Adaptive multi-resolution windowing technique for localized spatio-spectral analysis

This paper introduces an adaptive, multi-resolution windowing technique that can be used in conjunction with the spatially localized spherical harmonic transform (SLSHT) to process signals on the 2-sphere in the spatio-spectral domain. In contrast with the standard formulation, which uses a fixed window, the new windowing technique is able to respond locally to the signal under analysis, that is, be adaptive, and also is formulated to depend on the spectral degree to give it a multi-resolution character. We further enhance its simultaneous spatial and spectral localization by basing the window on a parametric band-limited Slepian maximum spatial concentration eigenfunction. The criterion for window design is to maximize the energy concentration in each spectral component in the spatio-spectral domain. A computationally efficient method is also developed to implement the adaptive window design. An example is also provided to demonstrate the superiority of the new adaptive, multiresolution window technique.