Commutative Anisotropic Convolution on the 2-Sphere

Abstract

We develop a new type of convolution between two signals on the 2-sphere. This is the first type of convolution on the 2-sphere which is commutative. Two other advantages, in comparison with existing definitions in the literature, are that 1) the new convolution admits anisotropic filters and signals and 2) the domain of the output remains on the sphere. Therefore, the new convolution well emulates the conventional Euclidean convolution. In addition to providing the new definition of convolution and discussing its properties, we provide the spectral analysis of the convolution output. This convolutional framework can be useful in filtering applications for signals defined on the 2-sphere.