an SO3FunRBF, a weighted sum of radial kernels, through values at orientations; MATLAB’s SO3FunRBF.interpolate

Syntax

SO3F = interpolateRBF(ori, y)

The defaults.

SO3F = interpolateRBF(ori, y, halfwidth=10*degree)

The halfwidth of the kernel.

SO3F = interpolateRBF(ori, y, kernel=psi, exact=True)

The kernels centred at the nodes.

SO3F = interpolateRBF(ori, y, kernel=psi, resolution=7*degree)

On an equispaced grid of that resolution.

SO3F = interpolateRBF(ori, y, kernel=psi, SO3Grid=S3G)

On a grid of your own.

SO3F, it = interpolateRBF(ori, y, density=True, withIterations=True)

A density, with the iterations.

Description

The function f(g) = sum_j w_j psi(angle(g, g_j)) has its kernels centred on an equispaced grid, on a grid of your own or, with exact=True, at the nodes. Its weights minimise the squared misfit sum_i |f(ori_i) - y_i|^2.

How the weights are found depends on the values:

density=True: non negative weights by the modified least squares method, mlsq; the result has mean one and drops the centres of negligible weight.
values of one sign: weights of that sign by mlsq, scaled to the mean given, or by default to the mean of the values over the Voronoi cells of the nodes. Above 10000 nodes without a mean, LSQR as below.
values of both signs: LSQR, the weights of any sign.

Values of shape (n, k) give a list of k functions, one per column, on the same centres, and withIterations a list of the iterations of each.

Input

`ori`	orientation, the nodes
`y`	the values at the nodes, (n,) or (n, k)

Keywords

keyword	default	description
`kernel`	`None`	SO3Kernel, the shape of the kernels instead of the de la Vallee Poussin kernel
`halfwidth`	`None`	the halfwidth of the de la Vallee Poussin kernel, by default the resolution or 5 degree
`resolution`	`None`	the resolution of the equispaced grid of the centres, by default the halfwidth of the kernel
`SO3Grid`	`None`	orientation, the centres instead of the equispaced grid
`density`	`False`	a non negative function of mean one
`mean`	`None`	the mean the result is scaled to, by default that of the values over the Voronoi cells of the nodes
`exact`	`False`	centre the kernels at the nodes themselves
`tol`	`0.001`	the tolerance of the iteration
`maxIter`	`None`	the largest number of iterations, by default 100 for mlsq and 30 for LSQR
`withIterations`	`False`	also return the number of iterations, MATLAB’s second output

Output

`SO3F`	SO3FunRBF, or a list of them for several columns
`it`	the number of iterations

References

H. Schaeben, F. Bachmann, J.-J. Fundenberger, Construction of weighted crystallographic orientations capturing a given orientation density function, J. Mater. Sci. 52 (2017) 2077-2090

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/interpolateRBF_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.

interpolateRBF