Geometry of the Phase Retrieval Problem: Graveyard of Algorithms

Alexander H. Barnett; Charles L. Epstein; Leslie Greengard; Jeremy Magland

doi:10.1017/9781009003919

Geometry of the Phase Retrieval Problem

Graveyard of Algorithms

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Steinerberger, Stefan and Wu, Hau-tieng 2022. Eigenvector Phase Retrieval: Recovering eigenvectors from the absolute value of their entries. Linear Algebra and its Applications, Vol. 652, Issue. , p. 239.

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Trefethen, Lloyd N. 2023. Numerical analytic continuation. Japan Journal of Industrial and Applied Mathematics, Vol. 40, Issue. 3, p. 1587.

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Bendory, Tamir and Edidin, Dan 2025. The generalized phase retrieval problem over compact groups. p. 1.

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Bendory, Tamir Dym, Nadav Edidin, Dan and Suresh, Arun 2025. A Transversality Theorem for Semi-Algebraic Sets with Application to Signal Recovery from the Second Moment and Cryo-EM. Foundations of Computational Mathematics,

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Alexander H. Barnett, Flatiron Institute, Charles L. Epstein, Flatiron Institute, Leslie Greengard, Courant Institute, Jeremy Magland, Flatiron Institute

Publisher:: Cambridge University Press
Online publication date:: April 2022
Print publication year:: 2022
Online ISBN:: 9781009003919
DOI:: https://doi.org/10.1017/9781009003919

Subjects:: Mathematics, Computational Science, Communications and Signal Processing, Engineering, Numerical Analysis and Computational Science
Series:: Cambridge Monographs on Applied and Computational Mathematics (39)

US$139.65

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Recovering the phase of the Fourier transform is a ubiquitous problem in imaging applications from astronomy to nanoscale X-ray diffraction imaging. Despite the efforts of a multitude of scientists, from astronomers to mathematicians, there is, as yet, no satisfactory theoretical or algorithmic solution to this class of problems. Written for mathematicians, physicists and engineers working in image analysis and reconstruction, this book introduces a conceptual, geometric framework for the analysis of these problems, leading to a deeper understanding of the essential, algorithmically independent, difficulty of their solutions. Using this framework, the book studies standard algorithms and a range of theoretical issues in phase retrieval and provides several new algorithms and approaches to this problem with the potential to improve the reconstructed images. The book is lavishly illustrated with the results of numerous numerical experiments that motivate the theoretical development and place it in the context of practical applications.

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Geometry of the Phase Retrieval Problem

Graveyard of Algorithms

This Book has been cited by the following publications. This list is generated based on data provided by Crossref.

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-xi

Acknowledgments
pp xii-xii

1 - Introduction
pp 1-36

Part I - Theoretical Foundations
pp 37-38

2 - The Geometry Near an Intersection
pp 39-64

3 - Well-Posedness
pp 65-84

4 - Uniqueness and the Nonnegativity Constraint
pp 85-102

5 - Some Preliminary Conclusions
pp 103-106

Part II - Analysis of Algorithms for Phase Retrieval
pp 107-108

6 - Introduction to Part II
pp 109-114

7 - Algorithms for Phase Retrieval
pp 115-146

8 - The Discrete, Classical, Phase Retrieval Problem
pp 147-190

9 - Phase Retrieval with the Nonnegativity Constraint
pp 191-204

10 - Asymptotics of Hybrid Iterative Maps
pp 205-220

Part III - Further Properties of Hybrid Iterative Algorithms and Suggestions for Improvement
pp 221-222

11 - Introduction to Part III
pp 223-225

12 - Statistics of Algorithms
pp 226-248

13 - Suggestions for Improvements
pp 249-291

14 - Concluding Remarks
pp 292-296

15 - Notational Conventions
pp 297-299

References
pp 300-304

Index
pp 263-308

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Geometry of the Phase Retrieval Problem

Graveyard of Algorithms

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