# Spherical Harmonics

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Spherical Harmonics are a set of orthonormal basis functions

## Contents

## Background

### Harmonic Function

Wikipedia Reference

A function is a harmonic function if it satisfies Laplace's equation:

- The Laplacian (or trace of the hessian) is zero.

## 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 functions for each band.

- for

- where are the associated Legendre Polynomials
- and
- l is the band, m is the function

For a real valued basis,

### 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