Optimal Window Design for Joint Spatial-Spectral Domain Filtering of Signals on the Sphere

We present the optimal design of an azimuthally symmetric window signal for carrying out joint spatial-spectral domain filtering of a spherical (source) signal contaminated by a realization of an anisotropic noise process. The resulting window is used in the computation of spatially localized spherical harmonic transform of the noise-contaminated signal. We formulate the window design problem using the joint spatial-spectral domain filtering framework and choose the optimality criterion which minimizes the mean square error between the (noise-free) source signal and its filtered estimate. The azimuthally symmetric optimal window signal is shown to be specified by the statistics of the source and noise processes. We illustrate the capability of the proposed window signal by applying the joint spatial-spectral domain filtering framework to the bandlimited Mars topography map and demonstrate improvements in the output signal to noise ratio (SNR) for different values of input SNR.