Image Recovery: Theory and Application
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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.
Table of Contents
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
9.4 Applications of the Inner Product Framework
9.5 Image Reconstruction from Incomplete Data
9.6 Conclusions
References
Chapter 10 Computer-Assisted Diffraction Tomography
10.1 Introduction
10.2 Transformations of the Wave Equation
10.3 Fourier Slice in Diffraction Tomography
10.4 Reconstruction Algorithms
10.5 Reconstruction from Limited Angular Data
10.6 Phase Determination
10.7 Experimental Results and Comparison of Born and Rytov Methods
10.8 Concluding Remarks
References
Chapter 11 Applications of Convex Projection Theory to Image Recovery in Tomography and Related Areas
11.1 Introduction
11.2 Demonstration of the Method of Projections onto Convex Sets: Spectral Extrapolation of Images
11.3 Review of the Direct Fourier Method
11.4 POCS-DFM Algorithm
11.5 A Case Study: Reconstruction of a Thorax Cross Section from Angularly Limited X-Ray Projection Data
11.6 Applications to Related Areas
11.7 Summary
References
Chapter 12 Image Synthesis: Discovery Instead of Recovery
12.1 Introduction
12.2 General Formulation of the Problem
12.3 Diffraction-Limited Systems
12.4 Binary Images through Diffraction-Limited Systems
12.5 Binary Images of Binary Masks
12.6 Complex Masks for Coherent Imaging Systems
12.7 Summary
References
Chapter 13 The Role of Analyticity in Image Recovery
13.1 Introduction
13.2 Multidimensional Bandlimited Functions
13.3 Consequences of Analyticity
13.4 Image Reconstruction from Limited Spectral Data
13.5 Image Reconstruction from Zeros
13.6 Image Reconstruction from Polynomial Representation of Two-Dimensional Bandlimited Functions
13.7 Analyticity and Phase Retrieval
13.8 Analyticity and Phase Unwrapping
13.9 Summary and Conclusions
References
Index
Product details
- No. of pages: 564
- Language: English
- Copyright: © Academic Press 1987
- Published: March 2, 1987
- Imprint: Academic Press
- eBook ISBN: 9780323145978
About the Editor
Henry Stark
Affiliations and Expertise
Illinois Institute of Technology
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