Spherical Harmonics

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Spherical Harmonics are a set of orthonormal basis functions

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