Nonlinear Filters for Image Processing

"This text covers key mathematical principles and algorithms for nonlinear filters used in image processing. Readers will gain an in-depth understanding of the underlying mathematical and filter design methodologies needed to construct and use nonlinear filters in a variety of applications. The 11 chapters explore topics of contemporary interest as well as fundamentals drawn from nonlinear filtering's historical roots in mathematical morphology and digital signal processing. This book examines various filter options and the types of applications for which they are best suited. The presentation is rigorous, yet accessible to engineers with a solid background in mathematics."

EDWARD R. DOUGHERTY, PhD, is Director of the Genomic Signal Processing Laboratory at Texas A&M University, where he holds the Robert M. Kennedy '26 Chair and is Professor in the Department of Electrical and Computer Engineering. He is also co-Director of the Computational Biology Division at the Translational Genomics Research Institute as well as Adjunct Professor in the Department of Bioinformatics and Computational Biology, M. D. Anderson Cancer Center at the University of Texas. Dr. Dougherty has published more than 300 peer-reviewed journal articles and book chapters.

MICHAEL L. BITTNER, PhD, is co-Director and Senior Investigator at the Computational Biology Division at the Translational Genomics Research Institute. Previously, he was associate investigator in the Cancer Genetics Branch of the National Human Genome Research Institute at the National Institutes of Health. Dr. Bittner holds a dozen patents and has published more than 100 articles.

Preface

xi

Logical Image Operators

1

(60)

Edward R. Dougherty

Junior Barrera

Boolean Functions

1

(4)

Morphological Representation

5

(5)

System Model

10

(2)

Optimal W-Operators

12

(6)

Estimation of Optimal W-Operators

18

(2)

Design Procedure

20

(9)

Constrained Optimization

29

(5)

Optimal Increasing Filters

34

(4)

Iterative Filters

38

(6)

Machine Learning Theory and Optimal Operator Design

44

(6)

Robustness

50

(11)

References

58

(3)

Computational Gray-Scale Operators

61

(38)

Edward R. Dougherty

Junior Barrera

Computational Functions

61

(4)

Representation of Increasing Computational Functions

65

(4)

Flat Computational Functions

69

(2)

Increasing Gray-to-Binary Image Operators

71

(3)

Representation of Increasing Gray-to-Gray Image Operators

74

(1)

Stack Filters

75

(5)

Nonflat Erosion

80

(1)

Representation of Generic Computational Functions

81

(4)

Representation of Generic Gray-to-Gray Image Operators

85

(3)

Optimal Gray-Scale Computational Filters

88

(2)

Application: Quantization Range Conversion

90

(2)

Comments on Gray-Scale Morphology

92

(7)

References

96

(3)

Translation-Invariant Set Operators

99

(22)

Edward Dougherty

Translation-Invariant Operators

99

(5)

Representation of Increasing Translation-Invariant Operators

104

(3)

Representation of Nonincreasing Translation-Invariant Operators

107

(2)

Openings and Closings

109

(7)

Representation of Openings and Closings

116

(2)

Convexity

118

(3)

References

119

(2)

Granulometric Filters

121

(42)

Edward R. Dougherty Yidong Chen

Granulometries

121

(2)

Representation of Euclidean Granulometries

123

(4)

Reconstructive Granulometries

127

(4)

Optimal Filtering by Reconstructive Granulometries

131

(5)

Adaptive Disjunctive Granulometric Filters

136

(5)

Size Distributions

141

(5)

Granulometric Classification

146

(4)

Discrete Granulometric Bandpass Filters

150

(2)

Continuous Granulometric Bandpass Filters

152

(11)

References

160

(3)

Esay Recipes for Morphological Filters

163

(44)

Henk J. A. M. Heijmans

Introduction

163

(2)

Morphology on Complete Lattices

165

(7)

Basic Theory

165

(3)

Application to Gray-Scale Functions

168

(4)

Openings and Closings

172

(3)

Basic Facts

172

(2)

Annular Opening

174

(1)

Adjunctional Filters

174

(1)

Annular Filters

175

(3)

Annular Filters for Binary Images

175

(2)

Annular Filters for Gray-Scale Images

177

(1)

AS-Filters

178

(1)

Overfilters and Inf-overfilters

179

(5)

Definitions and Basic Properties

180

(2)

Rank-max Openings

182

(2)

Generalized AS-Filters

184

(5)

Iteration

189

(2)

Convergence

189

(1)

Finite Window Operators

189

(1)

Iteration and Idempotence

190

(1)

Activity Ordering and Center Operator

191

(5)

Activity Ordering

191

(1)

Center Operator

192

(2)

Activity-Extensive Operators

194

(2)

Self-dual Filters

196

(11)

Self-dual Operators

197

(3)

Construction of Self-dual Filters

200

(4)

References

204

(3)

Introduction to Connected Operators

207

(30)

Henk J. A. M. Heijmans

Introduction

207

(1)

Connectivity and Reconstruction

208

(3)

Connected Operators

211

(6)

Grain Operators

217

(6)

Grain Operators and Grain Criteria

223

(6)

Gray-Scale Images

229

(4)

Concluding Remarks

233

(4)

References

234

(3)

Representation and Optimization of Stack Filters

237

(44)

Jaakko T. Astola

Pauli Kuosmanen

Representation of Stack Filters

237

(14)

Definition of Stack Filters

237

(3)

Continuous Stack Filters

240

(2)

Boolean Function Representations

242

(5)

Some Particular Stack Filters

247

(1)

Median and Order Statistic Filters

247

(3)

Morphological Filters

250

(1)

Optimization of Stack Filters

251

(30)

Optimization with Constant Ideal Signal and Known Noise Distribution

253

(5)

Constraints for the Numbers Ai

258

(5)

Lattice-Theoretic Representation of the Optimization Problem

263

(2)

An Algorithm to Minimize Second-Order Central Output Moment over Self-Dual Filters under Constraints

265

(3)

The Optimal Choice of the Minimal Elements of Ω1

268

(3)

Optimization by Simulated Annealing

271

(3)

Optimization by Genetic Algorithms

274

(3)

References

277

(4)

Invariant Signals of Median and Stack Filters

281

(18)

Jaakko T. Astola

Pauli Kuosmanen

Invariants of 1-D Median and Ranked-Order Filters

281

(18)

Invariants of Two-Dimensional Median Filters

293

(3)

References

296

(3)

Binary Polynominal Transforms and Logical Correlation

299

(56)

Karen O. Egiazarian

Jaakko T. Astola

Sos S. Agaian

Introduction

299

(1)

Binary Polynomial Functions and Matrices

300

(25)

Rademacher Functions and Matrices

300

(7)

(a, b, τ)-Polynomial Functions of I-Type

307

(8)

(a, b)-Polynomial Functions of II-Type

315

(1)

Binary Polynomial Logical Functions and Matrices. Constructions Using Two Operations

316

(5)

Binary Polynomial Logical Functions and Matrices. Extensions of Dimension

321

(4)

Binary Polynomial Transforms

325

(14)

(a, b)-Polynomial Transforms of II-Type as Binary Wavelet Transforms

325

(1)

(a, b)-Polynomial Functions of I-Type as Discrete Wavelet Packet Transforms

326

(1)

Efficient Computation Algorithms

326

(13)

Logical Correlations

339

(16)

Introduction

339

(1)

Arithmetic Auto- and Cross-Correlation Functions

339

(1)

Logical Auto-and Cross-Correlation Functions

340

(1)

General Correlation Function

341

(4)

Computation of General Cross-Correlation

345

(2)

Computation of Logical Cross-Correlation Based on any Boolean Operation

347

(2)

References

349

(6)

Applications of Binary Polynomial Transforms

355

(64)

Karen O. Egiazarian

Jaakko T. Astola

Sos S. Agaian

Rusen Oktem

Binary Polynomial Transforms in Nonlinear Filtering

355

(29)

Introduction

355

(2)

Stack Filters, Threshold Boolean Filters, and Extended Threshold Boolean Filters

357

(6)

Joint Distributions of Stack Filters

363

(12)

Selection Probabilities of Stack Filters

375

(9)

Binary Polynomial Transforms in Genetic Algorithms

384

(10)

Introduction

384

(1)

The Schema Theorem and the Walsh-Schema Transform

385

(2)

Average Fitness Transform Matrix and Cost Vector

387

(1)

Rectangular Wavelet Packets and Fitness Average Matrices

388

(4)

Rectangular Wavelet Packets and Fitness Average Cost Vectors

392

(2)

Binary Polynomial Transforms and Classification Problem

394

(13)

Classification using Generalized Tests

394

(10)

Evolutionary Heuristic Approach to Generalized Tests

404

(1)

Classification by Descriptors

405

(2)

Binary Polynomial Transforms in Compression of Binary Images

407

(12)

References

411

(8)

Random Sets in View of Image Filtering Applications

419

(30)

Ilya S. Molchanov

Sets Are Becoming Random

419

(3)

Capacities and Distributions

422

(3)

Averaging

425

(8)

Aumann Expectation

429

(1)

Doss Expectation

430

(1)

Radius-Vector Expectation

430

(1)

Fixed Points and Quantiles

430

(1)

Vorob'ev Expectation

430

(1)

Distance Average

431

(2)

Models or Priors

433

(4)

The Boolean Model

437

(5)

Distance between Distributions of Random Sets

442

(7)

References

444

(5)

Index

449

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Nonlinear Filters for Image Processing

Summary

Author Biography

Table of Contents

Supplemental Materials