von Mises Fisher kernel on the orientation space
The von Mises Fisher kernel \(\psi_{\kappa}\in L^2(\mathcal{SO}(3))\) is a nonnegative function depending on a parameter \(\kappa>0\) and is defined by its Chebyshev series
\[ \psi_{\kappa}(t) = \sum\limits_{n=0}^{\infty} \frac{\mathcal{I}_n(\kappa}-\mathcal{I}_{n+1}(\kappa)} {\mathcal{I}_0(\kappa)-\mathcal{I}_1(\kappa)} \, \mathcal U_{2n}(t)\]
or directly by
\[ \psi_{\kappa}(\cos\frac{\omega®}2) = \frac1{\mathcal{I}_0(\kappa)-\mathcal{I}_1(\kappa)} \, \mathrm{e}^{\kappa \cos\omega®}\]
while \(\mathcal I_n,\,n\in\mathbb N_0\) denotes the the modified Bessel functions of first kind
\[ \mathcal I_n (\kappa) = \frac1{\pi} \int_0^{\pi} \mathrm e^{\kappa \, \cos \omega} \, \cos n\omega \, \mathrm d\omega \].
Syntax
psi = SO3vonMisesFisherKernel(100)
psi = SO3vonMisesFisherKernel('halfwidth',5*degree)Input
| kappa | kernel parameter |
Output
| psi | SO3vonMisesFisherKernel |
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/SO3vonMisesFisherKernel.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.