Computing Harmonic Maps and Conformal Maps on Point Clouds

TL;DR

This paper introduces a meshless method for computing harmonic and conformal maps directly from point cloud data, enabling surface analysis without mesh construction.

Contribution

It presents a novel meshless approach using cubic lattices to approximate harmonic and conformal maps on point cloud surfaces, including algorithms and numerical examples.

Findings

Effective approximation of harmonic maps on point clouds

Successful computation of conformal maps and surface uniformization

Applicable to closed surfaces and topological disks

Abstract

We propose a novel meshless method to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space, using point cloud data only. Given a surface, or a point cloud approximation, we simply use the standard cubic lattice to approximate its $\epsilon$-neighborhood. Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices. The conformal map, or the surface uniformization, is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature. We propose algorithms and numerical examples for closed surfaces and topological disks.