the modified least squares solver of the pole figure inversion, an ODF as a weighted sum of unimodal components fitted to scattered pole figure intensities

Syntax

solver = MLSSolver(pf, resolution=5*degree, halfwidth=7.5*degree)

Set up the solver.

odf = solver.calcODF()

And run it.

solver = MLSSolver(pf, psi, S3G=grid)

A kernel and the centres given.

solver.alpha

The scaling of each pole figure.

Description

The ansatz functions have the shape of a kernel, by default the de la Vallee Poussin kernel, and sit on a grid of orientations, by default an equispaced grid of the resolution. The coefficients minimise the squared misfit between the pole figures of the ODF and the measured intensities, every pole figure with its own scaling alpha, under the constraint that they are non negative. calcODF(pf, …) builds the solver and runs it at once.

The kernel and the grid may also be passed positionally. The solver keeps its state: solver.S3G the centres, solver.psi the kernel, solver.c the coefficients, solver.alpha the scaling factors, solver.zrm the zero range method; solver.odf the ODF they describe.

Input

`pf`	PoleFigure
`psi`	SO3Kernel, the shape of the ansatz functions
`grid`	orientation, the centres of the ansatz functions

Keywords

keyword	default	description
`resolution`		the spacing of the grid the ansatz functions sit on, by default the resolution of the pole figures (MATLAB: 3/2 of it)
`halfwidth`		the halfwidth of the ansatz functions, by default the resolution of the grid (MATLAB: 2/3 of it)
`kernel`		SO3Kernel, the shape of the ansatz functions instead of the de la Vallee Poussin kernel of the halfwidth
`S3G`		orientation, the centres of the ansatz functions instead of the equispaced grid
`zeroRange`		apply the zero range method: drop the centres where some pole figure says the ODF vanishes; a zeroRangeMethod gives its parameters
`minIter`		the least number of iterations
`maxIter`		the largest number of iterations (MATLAB iterMax)
`c0`		the coefficients of the ansatz functions the iteration starts from, by default all equal
`ghostCorrection`		fix the uniform portion the data allows and reconstruct only the rest; MATLAB flag noGhostCorrection is ghostCorrection=False
`quadratureWeights`		weight the residuals with the quadrature weights of the measured directions (no specimen symmetry)
`intensityWeights`		weight the residuals with (1 + I)^(-1/2), the counting statistics of the intensities I
`regularisation`		the weight of a penalty on the Dirichlet energy, a discrete Laplacian on the grid; zero for none
`verbose`		report the fit of every pole figure in every iteration

Output

solver MLSSolver

References

R. Hielscher, H. Schaeben, A novel pole figure inversion method: specification of the MTEX algorithm, J. Appl. Cryst. 41 (2008) 1024-1037

Members

calcODF	the ODF the solver reconstructs, with the ghost correction
CS	the crystal frame of the pole figures the solver inverts
doIter	one step of the modified projected gradient
frameA	the frame the orientations map from, the crystal frame
frameB	the frame the orientations map to, the specimen or a second crystal frame
init	the operators on the grid the zero range method leaves
initIter	the state the iteration starts from: the normalisation directions a_i, the scalings alpha_i and the weighted residuals u_i
initRegularisation	sets the regularisation matrix of the solver, its weight times a discrete Laplacian on the grid
iterate	the iteration, stopped when the residual improves by less than five percent
MLSSolver	the modified least squares solver of the pole figure inversion, an ODF as a weighted sum of unimodal components fitted to scattered pole figure intensities
odf	the ODF the current coefficients describe
residual	the relative residual of each pole figure, and the norm of the vector of them
SS	the specimen frame of the pole figures the solver inverts

MATLAB’s MLSSolver_index

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/MLSSolver_index_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.

The class MLSSolver