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."

Author Notes

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.

1 Logical Image Operators	p. 1
1.1 Boolean Functions	p. 1
1.2 Morphological Representation	p. 5
1.3 System Model	p. 10
1.4 Optimal W-Operators	p. 12
1.5 Estimation of Optimal W-Operators	p. 18
1.6 Design Procedure	p. 20
1.7 Constrained Optimization	p. 29
1.8 Optimal Increasing Filters	p. 34
1.9 Iterative Filters	p. 38
1.10 Machine Learning Theory and Optimal Operator Design	p. 47
1.11 Robustness	p. 52
References	p. 60
2 Computational Gray-Scale Operators	p. 63
2.1 Computational Functions	p. 63
2.2 Representation of Increasing Computational Functions	p. 67
2.3 Flat Computational Functions	p. 71
2.4 Increasing Gray-to-Binary Image Operators	p. 73
2.5 Representation of Increasing Gray-to-Gray Image Operators	p. 76
2.6 Stack Filters	p. 77
2.7 Nonflat Erosion	p. 82
2.8 Representation of Generic Computational Functions	p. 83
2.9 Representation of Generic Gray-to-Gray Image Operators	p. 87
2.10 Optimal Gray-Scale Computational Filters	p. 90
2.11 Application: Quantization Range Conversion	p. 92
2.12 Comments on Gray-Scale Morphology	p. 94
References	p. 98
3 Translation-Invariant Set Operators	p. 101
3.1 Translation-Invariant Operators	p. 101
3.2 Representation of Increasing Translation-Invariant Operators	p. 106
3.3 Representation of Nonincreasing Translation-Invariant Operators	p. 109
3.4 Openings and Closings	p. 111
3.5 Representation of Openings and Closings	p. 118
3.6 Convexity	p. 120
References	p. 121
4 Granulometric Filters	p. 123
4.1 Granulometries	p. 123
4.2 Representation of Euclidean Granulometries	p. 125
4.3 Reconstructive Granulometries	p. 129
4.4 Optimal Filtering by Reconstructive Granulometries	p. 133
4.5 Adaptive Disjunctive Granulometric Filters	p. 138
4.6 Size Distributions	p. 144
4.7 Granulometric Classification	p. 147
4.8 Discrete Granulometric Bandpass Filters	p. 152
4.9 Continuous Granulometric Bandpass Filters	p. 154
References	p. 162
5 Easy Recipes for Morphological Filters	p. 165
5.1 Introduction	p. 165
5.2 Morphology on Complete Lattices	p. 167
5.2.1 Basic Theory	p. 167
5.2.2 Application to Gray-Scale Functions	p. 170
5.3 Openings and Closings	p. 174
5.3.1 Basic Facts	p. 174
5.3.2 Annular Opening	p. 176
5.3.3 Adjunctional Filters	p. 176
5.4 Annular Filters	p. 177
5.4.1 Annular Filters for Binary Images	p. 177
5.4.2 Annular Filters for Gray-Scale Images	p. 179
5.5 AS-Filters	p. 180
5.6 Overfilters and Inf-overfilters	p. 181
5.6.1 Definitions and Basic Properties	p. 182
5.6.2 Rank-max Openings	p. 184
5.7 Generalized AS-Filters	p. 186
5.8 Iteration	p. 191
5.8.1 Convergence	p. 191
5.8.2 Finite Window Operators	p. 191
5.8.3 Iteration and Idempotence	p. 192
5.9 Activity Ordering and Center Operator	p. 193
5.9.1 Activity Ordering	p. 193
5.9.2 Center Operator	p. 194
5.9.3 Activity-Extensive Operators	p. 196
5.10 Self-dual Filters	p. 198
5.10.1 Self-dual Operators	p. 199
5.10.2 Construction of Self-dual Filters	p. 202
References	p. 206
6 Introduction to Connected Operators	p. 209
6.1 Introduction	p. 209
6.2 Connectivity and Reconstruction	p. 210
6.3 Connected Operators	p. 213
6.4 Grain Operators	p. 219
6.5 Grain Operators and Grain Criteria	p. 225
6.6 Gray-Scale Images	p. 231
6.7 Concluding Remarks	p. 235
References	p. 236
7 Representation and Optimization of Stack Filters	p. 239
7.1 Representation of Stack Filters	p. 239
7.1.1 Definition of Stack Filters	p. 239
7.1.2 Continuous Stack Filters	p. 242
7.1.3 Boolean Function Representations	p. 244
7.1.4 Some Particular Stack Filters	p. 249
7.1.4.1 Median and Order Statistic Filters	p. 249
7.1.4.2 Morphological Filters	p. 252
7.2 Optimization of Stack Filters	p. 253
7.2.1 Optimization with Constant Ideal Signal and Known Noise Distribution	p. 255
7.2.1.1 Constraints for the Numbers Ai	p. 260
7.2.1.2 Lattice-Theoretic Representation of the Optimization Problem	p. 265
7.2.1.3 An Algorithm to Minimize Second-Order Central Output Moment over Self-Dual Filters under Constraints	p. 267
7.2.1.4 The Optimal Choice of the Minimal Elements of O1	p. 270
7.2.2 Optimization by Simulated Annealing	p. 273
7.2.3 Optimization by Genetic Algorithms	p. 276
References	p. 279
8 Invariant Signals of Median and Stack Filters	p. 283
8.1 Invariants of 1-D Median and Ranked-Order Filters	p. 283
8.1.1 Invariants of Two-Dimensional Median Filters	p. 295
References	p. 298
9 Binary Polynomial Transforms and Logical Correlation	p. 301
9.1 Introduction	p. 301
9.2 Binary Polynomial Functions and Matrices	p. 302
9.2.1 Rademacher Functions and Matrices	p. 302
9.2.2 (a, b; t)-Polynomial Functions of I-Type	p. 309
9.2.3 (a,b)-Polynomial Functions of II-Type	p. 317
9.2.4 Binary Polynomial Logical Functions and Matrices. Constructions Using Two Operations	p. 318
9.2.5 Binary Polynomial Logical Functions and Matrices. Extensions of Dimension	p. 323
9.3 Binary Polynomial Transforms	p. 327
9.3.1 (a,b)-Polynomial Transforms of II-Type as Binary Wavelet Transforms	p. 327
9.3.2 (a,b)-Polynomial Functions of I-Type as Discrete Wavelet Packet Transforms	p. 328
9.3.3 Efficient Computation Algorithms	p. 328
9.4 Logical Correlations	p. 341
9.4.1 Introduction	p. 341
9.4.2 Arithmetic Auto- and Cross-Correlation Functions	p. 341
9.4.3 Logical Auto- and Cross-Correlation Functions	p. 342
9.4.4 General Correlation Function	p. 343
9.4.5 Computation of General Cross-Correlation	p. 347
9.4.6 Computation of Logical Cross-Correlation Based on any Boolean Operation	p. 349
References	p. 351
10 Applications of Binary Polynomial Transforms	p. 357
10.1 Binary Polynomial Transforms in Nonlinear Filtering	p. 357
10.1.1 Introduction	p. 357
10.1.2 Stack Filters, Threshold Boolean Filters, and Extended Threshold Boolean Filters	p. 359
10.1.3 Joint Distributions of Stack Filters	p. 365
10.1.4 Selection Probabilities of Stack Filters	p. 377
10.2 Binary Polynomial Transforms in Genetic Algorithms	p. 386
10.2.1 Introduction	p. 386
10.2.2 The Schema Theorem and the Walsh-Schema Transform	p. 387
10.2.3 Average Fitness Transform Matrix and Cost Vector	p. 389
10.2.4 Rectangular Wavelet Packets and Fitness Average \ Matrices	p. 390
10.2.5 Rectangular Wavelet Packets and Fitness Average Cost Vectors	p. 394
10.3 Binary Polynomial Transforms and Classification Problem	p. 396
10.3.1 Classification using Generalized Tests	p. 396
10.3.2 Evolutionary Heuristic Approach to Generalized Tests	p. 406
10.3.3 Classification by Descriptors	p. 407
10.4 Binary Polynomial Transforms in Compression of Binary Images	p. 409
References	p. 413
11 Random Sets in View of Image Filtering Applications	p. 421
11.1 Sets Are Becoming Random	p. 421
11.2 Capacities and Distributions	p. 424
11.3 Averaging	p. 427
11.3.1 Aumann Expectation	p. 431
11.3.2 Doss Expectation	p. 432
11.3.3 Radius-Vector Expectation	p. 432
11.3.4 Fixed Points and Quantiles	p. 432
11.3.5 Vorob'ev Expectation	p. 432
11.3.6 Distance Average	p. 433
11.4 Models or Priors	p. 435
11.5 The Boolean Model	p. 439
11.6 Distance between Distributions of Random Sets	p. 444
References	p. 446

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