Fractional Riesz-Hilbert transforms and fractional monogenic signals

TL;DR

This paper introduces fractional and quaternionic fractional Riesz-Hilbert transforms, establishing their properties and using them to construct advanced monogenic signals with applications in optics and signal processing.

Contribution

It develops a novel eigenvalue decomposition-based framework for fractional Riesz-Hilbert transforms and constructs new fractional monogenic signals with specific invariance properties.

Findings

Proved shift and scale invariance of the transforms

Established orthogonality and semigroup properties

Constructed rotated and modulated monogenic signals

Abstract

The fractional Hilbert transforms plays an important role in optics and signal processing. In particular the analytic signal proposed by Gabor has as a key component the Hilbert transform. The higher dimensional Hilbert transform is the Riesz-Hilbert transform which was used by Felsberg and Sommer to construct the monogenic signal. We will construct fractional and quaternionic fractional Riesz-Hilbert transforms based on a eigenvalue decomposition. We will prove properties of these transformations such as shift and scale invariance, orthogonality and the semigroup property. Based on the fractional/quaternionic fractional Riesz-Hilbert transform we construct (quaternionic) fractional monogenic signals. These signals are rotated and modulated monogenic signals.