The Squared Singularity Kernel \(\psi_{\kappa}\in L^2(\mathcal{SO}(3))\) is a nonnegative function depending on a parameter \(\kappa\in(0,1)\) and is defined by its Chebyshev series
\[ \psi_{\kappa}(t) = \sum\limits_{n=0}^{\infty} \hat{f}_n(\kappa) \, \mathcal U_{2n}(t) \].
where the chebychev coefficients follows a 3-term recurrsion
\(\hat{f}_0 = 1\) \(\hat{f}_1 = \frac{1+\kappa^2}{2\kappa}-\frac1{\log\frac{1+\kappa}{1-\kappa}}\) \(\hat{f}_n = \frac{(2n-3)(2n+1)(1+\kappa^2)}{(2n-1)(n-1)2\kappa} \, \hat{f}_{n-1}(\kappa)-\frac{2\kappa(n-2)(2n+1)}{2n-3} \, \hat{f}_{n-2}(\kappa)\).
Syntax
psi = SO3SquareSingularityKernel(0.2)
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/SO3SquareSingularityKernel.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.