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Characterization of Analytic Wavelet Transforms and a New Phaseless Reconstruction Algorithm

Nicki Holighaus, Günther Koliander, Zdenĕk Průša, Luis Daniel Abreu

TL;DR

It is shown that the proposed method provides significant performance gains and a great flexibility regarding accuracy versus complexity and the existence of such relations is equivalent to analyticity of the WT up to a multiplicative weight and a scaling of the mother wavelet.

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.

Characterization of Analytic Wavelet Transforms and a New Phaseless Reconstruction Algorithm

TL;DR

It is shown that the proposed method provides significant performance gains and a great flexibility regarding accuracy versus complexity and the existence of such relations is equivalent to analyticity of the WT up to a multiplicative weight and a scaling of the mother wavelet.

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.

Paper Structure

This paper contains 19 sections, 4 theorems, 81 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Log-Magnitude and Phase Derivatives of the Wavelet Transform
  3. The Phase-Magnitude Relationship
  4. The Cauchy Wavelet Transform as Time-Frequency Representation
  5. Derivatives and Analyticity of the Continuous Wavelet Transform---Further Observations
  6. The Phase Around Zeros of the Wavelet Transform
  7. Scalogram Reassignment and Ridge Points
  8. Analytic Wavelets and the Analytic Wavelet Transform
  9. Application---Phaseless Reconstruction for the Discrete Continuous Wavelet Transform
  10. Discrete Continuous Wavelet Transform
  11. Application to Phaseless Reconstruction
  12. Experiments
  13. Experiment I---Comparison to Previous Methods
  14. Experiment II---Changing the Redundancy
  15. Conclusion
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\psi\in\mathbf L^2(\mathbb{R})$ with $\widehat{\psi}(\xi)=0$ for $\xi< 0$ and assume that $\psi$ is continuously differentiable with $\psi', \mathbf T\psi'\in\mathbf L^2(\mathbb{R})$, where $\mathbf T$, without subscript, denotes the time-weighting operator $(\mathbf T s)(t) = t s(t)$. Then, fo and

Figures (3)

  • Figure 1: 70 signals for $\alpha = 30$ (top), $\alpha = 300$ (middle), and $\alpha = 3000$ (bottom) sorted by R-FGLIM performance; the sort sequences are available on the manuscript webpage.
  • Figure 2: Comparison for different redundancies in WPGHI (top) and W-FGLIM (bottom) for 70 signals sorted by medhigh redundancy performance; the sort sequences are available on the manuscript webpage.
  • Figure 3: Top: Wavelet scalogram of a male German speech recording (derived from signal $54$ of the SQAM dataset). Other panels: Difference between the true WT phase the test signal and the phase estimate proposed by WPGHI. Gray level indicates the difference in the range $0$ (black) to $\pi$ (white). Whenever the WT magnitude is below a tolerance level, the phase difference is also set to zero. Phase difference fluctuation clearly increases with decreasing redundancy.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof