Sparse signal recovery on the sphere: Optimizing the sensing matrix through sampling

We propose a method for constructing a spherical harmonic sensing matrix that can be used to effectively recover a sparse signal on the sphere from limited measurements. For such a sensing matrix, in a compressed sensing setting, it is desirable that the restricted isometry property (RIP) holds. Our sensing matrix is obtained by drawing random samples from a grid derived from minimal discrete energy spiral distribution of sample points (spiral scheme). This sampling scheme and construction of sampling matrix is shown to be superior to the use of preconditioned equiangular samples from the literature (regular scheme). Our numerical results show that the success rate in near exact recovery of a sparse coefficient vector with our spiral scheme is superior to that of the regular scheme over a range of SNRs.