Spherical Harmonics

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

Background

Harmonic Function

Wikipedia Reference
A function \(\displaystyle f: \mathbb{R}^n \rightarrow \mathbb{R}\) is a harmonic function if it satisfies Laplace's equation:

• The Laplacian (or trace of the hessian) is zero.
• \(\displaystyle \Delta f = \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0\)

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
There are \(\displaystyle 2l+1\) functions for each band.

• \(\displaystyle Y_l^m(\theta, \varphi) = K_l^m e^{i m \varphi} P_l^{|m|} \cos(\theta)\) for \(\displaystyle -l \leq m \leq l\)

where \(\displaystyle P_l^m\) are the associated Legendre Polynomials

and \(\displaystyle K_l^m = \sqrt{\frac{(2l+1)(l-|m|)!}{4 \pi (l+|m|)!}}\)

l is the band, m is the function

For a real valued basis,

• \(\displaystyle y_l^m = \begin{cases} \sqrt{2}\operatorname{Re}(Y_l^m) & m \gt 0\\ \sqrt{2}\operatorname{Im}(Y_l^m) & m \lt 0\\ Y_l^0 & m = 0 \end{cases} = \begin{cases} \sqrt{2} K_l^m \cos(m \varphi) P_l^m(\cos\theta) & m \gt 0\\ \sqrt{2} K_l^m \sin(|m| \varphi) P_l^{|m|}(\cos\theta) & m \lt 0\\ K_l^0 P_l^0 (\cos \theta) & m = 0\\ \end{cases}\)

Visualizations

Below are distorted sphere visualizations where the radius corresponds to the value at each point.

• iq's 0-3
• Ruofei's 0-4

Applications

Saliency

• Ruofei's Website

Ruofei did a project on Saliency using Spherical Harmonics as part of his PhD dissertation.

Resources

• Stupid SH
• Wikipedia page