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

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. NOTE 1: This document is a re-publication of a subset of works originally published in my PhD thesis (I changed my last name from Fehr to Keuper): Fehr, Janis.