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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

archontis.politis@aalto.fi

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:

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/ 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 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.

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 any questions, comments, corrections, or general feedback, please contact archontis.politis@aalto.fi

For more details on the functions, check their help output in Matlab.

List of MATLAB files: