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Image Recovery: Theory and Application
1st Edition - March 2, 1987
Editor: Henry Stark
eBook ISBN:9780323145978
9 7 8 - 0 - 3 2 3 - 1 4 5 9 7 - 8
Image Recovery: Theory and Application focuses on signal recovery and synthesis problems. This book discusses the concepts of image recovery, including regularization, the… Read more
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Image Recovery: Theory and Application focuses on signal recovery and synthesis problems. This book discusses the concepts of image recovery, including regularization, the projection theorem, and the pseudoinverse operator. Comprised of 13 chapters, this volume begins with a review of the basic properties of linear vector spaces and associated operators, followed by a discussion on the Gerchberg-Papoulis algorithm. It then explores image restoration and the basic mathematical theory in image restoration problems. The reader is also introduced to the problem of obtaining artifact-free computed tomographic reconstruction. Other chapters consider the importance of Bayesian approach in the context of medical imaging. In addition, the book discusses the linear programming method, which is particularly important for images with large number of pixels with zero value. Such images are usually found in medical imaging, microscopy, electron microscopy, and astronomy. This book can be a valuable resource to materials scientists, engineers, computed tomography technologists, and astronomers.
Preface
Acknowledgments
Chapter 1 Signal Restoration, Functional Analysis, and Fredholm Integral Equations of the First Kind
1.1 Introduction
1.2 Hilbert Spaces and Linear Operators
1.3 Existence of Solutions
1.4 Least-Squares Solutions and the Operator Pseudoinverse
1.5 Regularization
1.6 The Truncated SVD Expansion and Filtering
1.7 The Iterative Algorithm of Landweber
1.8 Alternating Orthogonal Projections
1.9 Regularized Iterative Algorithms
1.10 Moment Discretization
1.11 Summary and Conclusions
References
Chapter 2 Mathematical Theory of Image Restoration by the Method of Convex Projections
2.1 Introduction
2.2 Some Properties of Convex Sets in Hilbert Space
2.3 Nonexpansive Maps and Their Fixed Points—Basic Theorems
2.4 Iterative Techniques for Image Restoration in a Hubert Space Setting
2.5 Useful Projections
2.6 Summary and New Developments
References
Chapter 3 Bayesian and Related Methods in Image Reconstruction from Incomplete Data
3.1 Introduction
3.2 Measurement Space—Null Space
3.3 Deterministic Solutions
3.4 The Bayesian Approach
3.5 Use of Other Kinds of Prior Knowledge
3.6 MAP Solutions
3.7 FAIR-Fit and Iterative Reconstruction
3.8 Comparison of MAP and FAIR Results
3.9 A Generalized Bayesian Method
3.10 Discussion
3.11 Summary
References
Chapter 4 Image Restoration Using Linear Programming
4.1 Image Restoration
4.2 Numerical Example of the Matrix Diagonalization of H
4.3 Linear Programming
4.4 Norms of the Error
4.5 Numerical Example of Minimum L1 Norm Method
4.6 Computation Considerations
4.7 Spatial Resolution
4.8 Results
4.9 Summary and Conclusions
References
Chapter 5 The Principle of Maximum Entropy in Image Recovery
5.1 Introduction
5.2 Frieden's Approach
5.3 Burch, Gull, and Skilling's Approach
5.4 A Differential Equation Approach to Maximum Entropy Image Restoration
5.5 Conclusion
References
Chapter 6 The Unique Reconstruction of Multidimensional Sequences from Fourier Transform Magnitude or Phase
6.1 Introduction
6.2 Fourier Synthesis from Partial Information
6.3 The Algebra of Polynomials in Two or More Variables
6.4 The Magnitude Retrieval Problem
6.5 The Phase Retrieval Problem
6.6 Summary and Other Problems
References
Chapter 7 Phase Retrieval and Image Reconstruction for Astronomy
7.1 Introduction
7.2 Uniqueness of Phase Retrieval from Modulus Data
7.3 Algorithms for Phase Retrieval from Modulus
7.4 Iterative Transform Algorithm
7.5 Solutions Specific to Measurement Techniques
7.6 Conclusions
References
Chapter 8 Restoration from Phase and Magnitude by Generalized Projections
8.1 Introduction
8.2 The Gerchberg-Saxton and Related Algorithms
8.3 The Method of Projections onto Convex Sets
8.4 Application of POCS to the Problem of Restoration from Phase
8.5 Computer Simulations of Restoration from Phase
8.6 The Method of Generalized Projections
8.7 Signal Recovery from Magnitude by Generalized Projections
8.8 Computer Simulations of Restoration from Magnitude
8.9 Conclusion
References
Chapter 9 Image Reconstruction from Limited Data: Theory and Applications in Computerized Tomography
9.1 Introduction
9.2 Review of Image Reconstruction
9.3 An Inner Product Framework for Image Reconstruction