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

Gabor wavelet analysis and the fractional Hilbert transform