a rotationally symmetric kernel represented by Chebyshev coefficients.

Syntax

psi = SO3Kernel(A)
values = psi(np.cos(omega/2))

Description

The series is sum(A[n]*U_(2*n)(t)), where t=cos(omega/2) and U is the Chebyshev polynomial of the second kind. A[0] is the mean with respect to uniform probability on SO(3); choose A[0]=1 for a mean-one density kernel. No positivity or normalisation is imposed on arbitrary coefficients.

Use an SO3Kernel family to construct a specified shape or angular halfwidth. For a callable formula, use SO3KernelHandle. The same kernel can be supplied to unimodalODF or calcDensity for orientation smoothing.

Input

`A`	one-dimensional real coefficient array, degrees zero through L

Output

psi SO3Kernel

Properties

`A`	Chebyshev coefficients through the stored bandwidth
`bandwidth`	largest stored harmonic degree

Members

calcFourier	the Chebyshev coefficients A_0 .
conv	convolution multiplies the coefficients and divides by 2n+1, on S2 when either factor is
eval	the kernel at the cosine argument t (the angle on S1), or at a rotation or direction, or between two of them
grad	the derivative with respect to the rotation angle omega at the cosine argument t = cos(omega/2)
minus
mrdivide
mtimes
norm	the L2 norm of the kernel, the Euclidean norm of its coefficients
plot	plot(x, …) draws x as its kind is drawn
plotPDF	pole figures, one per crystal direction: of an ODF, of a list of orientations or of a measured PoleFigure
plotSpektra	the Chebyshev coefficients of a kernel, or the harmonic power of a function, per degree
plus
radon	the Radon transform is the S2 kernel with the same coefficients
rdivide
SO3Kernel	a rotationally symmetric kernel represented by Chebyshev coefficients
times
uminus
volume	the share of orientations within a radius of a centre: the integral of f over the ball, int_{angle(g, c) < r} f(g) dg, relative to the integral over all orientations

Used on

SO3Kernels

MATLAB’s SO3Kernel_index

source

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/SO3Kernel_index_py.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.

The class SO3Kernel