Nonlinear Filters for Image Processing

by Dougherty, Edward R.; Astola, Jaakko

• ISBN13: 9780819430335
• ISBN10: 0819430331
• Format: Hardcover
• Copyright: 1999-03-01
• Publisher: Society of Photo Optical

Summary

Part of the SPIE/IEEE Series on Imaging Science & Engineering. 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 presentation is rigorous, yet accessible to engineers with a solid background in mathematics.

Table of Contents

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