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Sampling Density for Gabor Phase Retrieval

Ting Chen, Hanwen Lu, Wenchang Sun

TL;DR

This work determines sharp sampling-density thresholds for Gabor phase retrieval from discrete samples, covering regular and irregular sampling on square-root lattices and on line configurations. By transporting the problem to the Bargmann transform domain, it reduces phase retrieval to the distribution of zeros of entire functions and leverages canonical products and indicator diagrams to establish when sampling sets guarantee uniqueness up to a global phase. The authors obtain optimal densities for square-root lattices, three parallel lines, and two intersecting lines, and extend the theory to irregular sampling and multivariate functions, providing both positive results and explicit nonretrieval constructions. These results guide the design of sampling schemes in applications requiring phaseless Gabor measurements and broaden the theoretical understanding to higher dimensions and broader window classes.

Abstract

Gabor phase retrieval stands for recovering a square integrable function up to a global phase from absolute values of its Gabor transform. In this paper, we study Gabor phase retrieval from discrete samples. We consider three types of sampling sequences, which include square root lattices, square root sequences on two intersecting lines and on three parallel lines respectively. In all cases we give the optimal sampling density for a sequence to do Gabor phase retrieval.

Sampling Density for Gabor Phase Retrieval

TL;DR

This work determines sharp sampling-density thresholds for Gabor phase retrieval from discrete samples, covering regular and irregular sampling on square-root lattices and on line configurations. By transporting the problem to the Bargmann transform domain, it reduces phase retrieval to the distribution of zeros of entire functions and leverages canonical products and indicator diagrams to establish when sampling sets guarantee uniqueness up to a global phase. The authors obtain optimal densities for square-root lattices, three parallel lines, and two intersecting lines, and extend the theory to irregular sampling and multivariate functions, providing both positive results and explicit nonretrieval constructions. These results guide the design of sampling schemes in applications requiring phaseless Gabor measurements and broaden the theoretical understanding to higher dimensions and broader window classes.

Abstract

Gabor phase retrieval stands for recovering a square integrable function up to a global phase from absolute values of its Gabor transform. In this paper, we study Gabor phase retrieval from discrete samples. We consider three types of sampling sequences, which include square root lattices, square root sequences on two intersecting lines and on three parallel lines respectively. In all cases we give the optimal sampling density for a sequence to do Gabor phase retrieval.

Paper Structure

This paper contains 11 sections, 26 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. The Bargmann transform
  4. Density of a sequence of points
  5. The canonical product
  6. Indicators and supporting functions
  7. Regular sampling for Gabor phase retrieval
  8. Irregular sampling for Gabor phase retrieval
  9. A necessary condition
  10. Phase retrieval with sampling sequences of uniform density
  11. Phase retrieval for multivariate functions

Key Result

Theorem 1.1

Let $\Lambda=a\mathbb{Z}^{1/2}\times b\mathbb{Z}^{1/2}$, where $a,b>0$. If $a<1$ or $b<1$, then $\Lambda$ does Gabor phase retrieval. Moreover, if $a=b>1$, then $\Lambda$ does not do Gabor phase retrieval.

Figures (1)

  • Figure 1: Gabor phase retrieval over three parallel lines

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1: Boas1954
  • Proposition 2.2: Levin1996
  • Proposition 3.1: GrohsLiehr2025
  • Proposition 3.2: Wellershoff2024a
  • Proposition 3.3: Jaming2014
  • Proposition 3.4: Boas1954
  • Proposition 3.5: Boas1954
  • ...and 32 more