Analysis of variance in statistical image processing Analysis of variance in statistical image processing LUDWIK KURZ and M. HAFED BENTEFTIFA Polytechnic University CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521581820 © Ludwik Kurz and M. Hafed Benteftifa 1997 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1997 This digitally printed first paperback version 2006 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Kurz, Ludwik. Analysis of variance in statistical image processing / Ludwik Kurz, M. Hafed Benteftifa. p. cm. ISBN 0-521-58182-6 (hardback) 1. Image processing—Statistical methods. 2. Analysis of variance. I. Benteftifa, M. Hafed. II. Title. TA1637.K87 1997 96-22310 621.36'7'01519538—dc20 CIP ISBN-13 978-0-521-58182-0 hardback ISBN-10 0-521-58182-6 hardback ISBN-13 978-0-521-03196-7 paperback ISBN-10 0-521-03196-6 paperback To our families and Moncef in particular CONTENTS Preface xi 1 Introduction 1 1.1 Introductory remarks 1 1.2 Decision mechanism in image processing 3 1.2.1 Organization of the book 4 2 Statistical linear models 7 2.1 Introductory remarks 7 2.2 Linear models 8 2.2.1 Parameter estimation 8 2.2.2 Linear hypothesis testing 10 2.3 One-way designs 11 2.4 Two-way designs 14 2.5 Incomplete designs 17 2.5.1 Latin square design 17 2.5.2 Graeco-Latin square design 21 2.5.3 Incomplete block design 24 2.6 Contrast functions 29 2.6.1 Multiple comparisons techniques 30 2.6.2 Selection of a comparison method 31 2.6.3 Estimable functions 32 2.6.4 Confidence ellipsoids and contrasts 33 2.7 Concluding remarks 34 3 Line detection 35 3.1 Introductory remarks 35 3.2 Unidirectional line detectors 36 3.3 Bidirectional line detectors 39 3.4 Multidirectional line detectors 44 3.5 Multidirectional contrast detectors 46 vn viii Contents 3.6 Multidirectional detectors in correlated noise 50 3.7 Trajectory detection 55 3.7.1 Unidirectional detectors 55 3.7.2 Bidirectional trajectory detectors 60 3.8 Multidirectional adaptive detectors 62 3.9 Concluding remarks 65 4 Edge detection 66 4.1 Introductory remarks 66 4.2 Edge detection methodology 67 4.3 Unidirectional edge detectors 68 4.4 Bidirectional edge detectors 70 4.5 Multidirectional edge detectors 71 4.5.1 Latin square-based detector 72 4.5.2 Graeco-Latin square-based detector 76 4.6 Multidirectional detection in correlated noise 78 4.6.1 Sum of squares under Q 78 4.6.2 Sum of squares under the hypotheses 80 4.7 Edge reconstruction 81 4.7.1 Methodology 81 4.7.2 Gradient method 82 4.8 Concluding remarks 86 5 Object detection 90 5.1 Introductory remarks 90 5.2 Detection methodology 92 5.3 Transformation-based object detector 94 5.3.1 Uncorrelated data case 94 5.3.2 Correlated data case 96 5.3.3 Sum of squares under Q 97 5.3.4 Sum of squares under a) = H D Q 98 a a 5.3.5 Sum of squares under co = H n Q 99 b b 5.3.6 Determination of background and target correlation matrices 99 5.3.7 Background and target correlation matrices 100 5.3.8 Structure of the permutation matrices 101 5.3.9 A reduced algorithm 101 5.4 Partition-based object detector 104 5.4.1 Image representation 106 5.5 Basic regions and linear contrasts 108 5.5.1 Histogram approach 108 5.5.2 Linear contrasts 109 5.6 Detection procedure 109 5.6.1 Contrast estimate 110 5.6.2 Threshold selection 111 5.6.3 Contrast detection 111 Contents ix 5.7 Orthogonal contrasts—an extension 112 5.8 Form-invariant object detector 118 5.8.1 Invariant representation 119 5.9 Concluding remarks 124 A.I Contrast calculation 126 A.2 Contrast variance 126 Image segmentation 128 6.1 Introduction 128 6.2 Segmentation strategy 129 6.3 Nested design model 132 6.3.1 Gray level images 133 6.4 Logical predicates 135 6.5 Adaptive class formation 138 6.5.1 Test outcomes 139 6.5.2 Merging strategies 142 6.6 Concluding remarks 143 Radial masks in line and edge detection 147 7.1 Introductory remarks 147 7.2 Radial masks in one-way ANOVA design 147 7.3 Boundary detection procedure 149 7.4 Contrast-based detectors using radial masks 151 7.5 Power calculation 152 7.6 Concluding remarks 155 Performance analysis 156 8.1 Stochastic approximation in parameter estimation 156 8.1.1 Remarks 156 8.1.2 Stochastic approximation versus classical estimation 157 8.2 Stochastic approximation procedures 159 8.2.1 Deterministic case 159 8.2.2 Stochastic case 159 8.2.3 The scalar version of the Robbins-Monro procedure 162 8.2.4 The Kiefer-Wolfowitz procedure 163 8.3 Stochastic approximation in least square estimation 165 8.3.1 Scalar case 166 8.3.2 Vector case 167 8.3.3 Small sample theory 167 8.4 Robust recursive estimation 169 8.4.1 Rank statistic preprocessor 170 8.4.2 Huber's M-estimator 172 8.4.3 Robust SAMVLS 173 8.5 An illustrative example 174 8.6 Power calculations 176 Contents 8.6.1 F-statistic based detectors 176 8.6.2 Contrast-based detectors 178 8.7 Concluding remarks 179 Some approaches to image restoration 181 9.1 Introductory remarks 181 9.2 Edge detection preprocessors 181 9.3 Image restoration using linear regressions 186 9.4 Suppression of salt and pepper noise 189 9.5 Some results 192 9.6 Concluding remarks 196 References 197 Index 202