the Wigner coefficient transform: the coefficients of a Wigner series as the Fourier array of the same function in the Euler angles

Syntax

G = wignerTransform(fhat, L)

The full Fourier array, axes (l, k, j).

G = wignerTransform(fhat, L, real=True)

A real function, only l >= 0.

G = wignerTransform(fhat, L, folds=(4, 2))

A 4 fold about z in the crystal, 2 fold in the specimen.

Description

A Wigner series of bandwidth L is a trigonometric series in the Euler angles of the ZYZ convention (alpha = phi1 - pi/2, beta = Phi, gamma = phi2 + pi/2), f(alpha, beta, gamma) = sum G[l, k, j] e^{-i(k alpha + j beta + l gamma)}, with G[l, k, j] = sum_n sqrt(2n+1) i^{k-l} fhat_n^{kl} d_n^{jk}(pi/2) d_n^{jl}(pi/2). A NUFFT, or an FFT on a Clenshaw-Curtis grid, evaluates this array; SO3FunHarmonic.eval goes this way. Every index runs from -L to L and sits at its value plus L.

The coefficients are the port’s internal ones, SO3FunHarmonic._fhat, not the fhat a user reads. For a real function only l >= 0 is kept, the plane l = 0 halved, and f is twice the real part of the sum.

A function with an rA fold symmetry about z in the crystal and an rB fold in the specimen has coefficients only at l = 0 mod rA and k = 0 mod rB; with folds=(rA, rB) those axes are indexed by l / rA and k / rB.

Input

fhat the internal Wigner coefficients, a complex array of at least the degrees 0 to L
L the bandwidth

Keywords

keyword default description
real False the function is real, the array kept for l >= 0
folds (1, 1) (rA, rB), the folds about z in the crystal and the specimen
degrees None pairs (n, d_n(pi/2)) to sum over instead of the cached matrices, which selects the NumPy form

Output

G complex array of the shape foldedDims(L, real, folds)

See also

wignerTransformAdjoint, sphericalTransform, SO3FunHarmonic, SO3FourierPlan

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/wignerTransform_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.