k-a-mendoza / polatory

Fast, memory-efficient 3D spline interpolation and global kriging, via RBF (radial basis function) interpolation.

🚧 NOTE: The main branch is under development. Please check the v0 branch for the moment. 🚧

---

Polatory is a fast and memory-efficient framework for RBF (radial basis function) interpolation.

Features • Building • Contribution • Module Index • References

Features

• Fast spline surface reconstruction from 2.5D/3D point cloud
• Fast interpolation of 1D/2D/3D scattered data (kriging prediction)
• Meshing isosurfaces
• Supports large number (millions) of input points
• Supports inequality constraints
• List of available RBFs

Supported Compilers

Polatory requires a C++ compiler that supports C++20 and OpenMP 2.0.

Kriging via RBF Interpolation (Benchmark)

Polatory can perform kriging prediction via RBF interpolation (dual kriging). Although different terminologies are used, both methods produce the same results. Here is the correspondence between them:

Kriging	RBF interpolation
Prediction	Interpolation (fitting + evaluation)
Covariance function	RBF, positive definite
Generalized covariance function of order k	RBF, conditionally positive definite of order k + 1
Nugget effect model	Spline smoothing
Simple kriging	Interpolant with no polynomial
Ordinary kriging	Interpolant with polynomial of degree 0
Universal kriging	Interpolant with polynomial of degree ≥ 1
Weights	(Not computed)
Kriging variance	(Not computed)

A limited number of covariance functions are supported. See the list of available RBFs for details.

Building

On Windows • On macOS • On Ubuntu

Contribution

Contributions are welcome! You can contribute to this project in several ways:

Star the Repo

Just click the [★ Star] button on top of the page to show your interest!

File an Issue

Do not hesitate to file an issue if you have any questions, feature requests, or if you have encountered unexpected results (please include a minimal reproducible example).

Submit a Pull Request

You can fork the repo and make some improvements, then feel free to submit a pull request!

Module Index

Module	Description
common	Common utility functions and classes.
fmm	Fast multipole methods (wrapper of ScalFMM).
geometry	Geometric utilities.
interpolation	RBF fitting and evaluation.
isosurface	Isosurface generation.
kriging	Parameter estimation and validation for kriging.
krylov	Krylov subspace methods.
numeric	Numerical utilities.
point_cloud	SDF data generation from point clouds.
polynomial	Polynomial part of RBF interpolant.
preconditioner	The preconditioner used with Krylov subspace methods.
rbf	Definition of RBFs/covariance functions.

References

1. J. C. Carr, R. K. Beatson, J. B. Cherrie, T. J. Mitchell, W. R. Fright, B. C. McCallum, and T. R. Evans. Reconstruction and representation of 3D objects with radial basis functions. In Computer Graphics SIGGRAPH 2001 proceedings, ACM Press/ACM SIGGRAPH, pages 67–76, 12-17 August 2001. https://doi.org/10.1145/383259.383266

2. R. K. Beatson, W. A. Light, and S. Billings. Fast solution of the radial basis function interpolation equations: Domain decomposition methods. SIAM J. Sci. Comput., 22(5):1717–1740, 2000. http://doi.org/10.1137/S1064827599361771

3. G. M. Treece, R. W. Prager, and A. H. Gee. Regularised marching tetrahedra: improved iso-surface extraction. Computers and Graphics, 23(4):583–598, 1999. https://doi.org/10.1016/S0097-8493(99)00076-X