  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.
