On Invariance, Equivariance, Correlation and Convolution of Spherical Harmonic Representations for Scalar and Vectorial Data

TL;DR

This paper provides a comprehensive overview of spherical harmonic representations, including their invariance, equivariance, and convolution properties, extending from scalar to vectorial data for 3D applications.

Contribution

It introduces a unified theoretical framework for SH and VH representations, encompassing invariance, equivariance, and convolution, with practical implementation insights.

Findings

Generalized scalar SH methods to vectorial harmonics

Detailed analysis of rotation-invariant and equivariant features

Framework applicable to 3D vector fields on spheres

Abstract

The mathematical representations of data in the Spherical Harmonic (SH) domain has recently regained increasing interest in the machine learning community. This technical report gives an in-depth introduction to the theoretical foundation and practical implementation of SH representations, summarizing works on rotation invariant and equivariant features, as well as convolutions and exact correlations of signals on spheres. In extension, these methods are then generalized from scalar SH representations to Vectorial Harmonics (VH), providing the same capabilities for 3d vector fields on spheres