Spatially Varying Spectral Filtering of Signals on the Unit Sphere

This paper presents a general framework for spatially-varying spectral filtering of signals defined on the unit sphere, as an analogy to joint time-frequency filtering. For this purpose, we first map spherical signals from spatial domain into joint spatio-spectral domain, where a spatio-spectral signal transformation or modification is introduced. For mapping spatial signals into joint spatio-spectral domain, we use the spatially localized spherical harmonic transform (SLSHT) from the literature. We then propose a suitable scheme to transform the modified signal from the spatio-spectral domain back to an admissible signal in the spatial domain using the least squares approach. We also show that the overall action of the SLSHT and spatio-spectral signal modification can be described through a single transformation matrix, which is useful in practice. Finally, we discuss two specific and useful instances of spatially-varying spectral filtering, defined through multiplicative and convolutive modification of the SLSHT distribution, and show through numerical examples their effectiveness in selective spectral filtering of different spatial regions of the signal.