Calculate the L2-norm also known as texture index of a SO3FunHarmonic, by using Parseval's equality for the integral
\[ t = \sqrt{\frac1{8\pi^2}\int_{SO(3)} |f( R ) |^2 dR},\]
with \(vol(SO(3)) = \int_{SO(3)} 1 dR = 8\pi^2\). The Wigner-D functions (one Wigner-coefficient is 1 and all others are 0) are L2-normalized.
We can compute the Sobolev norm of an SO3FunHarmonic by
\[ t = \sqrt{ \sum_{n=0}^N (2n+1)^{2s} \, \sum_{k,l=-n}^n \abs{\hat{f}_n^{k,l}}^2 },\]
where \(s\) is the Sobolev index. (The default case \(s=0\) corresponds to the L2-norm.)
Syntax
t = norm(SO3F)
t = norm(SO3F,s)Input
| SO3F | SO3FunHarmonic |
| s | double (Sobolev index) |
Output
| t | double |
Options
| resolution | choose mesh width by calculation of mean |
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.norm.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.