S2DeLaValleePoussinKernel edit page

The spherical de la Vallee Poussin kernel is defined by

\[ K(t) = (1+\kappa)\,(\frac{1+t}{2})^{kappa}\]

for \(t\in[0,1]\). The de la Vallee Poussin kernel additionaly has the unique property that for a given halfwidth it can be described exactly by a finite number of Fourier coefficients. This kernel is recommended for Texture analysis as it is always positive and there is no truncation error in Fourier space.

Hence we can define the de la Vallee Poussin kernel \(\psi_{\kappa}\) depending on a parameter \(\kappa \in \mathbb N \setminus \{0\}\) by its finite Legendre polynomial expansion

\[ \psi_{\kappa}(t) = \sum\limits_{n=0}^{L} a_n(\kappa) \mathcal P_{n}(t)\].

We obtain the Legendre coefficients \(a_n(\kappa)\) by \(a_0=1\), \(a_1=\frac{\kappa}{2+\kappa}\) and the three term recurence relation

\[ (\kappa+l+2) a_{l+1} = -(2l+1)\,a_l + (\kappa-l+1)\,a_{l-1}\].

Syntax

psi = S2DeLaValleePoussinKernel(20)
psi = S2DeLaValleePoussinKernel('halfwidth',10*degree)

Input

kappa kernel parameter

Options

halfwidth angle at which the kernel function has reduced to half its peak value
bandwidth harmonic degree

See also

S2Kernel

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/S2DeLaValleePoussinKernel.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.