Implicit Surface Reconstruction with a Curl-free Radial Basis Function Partition of Unity Method

TL;DR

This paper introduces the curl-free partition of unity (CFPU) method for implicit surface reconstruction from point clouds, leveraging curl-free RBFs and local partitioning to handle noise and sharp features effectively.

Contribution

The paper proposes a novel CFPU method combining curl-free RBFs with partition of unity for efficient, noise-robust implicit surface reconstruction from point clouds.

Findings

Converges to the true surface as sampling density increases

Effectively handles noisy normal and point data

Performs well on various benchmark problems

Abstract

Surface reconstruction from a set of scattered points, or a point cloud, has many applications ranging from computer graphics to remote sensing. We present a new method for this task that produces an implicit surface (zero-level set) approximation for an oriented point cloud using only information about (approximate) normals to the surface. The technique exploits the fundamental result from vector calculus that the normals to an implicit surface are curl-free. By using a curl-free radial basis function (RBF) interpolation of the normals, we can extract a potential for the vector field whose zero-level surface approximates the point cloud. We use curl-free RBFs based on polyharmonic splines for this task, since they are free of any shape or support parameters. Furthermore, to make this technique efficient and able to better represent local sharp features, we combine it with a partition of unity (PU) method. The result is the curl-free partition of unity (CFPU) method. We show how CFPU can be adapted to enforce exact interpolation of a point cloud and can be regularized to handle noise in both the normal vectors and the point positions. Numerical results are presented that demonstrate how the method converges for a known surface as the sampling density increases, how regularization handles noisy data, and how the method performs on various problems found in the literature.