| Description |
1 online resource (322 p.) |
| Series |
Cambridge Monographs on Applied and Computational Mathematics Ser |
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Cambridge Monographs on Applied and Computational Mathematics Ser
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| Contents |
Cover -- Half-title -- Series information -- Title page -- Copyright information -- Contents -- Preface -- Acknowledgments -- 1 Introduction -- 1.1 Discrete, Phase Retrieval Problems -- 1.2 Conditioning and Ill-Posedness of the Discrete, Classical, Phase Retrieval Problem -- 1.3 Algorithms for Finding Intersections of Sets -- 1.4 Numerical Experiments -- 1.5 Comparison to the Continuum Phase Retrieval Problem -- 1.6 Outline of the Book -- 1.A Appendix: Factoring Polynomials in Several Variables -- 1.B Appendix: The Condition Number of a Problem -- Part I Theoretical Foundations |
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2 The Geometry Near an Intersection -- 2.1 The Tangent Space to the Magnitude Torus -- 2.2 The Intersection of the Tangent Bundle and the Support Constraint -- 2.3 Numerical Examples -- 2.A Appendix: The Tangent and Normal Bundles for Submanifolds of RN -- 2.B Appendix: Fast Projections onto the Tangent and Normal Bundles -- 3 Well-Posedness -- 3.1 Conditioning and Transversality -- 3.2 Examples of Ill-Posedness -- 4 Uniqueness and the Nonnegativity Constraint -- 4.1 Support and the Autocorrelation Image -- 4.2 Uniqueness for Nonnegative Images -- 4.3 Nonnegative Images and the 1-Norm |
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4.4 The 1-Norm on the Tangent Space -- 4.5 Transversality of mathbb A[sub(a)] cup ∂ B[sub(+)] and mahtbb A[sub(a)] cup ∂ B[sup(1)sub(r[sub(1)])] -- 5 Some Preliminary Conclusions -- Part II Analysis of Algorithms for PhaseRetrieval -- 6 Introduction to Part II -- 7 Algorithms for Phase Retrieval -- 7.1 Classical Alternating Projection -- 7.2 Hybrid Iterative Maps -- 7.3 Nonlinear Submanifolds -- 7.4 A Noniterative Approach to Phase Retrieval -- 7.A Appendix: Alternating Projection and Gradient Flows -- 8 The Discrete, Classical, Phase Retrieval Problem |
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8.1 Hybrid Iterative Maps in Model Problems -- 8.2 Linearization of Hybrid Iterative Maps Along the Center Manifold -- 8.3 Further Numerical Examples -- 9 Phase Retrieval with the Nonnegativity Constraint -- 9.1 Hybrid Iterative Maps Using Nonnegativity -- 9.2 Numerical Examples -- 9.3 Algorithms Based on Minimization in the 1-Norm -- 9.A Appendix: An Efficient Method for Projection onto a Ball in the 1-Norm -- 10 Asymptotics of Hybrid Iterative Maps -- 10.1 Stagnation -- 10.2 Numerical Examples -- Part III Further Properties of Hybrid Iterative Algorithms and Suggestions for Improvement |
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11 Introduction to Part III -- 12 Statistics of Algorithms -- 12.1 Statistics of Phases -- 12.2 Statistics of Ensembles -- 12.3 Averaging to Improve Reconstructions -- 12.4 Some Conclusions -- 13 Suggestions for Improvements -- 13.1 Use of a Sharp Cutoffs -- 13.2 External Holography -- 13.3 A Geometric Newton's Method for Phase Retrieval -- 13.4 Implementation of the Holographic Hilbert Transform Method -- 13.A Appendix: Proof of Theorem 13.6 -- 14 Concluding Remarks -- 15 Notational Conventions -- References -- Index |
| Summary |
This book provides a theoretical foundation and conceptual framework for the problem of recovering the phase of the Fourier transform |
| Notes |
Description based upon print version of record |
| Subject |
Geometry.
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Algorithms.
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geometry.
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algorithms.
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Algorithms
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Geometry
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| Form |
Electronic book
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| Author |
Epstein, Charles L
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Greengard, Leslie
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Magland, Jeremy
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| ISBN |
1009008552 |
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9781009008556 |
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