# Spherical Harmonics

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

## 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:

1. $$\displaystyle (l-m)P_l^m = x(2l-1)P_{l-1}^m - (l+1-1) P_{l-2}^m$$
2. $$\displaystyle P_m^m = (-1)^m(2m-1)!! (1-x^2)^{m/2}$$
3. $$\displaystyle P^m_{m+1} = x(2m+1)P^m_m$$

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 Sloan[1]. You can also find alternative equations in DLMF[2].

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,

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

## Properties

Copied from Green[3]

• Orthonormal
• Rotationally Invariant
• Integration of two spherical harmonic functions is a dot product of their coefficients
Proof

This follows from being orthonormal:
\displaystyle \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}

## Visualizations

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

## Applications

See Green[3].

### Saliency

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

## References

1. Peter-Pike Sloan, Stupid Spherical Harmonics (SH) Tricks
2. Digital Library of Mathematical Functions, 14.30. https://dlmf.nist.gov/14.30
3. Robin Green (2003). Spherical Harmonic Lighting URL: http://silviojemma.com/public/papers/lighting/spherical-harmonic-lighting.pdf