Generalized Sampling Expansion for Functions on the Sphere

Functions on the sphere appear in several applications, including geodesics, imaging and acoustics. Sampling of these functions may result in aliasing if the sampling condition is not met. The generalized sampling expansion introduced by Papoulis enables the reconstruction of a band-limited function sampled at a frequency lower than the Nyquist frequency using the outputs of several linear time-invariant systems. This paper formulates the generalized sampling expansion for functions on the sphere using spherical harmonics decomposition, facilitating sub-Nyquit sampling without aliasing error. An analysis of linear systems on the sphere and the aliasing phenomenon in the spherical harmonics domain is presented. Examples demonstrating the performance of the method and its limitations are studied.