the adjoint of the Wigner coefficient transform: the Fourier array of weighted values back to Wigner coefficients
Syntax
fhat = wignerTransformAdjoint(H, L)
The coefficients of the array, j = 0 at index L.
fhat = wignerTransformAdjoint(H, L, real=True)
A real function, only l >= 0 given.
fhat = wignerTransformAdjoint(H, L, centre=0)
The output of an FFT, negative orders wrapped.
Description
fhat_n^{kl} = sqrt(2n+1) i^{k-l} sum_j d_n^{jk}(pi/2) d_n^{jl}(pi/2) H[l, k, -j], the adjoint of wignerTransform. With H = sum_m w_m e^{i(k alpha_m + j beta_m + l gamma_m)}, the adjoint NUFFT of weighted values at the orientations, it gives the coefficients a quadrature takes, which is how the harmonic expansion of a function and calcDensity reach their coefficients.
For a real function only l >= 0 is given and the rest of every block follows from fhat_n^{-k,-l} = (-1)^{k-l} conj(fhat_n^{kl}). The folds step l by rA and k by rB, as in the forward transform.
Input
H |
complex array (l, k, j), the Fourier array of the values |
L |
the bandwidth |
Keywords
| keyword | default | description |
|---|---|---|
real |
False |
the function is real, H given 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 |
centre |
None |
the index of the order j = 0 on the last axis of H; L by default, 0 for the output of an FFT |
Output
fhat |
the internal Wigner coefficients, SO3FunHarmonic._fhat, of the degrees 0 to L |
See also
wignerTransform, sphericalTransformAdjoint, SO3FourierPlan, wignerSeriesAdjoint
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/wignerTransformAdjoint_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.