On Invariance, Equivariance, Correlation and Convolution of Spherical Harmonic Representations for Scalar and Vectorial Data
This is an incremental technical summary for researchers in machine learning working with spherical data, offering no new results but consolidating and generalizing known methods.
The report provides an in-depth introduction to Spherical Harmonic (SH) representations, summarizing existing methods for rotation invariance, equivariance, convolutions, and correlations on spheres, and extends these to Vectorial Harmonics (VH) for 3D vector fields.
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