Band-limited extrapolation on the sphere for signal reconstruction in the presence of noise

We investigate the problem of extrapolation of band-limited signals on the 2-sphere in the presence of noise. Specifically, given incomplete or spatially limited measurements subject to noise, find the unique extrapolation to the complete 2-sphere. We present an analytic solution to the extrapolation problem based on the expansion of a signal in Slepian basis corresponding to an orthogonal set of eigenfunctions of an associated energy concentration problem. An alternative equivalent iterative algorithm is also developed for practical implementation and guidelines are proposed to choose the parameters of the iterative algorithm. The capability of the proposed extrapolation is compared and demonstrated with the help of an illustration example.