A phase-sensitive method for filtering on the sphere

TL;DR

This paper introduces a phase-sensitive filtering method on the sphere, emphasizing the importance of phase over magnitude, and proposes FIR filters with desirable properties for spherical data processing.

Contribution

It demonstrates the significance of phase in spherical functions and develops FIR filters based on phase properties, enabling effective directional filtering on the sphere.

Findings

Phase is more influential than magnitude in spherical functions.

Proposed FIR filters are associative and preserve spherical structure.

Filters enable directional filtering on spherical and manifold data.

Abstract

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.