# 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. You can also find alternative equations in DLMF.

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

• 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.

### Saliency

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