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

翻译：球谐（SH）域中的数据数学表示近年来在机器学习社区中再度引起广泛关注。本技术报告深入介绍了SH表示的理论基础与实践实现，系统总结了旋转不变性与等变性特征、球面上信号的卷积与精确相关性等相关研究。在此基础上，将这些方法从标量SH表示推广至矢量谐波（VH），为球面上的三维矢量场提供了相同功能的实现方案。