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Phase retrieval in Fock space and perturbation of Liouville sets

Philipp Grohs, Lukas Liehr, Martin Rathmair

TL;DR

This work resolves the phaseless sampling problem in Fock space by showing that three perturbations of a Liouville set yield a finite-density uniqueness set for phase retrieval, provided an $f$-closeness and a noncollinearity angle condition hold. It situates Liouville sets between stable sampling and uniqueness, offers deterministic and random perturbation schemes (including almost-sure results) and connects these constructions to Gabor phase retrieval in subspaces of ${L^2(\mathbb{R})}$, achieving sharp density thresholds for real-valued and even-real-valued cases. The approach blends interpolation theory, Liouville-type arguments, and probabilistic geometry of random perturbations to extend phase retrieval solvability from lattices to irregular sampling patterns with practical density guarantees. Overall, the results provide rigorous foundations for uniquely recovering signals from phaseless time-frequency measurements, with implications for diffraction imaging and Ptychography where finite-density, nonlattice sampling is essential.

Abstract

We study the determination of functions in Fock space from samples of their absolute value, known as the phase retrieval problem in Fock space. An important finding in this research field asserts that phaseless sampling on lattices of arbitrary density renders the problem unsolvable. The present study establishes solvability when using irregular sampling sets of the form $A \cup B \cup C$, where $A, B,$ and $C$ constitute perturbations of a Liouville set, i.e., a set with the property that all functions in Fock space bounded on the set are constant. The sets $A, B,$ and $C$ adhere to specific geometrical conditions of closeness and noncollinearity. We show that these conditions are sufficiently generic so as to allow the perturbations to be chosen also at random. By proving that Liouville sets occupy an intermediate position between sets of stable sampling and sets of uniqueness, we obtain the first construction of uniqueness sets for the phase retrieval problem in Fock space having a finite density. The established results apply to the Gabor phase retrieval problem in subspaces of $L^2(\mathbb{R})$, where we derive additional reductions of the size of uniqueness sets: for the class of real-valued functions, uniqueness is achieved from two perturbed lattices; for the class of even real-valued functions, a single perturbation suffices, resulting in a separated set.

Phase retrieval in Fock space and perturbation of Liouville sets

TL;DR

This work resolves the phaseless sampling problem in Fock space by showing that three perturbations of a Liouville set yield a finite-density uniqueness set for phase retrieval, provided an -closeness and a noncollinearity angle condition hold. It situates Liouville sets between stable sampling and uniqueness, offers deterministic and random perturbation schemes (including almost-sure results) and connects these constructions to Gabor phase retrieval in subspaces of , achieving sharp density thresholds for real-valued and even-real-valued cases. The approach blends interpolation theory, Liouville-type arguments, and probabilistic geometry of random perturbations to extend phase retrieval solvability from lattices to irregular sampling patterns with practical density guarantees. Overall, the results provide rigorous foundations for uniquely recovering signals from phaseless time-frequency measurements, with implications for diffraction imaging and Ptychography where finite-density, nonlattice sampling is essential.

Abstract

We study the determination of functions in Fock space from samples of their absolute value, known as the phase retrieval problem in Fock space. An important finding in this research field asserts that phaseless sampling on lattices of arbitrary density renders the problem unsolvable. The present study establishes solvability when using irregular sampling sets of the form , where and constitute perturbations of a Liouville set, i.e., a set with the property that all functions in Fock space bounded on the set are constant. The sets and adhere to specific geometrical conditions of closeness and noncollinearity. We show that these conditions are sufficiently generic so as to allow the perturbations to be chosen also at random. By proving that Liouville sets occupy an intermediate position between sets of stable sampling and sets of uniqueness, we obtain the first construction of uniqueness sets for the phase retrieval problem in Fock space having a finite density. The established results apply to the Gabor phase retrieval problem in subspaces of , where we derive additional reductions of the size of uniqueness sets: for the class of real-valued functions, uniqueness is achieved from two perturbed lattices; for the class of even real-valued functions, a single perturbation suffices, resulting in a separated set.
Paper Structure (27 sections, 26 theorems, 189 equations, 6 figures)

This paper contains 27 sections, 26 theorems, 189 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Closeness and noncollinearity
  4. Deterministic perturbations of Liouville sets
  5. Sets of stable sampling, lattices, and Hardy's uncertainty principle
  6. Random perturbation of Liouville sets
  7. Gabor phase retrieval
  8. Previous work
  9. Square-root sampling
  10. Multi-window approach
  11. Restriction to subspaces
  12. Notation
  13. Liouville sets
  14. Uniform closeness and Lagrange-type interpolation
  15. Liouville sets in Fock space
  16. ...and 12 more sections

Key Result

Theorem 2.4

Let $\alpha > 0$, and let $\Lambda \subseteq {\mathbb C}$ be a Liouville set for ${\mathcal{F}}_{4\alpha}({\mathbb C})$. Further, let $f: {\mathbb C} \to [0,\infty)$ be given by $f(z) = e^{-\gamma |z|^2}$ with $\gamma > 2\alpha$. Suppose that $A=(a_\lambda)_{\lambda\in\Lambda},B=(b_\lambda)_{\lambda then is a uniqueness set for the phase retrieval problem in $\mathcal{F}_\alpha({\mathbb C})$. In

Figures (6)

  • Figure 1: Visualization of Theorem \ref{['thm:Main1']}: the sets $A,B$, and $C$ are uniformly noncollinear and $f$-close to a sufficiently dense (hexagonal) lattice $\Lambda$, which is a Liouville set for ${\mathcal{F}}_\alpha({\mathbb C})$. For simplicity, we consider $A$ to be the lattice itself, i.e., $A = \Lambda$. Theorem \ref{['thm:Main1']} establishes the unique determination (up to a global phase) of every function in the Fock space by its absolute values located on the union $A \cup B \cup C$. However, it is important to note that sampling only on the lattice $A=\Lambda$ does not guarantee uniqueness.
  • Figure 2: Visualization of Theorem \ref{['thm:Main4']}: the grey disks represent neighborhoods around points $\lambda$ of a sufficiently dense (hexagonal) lattice $\Lambda$. From each disk, three points are selected at random, resulting in three sets $A,B$, and $C$. With probability 1, the union $A \cup B \cup C$ forms a uniqueness set for the phase retrieval problem in Fock space.
  • Figure 3: This figure depicts an example related to Theorem \ref{['thm:density_lower_bound']}(2). The vertical and horizontal lines induce a (shifted) square lattice $\Lambda$ of density $>4$. The black points are perturbations of $\Lambda$. The red points depict a sublattice $\Lambda' \subseteq \Lambda$ of density $D(\Lambda') = \frac{1}{2}D(\Lambda)$. The union of the black and red points forms a uniqueness set for the Gabor phase retrieval problem in $L^2({\mathbb R},{\mathbb R})$, and has density $> 6$. The density can be as close to $6$ as we please.
  • Figure 4: This figure depicts an example related to Theorem \ref{['thm:density_lower_bound']}(3). The gray mesh induces a square lattice $\Lambda$ of density $>4$. The points $A,B,C$ are perturbations of points of $\Lambda$, with the property that their union, $A \cup B \cup C$, forms a uniqueness set for the Gabor phase retrieval problem in $L^2_e({\mathbb R},{\mathbb R})$, having density $>3$. This density can be as close to $3$ as we please. In addition, $A \cup B \cup C$ is separated.
  • Figure 5: Visualisation of the sets $Q_1,Q_2,Q_3$, and $Q_4$ used in the proof of Theorem \ref{['thm:density_lower_bound']}(3). The union $\Lambda' \coloneqq Q_1 \sqcup Q_2 \sqcup Q_3 \sqcup Q_4$ is a $1/4$-Liouville set for ${\mathcal{F}}_{4\alpha}({\mathbb C})$ (the intersections of the grey mesh are the points of the square-lattice $\Lambda$).
  • ...and 1 more figures

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 47 more