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Spherical Harmonics are a set of orthonormal basis functions defined over a sphere.<br>
<math>f: (\phi, \theta) \rightarrow f(\phi, \theta) \in \mathbb{R}</math>

==Background==
===Harmonic Function===
{{main | Wikipedia: Harmonic_function}}
A function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}</math> is a harmonic function if it satisfies Laplace's equation:
* The Laplacian (or trace of the hessian) is zero.
* <math>\Delta f = \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0</math>

===Associated Legendre Polynomials===
{{main | Wikipedia:Associated Legendre polynomials}}
Associated Legendre Polynomials are a set of orthogonal polynomials defined over \([-1, 1]\).  
Each is represented as \(P^m_l\) where \(0 \leq m \leq l\).  
I.e.  
\(P^0_0(x)\)  
\(P^0_1(x), P^1_1(x)\)  
\(P^0_2(x), P^1_2(x), P^2_2(x)\)  

The following 3 recurrance relations define the associated legendre polynomials:
# <math>(l-m)P_l^m = x(2l-1)P_{l-1}^m - (l+1-1) P_{l-2}^m</math>
# <math>P_m^m = (-1)^m(2m-1)!! (1-x^2)^{m/2}</math>
# <math>P^m_{m+1} = x(2m+1)P^m_m</math>

Notes: 
* Here <math>
(x)!! =
\begin{cases} 
(x)*(x-2)*...*(1) & x\text{ odd}\\
(x)*(x-2)*...*(2) & x\text{ even}
\end{cases}
</math>

==Definition==
Spherical Harmonics are a set of orthonormal basis functions defined on the sphere.  
Below are some explicit formulas for Laplace spherical harmonics stolen from Sloan<ref name="stupidsh">Peter-Pike Sloan, [http://www.ppsloan.org/publications/StupidSH36.pdf Stupid Spherical Harmonics (SH) Tricks]</ref>. You can also find alternative equations in DLMF<ref name="dlmf">Digital Library of Mathematical Functions, 14.30. [https://dlmf.nist.gov/14.30 https://dlmf.nist.gov/14.30]</ref>.

There are <math>2l+1</math> functions for each band.

* <math>Y_l^m(\theta, \varphi) = K_l^m e^{i m \varphi} P_l^{|m|} \cos(\theta)</math> for <math>-l \leq m \leq l</math>
: where <math>P_l^m</math> are the associated Legendre Polynomials
: and <math>K_l^m = \sqrt{\frac{(2l+1)(l-|m|)!}{4 \pi (l+|m|)!}}</math>
: l is the band, m is the function

For a real valued basis,
* <math>y_l^m = \begin{cases}
\sqrt{2}\operatorname{Re}(Y_l^m) & m > 0\\
\sqrt{2}\operatorname{Im}(Y_l^m) & m < 0\\
Y_l^0 & m = 0
\end{cases}
= \begin{cases}
\sqrt{2} K_l^m \cos(m \varphi) P_l^m(\cos\theta) & m > 0\\
\sqrt{2} K_l^m \sin(|m| \varphi) P_l^{|m|}(\cos\theta) & m < 0\\
K_l^0 P_l^0 (\cos \theta) & m = 0\\
\end{cases}</math>

==Properties==
Copied from Green<ref name="green2003lighting">Robin Green (2003). ''Spherical Harmonic Lighting'' URL: [http://silviojemma.com/public/papers/lighting/spherical-harmonic-lighting.pdf http://silviojemma.com/public/papers/lighting/spherical-harmonic-lighting.pdf]</ref>

* Orthonormal
* Rotationally Invariant
* Integration of two spherical harmonic functions is a dot product of their coefficients
{{hidden | Proof| 
This follows from being orthonormal:<br>
<math>
\begin{align}
\int_{S}\tilde{L}(s)\tilde{t}(s)ds &= \int_{S}\left(\sum_i L_i y_i(s)\right)\left(\sum_j t_j y_j(s)\right)ds \\
&= \sum_i\sum_j L_i t_j \int_{S} y_i(s) y_j(s)ds \\
&= \sum_i^{n^2} L_i t_i \int_{S}( y_i(s) y_i(s))ds\\
&\qquad\text{ because orthogonal}\\
&= \sum_i^{n^2} L_i t_i\\
&\qquad\text{ because orthonormal}
\end{align}
</math>
}}

==Visualizations==
Below are distorted sphere visualizations where the radius corresponds to the value at each point.
* [https://www.shadertoy.com/view/lsfXWH iq's 0-3]
* [https://www.shadertoy.com/view/4dsyW8 Ruofei's 0-4]

==Applications==
===Lighting===
See Green<ref name="green2003lighting></ref>.

===Saliency===
* [http://duruofei.com/Research/SphericalHarmonics Ruofei's Website]
Ruofei did a project on Saliency using Spherical Harmonics as part of his PhD dissertation.

==Resources==
* [http://www.ppsloan.org/publications/StupidSH36.pdf Stupid SH by Peter-Pike Sloan]
* [http://silviojemma.com/public/papers/lighting/spherical-harmonic-lighting.pdf Spherical Harmonics Lighting by Robin Green]
* [[Wikipedia:Spherical Harmonics]]

==References==
<references />