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 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. To make this technique efficient and able to better represent local sharp features, we combine it with a partition of unity 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 normals 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.
This is the peer reviewed version of the following article:
Drake, K.P., Fuselier, E.J., & Wright, G.B. (2022). Implicit Surface Reconstruction with a Curl-Free Radial Basis Function Partition of Unity Method. SIAM Journal on Scientific Computing, 44(5), A2895-C398,
which has been published in final form at https://doi.org/10.1137/22M1474485. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.
Drake, Kathryn P.; Fuselier, Edward J.; and Wright, Grady B.. (2022). "Implicit Surface Reconstruction with a Curl-Free Radial Basis Function Partition of Unity Method". SIAM Journal on Scientific Computing, 44(5), A3018-A3040. https://doi.org/10.1137/22M1474485