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Polatory is a fast and memory-efficient framework for RBF (radial basis function) interpolation, developed by GSI Co. Ltd.

NOTE: This is a pre-release version. APIs subject to change without notice.

Features | License | Building | Contribution | Module Index | Acknowledgements

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

Platforms

Polatory runs on x86-64 processors and continuously tested on the following platforms.

OS	Toolchain
Ubuntu 16.04 LTS	GCC 5.4 and Clang 3.8
Ubuntu 16.04 LTS	GCC 7.3 and Clang 6.0
Windows	Visual Studio 2017

Kriging via RBF Interpolation (Benchmark)

Polatory can perform global kriging prediction via RBF interpolation. Although different terminology is used, both methods produce the same results. Here is the correspondence between kriging and RBF interpolation:

Kriging	RBF interpolation
Prediction	Interpolation (fitting + evaluation)
Covariance function	RBF
Nugget effect	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)
Standard errors	(Not computed)

Rapidly decaying covariance functions are not supported in the current version, such as the spherical and the Gaussian models. The spherical model can be substituted by a similar one named cov_quasi_spherical9.

License

Polatory is available under two different licenses:

• GNU General Public License version 3 for non-commercial use
• Commercial license (please contact at mizuno(at)gsinet.co.jp)

Building

On Ubuntu

1. Install build tools

   On Ubuntu 16.04 LTS, CMake >= 3.9 must be installed manually.

   sudo apt install build-essential cmake git ninja-build

   If you use Clang, libomp-dev is required.

   sudo apt install clang libomp-dev

2. Download and install Intel(R) MKL.

   See https://software.intel.com/articles/installing-intel-free-libs-and-python-apt-repo for details.

   cd
   wget https://apt.repos.intel.com/intel-gpg-keys/GPG-PUB-KEY-INTEL-SW-PRODUCTS-2019.PUB -O - | sudo apt-key add -
   sudo sh -c 'echo deb https://apt.repos.intel.com/mkl all main > /etc/apt/sources.list.d/intel-mkl.list'
   sudo apt update
   sudo apt install intel-mkl-64bit-2018.3-051

3. Install Eigen

   sudo apt install libeigen3-dev

4. Install Google Test

   git clone https://github.com/google/googletest.git
   cd googletest
   mkdir build && cd build
   cmake .. -GNinja
   ninja
   sudo ninja install

5. Install Ceres Solver

   sudo apt install libgoogle-glog-dev
   cd
   git clone https://ceres-solver.googlesource.com/ceres-solver
   cd ceres-solver
   mkdir build && cd build/
   cmake .. -GNinja -DCMAKE_LIBRARY_PATH=/opt/intel/mkl/lib/intel64 -DGFLAGS=OFF -DLAPACK=ON
   ninja
   sudo ninja install

6. Install FLANN

   sudo apt install libflann-dev

7. Download and build Boost

   cd
   wget https://dl.bintray.com/boostorg/release/1.67.0/source/boost_1_67_0.tar.bz2
   tar xjf boost_1_67_0.tar.bz2
   cd boost_1_67_0
   ./bootstrap.sh
   ./b2 install -j8 --prefix=.

8. Build polatory

   cd
   git clone https://github.com/polatory/polatory.git
   cd polatory
   mkdir build && cd build
   cmake .. -GNinja -DBOOST_ROOT=~/boost_1_67_0
   ninja

On Windows

1. Install Visual Studio Community 2017

   https://www.visualstudio.com/

   From the Workloads tab, select the following item.

   • Desktop development with C++

   From the Individual components tab, select the following item.

   • Code tools > Git for Windows

2. Install libraries with vcpkg

   cd /d C:\
   git clone https://github.com/Microsoft/vcpkg.git
   cd vcpkg
   bootstrap-vcpkg.bat
   vcpkg install boost ceres flann eigen3 gtest --triplet x64-windows

   To update vcpkg and installed libraries, run the following commands:

   cd /d C:\vcpkg
   git pull
   bootstrap-vcpkg.bat
   vcpkg update
   vcpkg upgrade
   vcpkg upgrade --no-dry-run

3. Download and install Intel(R) MKL

4. Build polatory

   Open Start > Visual Studio 2017 > x64 Native Tools Command Prompt for VS 2017.

   cd /d %userprofile%
   git clone https://github.com/polatory/polatory.git
   cd polatory
   mkdir build
   cd build
   cmake .. -GNinja -DCMAKE_TOOLCHAIN_FILE=C:/vcpkg/scripts/buildsystems/vcpkg.cmake
   ninja

Contribution

We welcome your contributions! You can contribute to this project in several ways:

Add a Star

You can just click the ★Star button to show your interest.

File an Issue

Feel free to file an issue, if you have any questions, feature requests, or if you have found any unexpected results (please include a minimal reproducible example).

Create a Pull Request

You can fork the source tree and make some improvements to it. Then feel free to create a PR. When sending a PR for the first time, please review and sign the Individual Contributor License Agreement.

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.
third_party	Third party libraries.

Acknowledgements

Polatory utilizes the following libraries:

Library	License	Used
Boost	Boost Software License 1.0	In several modules
Ceres Solver	The 3-clause BSD license	In kriging module
Eigen	Mozilla Public License 2.0	In almost all modules
FLANN	The 2-clause BSD license	In point_cloud module
Google Test	The 3-clause BSD license	For unit testing
Intel(R) MKL	Intel Simplified Software License	As a backend for Ceres Solver, Eigen and ScalFMM
ScalFMM	The CeCILL-C license	In fmm module