Fundamentals of Nonlinear Digital Filtering

This book presents and evaluates current methods and applications in nonlinear digital filtering. Solid introductory material, balanced coverage of theoretical and practical aspects, and dozens of examples provide you with a self-contained, comprehensive information source on nonlinear filtering and its applications. Coverage of all major classes of filters from fundamentals to performance comparison the book covers dozens of classes of filters and, for each class, includes basic ideas, definitions, algorithms, and impulse and step responses that allow direct comparisons of filtering operations. This book is a major contribution of tremendous value to experimentalists, professional signal processing engineers, professors, and students.

PREFACE

v

(6)

List of Symbols and Abbreviations

xi

1 Nonlinear Signal Processing

1

(20)

1.1 Signal Processing Model

1

(2)

1.2 Signal and Noise Models

3

(1)

1.3 Fundamental Problems in Noise Removal

3

(15)

1.3.1 One-Dimensional Signals Corrupted by Additive White Gaussian Noise

4

(3)

1.3.2 One-Dimensional Signals Corrupted by Impulsive Noise

7

(3)

1.3.3 Mean and Median in Image Filtering

10

(8)

1.3.4 Linear Versus Nonlinear Filtering

18

(1)

1.4 Algorithms

18

(3)

2 Statistical Preliminaries

21

(28)

2.1 Random Variables and Distributions

21

(12)

2.1.1 One-Dimensional Random Variables

21

(7)

2.1.2 Random Vectors and Random Processes

28

(3)

2.1.3 Probability Elements

31

(1)

2.1.4 Transformations of Random Vectors

32

(1)

2.2 Signal and Noise Models

33

(3)

2.3 Estimation

36

(9)

2.3.1 Point Estimation

36

(3)

2.3.2 Maximum Likelihood Estimators

39

(2)

2.3.3 M-Estimators

41

(1)

2.3.4 L-Estimators

42

(1)

2.3.5 R-Estimators

43

(1)

2.3.6 Scale Estimation

44

(1)

2.4 Some Useful Distributions

45

(4)

3 1001 Solutions

49

(164)

3.1 Trimmed Mean Filters

52

(4)

3.1.1 Principles and Properties

52

(2)

3.1.2 Impulse and Step Response

54

(1)

3.1.3 Filtering Examples

54

(2)

3.2 Other Trimmed Mean Filters

56

(10)

3.2.1 Principles and Properties

56

(3)

3.2.2 Impulse and Step Response

59

(1)

3.2.3 Filtering Examples

60

(6)

3.3 L-Filters

66

(4)

3.3.1 Principles and Properties

66

(1)

3.3.2 Impulse and Step Response

67

(1)

3.3.3 Filtering Examples

68

(2)

3.4 C-Filters (Ll-Filters)

70

(3)

3.4.1 Principles and Properties

70

(2)

3.4.2 Impulse and Step Response

72

(1)

3.4.3 Filtering Examples

72

(1)

3.5 Weighted Median Filters

73

(5)

3.5.1 Principles and Properties

73

(3)

3.5.2 Impulse and Step Response

76

(1)

3.5.3 Filtering Examples

76

(2)

3.6 Ranked-Order and Weighted Order Statistic Filters

78

(10)

3.6.1 Principles and Properties

78

(3)

3.6.2 Impulse and Step Response

81

(1)

3.6.3 Filtering Examples

82

(6)

3.7 Multistage Median Filters

88

(9)

3.7.1 Principles and Properties

88

(3)

3.7.2 Impulse and Step Response

91

(1)

3.7.3 Filtering Examples

91

(6)

3.8 Median Hybrid Filters

97

(8)

3.8.1 Principles and Properties

97

(3)

3.8.2 Impulse and Step Response

100

(1)

3.8.3 Filtering Examples

100

(5)

3.9 Edge-Enhancing Selective Filters

105

(4)

3.9.1 Principles and Properties

105

(4)

3.9.2 Impulse and Step Response

109

(1)

3.9.3 Filtering Examples

109

(1)

3.10 Rank Selection Filters

109

(13)

3.10.1 Principles and Properties

109

(10)

3.10.2 Impulse and Step Response

119

(1)

3.10.3 Filtering Examples

119

(3)

3.11 M-Filters

122

(18)

3.11.1 Principles and Properties

122

(10)

3.11.2 Impulse and Step Response

132

(1)

3.11.3 Filtering Examples

133

(7)

3.12 R-Filters

140

(7)

3.12.1 Principles and Properties

140

(2)

3.12.2 Impulse and Step Response

142

(1)

3.12.3 Filtering Examples

143

(4)

3.13 Weighted Majority with Minimum Range Filters

147

(2)

3.13.1 Principles and Properties

147

(1)

3.13.2 Impulse and Step Response

147

(1)

3.13.3 Filtering Examples

148

(1)

3.14 Nonlinear Mean Filters

149

(14)

3.14.1 Principles and Properties

149

(3)

3.14.2 Impulse and Step Response

152

(1)

3.14.3 Filtering Examples

153

(10)

3.15 Stack Filters

163

(4)

3.15.1 Principles and Properties

163

(3)

3.15.2 Impulse and Step Response

166

(1)

3.15.3 Filtering Examples

167

(1)

3.16 Generalizations of Stack Filters

167

(5)

3.16.1 Principles and Properties

167

(5)

3.16.2 Impulse and Step Response

172

(1)

3.16.3 Filtering Examples

172

(1)

3.17 Morphological Filters

172

(7)

3.17.1 Principles and Properties

172

(5)

3.17.2 Impulse and Step Response

177

(1)

3.17.3 Filtering Examples

177

(2)

3.18 Soft Morphological Filters

179

(7)

3.18.1 Principles and Properties

179

(5)

3.18.2 Impulse and Step Response

184

(1)

3.18.3 Filtering Examples

184

(2)

3.19 Polynomial Filters

186

(3)

3.19.1 Principles and Properties

186

(3)

3.19.2 Impulse and Step Response

189

(1)

3.19.3 Filtering Examples

189

(1)

3.20 Data-Dependent Filters

189

(5)

3.20.1 Principles and Properties

189

(4)

3.20.2 Impulse and Step Response

193

(1)

3.20.3 Filtering Examples

193

(1)

3.21 Decision-Based Filters

194

(7)

3.21.1 Principles and Properties

194

(6)

3.21.2 Impulse and Step Response

200

(1)

3.21.3 Filtering Examples

200

(1)

3.22 Iterative, Cascaded, and Recursive Filters

201

(3)

3.22.1 Principles and Properties

201

(2)

3.22.2 Impulse and Step Response

203

(1)

3.22.3 Filtering Examples

203

(1)

3.23 Some Numerical Measures of Nonlinear Filters

204

(8)

3.24 Discussion

212

(1)

4 STATISTICAL ANALYSIS AND OPTIMIZATION OF NONLINEAR FILTERS

213

(34)

4.1 Methods Based on Order Statistics

213

(9)

4.1.1 Joint Distributions of Order Statistics

215

(1)

4.1.2 Analysis of L-Filters

216

(3)

4.1.3 Optimization of L-Filters

219

(3)

4.2 Stack Filters

222

(20)

4.2.1 Output Distribution of a Stack Filter

222

(5)

4.2.2 Joint Distribution of Two Stack Filters

227

(4)

4.2.3 Output Moments of Stack Filters

231

(2)

4.2.4 Rank Selection Probabilities of Stack Filters

233

(4)

4.2.5 Optimization of Stack Filters

237

(2)

4.2.6 Stack Filter Optimization in the Boolean Lattice

239

(3)

4.3 Multistage and Hybrid Filters

242

(3)

4.4 Discussion

245

(2)

5 Exercises

247

(16)

BIBLIOGRAPHY

263

(10)

INDEX

273

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Fundamentals of Nonlinear Digital Filtering

Summary

Table of Contents

Supplemental Materials