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This paper examines filtering on a sphere, by first examining the roles of spherical harmonic magnitude and phase. We show that phase is more important than magnitude in determining the structure of a spherical function. We examine the properties of linear phase shifts in the spherical harmonic domain, which suggest a mechanism for constructing finite-impulse-response (FIR) filters. We show that those filters have desirable properties, such as being associative, mapping spherical functions to spherical functions, allowing directional filtering, and being defined by relatively simple equations. We provide examples of the filters for both spherical and manifold data. [ABSTRACT FROM PUBLISHER]

Abstract: Let denote the symmetric group on and the stabilizer subgroup of . We derive algorithms for computing Fourier transforms of left and right -invariant signals that require a total of additions and scalar multiplications. Furthermore we show that the convolution of such signals can also be computed in time linear in . [Copyright &y& Elsevier]