Real/Complex Spherical Harmonic Transform, Gaunt Coefficients and Rotations

A small collection of routines for the Spherical Harmonic Transform and Gaunt coefficients.

# Spherical Harmonic Transform Library
#### A collection of MATLAB routines for the Spherical Harmonic Transform and related manipulations in the spherical harmonic spectrum.

---
> Archontis Politis, 2015
> Department of Signal Processing and Acoustics, Aalto University, Finland
> [e-mail]
---

This Matlab/Octave library was developed during my doctoral research in the [Communication Acoustics Research Group] (http://spa.aalto.fi/en/research/research_groups/communication_acoustics/), Aalto University, Finland. If you would like to reference the code, you can refer to my dissertation published [here](https://aaltodoc.aalto.fi/handle/123456789/22499):

Archontis Politis, Microphone array processing for parametric spatial audio techniques, 2016
Doctoral Dissertation, Department of Signal Processing and Acoustics, Aalto University, Finland

## Description

Both real and complex SH are supported. The orthonormalised versions of SH
are used. More specifically, the complex SHs are given by:

Y_{nm}(\theta,\phi) =
(-1)^m \sqrt{\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}} P_l^m(\cos\theta) e^{im\phi}

and the real ones as in:

R_{nm}(\theta,\phi) =
\sqrt{\frac{2n+1}{4\pi}\frac{(n-|m|)!}{(n+|m|)!}} P_l^{|m|}(\cos\theta) N_m(\phi)

where

N_m(\phi) = \sqrt{2} cos(m\phi}, m>0
N_m(\phi) = 1, m>0
N_m(\phi) = \sqrt{2} sin(|m|\phi}, m<0

Note that the Condon-Shortley phase of (-1)^m is not introduced in the code for
the complex SH since it is included in the definition of the associated
Legendre functions in Matlab (and it is canceled out in the code of the real SH).

The functionality of the library is demonstrated in detail in [http://research.spa.aalto.fi/projects/sht-lib/sht.html]
or in the included script TEST_SCRIPTS_SHT.m.

The SHT transform can be done by:

a) direct summation, for appropriate sampling schemes along with their
integration weights, such as the uniform spherical t-Designs, the Fliege-Maier
sets, Gauss-Legendre quadrature grids, Lebedev grids and others.

b) least-squares, weighted or not, for arbitrary sampling schemes. In this
case weights can be provided externally, or use generic weights based on the
areas of the spherical polygons around each evaluation point determined by
the Voronoi diagram of the points on the unit sphere, using the included
functions.

MAT-files containing t-Designs and Fliege-Maier sets are also included.
For more information on t-designs, see [http://neilsloane.com/sphdesigns/](http://neilsloane.com/sphdesigns/) and

McLaren's Improved Snub Cube and Other New Spherical Designs in Three
Dimensions, R. H. Hardin and N. J. A. Sloane, Discrete and Computational
Geometry, 15 (1996), pp. 429-441.

while for the Fliege-Maier sets see [http://www.personal.soton.ac.uk/jf1w07/nodes/nodes.html](http://www.personal.soton.ac.uk/jf1w07/nodes/nodes.html) and

The distribution of points on the sphere and corresponding cubature
formulae, J. Fliege and U. Maier, IMA Journal of Numerical Analysis (1999),
19 (2): 317-334

Some routines in the library evaluate Gaunt coefficients, which express the
integral of the three spherical harmonics. These can be evaluated either
through Clebsch-Gordan coefficients, or from the related Wigner-3j symbols.
Here they are evaluated through the Wigner-3j symbols through the formula
introduced in

Translational addition theorems for spherical vector wave functions,
O. R. Cruzan, Quart. Appl. Math. 20, 33:40 (1962)

which can be also found in [http://mathworld.wolfram.com/Wigner3j-Symbol.html, Eq.17.](http://mathworld.wolfram.com/Wigner3j-Symbol.html)

Finally, a few routines are included that compute coefficients of
rotated functions, either for the simple case of an axisymmetric kernel
rotated to some direction (\theta_0, \phi_0), or the more complex case of
arbitrary functions were full rotation matrices are constructed from Euler
angles. The algorithm used is according to the recursive method of Ivanic and
Ruedenberg, as can be found in

Ivanic, J., Ruedenberg, K. (1996). Rotation Matrices for Real
Spherical Harmonics. Direct Determination by Recursion. The Journal
of Physical Chemistry, 100(15), 6342?6347.

and with the corrections of

Ivanic, J., Ruedenberg, K. (1998). Rotation Matrices for Real
Spherical Harmonics. Direct Determination by Recursion Page: Additions
and Corrections. Journal of Physical Chemistry A, 102(45), 9099?9100.

Rotation matrices for both real and complex SH can be obtained.

For more details on the functions, check their help output in Matlab.
For any questions, comments, corrections, or general feedback, please
contact [e-mail]