the projection of the harmonic coefficients onto the functions invariant under a group, one matrix P_n per degree up to L

Syntax

P = invariantProjection(cs, 16)

Under the rotation parts of all elements.

P = invariantProjection(cs, 16, proper=True)

Under the proper elements alone.

P = invariantProjection(cs, 16, dtype=np.float32)

In single precision.

Description

P_n is the mean of the Wigner matrices of the group’s rotations; it is nonzero only where both indices are multiples of the order of the group’s axis along z (P.fold), and only that part is held. A group whose elements all have the identity as rotation part projects with the identity, no matrices at all. The projections are cached per group, precision and properness, and grow with the bandwidth asked; the symmetrisation of SO3FunHarmonic and S2FunHarmonic uses them.

Input

g pointGroup or a frame carrying one
L the bandwidth, the largest degree

Keywords

keyword default description
dtype np.float64 the real precision of the matrices, which are complex of that precision
proper False project onto the functions invariant under the proper elements alone, which the sphere needs beside the rotation parts of all elements

Output

P InvariantProjection, its blocks the matrices of the degrees 0 to L

See also

WignerD, SO3FunHarmonic.symmetrise, S2FunHarmonic.symmetrise

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/invariantProjection_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.