Long wavelength components in high degree spherical harmonics.

Spherical Harmonics

5 Spherical Harmonics 7 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.1 Harmonic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

3 Generalized Spherical Harmonics

Behavioral Spherical Harmonics for Long-Range Agents' Interaction

We introduce behavioral spherical harmonic (BSH), a novel approach to efficiently and compactly represent the directional-dependent behavior of agent. BSH is based on spherical harmonics to project the directional information of a group of multiple agents to a vector of few coefficients; thus, BSH drastically reduces the complexity of the directional evaluation, as it requires only few agent-gr...

Optimal Spherical Harmonics Projection

We introduce the reader to the mathematics behind projection of n-dimensional vectors into a basis on n-dimensional space, where n can be anything upto and including infinity. We show how ideally one wants to project into the duals of a basis if this basis is not orthonormal, and provide the mathematics to formulate this operation in matrix form. The second part of the article discusses spheric...

Importance Sampling Spherical Harmonics

In this paper we present the first practical method for importance sampling functions represented as spherical harmonics (SH). Given a spherical probability density function (PDF) represented as a vector of SH coefficients, our method warps an input point set to match the target PDF using hierarchical sample warping. Our approach is efficient and produces high quality sample distributions. As a...