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We obtain a characterization of all wavelets leading to analytic wavelettransforms (WT). The characterization is obtained as a by-product of thetheoretical foundations of a new method for wavelet phase reconstruction frommagnitude-only coefficients. The cornerstone of our analysis is an expressionof the partial derivatives of the continuous WT, which results inphase-magnitude relationships similar to the short-time Fourier transform(STFT) setting and valid for the generalized family of Cauchy wavelets. We showthat the existence of such relations is equivalent to analyticity of the WT upto a multiplicative weight and a scaling of the mother wavelet. Theimplementation of the new phaseless reconstruction method is considered indetail and compared to previous methods. It is shown that the proposed methodprovides significant performance gains and a great flexibility regardingaccuracy versus complexity. Additionally, we discuss the relation betweenscalogram reassignment operators and the wavelet transform phase gradient andpresent an observation on the phase around zeros of the WT.

Characterization of Analytic Wavelet Transforms and a New Phaseless Reconstruction Algorithm