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
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.
- \(\displaystyle \Delta f = \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0\)
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:
- \(\displaystyle (l-m)P_l^m = x(2l-1)P_{l-1}^m - (l+1-1) P_{l-2}^m\)
- \(\displaystyle P_m^m = (-1)^m(2m-1)!! (1-x^2)^{m/2}\)
- \(\displaystyle P^m_{m+1} = x(2m+1)P^m_m\)
Notes:
- Here \(\displaystyle (x)!! = \begin{cases} (x)*(x-2)*...*(1) & x\text{ odd}\\ (x)*(x-2)*...*(2) & x\text{ even} \end{cases} \)
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 [1]
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.
Applications
Saliency
Ruofei did a project on Saliency using Spherical Harmonics as part of his PhD dissertation.
Resources
References
- ↑ Peter-Pike Sloan, Stupid Spherical Harmonics (SH) Tricks