Efficient computation of commutative anisotropic convolution on the 2-sphere

Recently, the commutative anisotropic convolution has been defined for signals defined on the 2-sphere. Here, we present exact and efficient methods for computation of commutative convolution of two signals defined on the sphere. For fast computation of commutative convolution, we first review the use of existing efficient techniques developed to evaluate SO(3) convolution. By employing the factoring of a rotation into two rotations, followed by the separation of variables, we propose a fast algorithm for the efficient computation of commutative convolution. In terms of computational complexity, our proposed algorithm provides a saving of O(N) over the existing algorithms, where the convolution output is evaluated on O(N²) samples on the 2-sphere. Through numerical experiments, we also verify the improvement in the computational complexity.