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

TL;DR

This paper introduces a novel tangent radial basis function method to efficiently solve unbalanced optimal transport problems on point clouds by discretizing Poisson equations, demonstrated through numerical experiments.

Contribution

It presents a new TRBF-based approach for solving Poisson equations on point clouds within the UOT framework, simplifying computations and improving efficiency.

Findings

Effective solution for UOT on point clouds with complex geometries.

Simplified discretization requiring only point cloud data and normals.

Numerical experiments validate the method's robustness and accuracy.

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.