This C++ software package provides various tools for harmonic analysis on the special orthogonal group,
which is the configuration space for the attitude dynamics of a rigid body.
- Complex/Real Noncommutative Harmonic Analysis on SO(3)
- Complex/Real Spherical Harmonics on the Unit-Sphere
- Fast Fourier Transform on SO(3)
- Compute Wigner D-matrix
- Compute Clebsch-Gordon coefficients
- This packages utilizes OpenMP for accelerated computing for multicore processors.
- It implements Fast Fourier Transform (FFT) algorithms developed for SO(3).
- This is the only package that supports real harmonic analysis on SO(3).
- The computation in this package is verified by various unit-testing, utilizing GoogleTest.
- For convenience, it supports the indexing consistent with mathematical expressions. More specifically, harmonics on SO(3) is indexed by three integers varying from negative values to positive ones:
-
It is common that the indices
m,nis mapped to non-negative values. In this package, the above element can be directly accessed byF(l,m,n)without any conversion. Also, the (2l+1) by (2l+1) matrix can be accessed byF[l] -
This package provides routines for Clebsch-Gordon coefficients, or derivatives of harmonics that are not available elsewhere.
-
It is based on the Eigen library supporting vectorization.
This library is based on
- T. Lee, "Real Harmonic Analysis on the Special Orthogonal Group," arXiv, 2018 (Real harmonic analysis on SO(3))
It also utilizes the results of
- D. Varshalovich, A. Moskalaev, V. Khersonskii, "Quantum Theory of Angular Momentum," World Scientific, 1988 (Wigner-D function and spherical harmonics)
- G. Chirikjian, A. Kyatkin, "Engineering Applications of Noncommutative Harmonic Analysis," CPC Press, 2000 (Operational properties)
- D. Marinucci, G. Peccati, "Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications, Cambridge University Press, 2011 (Clebsch-Gordon coefficients)
The following libraries are required:
Please follow the instruction at each link to install the libraries, except the Eigen library that is included as a git submodule in this repository.
Tips for Mac: First, install XCode Command Line Tools for gcc compilers and git. For other libraries, it is recommended to install Homebrew, a software package manager for Mac. Then, the CMake and OpenMP can be installed by
brew install cmake
brew install llvm
brew install libomp
Tips for Linux: For Debian linux, such as Ubuntu, the above packages can be installed by
sudo apt-get install git-core
sudo apt-get install cmake
sudo apt-get install libomp-dev
Notes for Windows: While this package does not use any OS-specific command, it has not been tested in Windows.
Execute the following commands at a folder above this package will be installed:
git clone https://github.com/fdcl-gwu/FFTSO3.git
cd FFTSO3
git submodule update --init --recursive
cd build
cmake ..
make
../bin/fftso3_unit_test
The last command executes unit-testing, and the installation is succesful if it prints out the following message at the end
...
...
[ PASSED ] 12 tests.
Notes for Eigen library: If the Eigen library is already installed, the command git submodule update... can be skipped. Instead, CMakeList.txt should be modified accordingly. See Using Eigen in CMake Projets.
Four examples are provided.
example0.cppminimal working example for real harmonic analysis on SO(3)example1.cppelaborated example for real harmonic analysis on SO(3)example2.cppelaborated example for complex harmonic analysis on SO(3)example3.cppapplication to spherical image correlation introduced in this article
All of the example codes are under FFTSO3/exmaple, and once the above installation procedures are completed, the executable binary files are copied to the folder FFTSO3/bin
This example illustrates how to perform fast forward transform of the trace function, and compute the inverse transform at the identity, using real harmonic analysis on SO(3).
#include <iostream>
#include "fdcl_FFTSO3.hpp"
// define a real-valued function on SO(3)
double func(Eigen::Matrix3d R)
{
return R.trace();
}
int main()
{
int l_max=2; // the maximum order of Fourier transform
fdcl::FFTSO3_real RFFTSO3(l_max); // FFTSO3_real object for real harmonic analysis on SO(3)
fdcl::FFTSO3_matrix_real F(l_max); // FFTSO3_matrix_real object to save real-valued Fourier parameters
Eigen::Matrix3d R;
double f=0.;
F = RFFTSO3.forward_transform(func); // perform forward Fourier transform
std::cout << "Fourier parameter F = " << std::endl << std::endl << F << std::endl; // show Fourier parameters
R.setIdentity(); // R is set to the identity matrix
f = RFFTSO3.inverse_transform(F,R); // compute the inverse transform at the identity
cout << "f = " << f << std::endl;
return 0;
}Excecute FFTSO3/bin/example0 to get the followin results.
Fourier parameter F =
l=0
-4.16334e-17
l=1
0.333333 -4.53217e-17 7.28124e-17
-3.08183e-17 0.333333 8.25415e-18
-1.38104e-16 -5.13615e-18 0.333333
l=2
5.48551e-18 -1.81924e-17 1.05714e-18 -3.3415e-17 3.14623e-18
2.11998e-17 -4.16334e-17 -1.36215e-17 1.64077e-19 2.38253e-17
-7.2492e-19 -9.99741e-18 -1.38778e-17 5.84227e-18 2.16787e-18
-3.84623e-17 -3.84837e-18 -1.0536e-17 -9.71445e-17 1.59126e-17
-8.54102e-19 4.87669e-17 7.8598e-18 -1.92921e-17 9.78959e-20
f = 3
For other examples, see the comments within the source file.
More detailed user manual is available at Doxygen Manual for Class Members (HTML)
- The SOFT Package: FFTs on the Rotation Group: complex harmonic analysis on SO(3)
This library is developed by Flight Dynamics and Control Lab at The George Washington University, Washington DC. Contact [email protected] for question and comment.
This research has been supported in parts by NSF under the grant CMMI-1335008, and by AFOSR under the grant FA9550-18-1-0288.