Improving the spatial dimensionality of Gauss-Legendre and equiangular sampling schemes on the sphere

For the fast and exact computation of spherical harmonic transform (SHT) of a band-limited signal defined on the sphere from its samples, the Gauss-Legendre (GL) and equiangular sampling schemes on the sphere require asymptotically least number of samples. In comparison to the equiangular scheme, the GL scheme has larger spatial dimensionality, defined as the number of the samples required for the exact computation of SHT. In this work, we propose an efficient GL sampling scheme with spatial dimensionality equal to that of equiangular scheme. We also propose optimisation of samples along longitude to further reduce the spatial dimensionality of equiangular, GL and efficient GL sampling schemes. Furthermore, we demonstrate that the accuracy of the SHT is not affected with the proposed reduction in the spatial dimensionality.