Image Processing E-Book

Maria M. P. Petrou

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1127

Veröffentlichungsjahr: 2021

Beschreibung

The classic text that covers practical image processing methods and theory for image texture analysis, updated second edition

The revised second edition of Image Processing: Dealing with Textures updates the classic work on texture analysis theory and methods without abandoning the foundational essentials of this landmark work. Like the first, the new edition offers an analysis of texture in digital images that are essential to a diverse range of applications such as: robotics, defense, medicine and the geo-sciences.

Designed to easily locate information on specific problems, the text is structured around a series of helpful questions and answers. Updated to include the most recent developments in the field, many chapters have been completely revised including: Fractals and Multifractals, Image Statistics, Texture Repair, Local Phase Features, Dual Tree Complex Wavelet Transform, Ridgelets and Curvelets and Deep Texture Features. The book takes a two-level mathematical approach: light math is covered in the main level of the book, with harder math identified in separate boxes. This important text:

Contains an update of the classic advanced text that reviews practical image processing methods and theory for image texture analysis

Puts the focus exclusively on an in-depth exploration of texture

Contains a companion website with exercises and algorithms

Includes examples that are fully worked to enhance the learning experience

Written for students and researchers of image processing, the second edition of Image Processing has been revised and updated to incorporate the foundational information on the topic and information on the latest advances.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Preface to the Second Edition

Preface to the First Edition

Acknowledgements

About the Companion Website

1 Introduction

2 Binary Textures

2.1 Shape Grammars

2.2 Boolean Models

2.3 Mathematical Morphology

3 Stationary Grey Texture Images

3.1 Image Binarisation

3.2 Grey Scale Mathematical Morphology

3.3 Fractals and Multifractals

3.4 Image Statistics

3.5 Texture Features from the Fourier Transform

3.6 Markov Random Fields

3.7 Gibbs Distributions

3.8 Texture Repair

4 Non‐stationary Grey Texture Images

4.1 The Uncertainty Principle and its Implications in Signal and Image Processing

4.2 Gabor Functions

4.3 Prolate Spheroidal Sequence Functions

4.4 Local Phase Features

4.5 Wavelets

4.6 The Dual Tree Complex Wavelet Transform

4.7 Ridgelets and Curvelets

4.8 Where Image Processing and Pattern Recognition Meet

4.9 Laws' Masks and the “What Looks Like Where” Space

4.10 Local Binary Patterns

4.11 The Wigner Distribution

4.12 Convolutional Neural Networks for Textures Feature Extraction

Bibliographical Notes

References

Index

End User License Agreement

List of Tables

Chapter 2

Table 2.1 Pairs

when

is the distribution function of a Gaussian probabilit...

Table 2.2 The numbers on the left were drawn from a uniform probability densi...

Table 2.3 Perimeter and area of digital and continuous circles. The perimeter...

Table 2.4 Numerical values measured from the image for the aggregate paramete...

Table 2.5 Estimation of the aggregate and individual parameters of the images...

Table 2.6 Estimated parameters for the image in Figure 2.24b, assuming a 2D B...

Table 2.7 Aggregate parameters

and

and individual parameter

estimated fr...

Table 2.8 Aggregate parameters

and

and individual parameter

estimated fr...

Chapter 3

Table 3.1 Pixels with grey values in the ranges on the right are flagged in t...

Table 3.2 The last row shows the values of the various generalised fractal di...

Table 3.3 The numbers of each column have been fitted with the least squares ...

Table 3.4 The local connected fractal dimension of the pixels of Figure 3.41.

Table 3.5 Values of

and

estimated from the log–log plots of the average sq...

Table 3.6 The fractal dimension computed for the textures of Figure 3.1 using...

Table 3.7 The fractal dimensions and correlation coefficients with each

.

Table 3.8 The fractal dimensions for four different directions in the spectra...

Table 3.9 The multifractal spectra for the images using as measure the contra...

Table 3.10 The multifractal spectra for the images using as measure the sum o...

Table 3.11 The five point sets from the matrix of alpha values.

Table 3.12 The five point sets from the matrix of alpha values.

Table 3.13 The number of pixels with each one of the grey values and the rank...

Table 3.14 Number of pairs of samples with certain absolute difference in val...

Table 3.15 The range and sill of the variograms of images 3.1, as well as the...

Table 3.16 Features derived from the profiles of the autocorrelation function...

Table 3.17 Fitting the fractal model to the variograms of the texture images ...

Table 3.18 Best fitting of the fractal model to the variograms of the texture...

Table 3.19 The coordinates of the pixels that make up the digital circles in ...

Table 3.20 The coordinates of the pixels that will appear after the centre of...

Example Table 3.21 Contrast value for the four main directions for the image ...

Table 3.22 Contrast values for

different directions for the image in Figure ...

Table 3.23 The five strongest peaks in the signatures of the spectra of the i...

Table 3.24 The five strongest peaks in the signatures of the spectra of the i...

Table 3.25 Second column: values of function

. Third column: a Gaussian windo...

Table 3.26 Values of the power spectra of

,

, and

.

is a Gaussian mask th...

Table 3.27 The results of the fractal dimension

by calculating

.

Table 3.28 The data used for the estimation of the best set of Markov paramet...

Table 3.29 The data used for the estimation of the best set of Markov paramet...

Table 3.30 Parameters estimated for the textures in Figure 3.1 using the MLE ...

Table 3.31 Parameters estimated for the textures in Figure 3.1 using the maxi...

Table 3.32 Estimated MRF parameters using the LSE method for the images in Fi...

Table 3.33 The number of

horizontal

pairs

in the configurations of Figure 3....

Table 3.34 The number of

vertical

pairs

in the configurations of Figure 3.14...

Table 3.35 The number of

horizontal

pairs

in the configurations of Figure 3....

Table 3.36 The number of

vertical

pairs

in the configurations of Figure 3.14...

Table 3.37 The number of

horizontal

pairs

in the configurations of Figure 3....

Table 3.38 The number of

vertical

pairs

in the configurations of Figure 3.14...

Table 3.39 The number of

horizontal

pairs

in the configurations of Figure 3....

Table 3.40 The number of

vertical

pairs

in the configurations of Figure 3.14...

Table 3.41 The number of single pixels in each configuration of Figure 3.148 ...

Table 3.42 The number of single pixels with value

in each configuration prod...

Table 3.43 The number of single pixels with value

in each configuration prod...

Table 3.44 The number of single pixels with value

in each configuration prod...

Table 3.45 Most probable configurations for different sets of clique potentia...

Table 3.46 Values of probability

for the most probable configurations of exa...

Table 3.47 Number of iterations required for the greedy algorithm to converge...

Table 3.48 Value of

for number of grey levels

.

Table 3.49 Average grey value of each configuration of Figure 3.148.

Table 3.50 Average grey value of each configuration of Figure 3.148 if the tw...

Table 3.51 Average grey value of each configuration of Figure 3.148 if the tw...

Table 3.52 Average grey value of each configuration of Figure 3.148 if the tw...

Table 3.53 Probability of each configuration in Figure 3.148 arising, when ex...

Table 3.54 Probability of each configuration in Figure 3.148 arising if the t...

Table 3.55 Probability of each configuration in Figure 3.148 arising if the t...

Table 3.56 Probability of each configuration in Figure 3.148 arising if the t...

Table 3.57 Probability of having an image with average grey level

for differ...

Table 3.58 Free energy

for different structures of the configuration space....

Table 3.59 All possible grey values

of a pixel, their corresponding probabil...

Chapter 4

Table 4.2 Orientations of the angular bands of the Gaussian masks.

Table 4.1 Values of

,

,

, and

used to segment the image in Figure 4.33.

Table 4.3 Reconstruction errors for the three studied cases: complete Gaussia...

Table 4.4 Normalised values obtained for the low, band and high pass filters.

Table 4.5 If we use these ranges of indices in Equation 4.248 with

and the v...

Table 4.6 Some commonly used scaling filters. These filters are appropriate f...

Table 4.7 Two sets of wavelet filters that can be used in the dual tree compl...

Table 4.8 Classification results at ILSVRC (http://www.image‐net.org/).

Table 4.9 AND gate in two dimensional data.

Table 4.10 Perceptron training process (AND gate).

Table 4.11 OR gate in two‐dimensional data.

Table 4.12 OR training process.

Table 4.13 The step by step caluclation of the gradient method.

.

Table 4.14 XOR problem.

Table 4.15 The step by step calculation of three‐layer NN(1).

Table 4.16 The step by step calculation of three‐layer NN(2).

Table 4.17 Traditional methods for the development of texture analysis.

Table 4.18 Time‐line of deep‐learning based methods.

Table 4.19 The coordinates of the pixels that make up the digital circle acco...

Table 4.20 Results compared with other methods.

List of Illustrations

Chapter 1

Figure 1.1 Costas in bloom. Source: Maria Petrou.

Figure 1.2 (a) An original image. Source: Maria Petrou. (b) Manually extract...

Figure 1.3 (a) Blewbury from an aeroplane (size

). Source: Maria Petrou. (b...

Figure 1.4 (a) When a scanning window of size

was placed in Figure 1.3b wi...

Figure 1.5 A surface imaged from two different distances may create two very...

Figure 1.6 A surface imaged under different illumination directions may give...

Figure 1.7 (a) Original image of size

showing a town. Source: Maria Petrou...

Figure 1.8 Each tile in this figure represents a pixel. The black tiles repr...

Figure 1.9 (a) An original image of size

, showing the cross section of the...

Chapter 2

Figure 2.1 We can easily recognise the depicted objects, even from a binary ...

Figure 2.2 Four binary textures: which one is different from the other three...

Figure 2.3 (a) A random texture. (b) A texture with a regular primitive patt...

Figure 2.4 An example of a regular pattern.

Figure 2.5 There may be more than one primitive pattern that may be used to ...

Figure 2.6 The rules of a grammar that may be used to produce the texture of...

Figure 2.7 Successive application of the rules of Figure 2.6 allows us to re...

Figure 2.8 The rules of a grammar that may be used to characterise the textu...

Figure 2.9 Successive application of the rules of Figure 2.8 allows us to re...

Figure 2.10 The rules of a grammar that may be used to characterise the text...

Figure 2.11 Successive application of the rules of Figure 2.10 allows us to ...

Figure 2.12 An alternative pattern that may be produced by the successive ap...

Figure 2.13 Two semi‐stochastic textures.

Figure 2.14 A pattern that was produced by the application of the rules of F...

Figure 2.15 A one‐to‐one relationship between

and

.

Figure 2.16 A Gaussian probability density function with

and

and the his...

Figure 2.17 2D Boolean patterns created for different values of parameter

...

Figure 2.18 2D Boolean patterns created for different values of parameter

...

Figure 2.19 (a) A

binary image. (b)The boundary pixels of the image. These...

Figure 2.20 (a) Some digital circles. (b) Boundary pixels that share a commo...

Figure 2.21 A grain at distance

from the origin of the axes will have radi...

Figure 2.22 A circular image of radius

with two grains of radii

.

Figure 2.23 Estimating the aggregate parameters of a 2D Boolean model. (a) A...

Figure 2.24 Two semi‐stochastic textures.

Figure 2.25 The original image and two textures created using the 2D Boolean...

Figure 2.26 (a) An original binary image. The individual strings we can crea...

Figure 2.27 The first few steps of drawing a Hilbert curve.

Figure 2.28 At the top, an original binary image. In the middle, the individ...

Figure 2.29 (a) A raster scanning line that reads the rows of the image sequ...

Figure 2.30 An original image and the strings one can create from it by foll...

Figure 2.31 Function

estimated from image 2.24b using the scanning methods...

Figure 2.32 Function

estimated from image 2.24b using the image rows (a), ...

Figure 2.33 An

binary image and the string created from its pixels by read...

Figure 2.34 Some example structuring elements. The crossed lines mark the ce...

Figure 2.35 The amorous hippopotamus.

Figure 2.36 (a) The amorous hippopotamus about to be dilated with the struct...

Figure 2.37 (a) The amorous hippopotamus about to be eroded with the structu...

Figure 2.38 (a) The amorous hippopotamus about to be opened. All white pixel...

Figure 2.39 (a) The amorous hippopotamus about to be closed. All white pixel...

Figure 2.40 (a) The amorous hippopotamus about to be dilated with the struct...

Figure 2.41 (a) The amorous hippopotamus about to be eroded with the structu...

Figure 2.42 (a) The amorous hippopotamus about to be opened with the structu...

Figure 2.43 (a) The amorous hippopotamus about to be closed with the structu...

Figure 2.44 The object of Figure 2.34a dilated with the structuring element ...

Figure 2.45 (a) The amorous hippopotamus after his dilation with structuring...

Figure 2.46 (a) The amorous hippopotamus after his erosion with structuring ...

Figure 2.47 The image in (a) is dilated with structuring element 2.34c, to p...

Figure 2.48 (a) The amorous hippopotamus about to be dilated with structurin...

Figure 2.49 The dilated amorous hippopotamus of Figure 2.48b about to be dil...

Figure 2.50 (a) The amorous hippopotamus about to be eroded with structuring...

Figure 2.51 The eroded amorous hippopotamus of Figure 2.50b about to be erod...

Figure 2.52 Some openings of image 2.1 with structuring elements of size (a)...

Figure 2.53 The pattern spectrum for image 2.1.

Figure 2.54 (a) The original image of an object. (b) The

structuring eleme...

Figure 2.55 If the

neighbourhood of a pixel looks like any one of these co...

Figure 2.56 (a) The original image. (b) The delineated pixels will be remove...

Figure 2.57 Pairs of elements that should be used to produce the skeleton of...

Figure 2.58 Panels (e1) and (e2) are the hit‐or‐miss transforms of images (a...

Figure 2.59 Images (a1) and (a2) eroded with structuring elements (f1) and (...

Figure 2.60 Panels (b1) and (b2) are the complements of (a1) and (a2) and th...

Figure 2.61 Panels (e1) and (e2) are the hit‐or‐miss transforms that mark th...

Figure 2.62 The successive thinning stages of the second cycle of applying t...

Figure 2.63 Applying successively the elements of Figure 2.57 in turn to int...

Chapter 3

Figure 3.1 Some stationary grey texture images. Images (a), (c), (e), (f) an...

Figure 3.2 Splitting a grey image into a set of binary images by thresholdin...

Figure 3.3 Splitting an image into a set of binary images all of which have ...

Figure 3.4 Splitting an image into a set of binary images after histogram eq...

Figure 3.5 Splitting an image into a set of binary images by bit‐slicing. (a...

Figure 3.6 A binary sequence A may be dilated with the asymmetric structurin...

Figure 3.7 (a) A grey image. The highlighted

frame indicates the structuri...

Figure 3.8 (a) The original signal. (b) The structuring element. (c) The top...

Figure 3.9 (a) The original signal. (b) The flat structuring element of size...

Figure 3.10 (a) The original image. (b) The image eroded by subtracting at e...

Figure 3.11 Underneath each panel one can see which panels of Figure 3.10 we...

Figure 3.12 Underneath each panel one can see which panels of Figure 3.10 we...

Figure 3.13 Underneath each panel one can see which panels of Figure 3.10 we...

Figure 3.14 (a) Structuring element 3.10d with the bias removed. (b) Erosion...

Figure 3.15 (a) The signal of Figure 3.8a eroded with a flat structuring ele...

Figure 3.16 (a) The signal of Figure 3.8a dilated by a flat structuring elem...

Figure 3.17 (a) The signal of Figure 3.8a dilated and eroded by a flat struc...

Figure 3.18 Enlargement of the image in order to deal with boundary effects ...

Figure 3.19 Morphological operations applied to the image of Figure 3.1a wit...

Figure 3.20 Pattern spectra for some images of Figure 3.1. They may be used ...

Figure 3.21 To create a von Koch snowflake curve, start with a straight line...

Figure 3.22 To create a fractal surface from a

Sierpinski triangle

, start wi...

Figure 3.23 The line segments in (b) are assumed to be rigid measuring rods ...

Figure 3.24 The box‐counting method for the computation of the fractal dimen...

Figure 3.25 Calculating the fractal dimension of a circle by successive divi...

Figure 3.26 (a) The first stage of constructing the fractal surface of figur...

Figure 3.27 (a) The regular tetrahedron with which we replace triangle

of ...

Figure 3.28 The cross‐section of the surface created at the first stage of t...

Figure 3.29 The cross‐section of the surface created at the first four stage...

Figure 3.30 The first four stages of creating the curve of Example 3.21 and ...

Figure 3.31 The first bit of the indices of the pixels in each quadrant of a...

Figure 3.32 The largest font is used to indicate the first bit of the index ...

Figure 3.33 A

2‐bit image.

Figure 3.34 Treating the image as a landscape, we imagine that it exists in ...

Figure 3.35 A ganglion cell.

Figure 3.36 We cover the image with boxes of decreasing size and count the n...

Figure 3.37 The number of object pixels inside each box.

Figure 3.38 The multifractal spectrum of image 3.35.

Figure 3.39 A

area of the image in Figure 2.1. Source: Maria Petrou.

Figure 3.40 The multifractal spectrum of image 3.39.

Figure 3.41 (a) An original binary image. (b)–(f) The thick black frames ind...

Figure 3.42 The histogram of the local connected fractal dimensions, compute...

Figure 3.43 The histogram of the local connected fractal dimensions for imag...

Figure 3.44 Fractal dimension and self‐affinity.

Figure 3.45 Average square differences versus

for the textures of Figure 3...

Figure 3.46 The value of

, which appears in 3.114, is constant along each d...

Figure 3.47 Synthesising 1D fractals from a sequence of white noise with

....

Figure 3.48 Synthesising 2D fractals of size

. Source: Maria Petrou.

Figure 3.49 The pairs of points

fitted with a straight line for each of th...

Figure 3.50 A

‐bit image.

Figure 3.51 Lacunarity values using

windows displayed as images. (a) Food....

Figure 3.52 Histograms of the lacunarity values. (a) Food. (b) Plastic. Both...

Figure 3.53 Synthetic 1D multifractal

Figure 3.54 Synthetic 1D multifractal using a different random sequence at e...

Figure 3.55 Synthetic 2D multifractals. Source: Maria Petrou.

Figure 3.56 Synthetic 2D multifractals using a different random sequence at ...

Figure 3.57 The matrix of alpha values.

Figure 3.58 The five point sets from the matrix of alpha values.

Figure 3.59 The matrix of alpha values. Source: Maria Petrou.

Figure 3.60 The five point sets from the matrix of alpha values. Source: Mar...

Figure 3.61 Construction of a binomial multifractal distribution with

bins...

Figure 3.62 The first two stages of creating a 2D binomial multifractal.

Figure 3.63 Creating a 2D binomial multifractal image.

Figure 3.64 Raw (on the left) and smoothed (on the right) probability densit...

Figure 3.65 Histograms of the gradient magnitude for the textures of Figure ...

Figure 3.66 The orientations of the gradient vectors computed for each pixel...

Figure 3.67 (a) The orientation histogram of image 3.66. (b) The orientation...

Figure 3.68 The unit sphere and an infinitesimal surface element on it.

Figure 3.69 The generalisation of the Sobel filter to 3D. These are the thre...

Figure 3.70 The accumulator array we create in order to compute the 3D orien...

Figure 3.71 (a) A

‐bit grey image. (b) Its rank‐frequency plot.

Figure 3.72 A

‐bit

image.

Figure 3.73 Each matrix is

, as there are four different grey values and th...

Figure 3.74 Variograms for the images of Figure 3.1. (a) Using the whole ima...

Figure 3.75 The black dots represent data values. The solid line represents ...

Figure 3.76 The variograms of the two images in Figure 3.1.

Figure 3.77 Distorted versions of the 2D fractal in Figure 3.48 for

. The o...

Figure 3.78 Normalised auto‐covariance function (the autocorrelation functio...

Figure 3.79 Normalised auto‐covariance (top) and autocorrelation functions (...

Figure 3.80 Marginals of the auto‐covariance function for images food and pl...

Figure 3.81 Marginals of the auto‐covariance function for images cloth 2 and...

Figure 3.82 Marginals of the auto‐covariance function for images fabric 1 an...

Figure 3.83 Plots of the Weibull distribution for several values of paramete...

Figure 3.84 Fitting the histograms of the gradient magnitude for the texture...

Figure 3.85 An image with three grey levels.

Figure 3.86 Top row, co‐occurrence matrices. Bottom row their normalised ver...

Figure 3.87 Digital circles with radii

, from top to bottom and left to rig...

Figure 3.88 Rotationally invariant co‐occurrence matrices for the image bean...

Figure 3.89 Directional co‐occurrence matrices for the image fabric 1. Sourc...

Figure 3.90 Power spectra for four of the textures of Figure 3.1. The spectr...

Figure 3.91 Power spectral signatures for fabric 1, fabric 2, beans and sand...

Figure 3.92 The phase spectra of the first three images of Figure 3.1 with t...

Figure 3.93 Computing the power spectrum of an image. (a) Original image. (b...

Figure 3.94 (a)A 1D digital signal. (b) The discrete Fourier transform of th...

Figure 3.95 The contour along which we are going to integrate.

Figure 3.96 The black dots indicate poles of the integrand. In (a) the sum o...

Figure 3.97 Continuous and discrete Fourier transforms of a Gaussian. (a)

....

Figure 3.98 Two digital signals and their corresponding power spectra. (a)

Figure 3.99 (a) A digital signal. (b) The power spectrum of the digital sign...

Figure 3.100 (a) Digital signal 3.322 repeated three times. (b) The same sig...

Figure 3.101 Computing the power spectrum of an image using a Gaussian windo...

Figure 3.102 Computing the fractal dimension of an image from its power spec...

Figure 3.103 (a) The shaded region indicates the range of indices

and

ov...

Figure 3.104 Real and imaginary parts of the inverse Fourier transform after...

Figure 3.105 Image reconstruction using a fixed value for its Fourier transf...

Figure 3.106 Image reconstruction using a fixed value for its Fourier transf...

Figure 3.107 Inverse Fourier transform using

and

for different values of...

Figure 3.108 (a) Imagine that you start from point

and you move along the ...

Figure 3.109 At the top the true phase of a signal. Below it, the phase we c...

Figure 3.110 Phase‐unwrapped sequences obtained for two horizontal lines of ...

Figure 3.111 Four of the possible paths that lead from pixel A to pixel Q.

Figure 3.112 All paths that lead from A to Q when we can only move from left...

Figure 3.113 A

‐bit image may be written as the linear superposition of its...

Figure 3.114 The approximate representation of the original image.

Figure 3.115 The approximate representation of the original image. Source: M...

Figure 3.116 Some Markov neighbourhoods and the conditions that have to be s...

Figure 3.117 Some more exotic Markov neighbourhoods, where the neighbours th...

Figure 3.118 An empty grid. We have to choose grey values for these pixels t...

Figure 3.119 Four directional textures.

Figure 3.120 The parameters and pixels that influence the value of the centr...

Figure 3.121 Context allows us to determine the missing values.

Figure 3.122 Some Markov neighbourhoods. The numbers indicate the Markov par...

Figure 3.123 A random field created using a binomial distribution with

and...

Figure 3.124 Codings of a

image when the Markov neighbourhoods of figures ...

Figure 3.125 Creating textures using the Markov model with a binomial distri...

Figure 3.126 Creating textures using the Markov model with a binomial distri...

Figure 3.127 A random field created using the approximation of a binomial di...

Figure 3.128 Creating textures using the Markov model and approximating the ...

Figure 3.129 Two neighbourhoods with Markov parameters that retain the balan...

Figure 3.130 Creating textures using the Markov model and approximating the ...

Figure 3.131 Creating textures using the Markov model and approximating the ...

Figure 3.132 Creating textures using the Markov model with a normal distribu...

Figure 3.133 We have a different probability density function for each value...

Figure 3.134 An original texture and texture created using the parameters es...

Figure 3.135 (a) A

‐bit

image. (b) The two codings keeping only the pixel...

Figure 3.136 Synthesised textures using the parameters estimated by the leas...

Figure 3.137 Synthesised textures using the parameters estimated by least sq...

Figure 3.138 The way we construct Gaussian and Laplacian pyramids.

is the ...

Figure 3.139 The frequency domain of an image, with

and

the frequencies ...

Figure 3.140 (a) A

‐bit

image. (b) A smoothing Gaussian mask.

Figure 3.141 The smoothed and sub‐sampled images.

Figure 3.142 The smoothed image and sub‐sampled images.

Figure 3.143 The smoothed and sub‐sampled images.

Figure 3.144 The Laplacian pyramid.

Figure 3.145 The cliques that correspond to the Markov neighbourhoods shown ...

Figure 3.146 We mark with a different symbol the neighbourhood of each pixel...

Figure 3.147 The cliques that correspond to the Markov neighbourhoods shown ...

Figure 3.148 All possible binary

images we can have when the grey pixels c...

Figure 3.149 Results of the greedy algorithm applied to a starting configura...

Figure 3.150 Results of the greedy algorithm applied to random images with t...

Figure 3.151 Results of the greedy algorithm applied to starting configurati...

Figure 3.152Figure 3.152 Results of the modified greedy algorithm applied to...

Figure 3.153 Results of the modified greedy algorithm applied to a random im...

Figure 3.154 Probability density function for having an image with average g...

Figure 3.155 Free energy

as a function of the mean grey value

.

Figure 3.156 (a) A texture binary image. The circles indicate pixels with gr...

Figure 3.157 All islands of perimeter

one can draw around a pixel.

Figure 3.158 On the left some configurations

with the same island of 0s in...

Figure 3.159 The damaged image to be reconstructed. Source: Maria Petrou.

Figure 3.160 Each position, encoded here with a letter, in this

neighbourh...

Figure 3.161 The feature map of co‐occurrence matrix

. Source: Maria Petrou...

Figure 3.162 (a) An image patch that is to be repaired, by giving values to ...

Figure 3.163 A 2‐bit image that needs repair.

Figure 3.164 The first iteration of the in‐painting algorithm.

Figure 3.165 The second iteration of the in‐painting algorithm.

Figure 3.166Figure 3.166 The third iteration of the in‐painting algorithm.

Figure 3.167 Two low pass filters that may be used in normalised convolution...

Figure 3.168 (a) The image that is to be repaired with

values ...

Figure 3.169 The incomplete neighbourhood in (a) has to be compared with the...

Figure 3.170 (a) The distances from the centre of all positions of the neigh...

Chapter 4

Figure 4.1 A continuous function (a), and the real part (b), imaginary part ...

Figure 4.2 The effects of a rectangular window: windowed parts of the signal...

Figure 4.3 The effects of a rectangular window: windowed parts of the signal...

Figure 4.4 The effects of a Gaussian window: windowed parts of the signal (f...

Figure 4.5 The effects of a Gaussian window: windowed parts of the signal (f...

Figure 4.6 At the top an infinitely long signal with parts of it seen throug...

Figure 4.7 At the bottom an original signal with three different positions o...

Figure 4.8 Feature sequences obtained from the real (a) and the imaginary (b...

Figure 4.9 Feature sequences obtained from the magnitude of the short time F...

Figure 4.10 This figure shows a digital signal scanned by a window

samples...

Figure 4.11 Top: the signal of Equation 4.65. Bottom: an ideal feature that ...

Figure 4.12 Feature sequences obtained from the real (a) and imaginary (b) p...

Figure 4.13 (a) Feature sequences obtained from the magnitude of the short t...

Figure 4.14 Feature sequences obtained from the magnitude of the short Fouri...

Figure 4.15 Averaged feature sequences obtained from the magnitude of the sh...

Figure 4.16 The original signal of Figure 4.6 scanned by a narrow window. If...

Figure 4.17 Suppose that we are seeing a signal with fundamental frequency

Figure 4.18 Gabor function

may be used to create elementary signals in ter...

Figure 4.19 Gabor function

may be used to create elementary signals in ter...

Figure 4.20 (a) At the top the real and imaginary parts of

, the Fourier tr...

Figure 4.21 Inverse Fourier transform of all windowed Fourier transforms of ...

Figure 4.22 Gabor features produced by squaring and convolving the functions...

Figure 4.23 Gabor features produced by squaring and convolving the functions...

Figure 4.24 A schematic tessellation of the 2D frequency domain. The crosses...

Figure 4.25 A schematic representation of the analysis of an image into cont...

Figure 4.26 A polar coordinates‐based tessellation of the frequency domain. ...

Figure 4.27 A tessellation of the frequency domain where higher frequencies ...

Figure 4.28 The coordinates of point P in the large coordinate system are

,...

Figure 4.29 An octave is the interval between a frequency and its double. Th...

Figure 4.30 The ellipse represents an isocontour of a Gaussian window in the...

Figure 4.31 (a) When the number of azimuthal bands is even (e.g.

), pairing...

Figure 4.32 Function

for

,

, and

, from bottom to top respectively, whe...

Figure 4.33 (a) A Girona pavement to be segmented and (b) the magnitude of i...

Figure 4.34 Top row: magnitude of the Fourier transform of image 4.33a multi...

Figure 4.35 Top row: magnitude of the Fourier transform of image 4.33a multi...

Figure 4.36 Top row: magnitude of the Fourier transform of image 4.33a multi...

Figure 4.37 Top row: magnitude of the Fourier transform of image 4.33a multi...

Figure 4.38 Top row: magnitude of the Fourier transform of image 4.33a multi...

Figure 4.39 Energies of the 20 bands used in the Gabor expansion of the imag...

Figure 4.40 The central band containing the dc component of the image and th...

Figure 4.41 For each pair of panels: on the left the Gaussian filters that c...

Figure 4.42 For each pair of panels: on the left the Gaussian filters that c...

Figure 4.43 Reconstruction error as a function of the channels used for the ...

Figure 4.44 Energies of the

bands created by truncating the Gaussian masks...

Figure 4.45 For each pair of panels: on the left the truncated Gaussian filt...

Figure 4.46 For each pair of panels: on the left the truncated Gaussian filt...

Figure 4.47 Truncated Gaussian windows. The

and the corresponding local en...

Figure 4.48 Energies of the

bands into which the Fourier transform of the ...

Figure 4.49 (a) Reconstruction error as a function of the channels used for ...

Figure 4.50 For each pair of panels: on the left the flat filters that corre...

Figure 4.51 For each pair of panels: on the left the flat filters that corre...

Figure 4.52 Flat windows: The

and the corresponding local energy maps extr...

Figure 4.53 Top three lines: real part, imaginary part and energy of sequenc...

Figure 4.54 Top line: original signal. Next three lines: Fourier transform o...

Figure 4.55 We wish to have a filter with non‐zero values inside the two mar...

Figure 4.56 (a) Results after applying the filter obtained in Example 4.46 (...

Figure 4.57 (a) Results after applying the filter obtained in Example 4.51 (...

Figure 4.58 (a) Results after applying the filter obtained in Example 4.51 (...

Figure 4.59 Sequence

for the low pass convolution filter.

Figure 4.60 Sequence

for the band pass convolution filter.

Figure 4.61 The

sequence that corresponds to the high pass convolution fil...

Figure 4.62 (a) Results obtained for the segmentation of the signal defined ...

Figure 4.63 As Figure 4.62, but this figure concerns the band pass filter de...

Figure 4.64 (a) Results obtained for the segmentation of the signal defined ...

Figure 4.65 The squared values of

, shown as a grey image. Source: Maria Pe...

Figure 4.66 A tessellation of the 2D frequency domain of a

image into freq...

Figure 4.67 We shift the non‐zero values of array 4.269 in the grey area of ...

Figure 4.68 The tessellation of the 2D frequency domain of a

image shown i...

Figure 4.69 Left: tessellation of the 2D

image into sub‐images of size

. ...

Figure 4.70 (a) Basis functions on which the signal shown in (b) can be proj...

Figure 4.71 An image consisting of two different textures. (a) The pixel val...

Figure 4.72 Magnitude of the outputs of the nine convolution filters designe...

Figure 4.73 Real part of the outputs of the nine convolution filters designe...

Figure 4.74Figure 4.74 Imaginary part of the outputs of the nine convolution...

Figure 4.75 Averaged energies inside a window of size

of the outputs of th...

Figure 4.76 The numbers on the top and left of the grids identify the indice...

Figure 4.77 Real part of the nine filters designed. Each filter is of size

Figure 4.78 Imaginary part of the nine filters designed. Each filter is of s...

Figure 4.79 Real part of the

filters designed. Each filter is of size

an...

Figure 4.80 Imaginary part of the

filters designed. Each filter is of size...

Figure 4.81 The Girona pavement in size

. We wish to construct features tha...

Figure 4.82 Magnitude of the outputs of the nine‐filter bank. Source: Maria ...

Figure 4.83 Real part of the outputs of the nine‐filter bank. Source: Maria ...

Figure 4.84 Imaginary part of the outputs of the nine‐filter bank. Source: M...

Figure 4.85 Real part of the outputs of the 25‐filter bank. Source: Maria Pe...

Figure 4.86 Imaginary part of the outputs of the 25‐filter bank. Source: Mar...

Figure 4.87 Magnitude of the outputs of the 25‐filter bank. The two empty pa...

Figure 4.88 The last two panels of Figure 4.87. They correspond to low frequ...

Figure 4.89 (a) The Fourier transform of a symmetric band limited filter. (b...

Figure 4.90 A local window may isolate part of a signal. If the isolated seg...

Figure 4.91 An image that contains a symmetric and an antisymmetric feature ...

Figure 4.92 The negative of the second derivative of a Gaussian with

. In E...

Figure 4.93 The “what happens when” space. Consider a function that contains...

Figure 4.94 The “what happens when” space. As the sub‐space spanned by the s...

Figure 4.95 A wavelet vector for a 12‐sample long signal, for shifting param...

Figure 4.96 Wavelet and scaling vectors constructed from a six‐tap long filt...

Figure 4.97 A filter of odd length used to produce wavelet vectors for the d...

Figure 4.98 Schematic representation of the multi‐resolution analysis of a s...

Figure 4.99 Schematic representation of the multiresolution analysis of a si...

Figure 4.100 The tree wavelet analysis of a digital signal.

Figure 4.101 The packet wavelet analysis of a digital signal.

Figure 4.102 (a) Original signal. (b) Reconstructed signal after setting the...

Figure 4.103 The signal of Figure 4.102a reconstructed from only the stronge...

Figure 4.104 The packet wavelet representation of an image.

Figure 4.105 The tessellation of the frequency space that corresponds to the...

Figure 4.106 Zooming into a frequency band with the packet wavelet transform...

Figure 4.107 Tree wavelet representation of the image shown in Figure 4.81....

Figure 4.108 Packet wavelet representation of the image shown in Figure 4.81...

Figure 4.109 The structure tree for the texture image fabric 2.

Figure 4.110 The tessellation of the frequency space that corresponds to the...

Figure 4.111 Three levels of analysis with the maximum overlap algorithm. Th...

Figure 4.112 At the top, a band that has to be analysed. It is sub‐sampled i...

Figure 4.113 Maximum overlap structure tree of the image shown in Figure 4.8...

Figure 4.114 Averaged energies of the coefficients of the maximum overlap an...

Figure 4.115 The maximum overlap structure tree for the texture image food. ...

Figure 4.116 The tessellation of the frequency space that corresponds to the...

Figure 4.117 Time–frequency space showing schematically the resolution cells...

Figure 4.118 Gabor basis functions in the time and frequency domains for

,

Figure 4.119 Time–frequency space showing schematically the resolution cells...

Figure 4.120 Some wavelet basis functions in the time and frequency domains....

Figure 4.121 (a) A signal containing a pulse and its shifted version by one ...

Figure 4.122 In both panels, the black dots and the solid line corresponds t...

Figure 4.123 The magnitude plots of these Fourier transforms.

Figure 4.124 Filter

plotted against the real distance of its elements from...

Figure 4.125 An eight‐tap symmetric low pass filter, when sub‐sampled by kee...

Figure 4.126 The signal may be fully reconstructed from the scaling and wave...

Figure 4.127 The different grey tones identify the bands to which the 2D fre...

Figure 4.128 The application of the dual tree complex wavelet transform to a...

Figure 4.129 The feature maps for the image 4.71 using the dual tree complex...

Figure 4.130 The length

of the normal from the centre of the axes to the l...

Figure 4.131 At the top left, an image containing a line‐like structure. The...

Figure 4.132 For fixed

, Radon transform considers a batch of parallel line...

Figure 4.133 All lines that can be formed in a

image. Each panel shows the...

Figure 4.134 All lines of a certain fixed slope, made up from pixels, that c...

Figure 4.135 Lines of a certain slope created in an image with size not a pr...

Figure 4.136 A line and the normal to the line from the centre of the axes....

Figure 4.137 The numbers that have to multiply exponent

to produce the com...

Figure 4.138 These numbers have to multiply

to form the exponent of the fa...

Figure 4.139 As Figure 4.138, but for components 13–24 of the Fourier transf...

Figure 4.140 As Figure 4.138, but for components 25–36 of the Fourier transf...

Figure 4.141 As Figure 4.138, but for components 37–49 of the Fourier transf...

Figure 4.142 The thick black frame identifies the 2D DFT of the image. The d...

Figure 4.143 The numbers below the grid are the values of frequency index

....

Figure 4.144 The original Fourier frame of Figure 4.143 is repeated in all d...

Figure 4.145 The two slices one may consider as being cut in the 2D Fourier ...

Figure 4.146Figure 4.146 (a) All lines with the same slope (

), as in Exampl...

Figure 4.147Figure 4.147 The

vectors of the digital Radon lines as origina...

Figure 4.148 The lines with slope

, all marked with different symbols. Sort...

Figure 4.149 At the top, the digital lines we shall use for the Radon transf...

Figure 4.150 The nine finite ridgelet transforms of the corresponding

wind...

Figure 4.151 The numbers in each panel identify the pixels that make up the ...

Figure 4.152 Histograms of the absolute values of the wavelet coefficients f...

Figure 4.153 At the top, data points along the axis of a feature. In the mid...

Figure 4.154 If the features we are using are not good for class discriminat...

Figure 4.155 Histograms of the relative distances between the absolute value...

Figure 4.156 Histogram of the relative distances in the 10D feature space of...

Figure 4.157 Segmentation results for

using a Gaussian window to compute t...

Figure 4.158 (a) The reference hand segmentation of an image. (b) The result...

Figure 4.159 A region (A+B) in the reference segmentation partly overlaps wi...

Figure 4.160 An image in which we search to identify the

pattern shown on ...

Figure 4.161 The reference image and the three sub‐images with which it has ...

Figure 4.162 We must compute the mutual information of these two images by u...

Figure 4.163 Laws' masks of size

. On the left their names and on the right...

Figure 4.164 Laws' masks of size

. On the left their names and on the right...

Figure 4.165 Laws' masks of size

. On the left their names and on the right...

Figure 4.166 Outputs of the Laws' masks of size

. The

extreme values of e...

Figure 4.167 Segmentation results. Source: Maria Petrou.

Figure 4.168 Outputs of the Laws' masks of size

. The

extreme values of e...

Figure 4.169Figure 4.169 Segmentation results obtained using the determinist...

Figure 4.170 Segmentation results obtained using the deterministic annealing...

Figure 4.171 (a) The first six eigenvalues of matrix

. (b) All eigenvalues ...

Figure 4.172 First six feature maps obtained using PCA. Each principal compo...

Figure 4.173 Segmentation results obtained using the first three features co...

Figure 4.174 (a) If we process a

image with the inverse of the Walsh matri...

Figure 4.175 If we process a sub‐image of size

with the inverse Walsh matr...

Figure 4.176 (a) The nine elementary neighbourhoods in terms of which the

...

Figure 4.177 (a) The

elementary neighbourhoods in terms of which the

Wal...

Figure 4.178 (a) Convolution of an image along the vertical and horizontal d...

Figure 4.179 (a) Laws' masks of size

may be combined in all possible ways ...

Figure 4.180 The expansion of each

neighbourhood of the image in Figure 4....

Figure 4.181 Energies of the outputs of Figure 4.180 computed using a Gaussi...

Figure 4.182 Segmentation results obtained using the deterministic annealing...

Figure 4.183 Expanding the

neighbourhoods of the image in Figure 4.81 in t...

Figure 4.184 Segmentation results obtained using the deterministic annealing...

Figure 4.185 Expanding the

neighbourhoods of the image of Figure 4.81 in t...

Figure 4.186 Energies of the outputs of Figure 4.185 computed using a Gaussi...

Figure 4.187 Panels LI3–SI3 (left) and SI3–SI3 (right) of Figure 4.185, enla...

Figure 4.188 Segmentation results obtained using the deterministic annealing...

Figure 4.189 The pixels around the central pixel in (a) are given value

if...

Figure 4.190 Two circular neighbourhoods with radii

and

are considered a...

Figure 4.191 Local binary pattern maps for the image of Figure 4.81. Source:...

Figure 4.192 Segmentation of image 4.81 using the LBP histograms for radius

Figure 4.193 The stable patterns in a

neighbourhood. All rotational variat...

Figure 4.194 (a) Segmentation result using the LBP histograms of only the ro...

Figure 4.195 Segmentation results using the contrast values only for radius

Figure 4.196 (a) Segmentation result using the LBP histograms of the stable ...

Figure 4.197 Kaiser window for

, from top to bottom, respectively.

Figure 4.198 A train of delta functions

distance apart may be thought of a...

Figure 4.199 (a) A continuous band‐limited signal and its Fourier spectrum s...

Figure 4.200 If the teeth of the sampling comb in the real domain are closer...

Figure 4.201 (a) A continuous band‐limited signal

and its Fourier spectrum...

Figure 4.202 In (a) the 2D Kaiser window created by multiplying two 1D Kaise...

Figure 4.203 Wigner feature maps obtained for the image of Figure 4.81 using...

Figure 4.204 Segmentation results obtained using the deterministic annealing...

Figure 4.205 First six feature maps constructed using principal component an...

Figure 4.206 Segmentation results obtained using the deterministic annealing...

Figure 4.207 Segmentation results obtained using the deterministic annealing...

Figure 4.208 Segmentation results obtained using the deterministic annealing...

Figure 4.209 Energy features constructed from the components of the Wigner s...

Figure 4.210 Segmentation results obtained using the deterministic annealing...

Figure 4.211 The first six eigen‐feature maps produced by PCA for the

Wign...

Figure 4.212 Segmentation results obtained using the deterministic annealing...

Figure 4.213 Segmentation results obtained using the deterministic annealing...

Figure 4.214 Segmentation results obtained using the deterministic annealing...

Figure 4.215 A neuron model:

.

Figure 4.216 A simple Perceptron model [83].

Figure 4.217 A two ADALINE model (three‐layer MADALINE) [98].

Figure 4.218 Pre‐attentively distinguishable texture pairs by Juletz [41].

Figure 4.219 The principal component vectors. For each PCA result with size

Figure 4.220 Similarity definition of edges.

Figure 4.221 Similarity definition of lines.

Figure 4.222 A sigmoid function and the first derivative.

Figure 4.223 Three‐layer neural networks.

Figure 4.224 The three‐layer NN.

Figure 4.225 Error in the three‐layer NN.

Figure 4.226 Very large intra‐class variations caused by changes in illumina...

Figure 4.227 Inter‐class images may also appear to have a similar pattern du...

Figure 4.228 The data for Fisher discriminant analysis.

Figure 4.229 The result for Fisher discriminant analysis.

Figure 4.230 The framework of TCNN and FV‐CNN.

Figure 4.231 The calculation process of the co‐occurrence matrix.

Figure 4.232 Example of different angle

features for a texture image of di...

Figure 4.233 The visualization of GLCM features with parameter

, horizontal...

Figure 4.234 The framework and feature map dimension of Vgg‐16 Network[90]....

Figure 4.235 Examples of the DTD dataset.

Figure 4.236 Examples of the FMD dataset.

Figure 4.237 ResNet flowchart.

Figure 4.238 ResNet convergence.

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Image Processing

Dealing with Texture

Second Edition

(the late) Maria Petrouformerly, Imperial College, London, UK

Revising AuthorSei‐ichiro KamataWaseda University, Tokyo/Kitakyushu, Japan

This second edition first published 2021

© 2021 John Wiley & Sons, Ltd.

Edition History

John Wiley & Sons, Ltd (1e 2006)

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