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.