  We obtain a characterization of all wavelets leading to analytic wavelet
transforms (WT). The characterization is obtained as a by-product of the
theoretical foundations of a new method for wavelet phase reconstruction from
magnitude-only coefficients. The cornerstone of our analysis is an expression
of the partial derivatives of the continuous WT, which results in
phase-magnitude relationships similar to the short-time Fourier transform
(STFT) setting and valid for the generalized family of Cauchy wavelets. We show
that the existence of such relations is equivalent to analyticity of the WT up
to a multiplicative weight and a scaling of the mother wavelet. The
implementation of the new phaseless reconstruction method is considered in
detail and compared to previous methods. It is shown that the proposed method
provides significant performance gains and a great flexibility regarding
accuracy versus complexity. Additionally, we discuss the relation between
scalogram reassignment operators and the wavelet transform phase gradient and
present an observation on the phase around zeros of the WT.
