the function in harmonic form fitted by least squares to values at nodes: on SO(3) at orientations, on the sphere at directions, a vector field for vector values or tangent vectors
Syntax
SO3F = interpolate(ori, y)
An SO(3) function through values at orientations.
SO3F = interpolate(ori, y, bandwidth=8, weights='equal')
Of a given degree, the nodes weighted equally.
SO3F, (flag, residual, nIter) = interpolate(ori, y, withParameters=True)
With LSQR’s flag, residual and iterations.
sF = interpolate(v, y, bandwidth=16, lambda_=1e-4)
A function on the sphere through values at directions.
sVF = interpolate(v, w)
A vector field on the sphere through vectors w.
SO3VF = interpolate(SO3TangentVector(w, ori), bandwidth=8)
A vector field on SO(3) through tangent vectors.
SO3VF = interpolate(SO3TangentVector(w, ori), lambda_=0)
Without regularisation.
Description
The harmonic coefficients up to the bandwidth minimise the weighted squared misfit at the nodes plus lambda_ times the Sobolev norm of order s, solved by LSQR. Without a bandwidth it follows from the number of nodes, on SO(3) at most the preference ‘maxSO3Bandwidth’.
On SO(3) the fit is posed among the functions with the symmetries of the nodes. Nodes closer than 0.02 are merged into one carrying the mean of their values.
Values with further axes after the node axis give one function per component, and the LSQR parameters then arrays of one per component. A vector field on SO(3) is fitted componentwise, three SO3FunHarmonic, as MATLAB’s SO3VectorFieldHarmonic.interpolate; MATLAB passes the tangent vectors as values, here they carry their orientations.
Input
nodes |
orientation or rotation, vector3d, or SO3TangentVector for a vector field on SO(3) |
values |
ndarray of one value per node, or vector3d for a vector field on the sphere |
Keywords
| keyword | default | description |
|---|---|---|
bandwidth |
None |
the largest harmonic degree; set by hand it may undersample |
lambda_ |
None |
the regularisation parameter, 1e-8 on SO(3) by default, 0 for none; MATLAB’s ‘regularization’ |
s |
2 |
the order of the Sobolev norm; MATLAB’s ‘SobolevIndex’ |
weights |
None |
the weights of the nodes: by default their Voronoi volumes below 10000 nodes, else equal; ‘equal’, or one number per node |
maxIter |
100 |
the largest number of LSQR iterations |
tol |
None |
the tolerance of LSQR, 1e-3 on SO(3) and 1e-6 on the sphere by default |
withParameters |
False |
return also LSQR’s flag, 1 when the iterations ran out, the relative residual and the number of iterations, for a function on SO(3) or the sphere |
Output
SO3F |
SO3FunHarmonic, S2FunHarmonic, S2VectorFieldHarmonic or SO3VectorFieldHarmonic |
See also
SO3FunHarmonic.interpolate, S2FunHarmonic.interpolate, SO3FunRBF.interpolate, calcDensity, SO3FunVectorValued, HarmonicApproximationTheory
MATLAB’s SO3VectorFieldRBF.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/SO3VectorFieldRBF.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.