Efficient kernel-based formulations of spatio-spectral and related transformations on the 2-sphere

Abstract

In this paper we show that the spatially localized spherical harmonic transform (SLSHT), which represents a signal on the 2-sphere in the spatio-spectral domain, can be efficiently computed using new kernel-based formulations. In addition to the standard spatiospectral domain, we show there are three other related transforms that provide alternative representations in the spatio-spatial, spectrospatial and spectro-spectral domains. We provide inversion results that extend available results for the SLSHT. We show that for signals on the 2-sphere band-limited to degree L, the computational complexity using our class of kernel-based SLSHT transforms is O(L 4 ) and outperforms the previous best known fast methods, which have complexity O(L 5 ).Index Terms-2-sphere; unit sphere; spherical harmonic transform; spatio-spectral domain; spatially localized spherical harmonic transform; fast transforms.