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I • • •

• • • PRENTICE HALL INFORMATION AND SYSTEM SCIENCES SERIES Thomas Kailath, Editor • • • • ANDERSON & MOORE ANDERSON & MOORE ASTROM & WITIENMARK DICKINSON GARDNER GooDWIN & SIN GRAY & DAVISSON HAYKIN JAIN JOHNSON KAILATH KUMAR & VARAIYA Optimal Control: Linear Quadratic Methods Optimal Filtering Computer-Controlled Systems: Theory and Design, liE Systems: Analysis, Design and Computation Statistical Spectral Analysis: A Nonprobabilistic Theory Adaptive Filtering. Prediction. and Control Random Processes: A Mathematical Approach jar Engineers Adaptive Filter Theory. 21E Fundamentals of Digital Image Processing Lectures on Adaptive Parameter Estimation Linear Systems Stochastic Systems: Estimation, Indentification, and Adaptive • Control KUNG KUNG, WHITEHOUSE, & KAlLATH, EDS. KWAKERNAAK & SIVAN LANDAU L!UNG MACOVSKI MIDDLETON & GOODWIN NARENDRA & ANNASWAMY SASTRY & BODSON SOLIMAN & SRINATH SPILKER WILLIAMS • VLSI Array Processors VLSI and Modern Signal Processing • • Modem Signals and Systems System Identification and Control Design Using PlM+ Scftware System Identification: Theory for the User Medical imaging Systems Digital Control and Estimation: A Unified Approach Stable Adaptive Systems • Adaptive Control: Stability. Convergence, and Robustness Continuous and Discrete Signals and Systems Digital Communications by Sate/lite Designing Digital Filters . , •

• • • •• • Fundamentals of Digital Image Processing • . . • . , ANIL K. JAIN University of California, Davis • • • ., , - • • , • \ '. • c \ \ • • PRENTICE HALL, Englewood.Cliffs; NJ 07632 • , ,, i • • \ •

• Library of Congress Catulogtng-in-Publicmion Data • JAiN, Asn, K. Fundamentals of digital image processing. • Bibliography; p, Includes index. I. Image processing-e-Digital techniques. 1:1\1632.J35 ISBN 0-13-336165-9 1989 621.36'7 I. T.de. 88-U624 • • • • Editorial/production supervision: Colleen Brosnan Manufacturing buyer: Mary Noonan • Page layout: Martin Behan Cover design: Diane Saxe Logo design: A.M. Bruckstein Cover an: Halley's comet image by the author reconstructed from data gathered by NASA:$ Pioneer Venus Orbiter in 1986. e1989 bY Prentice-Hall, Inc. A Paramount Communications Company Englewood Cliffs, New Jersey 07632 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Printed in the United States of America 10 9 • • • ISBN 0-13-33b1b5-9 PRENTICE·HALL INTERIolATIONAL (UK) LIMITED, London PRENTICE-HALL OF AUSTRALIA PTY. LIMITED, Sydney PRE1'o'TICE·HALL CANADA INC., Toronto PRENTICE-HALL HISPANOAlI-IERICANA. S.A., Mexico PRENTICE-HALL OF INDIA PRIVATE LIMITED, New Delhi PREl'o'T!CE·HALL OF JAPAN, INC., Tokyo SIMON & SCHUSTER AsIA PTE. LTD., Singapore EDlTORA PRENTICE-HALL DO BRASIL, LTDA., Rio de Janeiro • 1 • • •

_ - - • • • 1ill'~ Contents • , PREFACE ACKNOWLEDGMENTS 1 INTRODUCTION 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Digital Image Processing: Problems and Applications 1 . Image Representation and Modeling 4 ' Image Enhancement 6 Image Restoration 7, Image Analysis 7 , Image Reconstruction from Projections 8 Image Data Compression, 9 Bibliography , 10 xix • XXI 1 •\, • • " 2 . / • TWO·DIMENSIONAL SYSTEMS AND MATHEMATICAL PRELIMINARIES • ; 2:1 2.2 2.3 2.4 • • Introduction 11 Notation and Definitions 11 Linear Systems and Shift Invariance 13 .. , ' , ' " The Fourier Transform 15 Properties of the Fourier Transform, 16 Fourier Transform of Sequences (Fourier ' Series), 18 . , , 11' • •

" 2.5 2.6 2.7 • 2.8 2.9 , 2.10 . 2.11 2.12 2.13 The Z·Transform or Laurent Series 20 Causality and Stability, 21 Optical and Modulation Transfer Functions 21 22 Matrix Theory Results VeL/ors and Matrices, 22 Rowand Column Ordering, 23 Transposition and Conjugation Rules, 25 Toeplitz and Circulant Matrices, 25 Orthogonal and Unitary Matrices, 26 Positive Definiteness and Quodralic Forms, 27 Diagonal Forms, 27 Block Matrices and Kronecker Products 28 , Block Matrices, 28 Kronecker Products, 30 " Separable Operations, 31 • Random Signals 31 Definitions, 31 Gaussian or Normal Distribution, 32 Gaussian Random Processes, 32 Stationary Processes, 32 Markov Processes, 33 Orthogonality and Independence, 34' The Karhunen Loeve (KL) Transform, 34 D ' Definitions. 35 "Separable and Isotropic Covariance iscrete cancom ~-lelU> T" "'_ 35 R rl Functions, 36 The Spectral Density Function 37 Properties of the SDF, 38 Some Results from Estimation Theory 39 Mean Square Estimates, 40 The Orthogonality Principle, 40 Some Results from Information Theory 41 Information, 42 Entropy, 42 The Rate Distortion Function, 43 Problems 44 Bibliography 47 • I • •, 3 IMAGE PERCEpTION • 3.1 3.2 Introduction 49 Light, Luminance, Brightness, and Contrast 49 Simultaneous Contrast, 51 Mach Bands, 53 • • • • • "49 • • • vi Contents .

3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 MTF of the Visual System 54 The Visibility Function 55 Monochrome Vision Models 56 Image Fidelity Criteria 57 60 Color Representation Color Matching and Reproduction 62 Laws of Color Matching, 63 Chromaticity Diagram, 65 Color Coordinate Systems 66 Color Difference Measures 71 Color Vision Model 73 Temporal Properties of Vision 75 Bloch's Law, 75 Critical Fusion Frequency (CFF), 75 Spatial versus Temporal Effects, 75 Problems 76 • Bibliography 78 • i 4 IMAGE SAMPLING AND QUANTIZATION • , .. • ., 80 •\ • • , . 4.1 i 4.2 -• , , • ;, • 4.3 Contents - 80 Introduction Image Scanning, 80 Television Standards, 81 Image Display and Recording, 83 • Two-Dimensional Sampling Theory 84 . Bandlimited Images, -84 Sampling Versus Replication, 85 Reconstruction of the Image from Its Samples, 85 Nyquist Rate, Aliasing, and Foldover Frequencies, 87 Sampling Theorem, 88 Remarks,89 • Extensions of Sampling Theory 89 Sampling Random Fields, 90 Sampling Theorem for Random Fields, 90 Remarks,90 Nonrectangular Grid Sampling and Interlacing, 91 Hexagonal Sampling, 92 Optima! Sampling, 92 • • • ., VII

Practical Limitations in Saznpling and Reconstruction 93 Sampling Aperture, 93 Display Aperturcllnterpolation Function, 94 Lagrange Interpolation, 98 Moire Effect and Flat Field Response, 99 . • Image Quantization 99 The Optimum Mean Square or Lloyd-Max Quantizer 101 The Uniform Optimal Quantizer, 103 Properties of the Optimum Mean Square Quantize~ '103 . Proof', 112 A Compandor Design Remarks, 114 113 • The Optimum Mean Square Uniform Quantizer for Nonuniform Densities 115 , Examples, Comparison, and Practical Limitations 115 Analytic Models for Practical Quantizers 118 Quantization of 'Complex Gaussian Random Variables 119 4.4 4.5 4.6 • 4.7 4.8 4.9 4.10 4.11 4.12 6 IMAGE TRANSFORMS 5.1 5.2 5.3 • viii Introduction 132 Two-Dimensional Orthogonal and Unitary Transforms Separable Unitary Transforms, 134 Basis Images, 135 Kronecker Products and Dimensionality, 137 Dimensionality of Image Transforms, 138 Transform Frequency, 138 Optimum Transform, 138 , 134 \ • ; Properties of Unitary Transforms Energy Conservation and Rotation, 138 Energy Compaction and Variances Qj Transform 138 Coefficients, 139 Decorrelation. 140 Other Properties, ]40 • • Contents 119 Visual Quantization Contrast Quantization, 120 Pseudorandom Noise Quantization, 120 Halftone Image Generation, 121 Color Quantization, 122 • •, Problems 124 Bibliography 128 • . , • • 132

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