The Hilbert Transform on the Two-Sphere: A Spectral Characterization

Abstract

The local analysis of signals arising on the sphere is a common task in earth sciences. On the real line the analytic signal turned out to be an important representation in local one-dimensional signal processing. Its generalization to two dimensions is the monogenic signal, and the properties of the analytic and the monogenic signal in the Fourier domain are well known. A generalization to the sphere is given by the Hilbert transform on the sphere known from Clifford analysis. To obtain a spectral characterization, the transform has to be decomposed into spherical harmonic functions. In this paper, we derive the spherical harmonic coefficients of the Hilbert transform on the sphere and give a series expansion. This will show that it acts as a differential operator on the spherical harmonic basis functions of the Laplace equation solution, analogously to the Riesz transform in two dimensions. This allows an interpretation of the Hilbert transform suitable for signal processing of signals naturally arising on the two-sphere. We show that the scale space naturally arising is a Poisson scale space in the unit ball. In addition, the obtained interpretation of the Hilbert transform is used for orientation analysis of plane waves. This representation is justified as a novel signal model on the sphere which can be used to construct intensity and rotation-invariant operators for local signal analysis in a scale-space concept.