a function on the sphere by its spherical harmonic series, or a list of such functions; the default representation of a function on the sphere

Syntax

sF = S2FunHarmonic(fhat)

From MTEX’s coefficients.

sF = S2FunHarmonic(fhat, cs, antipodal=True)

In a crystal frame, f(v) = f(-v).

sF = S2FunHarmonic(lambda v: v.z**2, bandwidth=16)

A callable expanded by quadrature.

sF = S2FunHarmonic(psi)

The series of a kernel on the sphere.

sF = S2FunHarmonic(sF, bandwidth=8)

Another function, truncated.

Description

The harmonic series is the form on which convolution, rotation, products and integrals are cheap. The coefficients fhat are MTEX’s, degree by degree, (L + 1)^2 of them for the bandwidth L; a vector ending inside a degree is padded with zeros. A list of functions carries its shape in front of the coefficients.

The first argument may also be:

a callable of vector3d: sampled on a quadrature grid of the bandwidth, by default the preference defaultS2Bandwidth; S2FunHarmonic.quadrature takes values at nodes
an S2Kernel: its series
another S2Fun: its harmonic form, truncated to the bandwidth if given

The function is made invariant under the group of its frame unless symmetrise=False, so a callable that is not invariant becomes its mean over the symmetric directions. A frame with an inversion makes the function antipodal.

Input

`fhat`	array of MTEX’s coefficients, an S2Fun, an S2Kernel or a callable of vector3d
`frame`	the crystalFrame or specimenFrame of the directions

Keywords

keyword	default	description
`real`		whether the function takes real values only; by default read off the coefficients or the sampled values
`antipodal`		f(v) = f(-v)
`symmetrise`		make the coefficients invariant under the group of the frame
`bandwidth`		the largest degree of the series
`dtype`		the complex type of the coefficients, complex128 unless given

Output

`sF`	S2FunHarmonic

Properties

`fhat`	spherical harmonic coefficients in MTEX’s public basis
`bandwidth`	largest spherical harmonic degree
`frame`	reference frame carrying the function symmetry
`CS`	alias for the spherical reference frame
`isReal`	whether the function is real-valued
`antipodal`	whether f(v)=f(-v)
`shape`	shape of a list of functions, excluding the coefficient axis

Members

abs	the absolute value \|f\| as a handle, evaluated when asked
adjoint	S2FunHarmonic.adjointNFSFT(v, values, bandwidth=32): the adjoint transform, the coefficients sum_j w_j y_j conj(Y(v_j)) with unit weights unless given, a grid’s own on a grid
adjointNFSFT	S2FunHarmonic.adjointNFSFT(v, values, bandwidth=32): the adjoint transform, the coefficients sum_j w_j y_j conj(Y(v_j)) with unit weights unless given, a grid’s own on a grid
approximate	the harmonic series of a function on the sphere by quadrature, from a callable or from values at nodes
bandwidth	the bandwidth, the highest degree of the coefficients
cat	the functions joined into one list along an axis, padded to the largest bandwidth
conj	the complex conjugate function
conv	convolve a spherical function with a kernel or another spherical function
CS	the frame of the function, MATLAB’s sF.CS
dot	the inner product with another function of the same frame, for the normalised surface measure: int conj(f) g dS / 4 pi
eig	eig(T) is T.eig(): the eigenvalues in ascending order and the eigenvectors
entropy	the entropy -int f log f of a function on SO(3) or on the sphere
eval	the function at directions, or at the nodes of a quadrature grid
even	return the even part (f(v)+f(-v))/2 of a spherical function
exp	the exponential of the function, expanded by quadrature, or at the bandwidth given
fhat	the coefficients in MTEX’s basis, as MATLAB’s sF.fhat reads and assigns them
grad	the surface gradient: a vector field, or its values at the directions x
horzcat	the functions joined into one list along an axis, padded to the largest bandwidth
how2plot	the plotting convention of the frame, or the session’s
interpolate	S2FunHarmonic.interpolate(v, y) is interpolate(v, y): the function of least squares through the values y at the directions v
invRadon	invert the Funk transform on the even spherical harmonics
isclose	whether the coefficients agree with those of g within the tolerances, relative to the norm of g
laplace	apply the spherical Laplace-Beltrami operator
log	the logarithm of the function, expanded by quadrature, or at the bandwidth given
mean	the mean over the sphere, int f dS / 4 pi, one per function
mtimes
norm	the L2 norm for the normalised surface measure, or the Sobolev norm, one per function
plot	plot(x, …) draws x as its kind is drawn
plotSpektra	the Chebyshev coefficients of a kernel, or the harmonic power of a function, per degree
plus
plus
power	the power of every degree, the norm of the coefficients of that degree, as an array of length bandwidth + 1 per function
power	the power of every degree, the norm of the coefficients of that degree, as an array of length bandwidth + 1 per function
quadrature	the harmonic series of a function on the sphere by quadrature, from a callable or from values at nodes
quadrature	the harmonic series of a function on the sphere by quadrature, from a callable or from values at nodes
radon	the Funk transform, the mean over the great circle perpendicular to every direction
rdivide
rdivide
reshape	the list of functions in another shape
rotate	the function rotated, f(rot^-1 v)
S2FunHarmonic	a function on the sphere by its spherical harmonic series, or a list of such functions
S2FunHarmonicSym	a function on the sphere by its spherical harmonic series, or a list of such functions
shape	the shape of a list of functions, () for one
size	the number of functions in a list, 1 for one function
sqrt	the square root of the function, expanded by quadrature, or at the bandwidth given
subsref
sum	the integral over the sphere, int f dS, MATLAB’s sum
symmetrise	the projection onto the functions invariant under the group of the frame, or under a list of rotations forming a group the frame does not name
T	the list transposed, a row of n functions becoming a column, and MATLAB’s ‘ as sF.conj().T
textureIndex	the texture index of a function on SO(3) or on the sphere, the squared L2 norm of its harmonic expansion
times
times
truncate	the coefficients up to degree L
uminus
unimodal	S2FunHarmonic.unimodal(v, psi): MATLAB’s other name of S2Fun.unimodal
vertcat	the functions joined into one list along an axis, padded to the largest bandwidth
volume	the share of orientations within a radius of a centre: the integral of f over the ball, int_{angle(g, c) < r} f(g) dg, relative to the integral over all orientations

Used on

SO3FunConvolution, S2FunHarmonicRepresentation, S2FunSampling, S2FunVectorValued, SphericalHarmonics

MATLAB’s S2FunHarmonic_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/S2FunHarmonic_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 S2FunHarmonic