A Spherical Harmonic Expansion of the Hilbert Transform on the Two-Sphere

• Main Authors: Fleischmann, Oliver, Wietzke, Lennart, Sommer, Gerald
• Format: Conference Proceeding
• Language: English
• ISSN: 0094-243X
• DOI: 10.1063/1.3241589

The classical Hilbert transform on the real line is a valuable tool in signal processing. It constitutes the analytic signal which allows the determination of the instantaneous phase and amplitude of a one dimensional signal. For signals in in the Euclidean plane its analogue is the monogenic signal based on the Riesz transform, a generalization of the Hilbert transform to the plane. In addition to the instantaneous phase and amplitude, the orientation of intrinsically one dimensional structures in the plane can be determined. Various disciplines like geosciences, omnidirectional vision or astrophysics have to deal with signals arising on the two-sphere. A Hilbert transform on the two-sphere is well known from Clifford analysis. Yet it lacks a suitable interpretation from a signal processing viewpoint, especially in the frequency domain. In this paper we derive a series expansion of the Hilbert transform on the two-sphere in terms of spherical harmonics. It provides an intuitive interpretation and turns out to be a gradient-like operator acting only on the angular parts of the signal. This leads to intensity and rotation invariant signal analysis techniques on the two-sphere in analogue to the Euclidean plane.