CiNii 図書 - Geometry of the phase retrieval problem : graveyard of algorithms

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.

Part I. Theoretical Foundations: 1. The geometry near an intersection
2. Well posedness
3. Uniqueness and the non-negativity constraint
4. Some preliminary conclusions
Part II. Analysis of Algorithms for Phase Retrieval: 6. Introduction to Part II
7. Algorithms for Phase Retrieval
8. Discrete classical phase retrieval
9. The non-negativity constraint
10. Asymptotics of hybrid iterative maps
Part III. Further Properties of Hybrid Iterative Algorithms and Suggestions for Improvement: 11. Introduction to Part III
12. Statistics of algorithms
13. Suggestions for improvements
14. Concluding Remarks
15. Notational conventions.

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Geometry of the phase retrieval problem : graveyard of algorithms

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