Nonlinear image processing

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Nonlinear Image Processing Nonlinear Image Processing Academic Press Series in Communications, Networking, and Multimedia EDITOR-IN-CHIEF Jerry D. Gibson Southern Methodist University This series has been established to bring together a variety of publications that represent the latest in cutting-edge research, theory, and applications of modern communication systems. All traditional and modern aspects of communications as well as all methods of computer communications are to be included. The se- ries will include professional handbooks, books on communication methods and standards, and research books for engineers and managers in the world-wide com- munications industry. Books in the Series: Handbook of Image and Video Processing, A1 Bovik, editor The E-Commerce Book, Steffano Korper and Juanita Ellis Multimedia Communications, Jerry Gibson, editor Nonlinear Image Processing, Sanjit K. Mitra and Giovanni L. Sicuranza, editors Nonlinear Image Pr O C e S S ing EDITORS SANJIT K. MITRA University of California Santa Barbara, California, USA GIOVANNI L. SICURANZA University of Trieste Trieste, Italy ACADEMIC PRESS A Harcourt Science and Technology Company SAN DIEGO / SAN FRANCISCO / NEW YORK / BOSTON / LONDON / SYDNEY /TOKYO This book is printed on acid-free paper. Copyright 9 2001 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any in- formation storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida, 32887-6777. ACADEMIC PRESS A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London, NWl 7BY, UK Library of Congress Catalog Number: 00-104376 ISBN: 0-12-500451-6 Printed in the United States of America 000102030405HP987654321 Contents Preface ix Analysis and Optimization of Weighted Order Statistic and Stack Filters 1 S. Peltonen, P. Kuosmanen, K. Egiazarian, M. Gabbouj, and J. Astola 1.1 Introduction 1 1.2 Median and Order Statistic Filters 1.3 Stack Filters 2 1.4 Image Processing Applications 21 1.5 Summary 22 Image Enhancement and Analysis with Weighted Medians G. Arce and J. Paredes 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Introduction 27 Weighted Median Smoothers and Filters 28 Image Denoising 45 Image Zooming 49 Image Sharpening 52 Optimal Frequency Selection WM Filtering Edge Detection 62 Conclusion 65 58 Spatial-Rank Order Selection Filters K. Barner and R. Hardie 3.1 Introduction 69 3.2 Selection Filters and Spatial-Rank Ordering 69 72 27 Vi CONTENTS 3.3 Spatial-Rank Order Selection Filters 3.4 Optimization 94 3.5 Applications 96 3.6 Future Directions 105 81 4 Signal-Dependent Rank-Ordered-Mean (SD-ROM) Filter E. Abreu 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Introduction 111 Impulse Noise Model 112 Definitions 113 The SD-ROM Filter 114 Generalized SD-ROM Method 116 Experimental Results 121 Restoration of Images Corrupted by Streaks Concluding Remarks 131 126 5 Nonlinear Mean Filters and Their Applications 6 in Image Filtering and Edge Detection C. Kotropoulos, M. Pappas, and I. Pitas 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 135 Introduction 135 Nonlinear Mean Filters 136 Signal-Dependent Noise Filtering by Nonlinear Means 140 Edge Detectors Based on Nonlinear Means 141 Grayscale Morphology Using s Mean Filters 142 Ultrasonic Image Processing Using s Mean Filters 147 Sorting Networks Using s Mean Comparators 157 Edge Preserving Filtering by Combining Nonlinear Means and Order Statistics 160 Summary 163 Two-Dimensional Teager Filters S. Thurnhofer 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 167 Introduction 167 Discrete Volterra Series and Properties 167 Interpretation of Frequency Responses 171 The Teager Algorithm and One-Dimensional Extensions Spectrum of the Output Signal 177 Mean-Weighted Highpass Filters 179 Least-Squares Design of Edge Extracting Filters 185 Summary 194 Appendix 195 172 111 CONTENTS vii 8 Polynomial and Rational Operators for Image Processing and Analysis 203 G. Ramponi 7.1 7.2 7.3 7.4 7.5 Introduction 203 Theoretical Survey of Polynomial and Rational Filters Applications of Polynomial Filters 208 Applications of Rational Filters 214 Conclusions and Remaining Issues 221 204 Nonlinear Partial Differential Equations in Image Processing 225 G. Sapiro 8.1 Introduction 225 8.2 Segmentation of Scalar and Multivalued Images 8.3 228 Nonlinear PDEs in General Manifolds: Harmonic Maps and Direction Diffusion 235 9 Region-Based Filtering of Images and Video Sequences: A Morphological Viewpoint P. Salembier 9.1 9.2 9.3 9.4 9.5 9.6 249 Introduction 249 Classical Filtering Approaches 251 Connected Operators 254 Connected Operators Based on Reconstruction Processes Connected Operators Based on Region-Tree Pruning 264 Conclusions 283 256 10 Differential Morphology P. Maragos 10.1 10.2 10.3 10.4 10.5 10.6 10.7 289 Introduction 289 2D Morphological Systems and Slope Transforms PDEs for Morphological Image Analysis 300 Curve Evolution 308 Distance Transforms 310 Eikonal PDE and Distance Propagation 318 Conclusions 323 294 11 Coordinate Logic Filters: Theory and Applications in Image Analysis 331 B. Mertzios and K. Tsirikolias 11.1 Introduction 331 V~ CONTENTS 11.2 11.3 11.4 11.5 11.6 11.7 Coordinate Logic Operations on Digital Signals 333 Derivation of the Coordinate Logic Filters 337 Properties of Coordinate Logic Filters 339 Morphological Filtering Using Coordinate Logic Operations on Quantized Images 340 Image Analysis and Pattern Recognition Applications Concluding Remarks 352 342 12 Nonlinear Filters Based on Fuzzy Models F. Russo 12.1 12.2 12.3 12.4 12.5 12.6 Introduction 355 FuzzyModels 356 Fuzzy Weighted Mean (FWM) Filters 359 FIRE Filters 363 Evolutionary Neural Fuzzy Filters: A Case Study Concluding Remarks and Future Trends 372 355 366 13 Digital Halftoning 375 D. Lau and G. Arce 13.1 Introduction 375 13.2 Halftone Statistics 381 13.3 Blue-Noise Dithering 385 13.4 Green-Noise Dithering 390 13.5 Conclusions 398 14 Intrinsic Dimensionality: Nonlinear Image Operators and Higher-Order Statistics 403 C. Zetzsche and G. Krieger 14.1 14.2 14.3 14.4 14.5 14.6 Introduction 403 Transdisciplinary Relevance of Imrinsic Dimensionality i2D-Selective Nonlinear Operators 413 Frequency Design Methods for i2D Operators 421 i2D Operators and Higher-Order Statistics 432 Discussion 437 406 Index 449 Preface In recent years, nonlinear methods and techniques have emerged as intensive re- search topics in the fields of signal and image processing. The reason for this increased interest in nonlinear methods of image processing is mainly due to the following observations. First, the human visual system (HVS) includes some non- linear effects that need to be considered in order to develop effective image pro- cessing algorithms. Therefore, to comply with the characteristics of the HVS and, thus, obtain better visual results, nonlinear algorithms are necessary. Moreover, the nonlinear behavior of optical imaging systems and their related image for- mation systems must be taken into account. Finally, images are signals that in general do not satisfy the widely used hypotheses of Gaussianity and stationarity that are usually assumed to validate linear models and filtering techniques. In this respect, it is well known, for example, that linear filters are not able to remove an impulsive noise superimposed on an image without blurring its edges and small details. Other situations in which linear filters perform poorly are those cases where signal-dependent or multiplicative noises are present in the images. Although linear filters continue to play an important role in signal processing because they are inherently simple to implement, the advances of computers and digital signal processors, in terms of speed, size, and cost, make the implementa- tion of more sophisticated algorithms practical and effective. These considerations are the basis for the increased interest in the develop- ment of new nonlinear techniques for image processing, with particular emphasis on the applications that benefit greatly from a nonlinear approach, such as edge preserving smoothing, edge enhancement, noise filtering, image segmentation, and feature extraction. An interesting aspect of the recent studies on nonlinear image processing is the fact that an attempt has been made to organize the previously scattered con- tributions in a few homogeneous sectors. While a common framework is far from being derived (or it is simply out of reach since nonlinearity is defined as the lack ix [...]... uncertainty that is typical of some image processing issues; and 9 Nonlinear operators modeled in terms of nonlinear partial differential equations (PDEs) All of these filter families are considered in this book, but with a different stress according to their popularity and impact on image processing tasks Another relevant aspect that constitutes at present a trend in the area of nonlinear filters is the search... morphology is the topic presented in Chapter 10 Morphological image processing has traditionally been based on modeling images as sets or as PREFACE xiii points in a complete lattice of functions and viewing morphological image transformations as set or lattice operations In parallel, there is a recently growing part of morphological image processing that is based on ideas from differential calculus... tool for processing gray-level images as a set of binary images In fact, these filters are coincident with morphological filters for binary images, while maintaining a similar functionality for gray-level images The remarkable advantage of coordinate logic filters is that their simplicity allows for very fast implementations because sorting operations are not required Typical applications for image enhancement... aspects of biological image processing conclude the chapter As might be clear from the discussion of its contents, our objective in editing this book has been to present both an overview of the state of the art and an exposition of some recent advances in the area of nonlinear image processing We have attempted to present a comprehensive description of the most relevant classes of nonlinear filters, even... partial/full/extended spatial and rank ordering information The developed filters are applied to various image processing tasks, such as noise smoothing, interpolation, and image restoration Chapter 4 contains another example of a combination of different kinds of operations to derive new sets of nonlinear filters In fact, the signal-dependent rank-ordered-mean filters exploit the rank ordering properties... Chapter 2 deals with an extended class of nonlinear filters derived from the median operator, that is, the local weighted median filter After reviewing the principles of weighted medians, smoothers allowing positive as well as negative weights are introduced These nonlinear tools are applied to image enhancement and analysis, with specific applications to image denoising and sharpening, zooming, and... particular, the rank-ordered differences provide information about the likelihood of corruption for the current pixel The resulting nonlinear algorithms are particularly efficient to remove impulse noises from highly corrupted images while preserving details and features Nonlinear mean filters, described in Chapter 5, can be considered as another alternative to median filters and their extension to remove... morphological filters, which are shown to be useful for removal of both Rayleigh and signal-dependent Gaussian speckle noises that usually affect ultrasonic images xii PREFACE Chapters 6 and 7 deal with polynomial filters and their application to image processing tasks The interest in such filters is mainly due to the fact that they can be considered to be the most natural extension of linear operators... the various classes of nonlinear operators while reducing their drawbacks This result can be achieved by combining different information to define new filter classes, as shown for example in some of the contributions contained in this book by the joint use of spatial and rank ordering information An alternative approach, especially useful for the solution of well-defined image processing tasks, is based...X PREFACE of a property, that is, linearity), suitable classes of nonlinear operators have been introduced A (not exhaustive) list of these classes includes: 9 Homomorphic filters, relying on a generalized superposition principle; 9 Nonlinear mean filters, using nonlinear definitions of means; 9 Morphological filters, based on geometrical rather than analytical properties; . Nonlinear Image Processing Nonlinear Image Processing Academic Press Series in Communications, Networking, and Multimedia. Handbook of Image and Video Processing, A1 Bovik, editor The E-Commerce Book, Steffano Korper and Juanita Ellis Multimedia Communications, Jerry Gibson, editor Nonlinear Image Processing, . Issues 221 204 Nonlinear Partial Differential Equations in Image Processing 225 G. Sapiro 8.1 Introduction 225 8.2 Segmentation of Scalar and Multivalued Images 8.3 228 Nonlinear PDEs