A method of SO3FunHarmonic.

the harmonic series of a function on SO(3) by quadrature; MATLAB’s approximate and quadrature

Syntax

SO3F = SO3FunHarmonic.quadrature(odf, bandwidth=48)

Another function to a bandwidth.

SO3F = SO3FunHarmonic.quadrature(lambda o: o.angle(), cs, bandwidth=16)

A callable and its frames.

SO3F = SO3FunHarmonic.quadrature(g, odf(g.nodes()))

Values on a quadratureSO3Grid.

SO3F = SO3FunHarmonic.quadrature(ori, values, bandwidth=8)

Values at orientations, weighted equally.

Description

A callable or another SO(3) function is expanded as SO3FunHarmonic(f, cs, ss, bandwidth=) does, exact up to the bandwidth; an SO3FunRBF reaches at most its kernel’s bandwidth.

Values at nodes are summed with the weights against the Wigner functions and made invariant under the frames. What that gives depends on the nodes:

  • a quadratureSO3Grid: the grid’s weights make the result exact up to its bandwidth, the default then. Values at its distinct nodes are spread onto all of them.
  • other nodes with the weights of a quadrature rule: the quadrature of that rule.
  • other nodes, equally weighted by 1/n: the series of the density of the nodes times the values, not a fit; for a fit see SO3FunHarmonic.interpolate.

SO3FunHarmonic.approximate is the same function.

Input

nodes a callable or SO3Fun, a quadratureSO3Grid, orientation or rotation
values array, the values at the nodes; none for a callable, whose crystal frame may stand here
cs, ss the frames of a callable or of rotations

Keywords

keyword default description
bandwidth None the largest degree of the series, by default the preference defaultSO3Bandwidth or the grid’s
weights None the quadrature weights of the nodes
real None whether the function takes real values only; by default whether the values are real
antipodal False f(g) = f(g^-1), for a misorientation function; taken from antipodal orientations as well

Output

SO3F SO3FunHarmonic

See also

SO3FunHarmonic, SO3FunHarmonic.interpolate, quadratureSO3Grid, calcDensity

MATLAB’s SO3FunHarmonic.approximate

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/SO3FunHarmonic.approximate_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.