Gabor wavelet analysis and the fractional Hilbert transform

TL;DR

This paper introduces an amplitude-phase representation of the dual-tree complex wavelet transform using fractional Hilbert transforms, linking wavelet analysis with windowed-Fourier analysis and extending it to directional, bivariate cases.

Contribution

It develops a novel amplitude-phase framework for DT-CWT based on fractional Hilbert transforms, enhancing interpretability and extending to directional bivariate analysis.

Findings

Explicit characterization of phase-shifting for Gabor-like wavelets

Extension of the framework to directional bivariate DT-CWT

Representation involving superposition of direction-selective wavelets

Abstract

We propose an amplitude-phase representation of the dual-tree complex wavelet transform (DT-CWT) which provides an intuitive interpretation of the associated complex wavelet coefficients. The representation, in particular, is based on the shifting action of the group of fractional Hilbert transforms (fHT) which allow us to extend the notion of arbitrary phase-shifts beyond pure sinusoids. We explicitly characterize this shifting action for a particular family of Gabor-like wavelets which, in effect, links the corresponding dual-tree transform with the framework of windowed-Fourier analysis. We then extend these ideas to the bivariate DT-CWT based on certain directional extensions of the fHT. In particular, we derive a signal representation involving the superposition of direction-selective wavelets affected with appropriate phase-shifts.