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Phase retrieval of bandlimited functions for the wavelet transform

Rima Alaifari, Francesca Bartolucci, Matthias Wellershoff

TL;DR

This paper advances wavelet phase retrieval by proving uniqueness results for real-valued, bandlimited signals from magnitude-only wavelet measurements when the wavelet has finitely many vanishing moments, even allowing complex-valued wavelets. It extends these results to sampling scenarios and shows that the analytic representation $f_+$ can be recovered up to a global phase from two-scale magnitude measurements of the Cauchy wavelet transform, tying into Mallat's prior work and the WSK sampling theorem. The contributions include the first uniqueness results for sampled wavelet phase retrieval with complex wavelets and for sampled Cauchy-wavelet-based measurements, addressing a long-standing conjecture about the well-posedness of wavelet phase retrieval for generic wavelet families. The results have practical implications for audio processing and signal reconstruction where only magnitude information is accessible, enabling stable recovery and analytic-signal interpretations from phaseless wavelet data.

Abstract

We study the recovery of square-integrable signals from the absolute values of their wavelet transforms, also called wavelet phase retrieval. We present a new uniqueness result for wavelet phase retrieval. To be precise, we show that any wavelet with finitely many vanishing moments allows for the unique recovery of real-valued bandlimited signals up to global sign. Additionally, we present the first uniqueness result for sampled wavelet phase retrieval in which the underlying wavelets are allowed to be complex-valued and we present a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements.

Phase retrieval of bandlimited functions for the wavelet transform

TL;DR

This paper advances wavelet phase retrieval by proving uniqueness results for real-valued, bandlimited signals from magnitude-only wavelet measurements when the wavelet has finitely many vanishing moments, even allowing complex-valued wavelets. It extends these results to sampling scenarios and shows that the analytic representation can be recovered up to a global phase from two-scale magnitude measurements of the Cauchy wavelet transform, tying into Mallat's prior work and the WSK sampling theorem. The contributions include the first uniqueness results for sampled wavelet phase retrieval with complex wavelets and for sampled Cauchy-wavelet-based measurements, addressing a long-standing conjecture about the well-posedness of wavelet phase retrieval for generic wavelet families. The results have practical implications for audio processing and signal reconstruction where only magnitude information is accessible, enabling stable recovery and analytic-signal interpretations from phaseless wavelet data.

Abstract

We study the recovery of square-integrable signals from the absolute values of their wavelet transforms, also called wavelet phase retrieval. We present a new uniqueness result for wavelet phase retrieval. To be precise, we show that any wavelet with finitely many vanishing moments allows for the unique recovery of real-valued bandlimited signals up to global sign. Additionally, we present the first uniqueness result for sampled wavelet phase retrieval in which the underlying wavelets are allowed to be complex-valued and we present a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements.

Paper Structure

This paper contains 14 sections, 16 theorems, 74 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Prior work
  3. Contributions
  4. Outline
  5. Notation
  6. Preliminaries
  7. Main results
  8. Examples
  9. The Morlet wavelet
  10. The linear-chirp wavelet
  11. Sampling Cauchy wavelet transform magnitudes
  12. Introduction
  13. The sampling result for analytic signals
  14. Acknowledgements

Key Result

Theorem 1

Let $\psi \in L^1(\mathbb R)$ be a wavelet with finitely many vanishing moments. Then, any real-valued bandlimited function $f \in L^2(\mathbb R)$ is uniquely determined by $\lvert \mathcal{W}_\psi f \rvert$ up to global sign.

Figures (3)

  • Figure 1: The Morlet wavelet $\psi$ for $\xi_0=5$ in time and frequency representation. Observe that the Fourier transform of $\psi$ is not identically zero on the negative frequencies but it is numerically small.
  • Figure 2: The linear-chirp wavelet for $\beta=1$ and $\xi_0=5$ in time and frequency representation.
  • Figure 3: The Cauchy wavelet $\widehat{\psi}(\xi) =\xi^2 \mathrm{e}^{-\xi} \boldsymbol{1}_{\xi > 0}$.

Theorems & Definitions (32)

  • Theorem 1: Cf. Theorem \ref{['thm:unquenessresultreal']}
  • Theorem 2: Cf. Theorem \ref{['thm:unquenessresultrealsampled2']}
  • Definition 3
  • Lemma 4
  • Lemma 5
  • proof
  • Definition 6
  • Proposition 7
  • Theorem 8
  • proof
  • ...and 22 more