Compute the SO(3)-Fourier/Wigner coefficients of an given SO3Fun or given evaluations on a specific quadrature grid.
Therefore we obtain the Fourier coefficients with numerical integration (quadrature), i.e. we choose a quadrature scheme of meaningful quadrature nodes \(R_m\) and quadrature weights \(\omega_m\) and compute
\[ \hat f_n^{k,l} = \int_{SO(3)} f®\, \overline{D_n^{k,l}(R)} \mathrm{d}\my® \approx \sum_{m=1}^M \omega_m \, f(R_m) \, \overline{D_n^{k,l}(R_m)}, \]
for all \(n=0,\dots,N\) and \(k,l=-n,\dots,n\).
Therefore this method evaluates the given SO3Fun on a with respect to symmetries fundamental Region. Afterwards it uses a inverse trivariate nfft/fft and an adjoint coefficient (Wigner) transform which is based on a representation property of Wigner-D functions.
Hence it do not use the NFSOFT (which includes a fast polynom transform) as in the older method SO3FunHarmonic.quadratureNFSOFT.
Syntax
SO3F = SO3FunHarmonic.quadrature(f)
SO3F = SO3FunHarmonic.quadrature(f,'bandwidth',bandwidth)
SO3F = SO3FunHarmonic.quadrature(f,'bandwidth',bandwidth,quadratureScheme)
SO3F = SO3FunHarmonic.quadrature(f,'bandwidth',bandwidth,'SO3Grid',S3G,'weights',w)
SO3F = SO3FunHarmonic.quadrature(nodes,values)
SO3F = SO3FunHarmonic.quadrature(nodes,values,'bandwidth',48,'weights',w)Input
| f | SO3Fun, function_handle in orientation (first dimension has to be the evaluations) |
| nodes | quadratureSO3Grid, rotation, orientation |
| values | double (first dimension has to be the evaluations) |
Options
| bandwidth | maximal harmonic degree (default: 64) |
| weights | quadrature weights |
| SO3Grid | quadrature nodes |
Flags
| quadratureScheme | ('ClenshawCurtis'|'GaussLegendre') --> default: CC |
Output
| SO3F | SO3FunHarmonic |
See also
SO3FunHarmonic.adjoint SO3FunHarmonic.approximate SO3FunHarmonic
Citing this page.
This page is part of the documentation of
MTEX, a free and open
source MATLAB toolbox for analyzing and modeling crystallographic textures.
It was written by The MTEX Developers and is published at
https://mtex-toolbox.github.io/SO3FunHarmonic.quadrature.html.
If you use MTEX, or reuse text or figures from this page, in your research,
please cite
F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63
BibTeX
@article{bachmann2010mtex,
author = {F. Bachmann and R. Hielscher and H. Schaeben},
title = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
journal = {Solid State Phenomena},
volume = {160},
pages = {63-68},
year = {2010},
doi = {10.4028/www.scientific.net/SSP.160.63},
url = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.