Minimum mean square error equalization on the 2-sphere

In this paper we consider the zero-forcing (ZF) and minimum mean square error (MMSE) criteria for signal recovery using linear operators as equalizers for signals observed on the 2-sphere that are subject to linear distortions and noise. The distortions considered are bounded operators and can include convolutions, rotations, spatial and spectral truncations, projections or combinations of these. Likewise the signal and noise are very general being modeled as anisotropic stochastic processes on the 2-sphere. In both the distortion model and signal model the findings in this paper are significantly more general than results that can be found in the literature. The MMSE equalizer is shown to reduce to the ZF equalizer when the distortion operator has an inverse and there is an absence of noise. The ability of the MMSE to recover a Mars topography map signal from a projection operator, which fails to have a ZF solution, is given as an illustration.