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

[email protected]

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 [email protected]

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

List of MATLAB files:

• getSH.m - Get SHs up to order N

• legendre2.m - Same as matlab Legendre extended for negative orders m<0

• leastSquaresSHT.m - SHT of function using least-squares

• weightedLeastSquaresSHT.m - SHT using weighted least-squares

• directSHT.m - Perform SHT directly by summation for special arrangements

• leastSquaresSHT.m - Perform SHT by least-squares, weighted or unweighted

• inverseSHT.m - Perform the inverse SHT

• getTdesign - Returns the spherical coordinates of T-designs up to t=21

• getFliegeNodes - Returns FliegeMaier point set, up to order N=29 SHT

• t_designs_1_21.mat - Tables of t-designs, up to t=21

• fliegeMaierNodes_1_30.mat - Tables of Fliege-MAier sets, up to 300 points

• complex2realCoeffs.m - Convert SH coeffs from the complex to real basis

• complex2realCoeffsMtx.m - Returns complex to real SH basis transformation matrix

• real2complexCoeffs.m - Convert SH coeffs from the real to complex basis

• rotateCoeffs.m - Get SH coefficients for a rotated axisymmetric pattern

• conjCoeffs.m - Get the complex SH coefficients of a conjugate function

• sphConvolution.m - Perform spherical convolution between a function and a filter, in the SH domain

• sphMultiplication.m - Computes coefficients of product of two spherical functions, through Gaunt Coefficients

• getSHrotMtx - Obtain rotation matrix for full rotation of the coordinate system, that when applied to the SH coefficients, returns the coefficients of the rotated function

• gaunt_mtx.m - Construct a 3D matrix of Gaunt coefficients up to three orders, each one a separate dimension of the matrix

• w3j - Evaluate the Wigner-3j symbol through the Racah formula

• w3j_stirling - Evaluate the Wigner-3j symbol through the Racah formula, using Stirling's large factorial approximation

• sym_w3j - Returns a Wigner-3j in symbolic form

• wignerD - Returns the Wigner-D and wigner-d matrices for rotation of complex spherical harmonics

• plotSphFunctionGrid - Plot easily spherical function defined on a regular grid

• plotSphFunctionTriangle - Plot easily spherical function defined on an irregular grid

• plotSphFunctionCoeffs - Plot spherical function with known SH coefficients

• Fdirs2grid.m - Helper function for plotting, used with grid2dirs

• grid2dirs.m - Construct a vector of regular grid points

• sphDelaunay.m - Computes the Delaunay triangulation on the unit sphere

• sphVoronoi.m - Computes the a Voronoi diagram on the unit sphere

• sphVoronoiAreas.m - Computes the areas of a voronoi diagram on the unit sphere

• getVoronoiWeights.m - Conveniently get voronoi weights for a sampling scheme

• checkCondNumberSHT.m - Computes the condition number of an sampling scheme for a least-squares SHT

• euler2rotationMatrix - Euler angles to rotation matrix

• unitCart2sph - Get directly azimuth and elevation from unit vectors in matrix form

• unitSph2cart - Get directly unit vectors from azimuth and elevation in matrix form

• replicatePerOrder - Replicate l^th element 2*l+1 times across specified dimension