Two-Dimensional Digital Signal Processing II. Transforms and Median Filters

228 Pages·2.622 MB·English

by  Huang T.S. (Ed.)

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
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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