Characterization of Analytic Wavelet Transforms and a New Phaseless Reconstruction Algorithm

Abstract: 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.