A method of S1FunHarmonic.
the Fourier series through noisy samples at angles on the circle, fitted by regularised least squares
Syntax
sF = S1FunHarmonic.interpolate(x, values)
Samples at the angles x.
sF = S1FunHarmonic.interpolate(x, values, bandwidth=10, regularisation=1e-3)
A bandwidth and a stronger smoothing.
sF = S1FunHarmonic.interpolate(x, values, weights=w)
The samples weighted.
Description
Equispaced samples, x = 2 pi k / n, are expanded by quadrature to the bandwidth (n - 1) / 2, or the one given. Other angles are taken modulo 2 pi, samples at coinciding angles merged to their mean, and the coefficients fitted by LSQR to the weighted samples, penalised by regularisation times the Sobolev norm sum |k|^(2 SobolevIndex) |c_k|^2; the bandwidth is round(n / 4) for n distinct angles, at most the preference maxS1Bandwidth. A bandwidth set by hand may undersample the data.
Input
x |
array of angles in radians |
values |
array, one value per angle |
Keywords
| keyword | default | description |
|---|---|---|
bandwidth |
None |
the largest degree of the series |
weights |
None |
one weight per sample, equal weights unless given |
regularisation |
1e-05 |
the factor lambda of the Sobolev penalty; 0 fits without it |
SobolevIndex |
2 |
the exponent s of the Sobolev penalty |k|^(2 s) |
tol |
1e-06 |
the stopping tolerance of LSQR |
maxIter |
100 |
the largest number of iterations of LSQR |
Output
sF |
S1FunHarmonic, real when the values are |
See also
S1FunHarmonic.quadrature, S1FunHarmonic, SO3FunHarmonic.interpolate
MATLAB’s S1FunHarmonic.interpolate
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/S1FunHarmonic.interpolate_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.