a vector field on SO(3) as a sum of radial kernels at centre orientations with weight vectors, exactly equivariant

Syntax

SO3VF = SO3VectorFieldRBF(center, weights, psi)

From centres, weight vectors and a kernel.

SO3VF = SO3VectorFieldRBF(center, weights, None, 'right')

In the right tangent space, the default kernel.

SO3VF = SO3VectorFieldRBF(t, kernel=psi)

Fitted through tangent vectors t.

SO3VF = SO3VectorFieldRBF(F, resolution=10*degree)

Another field fitted on a grid.

Description

The field is the sum over the centres of the kernel at the misorientation to the centre, times the centre’s weight vector. It is summed over the symmetric equivalents, so that it is equivariant under the frames.

Through tangent vectors t, an SO3TangentVector at reference orientations, the weights solve the dense linear system of the kernel between the references, which suits hundreds of centres. Another field is sampled on an equispaced grid of the resolution and fitted the same way, with a kernel of halfwidth the resolution unless one is given.

MATLAB stores the three components as SO3FunRBF; here the weights are one vector per centre.

Input

`center`	orientation, the centres; or an SO3TangentVector or SO3VectorField to fit
`weights`	(n, 3) array, one weight vector per centre
`psi`	SO3Kernel, a de la Vallee Poussin kernel of halfwidth 10 degree unless given

Keywords

keyword	default	description
`kernel`		the kernel of a fit
`resolution`		the grid spacing of the fit of another field, 5 degree unless given

Output

SO3VF SO3VectorFieldRBF

Members

bandwidth	the bandwidth of the kernel
CS	the crystal frame of the field, that of the centres
dot	the pointwise inner product: with another field a function on SO(3), with a constant vector a function on the component frames
eval	the tangent vectors of the field at orientations, the kernel sums over the symmetric equivalents of the centres
frameA	the frame the orientations map from, the crystal frame
frameB	the frame the orientations map to, the specimen or a second crystal frame
interpolate	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
norm	the length of the vectors as a function on SO(3), the same as abs
normalize	the field of unit vectors in the directions of the field, by quadrature at twice the bandwidth
plus
rotate	the field turned by a rotation, from the left or with side=’right’ from the right, the centres and weights moved
SO3VectorFieldRBF	a vector field on SO(3) as a sum of radial kernels at centre orientations with weight vectors, exactly equivariant
SS	the specimen frame of the field, that of the centres
times
uminus

MATLAB’s SO3VectorFieldRBF_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/SO3VectorFieldRBF_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 SO3VectorFieldRBF