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SHBox --- A matlab toolbox for spherical harmonics.
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Author: Ying Xiong.
Created: Jul 21, 2013.
Release: Jan 17, 2014 (v0.1).
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Quick start.
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>> addpath('Utils');
>> SHBoxTest;
>> demoSHBox;
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Definitions.
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The inner product in spherical space is defined as
/ \pi / \pi
<f, g> = | | f(\theta,\phi) g*(\theta,\phi) d\Omega,
/ \theta=0 / \phi=-pi
where
d\Omega = sin(\theta) d\phi d\theta.
The spherical harmonic functions are defined as
(2l+1) (l-m)!
Y_l^m(\theta,\phi) = sqrt{---------------} P_l^m(cos(\theta)) exp(im\phi)
4 pi (l+m)!
= (-1)^m sqrt{1/(2 pi)} N_l^m(cos(\theta)) exp(im\phi),
where P_l^m is associated Legendre function, and N_l^m is fully normalized
associated Legendre function. Therefore, we have
< Y_{l1}^{m1}, Y_{l2}^{m2} > = \delta_{l1,l2} \delta_{m1,m2}.
The real spherical harmonic functions are defined as
/ 1/sqrt(2) (Y_l^{-m} + (-1)^m Y_l^m) if m>0
Y_{l,m} = | Y_l^0 if m=0
\ 1/sqrt(2) (Y_l^m - (-1)^m Y_l^{-m}) if m<0.
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Representations.
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A spherical function f(\theta, \phi) is represented by a matrix 'F' with two
(optional) coordinate lists 'lTheta' and 'lPhi', such that
F(i,j) = f(lTheta(i), lPhi(j)).
If not specified, 'lTheta' and 'lPhi' are by default
lTheta = linspace(0, pi, size(F,1)),
lPhi = linspace(-pi, pi, size(F,2)).
If 'lTheta' and 'lPhi' are scalars, they will be converted to
lTheta = linspace(0, pi, lTheta),
lPhi = linspace(-pi, pi, lPhi).
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Features.
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* Generate real and complex spherical harmonic functions.
* Visualize spherical functions.
* Inner product in spherical space.
* Decompose spherical function into spherical harmonic components.
See 'SphericalHarmonics.pdf' for a more detailed documentation.
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Other notes.
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The 'pisa.png' file is a light probe image downloaded from http://gl.ict.usc.edu/Data/HighResProbes/ and modified (tone-mapped and resized) by Ying Xiong.