maria petrou, costas petrou - image processing the fundamentals 2nd

Image Processing: The Fundamentals Image Processing: The Fundamentals, Second Edition Maria Petrou and Costas Petrou © 2010 John Wiley & Sons, Ltd ISBN: 978-0-470-74586-1 Image Processing: The Fundamentals Maria Petrou Costas Petrou A John Wiley and Sons, Ltd., Publication This edition first published 2010 c 2010 John Wiley & Sons Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data Petrou, Maria Image processing : the fundamentals / Maria Petrou, Costas Petrou – 2nd ed p cm Includes bibliographical references and index ISBN 978-0-470-74586-1 (cloth) Image processing – Digital techniques TA1637.P48 2010 621.36 – dc22 2009053150 ISBN 978-0-470-74586-1 A catalogue record for this book is available from the British Library Set in 10/12 Computer Modern by Laserwords Private Ltd, Chennai, India Printed in Singapore by Markono This book is dedicated to our mother and grandmother Dionisia, for all her love and sacrifices Contents Preface xxiii Introduction Why we process images? What is an image? What is a digital image? What is a spectral band? Why most image processing algorithms refer to grey images, while most images we come across are colour images? How is a digital image formed? If a sensor corresponds to a patch in the physical world, how come we can have more than one sensor type corresponding to the same patch of the scene? What is the physical meaning of the brightness of an image at a pixel position? Why are images often quoted as being 512 × 512, 256 × 256, 128 × 128 etc? How many bits we need to store an image? What determines the quality of an image? What makes an image blurred? What is meant by image resolution? What does “good contrast” mean? What is the purpose of image processing? How we image processing? Do we use nonlinear operators in image processing? What is a linear operator? How are linear operators defined? What is the relationship between the point spread function of an imaging device and that of a linear operator? How does a linear operator transform an image? What is the meaning of the point spread function? Box 1.1 The formal definition of a point source in the continuous domain How can we express in practice the effect of a linear operator on an image? Can we apply more than one linear operators to an image? Does the order by which we apply the linear operators make any difference to the result? Box 1.2 Since matrix multiplication is not commutative, how come we can change the order by which we apply shift invariant linear operators? vii 1 1 2 3 6 7 10 11 11 12 12 12 12 12 13 14 18 22 22 22 viii Contents Box 1.3 What is the stacking operator? What is the implication of the separability assumption on the structure of matrix How can a separable transform be written in matrix form? What is the meaning of the separability assumption? Box 1.4 The formal derivation of the separable matrix equation What is the “take home” message of this chapter? What is the significance of equation (1.108) in linear image processing? What is this book about? H? 29 38 39 40 41 43 43 44 Image Transformations What is this chapter about? How can we define an elementary image? What is the outer product of two vectors? How can we expand an image in terms of vector outer products? How we choose matrices hc and hr ? What is a unitary matrix? What is the inverse of a unitary transform? How can we construct a unitary matrix? How should we choose matrices U and V so that g can be represented by fewer bits than f ? What is matrix diagonalisation? Can we diagonalise any matrix? 2.1 Singular value decomposition How can we diagonalise an image? Box 2.1 Can we expand in vector outer products any image? How can we compute matrices U , V and Λ needed for image diagonalisation? Box 2.2 What happens if the eigenvalues of matrix gg T are negative? What is the singular value decomposition of an image? Can we analyse an eigenimage into eigenimages? How can we approximate an image using SVD? Box 2.3 What is the intuitive explanation of SVD? What is the error of the approximation of an image by SVD? How can we minimise the error of the reconstruction? Are there any sets of elementary images in terms of which any image may be expanded? What is a complete and orthonormal set of functions? Are there any complete sets of orthonormal discrete valued functions? 2.2 Haar, Walsh and Hadamard transforms How are the Haar functions defined? How are the Walsh functions defined? Box 2.4 Definition of Walsh functions in terms of the Rademacher functions How can we use the Haar or Walsh functions to create image bases? How can we create the image transformation matrices from the Haar and Walsh functions in practice? What the elementary images of the Haar transform look like? Can we define an orthogonal matrix with entries only +1 or −1? Box 2.5 Ways of ordering the Walsh functions What the basis images of the Hadamard/Walsh transform look like? 47 47 47 47 47 49 50 50 50 50 50 50 51 51 54 56 56 60 61 62 62 63 65 72 72 73 74 74 74 74 75 76 80 85 86 88 Contents What are the advantages and disadvantages of the Walsh and the Haar transforms? What is the Haar wavelet? 2.3 Discrete Fourier transform What is the discrete version of the Fourier transform (DFT)? Box 2.6 What is the inverse discrete Fourier transform? How can we write the discrete Fourier transform in a matrix form? Is matrix U used for DFT unitary? Which are the elementary images in terms of which DFT expands an image? Why is the discrete Fourier transform more commonly used than the other transforms? What does the convolution theorem state? Box 2.7 If a function is the convolution of two other functions, what is the relationship of its DFT with the DFTs of the two functions? How can we display the discrete Fourier transform of an image? What happens to the discrete Fourier transform of an image if the image is rotated? What happens to the discrete Fourier transform of an image if the image is shifted? What is the relationship between the average value of the image and its DFT? What happens to the DFT of an image if the image is scaled? Box 2.8 What is the Fast Fourier Transform? What are the advantages and disadvantages of DFT? Can we have a real valued DFT? Can we have a purely imaginary DFT? Can an image have a purely real or a purely imaginary valued DFT? 2.4 The even symmetric discrete cosine transform (EDCT) What is the even symmetric discrete cosine transform? Box 2.9 Derivation of the inverse 1D even discrete cosine transform What is the inverse 2D even cosine transform? What are the basis images in terms of which the even cosine transform expands an image? 2.5 The odd symmetric discrete cosine transform (ODCT) What is the odd symmetric discrete cosine transform? Box 2.10 Derivation of the inverse 1D odd discrete cosine transform What is the inverse 2D odd discrete cosine transform? What are the basis images in terms of which the odd discrete cosine transform expands an image? 2.6 The even antisymmetric discrete sine transform (EDST) What is the even antisymmetric discrete sine transform? Box 2.11 Derivation of the inverse 1D even discrete sine transform What is the inverse 2D even sine transform? What are the basis images in terms of which the even sine transform expands an image? What happens if we not remove the mean of the image before we compute its EDST? 2.7 The odd antisymmetric discrete sine transform (ODST) What is the odd antisymmetric discrete sine transform? ix 92 93 94 94 95 96 99 101 105 105 105 112 113 114 118 119 124 126 126 130 137 138 138 143 145 146 149 149 152 154 154 157 157 160 162 163 166 167 167 x Contents Box 2.12 Derivation of the inverse 1D odd discrete sine transform What is the inverse 2D odd sine transform? What are the basis images in terms of which the odd sine transform image? What is the “take home” message of this chapter? expands an 171 172 Statistical Description of Images What is this chapter about? Why we need the statistical description of images? 3.1 Random fields What is a random field? What is a random variable? What is a random experiment? How we perform a random experiment with computers? How we describe random variables? What is the probability of an event? What is the distribution function of a random variable? What is the probability of a random variable taking a specific value? What is the probability density function of a random variable? How we describe many random variables? What relationships may n random variables have with each other? How we define a random field? How can we relate two random variables that appear in the same random field? How can we relate two random variables that belong to two different random fields? If we have just one image from an ensemble of images, can we calculate expectation values? When is a random field homogeneous with respect to the mean? When is a random field homogeneous with respect to the autocorrelation function? How can we calculate the spatial statistics of a random field? How we compute the spatial autocorrelation function of an image in practice? When is a random field ergodic with respect to the mean? When is a random field ergodic with respect to the autocorrelation function? What is the implication of ergodicity? Box 3.1 Ergodicity, fuzzy logic and probability theory How can we construct a basis of elementary images appropriate for expressing in an optimal way a whole set of images? 3.2 Karhunen-Loeve transform What is the Karhunen-Loeve transform? Why does diagonalisation of the autocovariance matrix of a set of images define a desirable basis for expressing the images in the set? How can we transform an image so its autocovariance matrix becomes diagonal? What is the form of the ensemble autocorrelation matrix of a set of images, if the ensemble is stationary with respect to the autocorrelation? How we go from the 1D autocorrelation function of the vector representation of an image to its 2D autocorrelation matrix? How can we transform the image so that its autocorrelation matrix is diagonal? 177 177 177 178 178 178 178 178 178 179 180 181 181 184 184 189 190 173 176 193 195 195 195 196 196 197 197 199 200 200 201 201 201 204 210 211 213 Contents How we compute the K-L transform of an image in practice? How we compute the Karhunen-Loeve (K-L) transform of an ensemble of images? Is the assumption of ergodicity realistic? Box 3.2 How can we calculate the spatial autocorrelation matrix of an image, when it is represented by a vector? Is the mean of the transformed image expected to be really 0? How can we approximate an image using its K-L transform? What is the error with which we approximate an image when we truncate its K-L expansion? What are the basis images in terms of which the Karhunen-Loeve transform expands an image? Box 3.3 What is the error of the approximation of an image using the KarhunenLoeve transform? 3.3 Independent component analysis What is Independent Component Analysis (ICA)? What is the cocktail party problem? How we solve the cocktail party problem? What does the central limit theorem say? What we mean by saying that “the samples of x1 (t) are more Gaussianly distributed than either s1 (t) or s2 (t)” in relation to the cocktail party problem? Are we talking about the temporal samples of x1 (t), or are we talking about all possible versions of x1 (t) at a given time? How we measure non-Gaussianity? How are the moments of a random variable computed? How is the kurtosis defined? How is negentropy defined? How is entropy defined? Box 3.4 From all probability density functions with the same variance, the Gaussian has the maximum entropy How is negentropy computed? Box 3.5 Derivation of the approximation of negentropy in terms of moments Box 3.6 Approximating the negentropy with nonquadratic functions Box 3.7 Selecting the nonquadratic functions with which to approximate the negentropy How we apply the central limit theorem to solve the cocktail party problem? How may ICA be used in image processing? How we search for the independent components? How can we whiten the data? How can we select the independent components from whitened data? Box 3.8 How does the method of Lagrange multipliers work? Box 3.9 How can we choose a direction that maximises the negentropy? How we perform ICA in image processing in practice? How we apply ICA to signal processing? What are the major characteristics of independent component analysis? What is the difference between ICA as applied in image and in signal processing? What is the “take home” message of this chapter? xi 214 215 215 215 220 220 220 221 226 234 234 234 235 235 235 239 239 240 243 243 246 246 252 254 257 264 264 264 266 267 268 269 274 283 289 290 292 xii Contents Image Enhancement 293 What is image enhancement? 293 How can we enhance an image? 293 What is linear filtering? 293 4.1 Elements of linear filter theory 294 How we define a 2D filter? 294 How are the frequency response function and the unit sample response of the filter related? 294 Why are we interested in the filter function in the real domain? 294 Are there any conditions which h(k, l) must fulfil so that it can be used as a convolution filter? 294 Box 4.1 What is the unit sample response of the 2D ideal low pass filter? 296 What is the relationship between the 1D and the 2D ideal lowpass filters? 300 How can we implement in the real domain a filter that is infinite in extent? 301 Box 4.2 z-transforms 301 Can we define a filter directly in the real domain for convenience? 309 Can we define a filter in the real domain, without side lobes in the frequency domain? 309 4.2 Reducing high frequency noise 311 What are the types of noise present in an image? 311 What is impulse noise? 311 What is Gaussian noise? 311 What is additive noise? 311 What is multiplicative noise? 311 What is homogeneous noise? 311 What is zero-mean noise? 312 What is biased noise? 312 What is independent noise? 312 What is uncorrelated noise? 312 What is white noise? 313 What is the relationship between zero-mean uncorrelated and white noise? 313 What is iid noise? 313 Is it possible to have white noise that is not iid? 315 Box 4.3 The probability density function of a function of a random variable 320 Why is noise usually associated with high frequencies? 324 How we deal with multiplicative noise? 325 Box 4.4 The Fourier transform of the delta function 325 Box 4.5 Wiener-Khinchine theorem 325 Is the assumption of Gaussian noise in an image justified? 326 How we remove shot noise? 326 What is a rank order filter? 326 What is median filtering? 326 What is mode filtering? 328 How we reduce Gaussian noise? 328 Can we have weighted median and mode filters like we have weighted mean filters? 333 Can we filter an image by using the linear methods we learnt in Chapter 2? 335 How we deal with mixed noise in images? 337 780 References [57] M Petrou and A Kolomvas, 1992 “The recursive implementation of the optimal filter for the detection of roof edges and thin lines” Signal Processing VI, Theory and Applications, 1489–1492 [58] M Petrou, 1995 “Accelerated optimisation in Image Processing via the Renormalisation Group Transformation” Complex Stochastic Systems and Engineering, Ed: D M Titterington, Clarendon Press, ISBN 19 853485 X, 105–120 [59] M Petrou and A Matrucceli, 1998 “On the stability of thresholding SAR images” Pattern Recognition, 31:1791–1796 [60] I Pitas and A N Venetsanopoulos, 1986 “Non-linear order statistic filters for image filtering and edge detection” Signal Processing, 10:395–413 [61] W K Pratt, 1978 Digital Image Processing, John Wiley & Sons, Inc, ISBN 0-471-01888-0 [62] R M Pringle and A A Rayner, 1971 Generalised inverse matrices with applications to Statistics, Being number twenty eight of Griffin’s Statistical monographs and courses, edited by A Stuart, ISBN 0-85264-181-8 [63] E Provenzi, M Fierro, A Rizzi, L De Carli, D Gadia and D Marini, 2007 “Random spray retinex: a new retinex implementation to investigate the local properties of the model” TIP-16:162–171 [64] S O Rice, 1945 “Mathematical analysis of random noise” Bell Systems Tech J., 24:46– 156 [65] A Rosenfeld and A C Kak, 1982 Digital Picture Processing, Academic Press, ISBN 0-12-597301-2 [66] M Samonas and M Petrou, 1999 “Multiresolution Restoration of Medical Signals using the Renormalisation Group and the Super-coupling Transforms” Computers in Biology and Medicine, 29:191–206 [67] S J Sangwine and R E N Horne, (eds), 1998 The Colour Image Processing Handbook Chapman and Hall, ISBN 0-412-80620-7 [68] W F Schreiber, 1970 “Wirephoto Quality Improvement by Unsharp Masking” Pattern Recognition, 2:117–121 [69] J Shi and J Malik, 2000 “Normalised cuts and image segmentation” PAMI-22:888–905 [70] L A Spacek, 1986 “Edge detection and motion detection” Image and Vision Computing, 4:43–53 [71] M Stokes, M Anderson, S Chandrasekhar and R Motta, 1996 “A standard default colour space for the Internet-sRGB”, http://www.w3org/Graphics/Color/sRGB.html [72] B A Wandell, 1995 Foundations of Vision Sinauer Associates, Inc Publishers, Sunderland, Massachusetts, ISBN 0-87893-853-2 [73] G Wyszecki and W S Stiles, 1982 Color Science ISBN 0-471-02106-7, John Wiley & Sons, Inc, New York [74] X Zhang and B A Wandell, 1996 “A spatial extension of CIELAB for digital color image reproduction” SID journal http://white.stanford.edu/∼brian/scielab/introduction.html Index D55 , 705, 706 D65 , 705, 706 for sRGB, 732 D75 , 705, 706 α-trimmed filter, 760 χ2 -test, 237, 239 k-means clustering, 767 z-transform, 294, 301 3D geometry, 721 algorithm k-means clustering, 767 anisotropic diffusion, 349 averaging angles, 629 colour constancy, 687 colour enhancement, 761 constrained matrix inversion, 464 DFT, 102 edge detection, 602 edge detection as zero crossings, 623 endmember spectra by ICA, 697 endmember spectra by PCA, 696 expectation maximisation, 537 Gauss-Seidel method, 485 gradient descent, 487 Haar transform, 89 histogram equalisation, 370 histogram equalisation with random additions, 372 histogram hyperbolisation, 373 histogram hyperbolisation with random additions, 374 Hough, 520 hysteresis edge linking, 621 ICA, 274 inverse filtering, 410 Jacobi’s method, 482 K-L transform, 214, 215 knee, 549 linear spectral unmixing, 692, 696, 697 local energy, 651 local variance, 339 metamer mapping, 757 monogenic signal, 660 normalised graph cuts, 589 pairwise image enhancement, 378 PCA, 674 PCA for material spectra, 709 power method, 682 reconstruction with unknown degradation matrix, 489 retinex, 360 Riesz transform, 660 simulated annealing, 539 simulated annealing with Gibbs sampler, 507 simulated annealing with Metropolis sampler, 506 spectral constancy, 687 sRGB to XYZ, 743 standard illuminant spectra, 705 successive doubling, 124 SVD, 69 thresholding a unimodal histogram, 549 toboggan enhancement, 387, 389 unsharp masking, 357 unsharp masking, adaptive, 358 Walsh transform, 91 watershed, 566 whirl transform, 469, 472 whirl transform, scrambling matrix for, 473 Wiener filtering, 431 alychne, 725 Image Processing: The Fundamentals, Second Edition Maria Petrou and Costas Petrou © 2010 John Wiley & Sons, Ltd ISBN: 978-0-470-74586-1 781 782 ambient light, 351 analytic signal, 659 anisotropic diffusion, 337, 342, 348 algorithm for, 349 approximation of image by DFT, 103, 176 by EDCT, 148, 176 by EDST, 165, 166, 176 by Haar transform, 79, 89, 176 by Hadamard transform, 91, 176 by K-L, 220 by ODCT, 156, 176 by ODST, 175, 176 by SVD, 63, 176 by Walsh transform, 91, 176 error of by DFT, 103, 176 by EDCT, 148, 176 by EDST, 165, 166, 176 by Haar transform, 79, 89, 176 by Hadamard transform, 91, 176 by K-L, 220 by ODCT, 156, 176 by ODST, 175, 176 by SVD, 63, 176 by Walsh transform, 91, 176 astronomical image, 402 attribute, 551 autocorrelation function, 211, 216, 313, 424 ensemble, 190 spatial, 196, 325 autocorrelation matrix, 211, 216 spatial, 215 autocovariance, 190 autocovariance matrix, 201 average angle, 629 axis of symmetry, 202 band, barrel distortion, 513 basis images, 75 DFT, 101 EDCT, 146 EDST, 163 Haar, 80, 93 Hadamard, 88 ICA, 281 Index independent, 292 K-L, 221 ODCT, 154 ODST, 173 orthogonal, 292 uncorrelated, 292 Walsh, 88 basis signals in ICA, 283 Bessel function, 298, 300 approximation of, 300 binary order, 86 blind source separation, 289 block circulant matrix, 437 diagonalisation of, 445 inversion of, 438 camera colour, sensor arrangement of, distortion due to lens, 513 multispectral, point spread function of, 12 Canny criteria, 606 Canny filter, 608 central limit theorem, 235, 264 central moments, 239 Chebyshev norm, 733 checkerboard effect, chroma, 738 chromaticity diagram, 718 definition of, 720 CIE, 704, 717 circulant matrix, 438 city block metric, 733 class of a pixel, 528 cocktail party problem, 234, 264, 283 colour calibration between different viewers, 730 comparison of, 733 definition of, 700 emulation of spatial dependence of, 752 enhancement, 761 factors influencing its perception, 740 spatial context of, 740 temporal context of, 740 colour camera, colour constancy, 713, 742 colour histogram, 767 Index colour matching experiments, 715 colour space Lab, 734, 742 Lab, errors in, 745 Lab, inverse transform, 736 Luv, 734, 742 Luv, definition of, 734 Luv, errors in, 743 Luv, inverse transform, 735 2D, 718 3D, 718 opponent, 741 perceptually uniform, 734, 742 colour system CIE RGB, 717 CIE RGB, chromaticity diagram for, 724 CIE RGB, colour matching functions of, 718 CIE RGB, definition of, 717 CIE RGB, definition of alychne for, 726 CIE RGB, tristimulus values of, 718 definition of, 715 S-CIELAB, 753 sRGB, 717, 742 sRGB, definition of, 732 transformation CIE RGB to XYZ, 727 transformation from Lab to RGB, 736 transformation from Luv to RGB, 735 transformation from sRGB to Lab, 742 transformation from sRGB to Luv, 742 tristimulus values of, 715 XYZ, 717, 718 XYZ, chromaticity diagram of, 728 XYZ, definition of, 726 XYZ, imaginary primaries of, 728 XYZ, tristimulus values of, 727 complete set of functions, 72 cones, 715, 741 confidence of χ2 -test, 239 conjugate transpose, 50 constrained matrix inversion, 436 algorithm for, 464 comparison with Wiener filtering, 462 filter derivation for, 459 filter for, 456 inhomogeneous degradation, 477 contrast, inhomogeneous, 375 783 convolution, 632 convolution theorem, 105, 108 convolution theorem, for z-transform, 302 cooling schedule, 506 correlated colour temperature, 705 cosine transform even symmetric, 137, 138 basis images of, 146 inverse of 1D, 143 inverse of 2D, 145 odd symmetric, 137, 149 basis images of, 154 inverse of 1D, 152 inverse of 2D, 154 cost function, 477, 481 for image restoration, 499 minimisation of, 503 minimisation of quadratic, 487 minimum of, 491 covariance, 184 cover proportions, 689 cross correlation, 193, 423 cross covariance, 193 data whitening, 265, 266 daylight, 703 daylight, variations of, 701, 703 degree, of a graph node, 577, 586 degrees of freedom of χ2 -test, 239 delta function, 95, 142, 144, 152, 405 Fourier transform of, 325 shifting property of, 15 DFT, 94, 176 algorithm for, 102 convolution theorem for, 105 dc component of, 118 display of, 112 for rotated image, 113 for scaled image, 119 for shifted image, 114 imaginary, 130 inverse of, 95 magnitude and phase of, 411 matrix for, 99 real valued, 126 dichromats, 717 differentiation with respect to a vector, 267 784 direct component, 118 discrete Fourier transform, see DFT distribution function, 178, 180 divergence, 345 dual grid, 569, 593 Earth observation, 688 EDCT, 138, 176 basis images of, 146 inverse 1D, 143 2D, 145 edge detection, 527, 591 algorithm, 602 and noise, 593, 605, 606 by linear filtering, 592, 602 by nonlinear filtering, 591 Canny’s criteria for, 606 for multispectral images, 769 in the first principal component, 769 statistical, 591 using phase congruency, 626 via local energy, 652 with Laplacian of Gaussian, 621, 623 edge preserving smoothness constraint, 502 edge, of a graph, 576 edgel, 593 EDST, 157, 176 basis images of, 163 inverse 1D, 160 2D, 162 eigenface, 292 eigenimage, 60, 222, 292 eigenvector, 222 elementary images DFT, 101 EDCT, 146 EDST, 163 Haar, 80, 93 Hadamard, 88 ICA, 281 independent, 292 K-L, 221 ODCT, 154 ODST, 173 orthogonal, 292 Index uncorrelated, 292 Walsh, 88 EM algorithm, 537 end members, 695, 697 energy, local, 660 ensemble autocorrelation function, 190 ensemble autocorrelation matrix, 210 ensemble of images, 190–192 ensemble statistics, 195, 292 entropy, 239, 243 of an image, 673 entropy of a Gaussian pdf, 243, 246 entropy of a uniform pdf, 244 equal energy spectrum, 715 ergodicity, 195, 197, 199, 200, 215 error due to wrong reference white, 743, 745 in image approximation by DFT, 103, 176 by EDCT, 148, 176 by EDST, 165, 166, 176 by Haar transform, 79, 89, 176 by Hadamard transform, 91, 176 by K-L, 220 by ODCT, 156, 176 by ODST, 175, 176 by SVD, 63, 176 by Walsh transform, 91, 176 least mean square, 226 least mean square in image approx., 292 least square in image approx., 292 error function, 237, 251 estimation least square error, 419 MAP, 490 maximum a posteriori, 490 Euclidean metric definition of, 733 use in measuring colour difference, 734 even antisymmetric sine transform, see EDST even symmetric cosine transform, see EDCT event, 178 probability of, 179 expectation maximisation algorithm, 537 expectation value, 181 false contouring, Index false maxima, 619 fast Fourier transform, 124 feature, 551 symmetry of, 661 feature detection antisymmetric, 650, 652 symmetric, 648, 650, 652 feature extraction, 44 feature space, 551 Fiedler vector, 586 filter α-trimmed, 337, 760 z-transform of, 294 Butterworth, 303 definition of, 294 edge adaptive, 574 edge preserving, 574 first derivative of Gaussian, 352 flat, 327, 328 frequency response function of, 294 Gaussian, 309, 332 high pass, 351 ideal 1D low pass, 296, 300 ideal 2D band pass, 299 ideal 2D high pass, 299 ideal 2D low pass, 296, 300 ideal low pass 1D-2D comparison, 301 IIR, 296 impulse response of, 294 infinite impulse response, 296 Laplace transform of, 294 Laplacian of Gaussian, 621 median, 326–328 mode, 326, 328, 574 nonlinear, 357 nonrecursive, 301 rank order, 326 real, frequency response of, 294 real, unit sample response of, 294 recursive, 301, 303 Robinson, 406 second derivative of Gaussian, 352 separable, 605 Sobel, 605 stability of, 294 statistical, 326 system transfer function of, 294 785 unit sample response of, 294 vector median, 760 weighted median, 333 weighted mode, 333 filtering averaging, 327, 328 lowpass, 328 mean shift, 574 median, 326, 327 mode, 328 first principal component for edge detection, 769 for maximum contrast, 672 fixed temperature annealing, 505 flatfielding, 364, 366, 407, 548 floor operator, 216 Fourier series, 625 Fourier slice theorem, 404 Fourier transform, 72, 94, 177, 293, 637 definition of, 637 duality of, 637, 638 fast, 124 inverse of, 637 of a constant, 637 of a delta function, 637 of an integral, 637 of derivative function, 637 Parseval’s theorem for, 637 properties of, 637 scaling property of, 637 shifting property of, 637 slice theorem of, 404 Wiener-Khinchine theorem of, 325 fourth order cumulant, 240 frequency convolution theorem, 108 frequency response function, 294, 396 Fresnel integrals, 400 asymptotic behaviour of, 401 function Γ, 183 ζ, 183 delta, 405 Gabor, 126 gradient vector of, 345 Haar, 73, 74 harmonic expansion of, 641 Laplacian of, 345 786 periodic, 141 Rademacher, 74, 86 Riemann, 183 step, 405 Walsh, 73, 74, 86 functions complete set of, 72 orthogonal, 72 orthonormal, 72 fuzzy logic, 200 Gabor functions, 126 Gamma function, 183 Gauss-Seidel method, 482, 485 Gaussian filter, 309 weights of, 329 Gaussian filtering, 342 Gaussian function Fourier transform of, 309 Gaussian mixture model, 537 Gaussian probability density function parameter estimation for, 537, 539 geometric image degradation global, 513 inhomogeneous, 515 geometric progression, 95 Gibbs sampler, 503 GMM, 537 good locality measure, 617 gradient, 345 magnitude of, 557 gradient descent algorithm, 487 gradient vector, 345 Gram matrix, 228 graph, 576 Laplacian matrix of, 577, 586 relational, 576 undirected, 576 weights, 576 Gray code, 86 grey level, grid, dual, 569, 593 Haar functions, 73, 74 discrete version of, 76 image basis from, 75 Haar transform, 74, 177 Index advantages of, 92 algorithm for, 76, 89 basis images of, 80 disadvantages of, 92 elementary images of, 80 Haar wavelet, 93 Hadamard matrices, 85 Hadamard transform, 74, 88 basis images of, 88 half width half maximum, 753 heat equation, 342, 348 Hermitian transpose, 50 Hilbert pair, 631 Hilbert transform, 631, 639 generalisation of, 660 hill in mathematical morphology, 568 histogram, 528 equalisation, 370 equalisation with random additions, 372 hyperbolisation, 373, 374 hyperbolisation with random additions, 374 manipulation, 367, 368 normalised, 236 stretching, 367 with upsampling, 553 homomorphic filter, 364 Hough transform, 520 hue, 733 in practice, 747 perceived differences in, 738 perceived, measurement of, 737 hue angle, 738 human vision, 700, 701 colour constancy of, 714 ganglion cells, 292 perception of brightness, 725 photopic, 733 physics of, 700 psychophysics of, 700 rods, 714, 733 scotopic, 733 hysteresis edge linking, 621 hysteresis thresholding, 528 ICA, 234 algorithm for, 274 Index blind source separation, 289 characteristics of, 289 differences in image and signal processing, 290 for linear spectral unmixing, 697 for signal processing, 283 in image processing, 264 independent components, 290 medical images, 260 sparse representation, 290 ideal white, 715, 716, 718 IIR, 296 image approximation of by DFT, 103, 176 by EDCT, 148, 176 by EDST, 165, 166, 176 by Haar transform, 79, 89, 176 by Hadamard transform, 91, 176 by K-L, 220, 226 by ODCT, 156, 176 by ODST, 175, 176 by SVD, 62, 63, 176 by Walsh transform, 91, 176 average of, 118 bit size of, classification of, 528 coding for optimisation algorithms, 505 colouring for optimisation algorithms, 505 compression, 44 contrast of, 10 dc component of, 118, 176 definition of, degradation, model of, 396 DFT of scaled, 119 digital, digital, formation of, dilation, 554 edge detection of, 527 eigenimages of, 60 enhancement, 43, 293, 358, 395 local, 375 pairwise, 377, 378 toboggan, 383 entropy, 673 erosion, 555 filtering, 293, 574 787 first difference of, 449 gradient magnitude of, 557 Haar transform of, 74, 76 Hadamard transform of, 74 histogram of, 528 information of, 673 inhomogeneous degradation of, 468 labelling of, 528 Laplacian of, 342, 449 mean value of, 220 morphological reconstruction of, 554 multispectral, 1, 669 multispectral, compression of, 767 multispectral, edge detection, 769 multispectral, restoration of, 767 multispectral, segmentation of, 767 of an ideal line, 403 panchromatic, quality of, registration, 395 representation by a graph, 576 resolution of, restoration, 44, 395 as MAP estimation, 490 constrained matrix inversion, 436 degradation matrix unknown, 489 geometric, 395, 513 grey value, 395 inverse filtering, 396 nonlinear, 490 Wiener filtering, 419 rotation of, 519 SAR, 326 scaling, 112 second difference of, 449 segmentation of, 527, 528 sharpening, 351 singular value decomposition of, 50, 51, 60 synthetic aperture radar, 326 thresholding, 528 transform, cosine even symmetric, 137, 138 transform, cosine odd symmetric, 137, 149 transform, DFT, 94, 95 transform, EDCT, 138 transform, EDST, 157 788 transform, Fourier, 94, 95 transform, Hadamard, 88 transform, ODCT, 149 transform, ODST, 167 transform, sine even antisymmetric, 137, 157 transform, sine odd antisymmetric, 137, 167 transform, unitary, 49 transform, Walsh, 88 transforms, comparison of, 176 Walsh transform of, 74, 76 whirl transform of, 468 image basis from DFT, 101 EDCT, 146 EDST, 163 Haar functions, 75 Haar transform, 80 Hadamard transform, 88 ICA, 281 K-L, 221 ODCT, 154 ODST, 173 SVD, 60 Walsh functions, 75 image noise, 311 image smoothing, 328 image thresholding and variable illumination, 545 images ensemble, 190 imaginary primaries, 728, 729 impulse response, 294 independent component analysis, see ICA independent components, 277 industrial inspection, 364, 366, 742 and metamers, 756 infinite impulse response, 296 information content, 673 information of a symbol, 673 interference, low frequency, 351 inverse filtering, 396 algorithm for, 410 comparison with Wiener filtering, 430 iterative conditional modes, 503 Index Jacobi’s method, 482 Jacobian matrix, 270 joint distribution function, 184 joint probability density function, 184 JPEG, 176 K-L transform, 200, 201, 214 as a first step of ICA, 265 basis images of, 221 comparison with SVD, 292 error of, 220 error of approximation with, 226 Karhunen-Loeve, see K-L kernel-based method, 553 knee algorithm, 549 Kronecker order, 86 Kronecker product, 38 kurtosis, 240 excess, 240 Pearson, 240 proper, 240 L cones, 715 label of a pixel, 528, 529 labelling, 528, 529 Lagrange multiplier, 269 Lagrange multipliers, 460 method of, 268 used in ICA, 269 Lambertian surface, 685 Laplace transform, 294 Laplacian, 345 as a smoothness constraint, 455 matrix operator, 450 matrix operator, eigenvalues of, 454 of an image, 342, 449 Laplacian matrix of a graph, 577, 586 Laplacian of Gaussian filter, 621 least square error estimation, 419 least square error solution, 692 Leibniz rule, 348, 531, 620 lens distortion algorithm for, 522 modelling of, 514 leptokurtic probability density function, 240 Index level set methods, 621 lexicographic order, 86 library spectra, 689 lightness for colour description, 738 line detection, 609 using phase congruency, 626 via local energy, 652 linear degradation process inversion of its matrix, 446 matrix of, 444 linear filter, 293 linear filtering, 293 high pass, 351 linear operator, 12 point spread function of, 12 linear spectral unmixing with ICA, 697 with library spectra, 692 with PCA, 695 without library spectra, 695 linear system of equations solution by Gauss-Seidel method, 482, 485 solution by Jacobi’s method, 482 local energy, 626, 647, 660 algorithm for, 651 filters for, 648 in 2D, 659 measurement of, 630 local image enhancement, 375 M cones, 715 Macbeth colour checker, 707, 708 magnetoencephalography, 288 manifold, 702 MAP estimation, 490 marginal probability density function, 188 Markov chain, 503 matrix block circulant, 22, 437, 600 circulant, 22, 438 conjugate transpose of, 50 diagonalisation of, 50 eigenvalues of, 587 Gram, 228 Hadamard, 85 Hermitian transpose of, 50 Jacobian, 270 789 Kronecker product of, 38, 445 norm of, 63, 64, 226 orthogonal, 50 partition of, 19 positive semidefinite, 228, 586 symmetric, 586 Toeplitz, 213 trace of, 64, 226 unitary, 49, 99 maximum a posteriori estimation, 490 Maxwell triangle, 720, 747 coordinates in, 722 MCMC minimisation, 503 mean distance of zero crossings, 619 mean shift, 339 for colour image segmentation, 767 for segmentation, 574 for smoothing, 337, 339 mean square error, 416 mean value, 181, 185 median, 326 medical images and ICA, 260 membrane model, 501 edge preserving, 502 metamerism, 756 metamers, 713, 756 in industrial inspection, 756 metric L1 norm, 733 L2 norm, 733 L∞ norm, 733 Chebyshev norm, 733 city block, 733 definition of, 733 Euclidean, 733 Metropolis sampler, 503 minimum error threshold, 530, 534, 535 drawbacks of, 541 mode, 328 mode filter, 574 edge adaptive, 574 edge preserving, 574 mode filtering, 326 modulus, 100 monochromats, 717 monogenic signal, 659, 660 Monte Carlo Markov chain, 503 790 Monte-Carlo method, 757 morphological image reconstruction, 557 morphological reconstruction, 554, 558, 560 motion blurring, 397 point spread function of, 398 MSE, 416 multiple illumination sources, 687 multiresolution, 511 via renormalisation group transform, 512 via super-coupling transform, 512 multiresolution optimisation, 512 multispectral camera, multispectral image, 1, 669 natural materials spectra of, 708 variation of, 706, 708 natural order, 86 near infrared, 704 negentropy, 239 approximation of, 246, 252, 254, 257 definition of, 243 maximisation of, 269 noise additive, 311, 337 additive Gaussian, 492 autocorrelation function of, 313 biased, 312, 314 blue, 313 coloured, 313, 314 dependent, 316, 318 filtered, 615 fixed pattern, 312 Gaussian, 311, 317, 326–328, 332, 337 homogeneous, 311 iid, 311, 313–315, 317, 492 impulse, 311, 328, 337 in multispectral images, 759 independent, 311, 312 independent identically distributed, 313 mixed, 337, 760 multiplicative, 311, 325 probability density function of, 235 salt and pepper, 311 shot, 311, 326 spec, 311 unbiased, 311, 312 Index uncorrelated, 311–313 uniform, 315 white, 311, 313–316, 318 zero-mean, 311–313, 318 non-Gaussianity measure of, 239, 240 non-maxima suppression, 601 norm L1 , 733 L2 , 733 L∞ , 733 Chebyshev, 733 of a matrix, 64, 226 normal colour vision, 717 normal distribution, 534 normal order, 86 normalised graph cuts algorithm, 576, 589 as an eigenvalue problem, 576 normalised histogram, 236 ODCT, 149, 176 basis images of, 154 inverse 1D, 152 2D, 154 odd antisymmetric sine transform, see ODST odd symmetric cosine transform, see ODCT ODST, 167, 176 basis images of, 173 inverse 1D, 171 2D, 172 opponent colour space, 741, 752 opponent colours, 741 optimisation, 503–505, 512 orthogonal matrix, 50 orthogonal set of functions, 72 orthonormal set of functions, 72 orthonormal vectors, 50 Otsu method, 541, 542 drawbacks of, 545 p-tile method, 530 pairwise image enhancement, 377, 378 Paley order, 86 panchromatic image, Parseval’s theorem, 496, 637, 647 Index path non-descending, 568 non-ascending, 568 PCA, 672 advantages of, 675 algorithm for, 674 disadvantages of, 675 for linear spectral unmixing, 695, 696 for material spectra, 708 of daylight spectra, 703 pel, perceptually uniform colour space, 742 period of a function, 141 phase congruency, 625 in 2D, 659 in practice, 630 measure of, 627 photometric stereo, 364, 366 photopic vision, 733 picture element, pin-cushion distortion, 513 pixel, platykurtic probability density function, 240 point source, 12, 14 point spread function, 12, 13, 396 from astronomical image, 402 deduction of, 27 from an ideal edge, 404 of a camera, 405 separable, 13 shift invariant, 13 Poisson distribution, 311 power method algorithm for, 682 for eigenvector estimation, 682 power spectrum, 313 primary lights, 715 principal component analysis, see PCA probability a posteriori, 491 a priori, 491 definition of, 179 of an event, 178 posterior, 491 prior, 491 probability density function, 181, 528, 530 fourth order moment of, 239 791 Gaussian, 235, 237, 240 Gaussian, entropy of, 243, 246 Gaussian, fourth moment of, 242 Gaussian, kurtosis of, 242 Gaussian, skewness of, 241 leptokurtic, 240 non-Gaussianity of, 235 normal, 238 of a function of a random variable, 320 of sum of two random variables, 545, 546 platykurtic, 240 second order moment of, 239 skewness of, 239 sub-Gaussian, 240 super-Gaussian, 240 third moment of, 239 third order moment of, 241 uniform, definition of, 244 uniform, entropy of, 244 uniform, fourth moment of, 247 uniform, negentropy of, 244, 247, 251 uniform, variance of, 244 variance of, 239 probability theory, 200 pseudo-convolution, 632 pseudorandom, 178 purple line, 725 quad tree, 554 Rademacher function, 74, 86 radiant power, 718 radiometric correction, 689 ramp detection, 609 random experiment, 178 random field, 177, 178, 189, 190 autocorrelation function of, 424 autocorrelation of, 190 autocovariance, 190 ergodic, 195, 197, 428 homogeneous, 195, 424 spatial autocorrelation function of, 325 spatial statistics of, 196 stationary, 195 random fields cross correlation of, 193, 423 cross covariance of, 193 792 uncorrelated, 193 random number generator, 178 according to given probability density function, 510 random variable, 177, 178 central moments of, 239 distribution function of, 180 entropy of, 243 expectation value of, 181, 190 function of, 320 marginal probability density function of, 188 mean value of, 181, 185 moments of, 239 probability density function of, 181 standard deviation of, 181 variance of, 181 zero-mean, 234 random variables, 190 covariance of, 184 expectation value of, 187 independent, 184, 234 joint distribution function of, 184 joint probability density function of, 184 orthogonal, 184, 234 uncorrelated, 184, 234 rank order filtering, 326 ranking vectors, 760 Rayleigh quotient, 578, 682 minimisation of, 585 recursive filter, 303 reference white, 730, 742 and transformations to Luv and Lab, 735 error due to, 743, 745 reflectance function, 545, 684, 700 region growing, 553 regularisation term, 501 relational graph, 576 remote sensing, 671, 675, 688, 689, 770 renormalisation group transform, 512 retinex algorithm, 357, 360 ridges in the watershed algorithm, 568 Riemann’s ζ function, 183 Riesz transform, 659, 660 Robinson operators, 406 rods, 733 Index rotation transformation, 519 S cones, 715 SAD, 693 saturation, 733 in practice, 747 perceived, measurement of, 737 scale space, 342 scaling function, 93 scotopic vision, 733 seed of a random number generator, 178 seed pixel, 554, 557 segmentation, 527, 528 by filtering, 574 by normalised graph cuts, 576 via edge detection, 620 with mean shift, 574 with region growing, 553 with split and merge, 554 with watershed, 553 sensitivity function of multispectral camera, sensor sensitivity function, 711 sequency order, 86 sharpening, 351 signal analytic, 659 local energy of, 626, 647 monogenic, 659, 660 projection of, 632 signal to noise ratio, 616 simulated annealing, 503, 539 accelerated, 511 cooling schedule for, 506 fixed temperature, 505 properties of, 511 termination criterion for, 505 with Gibbs sampler, algorithm for, 507 with Metropolis sampler, algorithm for, 506 sine transform even antisymmetric, 137, 157 basis images of, 163 inverse of 1D, 160 inverse of 2D, 162 odd antisymmetric, 137, 167 basis images of, 173 Index inverse of 1D, 171 inverse of 2D, 172 singular value decomposition, see SVD skewness, 239 smoothing, edge adaptive, 337 smoothness constraint, 455, 493, 501 snakes, 621 Sobel filter, 596, 605 source signal, 290 sparse representation, 290 spatial autocorrelation function, 196 spatial autocorrelation matrix, 215, 222 spatial mean, 196 spatial statistics, 195, 196 spectral angular distance, 693 spectral band, spectral constancy, 669, 684, 742 algorithm for, 687 assumptions for, 687 spectral density, 427 spectral histogram, 767 spectral signature of a pixel, 684 spectral unmixing, 671, 688 linear, 688, 689 spectrum locus, 724 split and merge algorithm, 554 stacking operator, 29 standard deviation, 181 standard illuminant, 704 as reference white, 735 standard illuminants chromaticity coordinates of, 705 correlated colour temperature of, 705 spectral radiant power of, 705, 706 standard observer, 718 statistics ensemble, 190, 195 spatial, 195, 196 step function, 405 structuring element, 554, 555, 560 successive doubling, 124 super-coupling transform, 512 SVD, 50, 51, 60, 176 algorithm for, 69 comparison with K-L, 292 intuitive explanation of, 62 symmetry axis, 202 793 system transfer function, 294 temperature parameter, 501 theorem of the three perpendiculars, 721 thin plate model, 501 threshold and variable illumination, 545, 548 minimum error, 530, 534, 535, 541 Otsu, 541, 542, 545 thresholding, 528 a unimodal histogram, 549 drawbacks of, 550 hysteresis, 528 minimum error, 530, 534 Otsu method for, 541, 542, 545 p-tile method, 530 under variable illumination, 548 time convolution theorem, 108 toboggan enhancement, 383 Toeplitz matrix, 213 trace of a matrix, 64, 226 transform DFT, 176 EDCT, 176 EDST, 176 Fourier, 177 Haar, 176, 177 Hilbert, 639, 660 K-L, 200, 201, 214 ODCT, 176 ODST, 176 Riesz, 660 Walsh, 176, 177 triangle inequality, 733 trichromatic theory of colour vision, 715 trichromats, 717 tristimulus values, 715 negative, 715 ultraviolet, 704 unit sample response, 294 unitary matrix, 49 unitary transform, 49 unsharp masking, 357 local, 357 locally adaptive, 358 value in colour representation, 761 794 variable illumination, 351, 364, 407, 545, 548 as low frequency interference, 351 variance, 181 efficient computation of, 339 interclass, 541, 544 intraclass, 541, 544 vector α-trimmed median filter, 760 vector median filter, 760 vector outer product, 47 vector ranking, 760 vectors orthonormal, 50 outer product of, 47 vertex of a graph, 576 Walsh functions, 73, 74, 86 Paley order of, 86 binary order of, 86 discrete version of, 76 dyadic order of, 86 from Hadamard matrices, 85 from Rademacher functions, 74 image basis from, 75 Kronecker order of, 86 lexicographic order of, 86 natural order of, 86 normal order of, 86 ordering of, 86 sequency order of, 86 Walsh order of, 86 Walsh-Kaczmarz order of, 86 Walsh order, 86 Walsh transform, 74, 88, 177 advantages of, 92 algorithm for, 76, 81, 91 basis images of, 88 disadvantages of, 92 Walsh-Kaczmarz order, 86 waterhole in mathematical morphology, 568 waterlines for the watershed algorithm, 568 watershed algorithm, 566 drawbacks of, 568 watershed segmentation, 553 wavelets, 93 Weber-Fechner law, 360 whirl transform, 468 algorithm for, 469, 472, 473 Index Wiener filter, 417, 420, 429 comparison with constrained matrix inversion filter, 462 derivation of, 428 in practice, 430 relationship with inverse filter, 430 Wiener filtering, 419 algorithm for, 431 Wiener-Khinchine theorem, 313, 325, 427 ... Cataloging-in-Publication Data Petrou, Maria Image processing : the fundamentals / Maria Petrou, Costas Petrou – 2nd ed p cm Includes bibliographical references and index ISBN 97 8-0 -4 7 0-7 458 6-1 (cloth)... 2) f (N, N ) Image Processing: The Fundamentals, Second Edition Maria Petrou and Costas Petrou © 2010 John Wiley & Sons, Ltd ISBN: 97 8-0 -4 7 0-7 458 6-1 Image Processing: The Fundamentals with... spatial frequency (see page 740) Image Processing: The Fundamentals, Second Edition Maria Petrou and Costas Petrou © 2010 John Wiley & Sons, Ltd ISBN: 97 8-0 -4 7 0-7 458 6-1 772 Plates Plate IV: Colour