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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

### Henry Stark

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1st Edition - March 2, 1987

Editor: Henry Stark

Language: EnglisheBook ISBN:

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

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

- No. of pages: 564
- Language: English
- Edition: 1
- Published: March 2, 1987
- Imprint: Academic Press
- eBook ISBN: 9780323145978

HS

Affiliations and expertise

Illinois Institute of Technology