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E-book: Geometry of the Phase Retrieval Problem: Graveyard of Algorithms

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Recovering the phase of the Fourier transform is a ubiquitous problem in imaging applications. This book introduces a conceptual, geometric framework for the analysis of these problems, and, using this framework, both analyzes well-known algorithms and introduces new approaches for phase retrieval.

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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This book provides a theoretical foundation and conceptual framework for the problem of recovering the phase of the Fourier transform.
Preface ix
Acknowledgments xii
1 Introduction
1(36)
1.1 Discrete, Phase Retrieval Problems
8(5)
1.2 Conditioning and Ill-Posedness of the Discrete, Classical, Phase Retrieval Problem
13(3)
1.3 Algorithms for Finding Intersections of Sets
16(5)
1.4 Numerical Experiments
21(7)
1.5 Comparison to the Continuum Phase Retrieval Problem
28(3)
1.6 Outline of the Book
31(1)
1.A Appendix: Factoring Polynomials in Several Variables
31(2)
1.B Appendix: The Condition Number of a Problem
33(4)
PART I THEORETICAL FOUNDATIONS
37(70)
2 The Geometry Near an Intersection
39(26)
2.1 The Tangent Space to the Magnitude Torus
42(6)
2.2 The Intersection of the Tangent Bundle and the Support Constraint
48(6)
2.3 Numerical Examples
54(6)
2.A Appendix: The Tangent and Normal Bundles for Submanifolds of Rn
60(3)
2.B Appendix: Fast Projections onto the Tangent and Normal Bundles
63(2)
3 Well-Posedness
65(20)
3.1 Conditioning and Transversality
68(8)
3.2 Examples of Ill-Posedness
76(9)
4 Uniqueness and the Nonnegativity Constraint
85(18)
4.1 Support and the Autocorrelation Image
87(3)
4.2 Uniqueness for Nonnegative Images
90(3)
4.3 Nonnegative Images and the 1-Norm
93(1)
4.4 The 1-Norm on the Tangent Space
94(3)
4.5 Transversality of Aa & δ B+ and Aa %% dδBxr11
97(6)
5 Some Preliminary Conclusions
103(4)
PART II ANALYSIS OF ALGORITHMS FOR PHASE RETRIEVAL
107(114)
6 Introduction to Part II
109(6)
7 Algorithms for Phase Retrieval
115(32)
7.1 Classical Alternating Projection
116(2)
7.2 Hybrid Iterative Maps
118(11)
7.3 Nonlinear Submanifolds
129(5)
7.4 A Noniterative Approach to Phase Retrieval
134(6)
7.A Appendix: Alternating Projection and Gradient Flows
140(7)
8 The Discrete, Classical, Phase Retrieval Problem
147(44)
8.1 Hybrid Iterative Maps in Model Problems
149(16)
8.2 Linearization of Hybrid Iterative Maps Along the Center Manifold
165(5)
8.3 Further Numerical Examples
170(21)
9 Phase Retrieval with the Nonnegativity Constraint
191(14)
9.1 Hybrid Iterative Maps Using Nonnegativity
192(2)
9.2 Numerical Examples
194(5)
9.3 Algorithms Based on Minimization in the 1-Norm
199(4)
9.A Appendix: An Efficient Method for Projection onto a Ball in the 1-Norm
203(2)
10 Asymptotics of Hybrid Iterative Maps
205(16)
10.1 Stagnation
206(5)
10.2 Numerical Examples
211(10)
PART III FURTHER PROPERTIES OF HYBRID ITERATIVE ALGORITHMS AND SUGGESTIONS FOR IMPROVEMENT
221(79)
11 Introduction to Part III
223(3)
12 Statistics of Algorithms
226(23)
12.1 Statistics of Phases
228(2)
12.2 Statistics of Ensembles
230(9)
12.3 Averaging to Improve Reconstructions
239(8)
12.4 Some Conclusions
247(2)
13 Suggestions for Improvements
249(40)
13.1 Use of a Sharp Cutoffs
250(5)
13.2 External Holography
255(9)
13.3 A Geometric Newton's Method for Phase Retrieval
264(5)
13.4 Implementation of the Holographic Hilbert Transform Method
269(20)
13 A Appendix: Proof of Theorem 13.6
289(3)
14 Concluding Remarks
292(5)
15 Notational Conventions
297(3)
References 300(5)
Index 305
Alexander H. Barnett is Group Leader for Numerical Analysis at the Center for Computational Mathematics in the Flatiron Institute. He has published around 60 papers on partial differential equations, waves, fast algorithms, integral equations, neuroscience, imaging, signal processing, inverse problems, and physics, and received several research grants from the National Science Foundation. Charles L. Epstein is the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, where he founded the graduate group in Applied Mathematics and Computational Science. He has worked on a wide range of problems in pure and applied analysis and is the author of a widely used textbook An Introduction to the Mathematics of Medical Imaging (SIAM 2008). He shared the Bergman Prize in 2016 with Francois Treves and is a fellow of the AAAS and the AMS. Leslie Greengard is Silver Professor of Mathematics and Computer Science at the Courant Institute, New York University and Director of the Center for Computational Mathematics at the Flatiron Institute. His is co-inventor of several widely used fast algorithms and a member of the American Academy of Arts and Sciences, the National Academy of Sciences, and the National Academy of Engineering. Jeremy Magland is a Senior Data Scientist at the Flatiron Institute. He received his PhD in Mathematics from the University of Pennsylvania. Prior to joining the Flatiron Institute in 2015, he worked for about a decade as a research scientist in the Radiology Department of the Hospital of the University of Pennsylvania, where he developed software systems that dramatically streamlined experimental work on MR-scanners.
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