Gabor Wavelet Analysis and the Fractional Hilbert Transform

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.