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Phasebook: A Survey of Selected Open Problems in Phase Retrieval

Marc Allain, Selin Aslan, Wim Coene, Sjoerd Dirksen, Jonathan Dong, Julien Flamant, Mark Iwen, Felix Krahmer, Tristan van Leeuwen, Oleh Melnyk, Andreas Menzel, Allard P. Mosk, Viktor Nikitin, Gerlind Plonka, Palina Salanevich, Matthias Wellershoff

TL;DR

This survey synthesizes open research problems in phase retrieval, drawing from the PRiMA 2024 workshop to bridge theory and practice. It organizes and analyzes intrinsic difficulty, physics-informed object/probe modeling, generative-prior regularization (PtyGenography), structured random measurement designs, one-bit quantization for low-dose scenarios, and the mathematical underpinnings of Wigner Distribution Deconvolution. Key contributions include formal complexity measures, stability and transversality insights, parametric object representations, unified generative-conventional reconstruction frameworks with bias-detection strategies, thresholds for structured randomness, and foundational questions for practical algorithms with theoretical guarantees. The work highlights practical impacts for CDI, ptychography, and 4D STEM, guiding both rigorous analysis and the design of implementable measurement schemes with improved reconstruction reliability.

Abstract

Phase retrieval is an inverse problem that, on one hand, is crucial in many applications across imaging and physics, and, on the other hand, leads to deep research questions in theoretical signal processing and applied harmonic analysis. This survey paper is an outcome of the recent workshop Phase Retrieval in Mathematics and Applications (PRiMA) (held on August 5--9 2024 at the Lorentz Center in Leiden, The Netherlands) that brought together experts working on theoretical and practical aspects of the phase retrieval problem with the purpose to formulate and explore essential open problems in the field.

Phasebook: A Survey of Selected Open Problems in Phase Retrieval

TL;DR

This survey synthesizes open research problems in phase retrieval, drawing from the PRiMA 2024 workshop to bridge theory and practice. It organizes and analyzes intrinsic difficulty, physics-informed object/probe modeling, generative-prior regularization (PtyGenography), structured random measurement designs, one-bit quantization for low-dose scenarios, and the mathematical underpinnings of Wigner Distribution Deconvolution. Key contributions include formal complexity measures, stability and transversality insights, parametric object representations, unified generative-conventional reconstruction frameworks with bias-detection strategies, thresholds for structured randomness, and foundational questions for practical algorithms with theoretical guarantees. The work highlights practical impacts for CDI, ptychography, and 4D STEM, guiding both rigorous analysis and the design of implementable measurement schemes with improved reconstruction reliability.

Abstract

Phase retrieval is an inverse problem that, on one hand, is crucial in many applications across imaging and physics, and, on the other hand, leads to deep research questions in theoretical signal processing and applied harmonic analysis. This survey paper is an outcome of the recent workshop Phase Retrieval in Mathematics and Applications (PRiMA) (held on August 5--9 2024 at the Lorentz Center in Leiden, The Netherlands) that brought together experts working on theoretical and practical aspects of the phase retrieval problem with the purpose to formulate and explore essential open problems in the field.

Paper Structure

This paper contains 29 sections, 3 theorems, 58 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Purpose of the Survey
  3. Organization of the Paper
  4. Notation
  5. Problem: Intrinsic Difficulty of Phase Retrieval
  6. Complexity measures
  7. Theoretical background
  8. Complexity and reconstruction
  9. Confidence intervals
  10. Problem: Physics-Informed Modeling of Object and Probe
  11. Object Parameterization
  12. Sparse Crystallography
  13. PtyGenography: Using Generative Model for Regularization with Varying Noise Levels
  14. Methods
  15. Characterizing the reconstruction error
  16. ...and 14 more sections

Key Result

Lemma 5.2

Let ${\bf \tilde{f}} = \arg\min_{{\bf f}} \|\mathcal{A}({\bf f})-{\bf y}\|_2$ with ${\bf y} = \mathcal{A}({\bf f_0}) + \boldsymbol{\varepsilon}$. Then the reconstruction error is given by

Figures (8)

  • Figure 1: Examples of objects.
  • Figure 2: (a) Example of an object ${\bf f_0} = {\bf f} + \boldsymbol{\eta}$, where ${\bf f}$ is the letter 'a' and the imperfection $\boldsymbol{\eta}$ is the dot accent. (b) Imperfection ${\boldsymbol{\eta}}$ retrieved from correlating the generative reconstruction ${\bf f}$ with the given phaseless measurements.
  • Figure 3: Samples of the data set on which the generative model was trained ($n=64$). The top row displays the real part while the bottom row displays the imaginary part.
  • Figure 4: Samples generated by the generative model ($k=30$) The top row displays the real part while the bottom row displays the imaginary part.
  • Figure 5: Examples of the binary masks used to generate the measurements.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Remark 4.5
  • Remark 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Lemma 5.5
  • Remark 6.1
  • Remark 6.3
  • Remark 6.4