Geometry of the Phase Retrieval Problem

Graveyard of Algorithms

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

• Publisher: Cambridge University Press
• Year: 2022
• Print ISBN: 9781316518878
• eBook ISBN: 9781009008556
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

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.

Table of contents

• Cover
• Half-title
• Series information
• Title page
• Copyright information
• Contents
• Preface
• Acknowledgments
• 1 Introduction
• Part I Theoretical Foundations
• Part II Analysis of Algorithms for PhaseRetrieval
• Part III Further Properties of Hybrid Iterative Algorithms and Suggestions for Improvement
• References
• Index