One of the most basic operations on signals in any domain, including 2-sphere, is linear filtering or convolution. Yet, unlike conventional time-domain signals, for signals on the 2-sphere this is not consistently well defined and there exist competing definitions.
So the first aim of this chapter is to study existing definitions of convolution on 2-sphere in the literature and identify their properties, advantages and shortcomings. We determine the relationship between various definitions and show that two seemingly different definitions are essentially the same (which stems from the azimuthally symmetric or isotropic convolution property inherent in those definitions) (Kennedy et al., 2011). Our framework reveals that none of the existing formulations are natural extensions of convolution in time domain. For example, they are not commutative. Changing the role of signal and filter would change the outcome of convolution. Moreover, in one definition, the domain of convolution output does not remain on the 2-sphere.
Recognizing that a well-posed general definition for convolution on 2-sphere which is anisotropic and commutative is more difficult to formulate and that a true parallel with Euclidean convolution may not exist, in the second part of this chapter we use the power of abstract techniques and the tools we have studied to show that a commutative anisotropic convolution on the 2-sphere can indeed be simply constructed. We discuss additional properties of the proposed convolution, especially in the spectral domain, and present some clarifying examples.
We begin with the familiar case of convolution in time domain or convolution on the real line, which shall guide our subsequent developments for convolution on the 2-sphere.