Spherical Harmonics

Spherical Harmonics are a set of orthonormal basis functions

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

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