol{x}_c)$ are marked in red. The remaining black points $\boldsymbol{x}$ on the manifold are shown for reference. The projected position on the tangent space is marked in blue $\Tilde{\boldsymbol{x}}\in\Lambda_P(\boldsymbol{x}_c)$. The manifold is shown in black, while the tangent to the center point is shown as a dashed blue line.
• Figure 5.2.1: SOT examples ($\beta=1$) and the calculation times are 442s, 156s, 227s, 278s and 186s respectively.
• Figure 5.2.2: SUOT examples ($\beta=1.5$) and the calculation times are 450s, 160s, 236s, 287s and 192s respectively.
• Figure 5.2.3: SUOT test on point cloud ($\beta=1.5$): source item $f$.
• Figure 5.2.4: SUOT test on point cloud for $S1$, $S2$ and the calculation times are both 452s.

Theorems & Definitions (1)

• Remark 1