Vector Space Projections A Numerical Approach

This book deals with the theory and applications of a mathematical technique known as "projections onto convex sets" which is used for solving problems in science and engineering. The material applies to all branches of science and engineering.

HENRY STARK is Bodine Professor and former chairman of the Department of Electrical and Computer Engineering at the Illinois Institute of Technology. He has written over 130 papers and book chapters on topics ranging from image recovery and signal processing to neural nets and sampling theory. Professor Stark is coauthor of Modern Electrical Communications and Probability, Random Processes, and Estimation Theory for Engineers and coeditor of Signal Processing Methods for Audio, Images, and Telecommunications. YONGYI YANG is Assistant Professor at the Illinois Institute of Technology. He is a principal contributor to Signal Processing Methods for Audio, Images, and Telecommunications, coedited by Henry Stark.

Preface

xi

1 Vector Space Concepts

1

(32)

1.1 Introduction

1

(1)

1.2 Vector Spaces

1

(5)

1.3 Normed Vector Spaces

6

(7)

1.4 Inner Product Spaces

13

(8)

1.5 Hilbert Spaces

21

(8)

1.6 Summary

29

(1)

References

29

(4)

2 Projections Onto Convex Sets

33

(58)

2.1 Introduction

33

(1)

2.2 Convex Sets

33

(2)

2.3 Projection

35

(7)

2.4 Properties of Projections

42

(7)

2.5 Fundamental Theory of POCS

49

(8)

2.6 The POCS Algorithm and Its Numerical Aspects

57

(10)

2.7 Application of POCS Method

67

(5)

2.8 Proof of Opial's Theorem

72

(6)

2.9 POCS Algorithm Formulated in a Product Space

78

(4)

2.10 Projection Algorithms in the Presence of Nonintersecting Sets

82

(4)

2.11 Summary

86

(1)

References

87

(4)

3 Elementary Projectors

91

(42)

3.1 Introduction

91

(1)

3.2 Linear Type Constraint

91

(6)

3.3 Soft Linear Type Constraint

97

(5)

3.4 Error Type Constraint

102

(7)

3.5 Similarity Constraint

109

(5)

3.6 Time-Domain Constraints

114

(5)

3.7 Frequency-Domain Constraints

119

(6)

3.8 Summary

125

(1)

Appendix

126

(4)

References

130

(3)

4 Solutions of Linear Equations

133

(48)

4.1 Introduction

133

(3)

4.2 Convex-Set Formulation

136

(6)

4.3 Convergence Analysis

142

(6)

4.4 Convergence Acceleration

148

(6)

4.5 Minimum Distance Property

154

(3)

4.6 A Hybrid Algorithm

157

(5)

4.7 Application Example: Computed Tomography

162

(8)

4.8 Application Example: Resolution Enhancement

170

(6)

4.9 Summary

176

(1)

References

177

(4)

5 Generalized Projections

181

(24)

5.1 Introduction

181

(1)

5.2 Convex versus Non-Convex Sets

182

(2)

5.3 Examples of Non-convex Sets

184

(1)

5.4 Theory of Generalized Projection

185

(7)

5.5 Traps and Tunnels

192

(1)

5.6 Proof of Theorem 5.4-2

193

(3)

5.7 Generalized Projection Algorithm in a Product Vector Space

196

(6)

5.8 Summary

202

(1)

References

202

(3)

6 Applications to Communications

205

(76)

6.1 Introduction

205

(1)

6.2 Non-uniform Sampling

205

(6)

6.3 Digital Communications

211

(10)

6.4 Digital Filters

221

(18)

6.5 Estimation of Probability Density Functions

239

(10)

6.6 Spread Spectrum

249

(13)

6.7 Image Compression

262

(10)

6.8 Summary

272

(3)

References

275

(6)

7 Applications to Optics

281

(38)

7.1 Introduction

281

(1)

7.2 Image Sharpening

282

(3)

7.3 The Phase Retrieval Problem

285

(3)

7.4 Beam Forming

288

(3)

7.5 Color Matching

291

(4)

7.6 Blind Deconvolution

295

(7)

7.7 Design of Diffractive Optical Elements

302

(10)

7.8 Summary

312

(1)

References

312

(7)

8 Applications to Neural Nets

319

(44)

8.1 Introduction

319

(1)

8.2 The Brain as a Neural Network

319

(2)

8.3 The Hopfield Net

321

(7)

8.4 POCS and the Continuous Hopfield ACAM

328

(3)

8.5 A Hopfield-Net Based Classifier

331

(6)

8.6 Learning in Multi-Layer Nets

337

(2)

8.7 Back-Propagation Rule for the Two-Layer Net

339

(4)

8.8 Projection Method Learning Rule

343

(3)

8.9 Experiments and Results

346

(11)

8.10 Summary

357

(1)

References

358

(5)

9 Applications to Image Processing

363

(38)

9.1 Introduction

363

(1)

9.2 Noise Smoothing

364

(5)

9.3 Image Synthesis

369

(6)

9.4 Restoration of Blurred and Noisy Images

375

(7)

9.5 High-Resolution Image Restoration

382

(7)

9.6 Restoration of Quantum-Limited Images

389

(6)

9.7 Summary

395

(2)

References

397

(4)

Index

401

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Vector Space Projections A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics

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