Spatio-Spectral Analysis on the Sphere Using Spatially Localized Spherical Harmonics Transform

This correspondence studies a spatially localized spectral transform for signals on the unit sphere, which we call spatially localized spherical harmonics transform (SLSHT). For a systematic treatment, we explicitly express the transform in terms of rotated versions of an azimuthally symmetric window function and introduce the spatio-spectral SLSHT distribution with a succinct matrix representation. We present guidelines for the choice of the window function in the SLSHT, based on the inherent tradeoff between the spatial and spectral resolution of different window functions from the perspective of the uncertainty principle. We demonstrate the use of an eigenfunction window, obtained from the Slepian concentration problem on the sphere, as a good choice for window function. As an illustration, we apply the transform to the topographic map of Mars, which can reveal spatially localized spectral contributions that were not obtainable from traditional spherical harmonics analysis.