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Two -Dimensional Digital Signal Processing II Transforms and Median Filters Edited by .T .S Huang With Contributions by J.-O. Eklundh T.S. Huang B.I. Justusson H. .J Nussbaumer S.G. Tyan .S Zohar With 94 serugiF galreV-regnirpS Berlin Heidelberg New York 1891 Professor Thomas .S Huan9, Ph.D. tnemtrapeD of lacirtcelE gnireenignE dna detanidrooC Science ,yrotarobaL ytisrevinU of sionillI ,anabrU LI ,10816 ASU ISBN 3-540-10359-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10359-7 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in Publication Data. Main entry under title: Two-dimensional digital signal processing II. (Topics in applied physics; v. 43) Bibliography: p. Includes index. 1. Image processing-Digital techniques. 2. Digital f'dters (Mathematics) 3. Transformations (Mathematics) I. Huaug, Thomas S., 1936 TA1632.T9 621.38'0433 80-24775 This work is subject to copyright. All rightsa re reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under }{ 54 of the German Copyright Law, where copies are made for other than private use, a fee is payable to 'Verwertungsgesellschaft Wort', Munich. © by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement° that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Monophoto typesetting, offset printing and bookbinding: Brtihlsche Universit~itsdruckerei, Giessen 2153/3130-543210 Preface Activities in digital image processing have been increasing rapidly in the past decade. This is not surprising when one realizes that in a broad sense image processing means the processing of multidimensional signals and that most signals in the real world are multidimensional. In fact, the one-dimensional signals we work with are often collapsed versions of multidimensional signals. For example, speech is often considered as a one-dimensional signal, viz., a function of a single variable (time). However, speech originally exists in space and therefore is a function of 4 variables (3 spatial variables and time). There are analog (optical, electro-optical) as well as digital techniques for image processing. Because of the inherent advantages in digital techniques (flexibility, accuracy), and because of the rapid progress in computer and related technologies, such as LSI, and VLSI, it is fair to say that except for some specialized problems, digital techniques are usually preferred. The purpose of this book and its companion (Two-Dimensional Digital Signal Processing I: Linear Filters) is to provide in-depth treatments of three of the most important classes of digital techniques for solving image processing problems: Linear filters, transforms, and median filtering. These two books are related but can be used independently. In the earlier volume 6 of this series, Picture Processin 9 and Digital Filtering (first edition, 1975), selected topics in two-dimensional digital signal processing including transforms, filterd esign, and image restoration, were treated in depth. Since then, a tremendous amount of progress has been made in these areas. In 1978 when we were planning on a second edition of that book (published in 1979), a decision was made not to make significant revisions but only to add a brief new chapter surveying the more recent results. And we projected that in- depth treatments of some of the important new results would appear in future volumes of the Springer physics program. These two present books on two-dimensional digital signal processing represent the first two of these projected volumes. The material is divided into three parts. In thef irst part on linear filters, which is contained in the companion volume, major recent results in the design of two-dimensional nonrecursive and recursive filters, stability testing, and Kalman filtering (with applications to image enhancement and restoration) are presented. Among the highlights are thed iscussions on the design and stability testing of half-plane recursive filters, a topic of great current interest. IV ecaferP The second and third parts are contained in thisv olume. In the second part on transforms, two topics are discussed: algorithms for transposing large matrices, and number-theoretic techniques in transforms and convolution. Here we have a detailed derivation of the Winograd Fourier transform algorithm. In the first and the second parts, the main concern si linear processing. In the third part on median filtering, a particular nonlinear processing technique si studied. Median filtering has become rather popular in image and speech processing. However, published results on it have been scarce. Thet wo chapters of the third part contain new results most of which are published here for the first time. The chapters int his volume are tutorial in nature, yet they bring the readers to the very forefront of current research. It will be useful as a reference book for working scientists and engineers, and as a supplementary textbook in regular or short courses on digital signal processing, image processing, and digital filtering. Urbana, Illinois, September 1980 Thomas S. Huang Contents 1. Introduction. By .T S. Huang (With 3 Figures) . . . . . . . . . . . 1 1.1 Transforms . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Median Filters . . . . . . . . . . . . . . . . . . . . . . 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Efficient Matrix Transposition. By J.-O. Eklundh . . . . . . . . . 9 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Methods for Transposing Externally Stored Matrices ..... 12 2.2.1 Definition of Performance Criteria . . . . . . . . . . . 21 2.2.2 A Simple Block-Transposition Method . . . . . . . . . 12 2.2.3 Transposition Using Square Partitions . . . . . . . . . 31 2.2.4 Floyd's Algorithm . . . . . . . . . . . . . . . . . . 71 2.2.5 Row-in/Column-out Transposition . . . . . . . . . . . 81 2.2.6 The Rectangular Partition Algorithm . . . . . . . . . . 20 2.3 Optimization of the Performance of the Algorithms ...... 21 2.3.1 Two Lemmas . . . . . . . . . . . . . . . . . . . . 21 2.3.2 The Square Partition Algorithm . . . . . . . . . . . . 22 2.3.3 The Rectangular Partition Algorithm . . . . . . . . . . 26 2.3.4 On the Advantages of Inserting the Factor 1 . . . . . . . 28 2.4 Optimizing the Square Partition and the Row-in/Column-out Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 An Example . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Anderson's Direct Method for Computing the FFT in Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . 32 2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 33 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3. Two-Dimensional Convolution and DFT Computation By H. J. Nussbaumer (With 8 Figures) . . . . . . . . . . . . . . 37 1.3 Convolutions and Polynomial Algebra . . . . . . . . . . . . 37 3.1.1 Residue Polynomials . . . . . . . . . . . . . . . . . 38 3.1.2 Convolution and Polynomial Product Algorithms in Polynomial Algebra . . . . . . . . . . . . . . . . . 39 3.2 Two-Dimensional Convolution Using Polynomial Transforms. 42 3.2.1 Polynomial Transforms . . . . . . . . . . . . . . . . 43 3.2.2 Composite Polynomial Transforms . . . . . . . . . . . 47 VIII Contents 3.2.3 Computation of Polynomial Transforms and Reductions 51 3.2.4 Computation of Polynomial Products and One-Dimensional Convolutions . . . . . . . . . . . . . . . . . . . . 54 3.2.5 Nesting Algorithms . . . . . . . . . . . . . . . . . 60 3.2.6 Comparison with Conventional Computation Methods . 63 3.3 Computation of Two-Dimensional DFTs by Polynomial Transforms . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.1 Reduced DFT Algorithm . . . . . . . . . . . . . . . 66 3.3.2 Nesting and Prime Factor Algorithms . . . . . . . . . 74 3.3.3 Computation of Winograd Fourier Transforms by Polynomial Transforms . . . . . . . . . . . . . . . . 76 3.3.4 Relationship Between Polynomial Transforms and DFTs . 80 3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 80 3.5 Appendix - Short Polynomial Product Algorithms . . . . . . . 81 3.5.1 Polynomial Product Modulo (Z + 2 1) . . . . . . . . . . 81 3.5.2 Polynomial Product Modulo (Z 3- 1)/(Z- 1) . . . . . . . 81 3.5.3 Polynomial Product Modulo (Z4+ 1) . . . . . . . . . . 82 3.5.4 Polynomial Product Modulo (Z 5- 1)/(Z- 1) . . . . . . . 82 3.5.5 Polynomial Product Modulo (Z ~- 1)/(Z 3- 1) . . . . . . 83 3.5.6 Polynomial Product Modulo (Z 7- 1)/(Z- 1) . . . . . . . 84 3.5.7 Polynomial Product Modulo (Z~+ 1) . . . . . . . . . . 85 3.6 Appendix - Reduced DFT Algorithms for N= 4, 8, 9, 16 .... 86 3.6.1 N=4 . . . . . . . . . . . . . . . . . . . . . . . . 86 3.6.2 N=8 u=rc/4 . . . . . . . . . . . . . . . . . . . . 86 3.6.3 N= 16u=2g/16 . . . . . . . . . . . . . . . . . . . 87 3.6.4 N=9 u=2rc/9 . . . . . . . . . . . . . . . . . . . . 87 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4. Winograd's Discrete Fourier Transform Algorithm By S. Zohar (With 21 Figures) . . . . . . . . . . . . . . . . . 89 4.1 An Overview . . . . . . . . . . . . . . . . . . . . . . 89 4.2 Strategy Development . . . . . . . . . . . . . . . . . . . 91 4.3 The Basic LCT Algoritms . . . . . . . . . . . . . . . . . 99 4.3.1 Left-Circulant Transformation of Order 2 . . . . . . . . 100 4.3.2 Left-Circulant Transformation of Order 4 . . . . . . . . 102 4.3.3 Left-Circulant Transformation of Order 6 . . . . . . . . 105 4.4 The Basic DFT Algorithms for Prime N . . . . . . . . . . . 109 4.4.1 DFT of Order 3 . . . . . . . . . . . . . . . . . . . 112 4.4.2 DFT of Order 5 . . . . . . . . . . . . . . . . . . . 113 4.4.3 DFT of Order 7 . . . . . . . . . . . . . . . . . . . 114 4.5 The Basic DFT Algorithms for N= 4, 9 . . . . . . . . . . . 115 4.5.1 DFT of Order 4 . . . . . . . . . . . . . . . . . . . 116 4.5.2 DFT of Order 9 . . . . . . . . . . . . . . . . . . . 119 Contents IX 4.6 The Basic DFT Algoritms for N=8, 16 . . . . . . . . . . . 124 4.6.1 DFT of Order 8 . . . . . . . . . . . . . . . . . . . 126 4.6.2 DFT of Order 16 . . . . . . . . . . . . . . . . . . 130 4.6.3 Implementation of ~,=&l~l . . . . . . . . . . . . . . 132 4.6.4 Implementation of 62=~/32 . . . . . . . . . . . . . . 133 4.7 The Overall Algorithm . . . . . . . . . . . . . . . . . . . 138 4.8 Speed Analysis . . . . . . . . . . . . . . . . . . . . . . 150 4.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . 156 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5. Median Filtering: Statistical Properties By B. I. Justusson (With 15 Figures) . . . . . . . . . . . . . . 161 5.1 Definition of Median Filters . . . . . . . . . . . . . . . . 162 5.1.1 One-Dimensional Median Filters . . . . . . . . . . . . 162 5.1.2 Two-Dimensional Median Filters . . . . . . . . . . . 163 5.1.3 Edge Preservation . . . . . . . . . . . . . . . . . . 164 5.2 Noise Reduction by Median Filtering . . . . . . . . . . . . 164 5.2.1 White Noise . . . . . . . . . . . . . . . . . . . . . 164 5.2.2 Nonwhite Noise . . . . . . . . . . . . . . . . . . . 168 5.2.3 Impulse Noise and Salt-and-Pepper Noise . . . . . . . . 169 5.3 Edges Plus Noise . . . . . . . . . . . . . . . . . . . . . 173 5.3.1 Comparison of Median Filters and Moving Averages.. 173 5.3.2 Distribution of Order Statistics in Samples from Two Distributions . . . . . . . . . . . . . . . . . . . . 176 5.4 Further Properties of Median Filters . . . . . . . . . . . . . 177 5.4.1 Covariance Functions; White-Noise Input . . . . . . . . 177 5.4.2 Covariance Functions; Nonwhite-Noise Input . . . . . . 180 5.4.3 Frequency Response . . . . . . . . . . . . . . . . . 182 5.4.4 Sample-Path Properties . . . . . . . . . . . . . . . . 185 5.5 Some Other Edge-Preserving Filters . . . . . . . . . . . . . 186 5.5.1 Linear Combination of Medians . . . . . . . . . . . . 186 5.5.2 Weighted-Median Filters . . . . . . . . . . . . . . . 187 5.5.3 Iterated Medians . . . . . . . . . . . . . . . . . . . 188 5.5.4 Residual Smoothing . . . . . . . . . . . . . . . . . 190 5.5.5 Adaptive Edge-Preserving Filters . . . . . . . . . . . . 190 5.6 Use of Medians and Other Order Statistics in Picture Processing Procedures . . . . . . . . . . . . . . . . . . . . . . . 191 5.6.1 Edge Detection . . . . . . . . . . . . . . . . . . . 191 5.6.2 Object Extraction . . . . . . . . . . . . . . . . . . 193 5.6.3 Classification . . . . . . . . . . . . . . . . . . . . 194 5.6.4 General Order Statistics . . . . . . . . . . . . . . . . 195 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 X Contents 6. Median Filtering: Deterministic Properties By S. G. Tyan (With 2 Figures) . . . . . . . . . . . . . . . . 197 6.1 Fixed Points of One-Dimensional Median Filters . . . . . . . 197 6.2 Some Generalized Median Filters . . . . . . . . . . . . . . 201 6.3 Fixed Points of Two-Dimensional Median Filters . . . . . . . 205 6.4 A Fast Median Filtering Algorithm . . . . . . . . . . . . . 209 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 211 Appendix 6.A . . . . . . . . . . . . . . . . . . . . . . . . . 212 Appendix 6.B . . . . . . . . . . . . . . . . . . . . . . . . . 214 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Additional References with Titles . . . . . . . . . . . . . . . . . 219 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Contributors Eklundh, Jan-Olof National Defense Research Institute (FOA), P.O. xoB 1t65 185-S 11 Link6ping, Sweden Huang, Thomas .S Department of Electrical Engineering and Coordinated Science Laboratory, University of Illinois Urbana, IL ,10816 USA Justusson, Bo I. Department of Mathematics, Royal Institute of Technology 001-S 44 Stockholm ,07 Sweden Nussbaumer, Henri .J IBM Centre d'Etudes et Recherches F-06610 LaGaude, France Tyan, Shu-Gwei M/A-COM Laboratories, 71711 Exploration Lane Germantown, MD ,76702 USA Zohar, Shalhav Jet Propulsion Laboratory California Institute of Technology, 4800 Oak Grove Drive Pasadena, CA ,30119 USA