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Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$

Rima Alaifari, Francesca Bartolucci, Matthias Wellershoff

TL;DR

The paper investigates the fundamental limits of uniqueness in sampled Gabor phase retrieval for $f\in L^2(\mathbb{R})$, recovering from $|\mathcal{G} f|$ on a lattice. It leverages the Bargmann transform to recast the problem as phase retrieval for entire functions in the Fock space, constructing counterexamples via $H_\delta^{\pm}$ and polynomial multipliers to show that the counterexample set is dense in $L^2(\mathbb{R})$ for lattices or equidistant parallel lines. It further shows that the Gaussian is not a counterexample for quadratic lattices $\Lambda = a\mathbb{Z}^2$ with $a\in(0,1)$, indicating the density of non-uniqueness does not extend to all signals. Overall, the work clarifies that while non-uniqueness is pervasive under certain sampling schemes, it is not universal, guiding future efforts toward establishing uniqueness or stability under additional structure.

Abstract

Sampled Gabor phase retrieval - the problem of recovering a square-integrable signal from the magnitude of its Gabor transform sampled on a lattice - is a fundamental problem in signal processing, with important applications in areas such as imaging and audio processing. Recently, a classification of square-integrable signals which are not phase retrievable from Gabor measurements on parallel lines has been presented. This classification was used to exhibit a family of counterexamples to uniqueness in sampled Gabor phase retrieval. Here, we show that the set of counterexamples to uniqueness in sampled Gabor phase retrieval is dense in $L^2(\mathbb{R})$, but is not equal to the whole of $L^2(\mathbb{R})$ in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval.

Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$

TL;DR

The paper investigates the fundamental limits of uniqueness in sampled Gabor phase retrieval for , recovering from on a lattice. It leverages the Bargmann transform to recast the problem as phase retrieval for entire functions in the Fock space, constructing counterexamples via and polynomial multipliers to show that the counterexample set is dense in for lattices or equidistant parallel lines. It further shows that the Gaussian is not a counterexample for quadratic lattices with , indicating the density of non-uniqueness does not extend to all signals. Overall, the work clarifies that while non-uniqueness is pervasive under certain sampling schemes, it is not universal, guiding future efforts toward establishing uniqueness or stability under additional structure.

Abstract

Sampled Gabor phase retrieval - the problem of recovering a square-integrable signal from the magnitude of its Gabor transform sampled on a lattice - is a fundamental problem in signal processing, with important applications in areas such as imaging and audio processing. Recently, a classification of square-integrable signals which are not phase retrievable from Gabor measurements on parallel lines has been presented. This classification was used to exhibit a family of counterexamples to uniqueness in sampled Gabor phase retrieval. Here, we show that the set of counterexamples to uniqueness in sampled Gabor phase retrieval is dense in , but is not equal to the whole of in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval.
Paper Structure (6 sections, 5 theorems, 35 equations, 1 figure)

This paper contains 6 sections, 5 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Prior arts: Counterexamples to uniqueness in sampled Gabor phase retrieval
  3. Our contributions
  4. The relation between the Bargmann and Gabor transform
  5. The set of counterexamples is dense
  6. The Gaussian is not a counterexample

Key Result

Theorem 3.1

Let $a > 0$. Then, $\mathfrak{C}(\mathbb{R} \times a \mathbb{Z})$ is dense in $L^2(\mathbb{R})$.

Figures (1)

  • Figure 1: The Gabor magnitude of the fifth Hermite function (Figure \ref{['fig:Gabor_Hermite']}) and of a counterexample $g_\delta^+$ to sampled Gabor phase retrieval on $\mathbb{R} \times \tfrac{1}{4} \mathbb{Z}$ (Figure \ref{['fig:counterexample']}).

Theorems & Definitions (11)

  • Definition 1.1: Counterexamples
  • Theorem 3.1
  • proof
  • Remark 3.2: Some explanations on the proof
  • Theorem 3.3
  • Example 3.4
  • Theorem 4.1
  • Lemma 4.2
  • Theorem 4.3
  • proof : Proof of Lemma \ref{['lem:restated']}
  • ...and 1 more