Virus Recognition in Electron Microscope Images using Higher
Order Spectral Features
by
Hannah Ong Chien Leing, BEng Medical (Hons)
PhD Thesis
Submitted in Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
at the
Queensland University of Technology
Speech and Image, Video Technology (SAIVT)
School of Electrical and Electronic Systems Engineering
February 2006
Keywords
Virus recognition, electron micrograph, higher order spectra, bispectrum, invariant
features, feature averaging, texture and contour analysis.
Abstract
Virus recognition by visual examination of electron microscope (EM) images is time
consuming and requires highly trained and experienced medical specialists. For these
reasons, it is not suitable for screening large numbers of specimens. The objective of this
research was to develop a reliable and robust pattern recognition system that could be
trained to detect and classify different types of viruses from two-dimensional images
obtained from an EM.
This research evaluated the use of radial spectra of higher order spectral invariants to
capture variations in textures and differences in symmetries of different types of viruses
in EM images. The technique exploits invariant properties of the higher order spectral
features, statistical techniques of feature averaging, and soft decision fusion in a unique
manner applicable to the problem when a large number of particles were available for
recognition, but were not easily registered on an individual basis due to the low signal to
noise ratio. Experimental evaluations were carried out using EM images of viruses, and a
high statistical reliability with low misclassification rates was obtained, showing that
higher order spectral features are effective in classifying viruses from digitized electron
micrographs. With the use of digital imaging in electron microscopes, this method can be
fully automated.
iii
Contents
Abstract i
List of Tables ix List of Figures xi
Acronyms & Abbreviations xviii
Certification of Thesis xix
Acknowledgments xx
Chapter 1 Introduction 1
1.1 Motivation and Overview 1 1.2 Aims and Objectives 5
1.3 Thesis Outline 6
1.4 Original Contributions of the Thesis 9
1.5 Publications Resulting from this Research 11
iv CONTENTS
Chapter 2 Virus Morphology and Electron Microscopy (EM) 13
2.1 Introduction 13 2.2 Morphology of Virus 14
2.3 Preparation for Examination by EM 16
2.3.1 Negative Staining 16
2.4 Role of Electron Microscopy in Virology 17
2.5 Summary 19
Chapter 3 Towards Automated Virus Recognition 21
3.1 Introduction 21 3.2 Segmentation 22
3.3 Feature Extraction 25
3.3.1 Co-occurrence Matrices 27 3.3.2 Autocorrelation 28
3.3.3 Wavelet Features 28
3.3.4 Gabor Features 30
3.3.5 Higher Order Spectra 31
3.3.5.1 Time Domain Definition and Properties 31
3.3.5.2 Frequency Domain Definition and Properties 35
3.3.5.3 Motivations of using HOS in Feature Extraction 38
3.3.5.4 Rotation, Translation and Scaling Invariants Generation 40
3.3.5.4.1 Feature Extraction from 2-D Images 43
CONTENTS ___________ __________________________________________v 3.4 Classifiers 45
3.4.1 Parametric and Non-parametric Classifiers 46 3.4.2 Gaussian Mixture Model (GMM) 47
3.4.3 Support Vector Machines (SVMs) 50
3.5 Summary 54 Chapter 4 A Study of Texture and Contour 55
4.1 Introduction 55 4.2 The Relative Importance of Phase and Magnitude in Texture 57
4.2.1 Experimental Procedure 58 4.2.2 Results and Discussion 60
4.3 Texture and Contour Analysis using Indirect Method for HOS Feature
Extraction 63
4.3.1 Introduction 63
4.3.2 Experiment with a Large Database of Different Textures and
Contours 64
4.3.3 Experiment with Images of the Same Texture of Different
Contours 67
4.3.4 Experiment with Various Level of Noise 70 4.3.5 Experiment with Averaging of Features and Input Fusion of
Noisy Images 73
4.4 Summary 76
vi CONTENTS Chapter 5 Virus Recognition using Higher Order Spectral Features 79
5.1 Introduction 79 5.2 Higher Order Spectra 81
5.2.1 Illustration of the Indirect Method in obtaining Bispectral Features
using Synthetic Images 84
5.3 Gastroenteric Viruses 91 5.4 Image Analysis 92 5.5 GMM Modeling 96
5.5.1 Experimental Evaluation of Model Order and Dimensionality of
Training Observations 97
5.6 GMM Verification 100 5.7 Experiments 101
5.7.1 Experiment on a Pooled Population 104 5.7.2 Experiment on a Single Image Population 107
5.8 Virus of Similar Size 110 5.9 Evaluation using Support Vector Machine Classifier 112
5.9.1 Introduction 112 5.9.2 Experimental Procedure 113 5.9.3 Results and Discussion 114
5.10 3D Virus Reconstruction 115 5.11 Summary 116
Chapter 6 Relative Performance Evaluation 119
6.1 Introduction 119
CONTENTS vii 6.2 Gabor Filter 121 6.3 Discriminant Analysis and Results 122 6.4 Training and Classification using Support Vector Machine 124
6.4.1 Results and Discussion 125
6.5 Summary 126
Chapter 7 Conclusion and Further Work 129
7.1 Conclusion 129 7.2 Further Work 131
Bibliography 133 Appendix A A Study of Texture and Contour 143
A.1 Experimental Results of the Relative Importance of Phase and Magnitude
in Texture Analysis 143
A.2 Classification Results of a Large Database of Different Textures and
Contours using Higher Order Spectral Features 153
A.3 Classification Results of Images of the Same Texture but Different
Contours using Higher Order Spectral Features 155
A.4 Classification Results of Images with Various Level of Noise using Higher
Order Spectral Features 156
A.5 Classification Results with Averaging of Features and Input Fusion of
Noisy Images using Higher Order Spectral Features 157
Glossary 159
ix
List of Tables Table 3.1 Some commonly used kernels 53
Table 4.1 Misclassification rate (%) of the first 10 textures out of the 40 that were chosen
from the Brodatz album. The overall performance over forty textures shows an
average misclassification of 1.37%. 66
Table 4.2 Misclassification rate (%) of four different contours of (a) D5 and (b) D21 texture
images using higher order spectral features 69
Table 4.3 Misclassification rate (%) of four textures with added white Gaussian noise of
SNR equal to (a) -50 dB (b) -30 dB (c) -20 dB using higher order spectral
features 72
Table 4.4 Misclassification rate (%) when feature averaging of (a) 10 images (b) 20 images
were used for classification using higher order spectral features 74
Table 4.5 Misclassification rate (%) when M = (a) 4 and (b) 8 sets and N=5 images were
used for classification using higher order spectral features. Gaussian noise of
SNR = -30 dB was added to the image. 75
Table 5.1 Efficiency (%) of different combinations of number of mixture, M and
dimensionality of training observations, D used to examine the performance of
GMM 100
x LIST OF TABLES Table 6.1 Comparison of the Feature Space Separability of Gabor and HOS features using
FLD 123
Table 6.2 Efficiency (%) of higher order spectral features and Gabor features in verification
of Rotavirus, Calicivirus and Adenovirus 127
Table A.2 Misclassification rate (%) of a large database of different textures and contours
images that were chosen from the Brodatz album using higher order spectral
features. The overall performance over forty textures shows an average
misclassification of 1.37% 153
Table A.3 Misclassification rate (%) of four different contours of (a) D8 and (b) D17 texture
images using higher order spectral features 155
Table A.4 Misclassification rate (%) of four textures with added white Gaussian noise of
SNR equal to (a) -80 dB (b) -10 dB (c) 0 dB (d) 10dB using higher order spectral
features 156
Table A.5 Misclassification rate (%) when M = (a) 2 (b) 5 and (c) 10 sets and N=5 images
were used for classification using higher order spectral features. Gaussian noise
of SNR = -30 dB was added to the image. 157
xi
List of Figures
Figure 2.1 A basic structure of virus is the icosahedron which composed of 20 facets, each
an equilateral triangle and 12 vertices. Due to rotational symmetry it possesses
axes of (a) 5 fold (b) 3 fold and (c) 2 fold symmetry 15
Figure 2.2 Adenovirus after negative stain electron microscopy 16
Figure 3.1 An EM image of Adenovirus. A significant number of complex artificial objects
that normally present and non-uniform illumination of the image makes
segmentation of the virus particle a challenging task 23
Figure 3.2 The symmetry regions of the bispectrum (labeled 1). The region labeled 1
contains a unique set of values and those in the other labeled regions can be
mapped to these 38
Figure 3.3 Owing to symmetries of the bispectrum in equation (3.27), the bispectrum
possess redundancy and needs to be only computed for the triangular region
shown above. Features are extracted by integrating the bispectrum along a radial
line as shown and taking the phase of the complex-valued integral. 1f and 2f are
frequencies normalized by one half of the sampling frequency 42
Figure 3.4 The direct and indirect methods for computation of invariant parameters for a
1-D sequence 44
xii LIST OF FIGURES Figure 3.5 The Radon transform of a 2-D image yields 1-D parallel beam projections,
)n(x at various angles, θ 45
Figure 3.6 SVM Margin 53
Figure 4.1 A single virus particle of (a) Adenovirus (b) Astrovirus (c) Calicivirus. Note that
although there are small differences in texture and contour, it is difficult to tell
them apart by visual examination 57
Figure 4.2 Flow chart of analysis of two different textures to evaluate the importance of
phase or magnitude in representing the texture 59
Figure 4.3 Example of (a) homogeneous textures (b) inhomogeneous textures used in the
experiment. Two of the images above are randomly chosen and subjected to steps
in Figure 4.2 60
Figure 4.4 Results when images of (a) 2 inhomogeneous textures (b) 2 homogeneous
textures (c) a homogenous and inhomogeneous texture were used as input
textures after subjected to steps in Figure 4.2. The top images of each subfigure
are the input and the bottom images are the output 62
Figure 4.5 Four different texture images with four different contours (a) D5 with 5 fold
symmetry contour (b) D17 with 6 fold symmetry contour (c) D21 with 7 fold
symmetry contour (d) D33 with circular contour chosen out of the forty Brodatz
images that were used for classification 65
Figure 4.6 Four texture images from the Brodatz album (a) D5 (b) D8 (c) D17 (d) D21
used in the experiment. 4 different contours were applied to each of these texture
images 67
Figure 4.7 The 512 X 512 texture is tiled into 64 X 64 sized images. The texture
inhomogeneity between the tiles causes higher classification inaccuracy 70
LIST OF FIGURES xiii Figure 4.8 Gaussian noise added to texture D5 with SNR equal to (a) -50 dB (b) -30 dB (c)
-20 dB 71
Figure 4.9 Noise performance of bispectral features. Misclassification shown is an average
misclassification of textures D5, D8, D17 and D21 74
Figure 4.10 Average misclassification rate in percentage by input fusion where the features
are combined by averaging of M sets where each set consists of 5 images 77
Figure 5.1 Illustration of the inability of the power spectrum to retain phase information.
Figure 5.1(a) Non minimum phase of input sequence, h(k), Figure 5.1(b) Zeros of
the non-minimum phase, Figure 5.1(c) Power spectrum of the non-minimum
phase sequence. The zeros outside the unit circle in Figure 5.1(b) are inversed
conjugate to produce a minimum phase sequence. Figure 5.1(d) 1-D plot of the
minimum phase sequence, Figure 5.1(e) Zeros of the minimum phase sequence,
Figure 5.1(f) Power spectrum of the minimum phase sequence 83
Figure 5.2 The left top and bottom figures show the imaginary and real parts of the
bispectrum of Figure 5.1(a) and the right top and bottom figures show the
imaginary and real parts of the bispectrum of Figure 5.1(d). The differences
observed in the plots show the ability of bispectrum to retain the phase
information 84
Figure 5.3 Flow chart of computation of invariant parameters. ),( θaP is invariant to
scaling and translation and ),( θωaP is invariant to scaling, translation and
rotation. The algorithm was tested on a 5 fold symmetry and a 7 fold symmetry
image and results are presented in Figures 5.4, 5.5 and 5.6 86
Figure 5.4 Figure 5.4(c) and 5.4(d) show the Radon transform projection at 45 degree angle
of the 7 fold symmetry image, Figure 5.4(a) and the 5 fold symmetry image,
Figure 5.4(b) 87
xiv LIST OF FIGURES
Figure 5.5 Figure 5.5(a) and Figure 5.5(c) show the real and imaginary parts of the
bispectrum of the Radon transform projection at 45 degree angle of Figure 5.4(a).
Figure 5.5(b) and Figure 5.5(d) show the real and imaginary parts of the
bispectrum of Figure 5.4(b). The bispectrum is a triple product of Fourier
coefficients and is a complex valued function of two frequencies, f1 and f2, where
f1 and f2 are frequencies normalized by one half of the sampling frequency.
Different shaped projections result in different bispectra. Invariant features are
extracted by integrating along radial lines and taking the phase. The scale shown
at the colorbar above is the log of the absolute value of the real and imaginary
parts of the bispectrum. The above plots show that features, )a(P close to
21a = may capture differences as well 88
Figure 5.6 Figure 5.6(a) and 5.6(b) show plot of ))(2/1( θωP as a function of θω , (where
θω is a frequency in cycles per 180 degrees) of Figure 5.4(a) and Figure 5.4(b).
))(2/1( θωP is invariant to scaling, translation and rotation. Note that Figure
5.6(a) shows a dominant symmetry at 7 cycles per 180 degrees whereas Figure
5.6(b) shows a dominant symmetry at 5 cycles per 180 degrees 88
Figure 5.7 Figure 5.7(a) shows the radial spectrum of the bispectral features, ))(2/1( θωP
as a function of θω , ( θω is a frequency in cycles per 180 degrees) of a 7 fold
symmetry. White Gaussian noise has been added and SNR = 0dB to the image. In
Figure 5.7(b), the individual spectra are accumulated over 75 such images. Note
that Figure 5.7(a) does not demonstrate a dominant symmetry at 7 cycles per 180
degrees due to the low signal to noise ratio. As an ensemble of images is taken
and the spectra are accumulated, it eventually converges to a shape. A peak at 7
cycles per 180 degrees can be seen in Figure 5.7(b). Robustness to noise can thus
be achieved by averaging these features 89
Figure 5.8 Comparison between a 5 fold symmetry image and a 3D reconstructed virus
image using plots of radial spectra of the bispectral features, P(1/2). These
features which are invariant to translation, rotation and scaling contain
information from the contour and texture that are useful for verification 90
LIST OF FIGURES xv Figure 5.9 A sample image of each type of virus used for testing. These images are
different in magnification and resolution. (a) Adenovirus (b) Astrovirus (c)
Rotavirus (d) Calicivirus 93
Figure 5.10 A single virus of each type. (a) Adenovirus (b) Astrovirus (c) Rotavirus (d)
Calicivirus. These subimages are extracted from portions shown by square boxes
in figure 5.9 and a circular mask is applied to each. Note that although there are
small differences in texture, it is difficult to tell them apart by visual examination.
Pseudo colouring could be used to emphasize the differences in texture but the
difficulty arises when there is some variation in texture within the same virus
type on images that are obtained from various sources of different background,
scale, contrast and noise 94
Figure 5.11 Cluster plot of features from three different sets of Rotavirus images with
different backgrounds, contrast and scale. Each point is an average feature from a
subpopulation of 10 viral particles. The plot shows quite compact and isolated
clusters or modes in feature space 102
Figure 5.12 Illustration of selection of viral particles from different sets of EM images used
for testing and training in pooled population and single image population. In the
single image population, the testing and training is done on populations derived
from separate images. In a pooled population, the virus images pooled from all
the EM images of that type obtained from various sources 103
Figure 5.13 DET curve using the bispectral features from subpopulations of 5, 8, 10, 13 and
15 viral particles on a pooled population. As we increase the test ensemble size
for feature averaging, the EER drops. EER is the point on the DET curve where
the false alarm probability is equal to the miss probability. Refer to Figure 5.14
for EER values of each subpopulation size 105
Figure 5.14 Plot of EER versus ensemble size shows that as we increase the test ensemble for
feature averaging, the EER drops for N= 15 to 2.5% 107
xvi LIST OF FIGURES Figure 5.15 DET curve using the bispectral features from subpopulation of N=15 viral
particles on a pooled population. The solid line shows an average of 2 sets of
subpopulation of 15 viral particles while the dotted line shows the case for M = 1
(M, N, refer to equations 5.2, 5.3). The EER drops to less than 0.2%. The darker
solid line shows an output fusion of two test populations. In this case, false
acceptance rate will be the product of individual false acceptance rates and the
false rejection rate will be the sum of the individual ones. This shows that the
averaged scores yield better performance than output fusion in this case 108
Figure 5.16 DET curve using the bispectral features from subpopulations of 5, 8, 10, 13, 15
and 18 viral particles on a single image population. The EER drops as we
increase the feature averaging of the subpopulation size. Refer to Figure 5.17 for
EER values of each subpopulation size 109
Figure 5.17 Plot of EER versus ensemble size shows that as we increase the test ensemble for
feature averaging, the EER drops for 18 particles to 2% 110
Figure 5.18 A sample image of (a) Astrovirus and (b) Hepatitis A virus. The magnifications
of these images are the same 111
Figure 5.19 DET curve using the bispectral features from subpopulation of N = 5, 10, 15, 20
particles, M =1 and subpopulation of N = 20, M=2 on a pooled population. The
EER drops to less than 2% when features of 2 sets of subpopulation of 20
particles were averaged 112
Figure 5.20 DET curve using the bispectral features from subpopulations of 5, 10 and 15 viral
particles on a pooled population using SVM and GMM classifier 114
Figure 6.1 The plot of efficiency versus number of features for a subpopulation of 10 viral
particles trained using SVM classifier. A choice of 3 kernel widths were used,
0.001, 0.1 and 10 to train the features. The efficiency is calculated from miss and
false alarm probabilities 126
LIST OF FIGURES xvii Figure A.1(a) Subfigures of (a)-(l) Results when images of 2 homogeneous textures used as
input textures after subjected to steps in Figure 4.2. The top images of each
subfigure are the input and the bottom images are the output 143
Figure A.1(b) Subfigures (a)-(l) Results when images of 2 inhomogeneous textures used as
input textures after subjected to steps in Figure 4.2. The top images of each
subfigure are the input and the bottom images are the output 146
Figure A.1(c) Subfigures (a)-(l) Results when images of a homogenous and inhomogeneous
texture used as input textures after subjected to steps in Figure 4.2. The top
images of each subfigure are the input and the bottom images are the output 149
Acronyms & Abbreviations DFT Discrete Fourier Transform EER Equal Error Rate ELISA Enzyme-Linked Immunosorbent Assay EM Electron Microscopy E-M Expectation Maximization FLD Fisher Linear Discriminant GLCM Gray level Co-occurrence matrices GMM Gaussian Mixture Model HAV Hepatitis A Virus HOS Higher Order Spectra IFA Immunofluorescence Assay ML Maximum Likelihood OSH Optimal Separating Hyperplane PCA Principal Component Analysis PCR Polymerase Chain Reaction SNR Signal to Noise Ratio SRM Structural Risk Minimization SVM Support Vector Machine TEM Transmission Electron Microscope
Certification of Thesis This work contained in this thesis has not been previously submitted for a degree or
diploma at any higher educational institution. To the best of my knowledge and belief,
the thesis contains no material previously published or written by another person except
where due reference is made.
Signed: ________________________ Date: ________________________
Acknowledgments
First and foremost, my most sincere thanks must go to my principal supervisor Assoc.
Prof. Vinod Chandran, for his guidance, support, and most of all his infinite patience and
understanding without which this thesis may not have been completed at all. Honestly, I
could not think of a better supervisor. I also wish to acknowledge the support my
associate supervisors Professor Sridha Sridhran and Assoc. Prof. John Aaskov have
provided.
I would also like to thank the QUT super-computing services for their assistance in
providing their computing facilities, as well as all the members in SAIVT Laboratory, for
creating an enjoyable and fun atmosphere during the course of my study. In particular, to
Ronald Elunai for his invaluable help with a significant portion of work presented in this
thesis. I am also grateful to the Centre of Disease Control, USA and The Universal Virus
Database of the International Committee on Taxonomy of Viruses for their public domain
virus images and also to Prof. Hans Gelderblom for providing some of the electron
micrographs.
xxi ACKNOWLEDGMENTS
I’m very grateful to my family for their unconditional love and supporting me financially,
spiritually and emotionally the whole way through. I would also like to express my
gratitude to my friends here in Brisbane, who have been a constant source of support for
me while completing this work, through both the good times and the bad.
And finally to the most important person in my life, for You are the ultimate source of
our success, love, hope, dreams and future. I’m grateful for all that You have done not
only during the course of my study but my entire life.
Hannah Ong Chien Leing
Chapter 1 Introduction 1.1 Motivation and Overview Parallel to its technical development starting in the 1930s, electron microscopy (EM)
emerged as an important tool in basic and clinical virology [1] . EM has allowed a direct
demonstration of the particulate nature of viruses, as well as the description of their size
and morphology. The introduction of negative staining in the late 1950s and wider
availability of electron microscopes have encouraged broad application of EM to routine
viral diagnosis using cell cultures and clinical samples, such as stool, urine and biopsy
specimens. Within a short period of time, many viruses and bacteriophages were
characterized morphologically, and the differences observed in morphology were used as
criteria for virus classification [2].
2 1.1 Motivation and Overview
The value of EM in viral diagnosis soon became apparent, but over the years due to some
limitations of EM, other techniques and new kits have appeared in the market. Virus
recognition by visual examination of EM images is quite time consuming. To interpret a
micrograph, highly skilled and experienced medical specialists are needed. Success is
determined not only by the ability to carry mental images and compare them with what is
visible on the screen of the microscope, but also to see whether the observed image is
compatible with what is known of the particular virus, or one very like it, and, just as
importantly, when it is incompatible [3].
For these reasons, EM is not suitable for screening large numbers of specimens. Many
alternate methods have been developed to overcome this issue, including (a) detection of
viral antigen, e.g immunofluorescence assay (IFA), (b) detection of virus nucleic acid,
e.g polymerase chain reaction (PCR), and (c) detection of anti-viral antibody, e.g
enzyme-linked immunosorbent assay (ELISA).
Even though these techniques allow mass screening, they have several limitations
compared to EM. One of the primary disadvantages of immunologic tests such as ELISA
and IFA is that detectable antibodies may not be detectable in immunosuppressed patients
or early in the course of infection. Besides, the reagents may not exist that would permit
complete immunologic testing. Even when it is appropriate for the etiologic agent, the
sensitivity may only equal that of EM [4, 5].
This also is true for nucleic acid amplification techniques such as PCR that are capable
only of identifying genomic material for previously identified agents. Besides, nucleic
1.1 Motivation and Overview 3 acid amplification techniques will not identify subviral components such as empty virions
(viral particle) that may be present late in an infection. Furthermore, mutations in viruses
may cause this method to be less effective [2].
Although these techniques have taken over much of the diagnostic EM, they are not a real
substitute for, and therefore not comparative with EM diagnosis, because their objective,
e.g. viral antigen or nuclei acids, or antiviral antibodies, differs from that of EM, the virus
particle itself. Thus, so far, these modern diagnostic techniques can act only as
complementary and not alternatives to EM [3].
Automated recognition of viruses from EM images can make it possible to screen large
numbers of samples from various parts of the world. Besides, electron microscope images
can be transmitted without the risk of spreading a virus. With the advance and increase in
technology, the use of digital imaging with CCD cameras and powerful computers are
becoming common for Transmission Electron Microscope (TEM). This development
makes it possible to have a fully automated system, and thus it can be a useful tool in
assisting virus recognition.
To date, research has been done in this area to extract the individual virus cell from
electron microscope images for classification [6] and to automatically recognize the virus
extracted using methods such as topology [7] , template matching [8] and iterative
Bayesian approach [9]. Matuszewski [9] suggested classification of viruses based on
4 1.1 Motivation and Overview topology has a major disadvantage because the introduction of a new virus type to the
‘library’ of the recognizable viruses requires revision of the topological measures of all
the viruses in the library. Template matching has poor classification performance in
identifying a large number of different virus types.
In the iterative Bayesian approach, the multi-category multi-feature classification
problem is decomposed into a set of two-category classification sub-problems, with each
classification sub-problem solved based on this approach. However, this method
normalizes viral particles in size before feature extraction. Normalisation may not work
well for non-circular objects and often fails when the signal to noise ratio is low.
Registration of the image or its spectrum over a corresponding prototype is quite
computationally demanding and also error-prone. This method is not able to classify
viruses from different sources.
Viruses are the smallest infectious biological entities that depend on their host for
replication. They are classified on the basis of fundamental characteristics, the principal
ones being the nature of their genetic material or genome, either DNA or RNA, and their
morphology in terms of size, shape, and general appearance. The International
Committee on Taxonomy of Viruses (ICTV) recognizes about 1,550 virus species which
belong to 56 families, 9 subfamilies, and 233 genera [10].
Viruses in EM images vary in orientation, position and size. EM images of different
viruses also exhibit fine differences in texture that arise from differences in their 3D
surfaces and internal structures. Symmetries are commonly observed in biological
1.2 Aims and Objectives 5
reproduction processes. Textures on images of viral particles tend to exhibit different
rotational symmetries as well. However, the variation in texture and any differences in
symmetry are difficult to visualize in any single specimen because images of virus taken
from TEM often are noisy due to the high resolution and the low dose of electrons used in
the microscope. Consequently not much information can be extracted by simple visual
inspection of a single particle or image. Additional processing such as averaging an
ensemble of virus particles is necessary to obtain a better result. Averaging can be done
only if the features are robust to translation, size, rotation and noise. Thus, features that
have these invariant properties are important in this application. In addition, these
features need to capture contour and texture properties.
Automated virus recognition using EM will speed up the diagnostic process, release
medical specialists’ time by de-skilling the diagnostic task to allow the use of non-
specialist medical staff, and allow screening a large number of specimens. This work
attempted to address this vision by developing a method that could be used to detect and
classify different types of viruses from negative stained EM images.
1.2 Aims and Objectives The general aim of this project, as a result of the discussion in Section 1.1, is to develop a
reliable and robust pattern recognition system that can be trained to detect and classify
different types of viruses, in particular viruses that are difficult to distinguish visually
from two-dimensional images obtained from an electron microscope. The methods
6 1.3 Thesis Outline
proposed are based on higher order spectral features that capture contour and texture
information, while providing robustness to shift, rotation, changes in size and noise. This
method also can be applied to other microorganisms such as bacteria and biological cells.
The specific goals of this work are to:
(i) Conduct a study on contour and texture using higher order spectral features to
have a better understanding of this application in virus classification from EM
images.
(ii) Investigate the use of higher order spectral features in automated virus recognition
system from 2-D EM images.
(iii) Evaluate and compare other texture and contour based features in the application
of virus classification.
(iv) Evaluate and seek improvement of results by using different classifiers and
additional processing such as fusion of features and fusion of classifier scores.
1.3 Thesis Outline The remainder of the thesis is organized as follows: Chapter 2 discusses the morphology of virus, preparation method, and the role of
electron microscopy in virus recognition, including a brief history of diagnostic electron
microscopy. The use of EM as a diagnostic tool was compared with current
1.3 Thesis Outline 7 diagnostic kits such as enzyme-immunoassay and latex agglutination tests, and various
molecular techniques.
Chapter 3 covers the basic recognition steps such as segmentation, feature extraction
and classification. The feature extraction methods that are fundamental and used in many
pattern recognition systems, particularly in texture analysis are presented. The classifiers
that will be used in this thesis also are reviewed. This chapter also provides a brief
background on Higher Order Spectra (HOS), highlighting the time and frequency domain
definitions, properties, and naming convention and notations used. The applications of
HOS and motivations of using HOS in feature extraction also are discussed, followed by
feature extraction from a 2-D image and generation of the bispectral invariant features.
Chapter 4 studies the relative importance of phase and magnitude in texture and
contour using a set of images from the Brodatz album. Subsequently, further analysis of
texture and contour was carried out on these images using the indirect method for HOS
feature extraction as outlined in Chapter 3. Noise was added to the images to reflect EM
images that are normally noisy due to the high magnification used to capture the virus.
Averaging of features was used to improve the classification results. The evaluation of
texture and contour features is important for virus recognition because different viruses
appear differently on electron micrographs. The morphological (texture and contour)
8 1.3 Thesis Outline differences appear on the micrograph are useful in distinguishing one virus group from
another.
Chapter 5 presents the verification results of gastroenteric viruses using higher order
spectral features. Experiments were conducted to identify virus population from digitized
EM images of four types of viruses; rotavirus, adenovirus, astrovirus and calicivirus,
whose morphologies are quite similar. Experiments were also conducted on viruses with
similar size. Results are presented in detection error trade-off (DET) plots. Gaussian
Mixture Model (GMM) and Support Vector Machine (SVM) classifiers were used
separately to train for viruses and comparison of results were made. This chapter also
explores the possibility of using higher order spectral invariant features in the application
of 3D virus reconstruction from 2D images.
Chapter 6 compares the bispectral features with Gabor features in virus recognition.
SVM was used to train and classify detectors for virus population from EM images of
Rotavirus, Adenovirus, Astrovirus and Calicivirus. Optimal results were achieved by
trying different kernel widths and number of training features.
Chapter 7 concludes with a discussion of contributions made in this thesis. The
possibilities for future work, which extend upon this thesis, also are included.
1.4 Original Contributions 9 1.4 Original Contributions of the Thesis This research provides original contributions in these areas; (i) A new method of identifying viruses from negative stained EM images.
(ii) New methodologies that take advantage of the large numbers of viral particles in
these images to improve classification accuracy.
A methodology has been developed to recognize viruses from negative stained images
using higher order spectral features. This methodology exploits invariant properties of the
higher order spectral features, statistical techniques of feature averaging, and soft
decision fusion in a unique manner applicable when a large number of particles are
available for recognition, but are not easily registered on an individual basis. Higher order
spectral features that are invariant to similarity transformations and robust to noise have
been defined and applied in previous pattern recognition work, but have not been used in
this application before. In this thesis, the radial spectra of higher order spectral features
were used for classification of viral populations. There are no pervious reports of batch
training and testing over populations of viral particles.
More specifically, the detailed smaller contributions that are associated with the broad
ones listed above are as stated below:
10 1.4 Original Contributions (i) Texture and contour analysis
The evaluation of texture and contour features is important for virus recognition. This
forms a basis for creating an automated virus recognition system because different
viruses appear differently on electron micrographs. The morphological (texture and
contour) differences appearing on the micrograph are useful in distinguishing one virus
group from another. The analysis was carried out using a set of synthetic textures from
the Brodatz album with noise added to the images to reflect the EM images that are
normally noisy. The analytical and empirical evaluations show that the indirect method in
obtaining higher order spectral features as proposed by Chandran et. al. [11] is suitable in
classifying homogeneous texture. This analysis also shows the ability of higher order
spectral features in discriminating a large database of different textures and contours and
averaging of features and input fusion for improvement of classification accuracy.
These experiments have enhanced our understanding of texture and contour classification
of noisy images using higher order spectral features. This knowledge is useful and
provides a foundational work in virus recognition.
(ii) Comparative studies
This study indicates that higher order spectral features perform better than Gabor features
in the application of virus recognition. Comparison of Support Vector Machine (SVM)
and Gaussian Mixture Model (GMM) classifiers to train and classify the viruses shows
1.5 Publications 11
that there is not much difference between these two classifiers in terms of Equal Error
Rate (EER) in classifying subpopulations of 5, 10 and 15 viral particles.
1.5 Publications Resulting from this Research Conference Publications
(i) V. Chandran and H.Ong , “Identification and Classification of viruses in electron
microscope images using higher order spectral features,” Proceedings of Fourth
Australasian Workshop on Signal Processing and Applications (WOSPA), Brisbane,
Australia, 17-18 December, pp. 55-59, 2002.
(ii) H.Ong and V.Chandran, ‘Recognition of viruses by electron microscopy using
higher order spectral features,’ Proceedings of the International Society for Optical
Engineering (SPIE) - Medical Imaging, San Diego, USA, 15-20 February, pp 234-42,
2003.
Journal Publication (i) H.Ong and V.Chandran, ‘Identification of gastroenteric viruses by electron
microscopy using higher order spectral features’ Journal of Clinical Virology,
vol. 34, pp.195-206, 2005.
Chapter 2 Virus Morphology and Electron Microscopy 2.1 Introduction This chapter begins by discussing the morphology of viruses, preparation methods and
the role of electron microscopy in virus identification. The use of electron microscopy as
a diagnostic tool was compared with other current diagnostic kits such as enzyme-
immunoassay and latex agglutination tests, and various molecular techniques. The
advantages and disadvantages of using electron microscopy as a virus recognition tool are
also discussed.
14 2.2 Morphology of Virus 2.2 Morphology of Virus
Viruses come in two basic structures; icosahedral structure of isometric viruses and
helical structure [12]. All known animal viruses, except poxviruses, belong to either of
these two structural types. Icosahedral structure is also known as 5:3:2 symmetry because
it possesses axes of 5-fold, 3-fold and 2-fold symmetry (Figure 2.1). Icosahedral structure
is found in both DNA and RNA viruses. In some virus groups the icosahedral structure
exists as a naked nucleocapsid, while in others it is surrounded by an envelope studded
with projections. The morphological units, or capsomers, of icosahedral viruses are
arranged such that they result in a rigid geometric configuration, an icosahedron, which
has 12 vertices (points), 20 faces (flat sides), and 30 edges. Each vertex, face or edge is
an axis of symmetry. The icosahedral viruses have a relatively constant size and shape
within each genus [13] .
In most icosahedral viruses, the protomers are arranged in oligomeric clusters
(capsomeres), that are readily delineated by negative staining electron microscopy and
form the closed capsid shell as shown in Figure 2.2. The arrangement of capsomeres into
an icosahedral shell permits the classification of such viruses by capsomere number and
pattern. This requires the identification of the nearest pair of vertex capsomeres (called
penton: those through which the fivefold symmetry axes pass) and the distribution of
capsomeres between them. Today, the International Committee on Taxonomy of Viruses
(ICTV) recognizes about 1,550 virus species comprising of 56 separate families and
humans have been found to host 21 of the 26 families specific for vertebrates [10].
2.2 Morphology of Virus 15
Viral morphology has provided the basis for grouping viruses into families. Most viruses
have sufficiently distinctive morphology such that this property can be used to distinguish
one virus group from another.
(a) (b) (c)
Figure 2.1: A basic structure of virus is the icosahedron which composed of 20 facets, each an equilateral triangle and 12 vertices. Due to rotational symmetry it possesses axes of (a) 5 fold symmetry (b) 3 fold symmetry and (c) 2 fold symmetry.
The helical viruses, another virus structure contain RNA which is assembled with protein
subunits into a helical nucleocapsid with the RNA located in a channel in the centre of
the helix. The single axis of symmetry passes longitudinally down this helical centre [13].
Whereas the icosahedral viruses have a relatively constant size and shape within each
family, the helical viruses tend to be heterogeneous in size and are frequently
pleomorphic.
16 2.3 Preparation for Examination by EM
Figure 2.2: Adenovirus after negative stain electron microscopy.
2.3 Preparation for Examination by EM
There are numerous ways to prepare biological material for examination or identification
in the TEM. Technical methodology has reached a point where reproducibility of
specimen preparation is possible. A basic aim is to obtain morphological information
by methods that can be repeated anywhere in the world. Reproducibility strengthens the
belief that micrographs are faithful reflections of the native state of the specimen. One
type of preparation that is used for routine identification of viruses in EM is negative
staining.
2.3.1 Negative Staining
The introduction of negative staining [14] revolutionized the field of electron microscopy
of viruses. Within just a few years, much new and exciting information about the
architecture of virus particles was acquired. The negative stain moulded round the virus
particle, outlining its structure, and also is able to penetrate between small surface
projections and to delineate them. If there are cavities within the virus particle that are
accessible to the stain, they will be revealed and some of the internal structure of the virus
2.4 Role of Electron Microscopy in Virology 17 may be disclosed. Thus, it does not only reveal the overall shapes of particles, but also the
symmetrical arrangement of their components. This information is important to
distinguish one group of viruses from another.
The negative staining technique is simple, rapid and requires a minimum of experience
and equipment. A negative staining technique uses heavy metal salts to enhance the
contrast between the background and the virion’s image. A variety of heavy metal
compounds are available for negative staining. Among the most commonly used stains
are uranyl and tungstate stains and ammonium molybdate [15].
Besides the simplicity, rapidity and minimum of experience needed, negative staining
also has other benefits over the more recent techniques such as vitrification. Viruses
prepared by vitrification usually adopt random orientations in the amorphous ice layer.
Whereas negative staining tends to induce preferred orientations of the molecules on the
carbon support film [16]. This is an advantage because less heterogeneity among the
same virus will increase the classification accuracy when averaging of particles is made
to improve the signal to noise ratio in automated virus recognition.
2.4 Role of Electron Microscopy in Virology
Since the 1930s, electron microscopy (EM) has become an important tool in basic and
clinical virology. The first electron micrograph of poxvirus was published in 1938 [17].
In the early 1940s, immune EM techniques were developed and used in electron
microscopic study of the tobacco mosaic virus [18]. Within a short period of time, EM
18 2.4 Role of Electron Microscopy in Virology was successfully introduced in the differential diagnosis of smallpox and chickenpox
infections [19, 20].
The introduction of negative staining and wider availability of electron microscopes has
encouraged a broad application of EM. In the following years, a great number of
clinically important, previously undescribed agents such as adeno-, entero-, myxo-,
paramyxo-, and reoviruses were indentified. Virus diagnosis by electron microscopy that
relies on the detection and identification of viruses on the basis of size and particle
morphology leads to rapid identification of infection agents. The initial classification of
many agents was therefore based on a combination of morphology and genome structure.
The virions of each of these families display distinct morphologies and this feature can be
used to group them accordingly. In fact, in most cases, this morpho-diagnosis combined
with clinical information is sufficient to permit a provisional diagnosis or rule out a more
serious infection and to initiate treatment without waiting for other test results [2].
EM is not suitable for a mass screening of clinical specimens because virus identification
by visual examination is quite time consuming. In addition, the identification of the
viruses is critically dependent on the skill and experience of the microscopist or
virologist. Therefore, in the following years a number of ‘modern’ techniques, such as
immunologic and molecular methods based on nucleic acid amplification have been
developed. Both of these techniques have several limitations compared to EM. (discussed
in Section 1.1). One of the primary limitations is that a priori notion of the virus identity
must be known to be able to select the appropriate reagents.
2.5 Summary 19
When compared to these diagnostic tests, diagnostic EM differs in its rapidity and its
undirected ‘open view’ [2]. A specimen can be ready for examination and an
experienced virologist or technologist can identify a viral pathogen morphologically
within 10 minutes. Nucleic acid amplification and enzyme assays normally have to wait
overnight.
Nevertheless, there are several limitations that exist in the identification of viral agents
using negative stain electron microscopy. First is the need for high particle concentrations
(106 ml-1), which means this method might not suit all viruses. It has been widely
accepted that this threshold of virus concentration is too high and may be reduced [3].
Many researchers have tried different methods such as using different coating on the
grids [21] to improve virus concentration on the grids. Other techniques include agar
filtration [22], sedimentation [23] and bioaffinity [24] which have successfully increased
the concentration of virus particles on the grid.
Other disadvantages of EM include the high maintenance cost of the equipment, other
capital expenses such as room or space, labor intensive nature and lack of career structure
for operators. Biel and Madeley [3] have discussed and addressed most of these issues.
2.5 Summary This chapter explores the morphology of virus, which comes in two basic structures;
icosahedral structure of isometric viruses and helical structure. Some of the current
diagnostic techniques also were discussed and compared with EM. One of the limitations
20 2.5 Summary
of EM includes the skill and experience needed by the microscopist or virologist in
interpreting these images. This aspect of EM (ultrastructural interpretation) can take
many years of training to master. To examine every specimen individually is quite time
consuming and can be a burdensome task. Thus, having an automated virus recognition
that can identify viruses from EM images can help to overcome these issues.
Chapter 3
Towards Automated Virus
Recognition 3.1 Introduction
Virus recognition is the process of recognizing a virus particle or population based on
their morphology (texture and contour). As stated in Chapter 2, the morphological
differences arise from the arrangement of protomers in oligomeric clusters or capsomeres
that are evidenced through negative staining of the viral particle.
Virus recognition may be classified into two main areas, that is, virus identification and
verification. Virus identification is the means of identifying a virus from a group of
viruses. Virus verification is the systematic process of accepting or rejecting (as a binary
decision) the claimed identity of a virus based on the sample morphology.
22 3.2 Segmentation
Automated virus recognition system involves steps such as segmentation of the viral
particles, feature extraction and also classification. In this thesis, the major focus will be
on feature extraction methods and classification. Segmentation methods proposed by
Shark [6] and Utagawa [25] that automatically segment the individual virus particles can
be incorporated to make the system fully automated. In this chapter, feature extraction
methods that are fundamental and used in many pattern recognition systems, particularly
in texture analysis are presented. The core method of virus recognition system developed
for this thesis is also addressed in this chapter.
A brief background on Higher Order Spectra (HOS), highlighting the time and frequency
domain definitions, properties, and naming convention and notations used are discussed.
This chapter also discusses the applications of HOS and motivations of using HOS in
feature extraction. This is followed by feature extraction from 2-D images and generation
of the bispectral invariant features using the direct and indirect method as proposed by
Chandran et. al. [11]. Classifiers that will be used in this thesis are also reviewed.
3.2 Segmentation Segmentation refers to the process of differentiating between the objects of interest (the
foreground objects) and the background. Segmentation of the negative stained EM
images of viruses is complicated because a significant number of complex artificial
objects usually are present and the images have non-uniform illumination. This can be
3.2 Segmentation 23
illustrated by a typical virus image taken from an electron microscope as shown in Figure
3.1.
It can be seen that a virus image contains not only good virus cells (approximately
circular in shape with a non-uniform internal structure consisting of a characteristic
pattern or texture), but also some corrupted virus particles with no diagnostic value, and
some unwanted artificial objects. The image also reveals non-uniform illumination. All
these factors must be taken into consideration to ensure that the virus particles can be
distinguished from the dark background and from the unwanted objects in the image, as
well as ensuring the size of the segmented viruses is accurate [6].
Figure 3.1: An EM image of Adenovirus. A significant number of complex artificial objects that normally present and non-uniform illumination of the image makes segmentation of the virus particle a challenging task.
24 3.2 Segmentation Researchers [6] in this area have developed a method of segmentation of viruses from
electron microscope images based on two main strategies a) Image histogram shaping by
a combination of filtering, edge detection and morphological operations, thereby
enabling regional threshold values to be accurately estimated via model fitting with the
initial parameters derived from the fuzzy c-means clustering technique, and b) image
segmentation based on a threshold value adaptive to local image brightness, thereby
resulting not only in the removal of artificial objects, but also in the accurate estimation
of the size and the position of virus cells. The effectiveness of this method was tested on
actual images of viruses such as Rota-, Adeno-, Astro- and Calicivirus. All the viruses
were extracted correctly with accurate size estimation except for one virus in one of the
images tested.
Recently an automated specimen search system in which a microtracing device is
installed on a Transmission Electron Microscope (TEM) has been developed and is
commercially available. Utagawa [25] demonstrated the possibility of applying the
automated specimen search system installed in an electron microscope to virological
studies, especially the detection of caliciviruses in semipurified stool samples.
In this system, TEM images of inspection specimens are recorded using a built-in TV
camera. The particles are automatically detected from digitized images and input
parameters such as diameters and the roundness of particles of interest. The detected
images are automatically stored in memory with their specimen positions (X,Y
coordinates), operating magnifications, accelerating voltages and other TEM operating
3.3 Feature Extraction 25
conditions put into a database. The result shows an accuracy of more than 95% for
detection of a single Calicivirus particle in a purified virus fraction. This system is useful
for clinical diagnosis without the need for operator intervention. These segmentation
methods can be applied and incorporated with our system to have a fully automated virus
recognition system in the future.
3.3 Feature Extraction Feature extraction is an essential component in any recognition system. The goal of a
feature extractor is to characterize an object to be recognized by measurements whose
values are similar for objects in the same category, and different for objects in different
categories. In a broad sense, features may include both text-based descriptions
(keywords, annotations, etc) and visual features (color, texture, shape, spatial
relationships etc). This section overviews some of the feature extraction methods that are
fundamental and used in many pattern recognition systems, particularly in texture
analysis. The core approach or method used in this thesis will be largely addressed in this
section as well.
According to Tuceryan and Jain [26], methods for texture feature extraction are
categorized into four classes i.e statistical, model-based, geometrical, and signal
processing or filtering methods. Statistical and signal processing methods are the ones
that are widely used. Statistical methods are comprised of techniques such as
co-occurrence matrices and autocorrelation (or power spectrum) function features. The
26 3.3 Feature Extraction co-occurrence matrix estimates the image properties based on the orientation and the
distance between the pixels and summarizes them into meaningful statistics.
The autocorrelation method is based on finding the linear spatial relationships between
primitives. If the primitives are large, the function decreases slowly with increasing
distance, whereas it decreases rapidly if texture consists of small primitives. For periodic
primitives, the autocorrelation increases and decreases periodically with distance. The
power spectrum, which is the Fourier transform of the autocorrelation function, shows the
directionality of the texture. Bispectral (or bicorrelation) features also were used in
texture analysis [27].
Most techniques in the signal processing method try to compute certain features from the
transformed images which are then used in classification or segmentation tasks. In the
early 90’s, the wavelet transform was investigated for texture representation [28] by
decomposing the texture into its frequency components. Another filtering technique,
Gabor filters also was applied in texture analysis. Gabor filters are frequency and
orientation selective and have some desirable optimization properties.
Following is a detailed review of some of the techniques for feature extraction that are
commonly used in texture analysis.
3.3 Feature Extraction 27 3.3.1 Co-occurrence Matrices
To capture the spatial dependencies in the image gray levels, a simple histogram is not
adequate because the image is 2-D [29]. This is the motivation behind the development of
Gray Level Co-occurrence Matrices (GLCM), which along with the autocorrelation
function is the most extensively used statistical method for texture
analysis. The co-occurrence matrix ),( jiPd of an N x N image I is defined mathematically
as the probability of two pixels, (r, s) and (t, v) that are separated by distance d, having
grey level values of i and j respectively. This can be expressed as
NNjvtIisrIvtsrjiPd
|}),(,),(:)),(),,{((|),( === (3.1)
where | | represents the cardinality of a set. Although GLCM features have proven to be
useful in many texture classification tasks, they have a number of disadvantages [26]. The
features extracted depend heavily on the choice of the displacement parameter and
currently there is no established method of choosing these parameters for optimal texture
characterization. Calculating the GLCM for a large number of distances and angles is
computationally expensive, and leads to an excessive number of total features.
28 3.3 Feature Extraction 3.3.2 Autocorrelation
The autocorrelation of an image ),( yxI can be given by
∑ ∑∑ ∑
= =
= = ++= N
0uN
0v2
N0u
N0v
vuyvxuIvuIyx
I ),(),(),(),(ρ (3.2)
Autocorrelation has been used in a wide range of applications such as character
recognition [30], affine-invariant texture classification [31], time series classification [32]
and face detection and recognition [33]. However, in most cases, the applicability of the
autocorrelations have been limited to first or second order due to high computational
costs. In texture analysis, the autocorrelation function is useful for capturing
repetitiveness in texture patterns. It can also be used to measure the scale of the texture
primitives from spacing of the repetitiveness.
3.3.3 Wavelet Features
Wavelet features have demonstrated good discriminance in the analysis and classification
of textures [34, 35]. There are a number of ways to approach wavelet transform [36]. It
can be approached from the mathematical standpoint of scaling functions and wavelets
[37], as well as in terms of averaging and detail of discrete sequences [38], filtering
operations [39] or multiresolution analysis [34]. Even though a researcher may focus on a
particular approach in interpreting wavelets, all the other remaining approaches are still
evident, as they are all interrelated. The concept of multiresolution for instance, is
3.3 Feature Extraction 29 closely related to the space spanned by scaling functions and wavelets, which is related to
the filter coefficients of these functions.
The wavelet transform is defined as decomposition of a signal )()( 2 RLtf ∈ into a family
of functions )(, tnmψ obtained through translation and dilation of a kernel function
)(tψ known as mother wavelet:
(3.3)
where m and n are the scale and translation indices, respectively. The mother wavelet is
constructed from the scaling functions )(tφ as follows:
)2()(2)( ktkhtk
o −= ∑∞
−∞=
φφ (3.4)
)2()(2)( 1 ktkhtk
−= ∑∞
−∞=
φψ (3.5)
where )(kho and )(1 kh are coefficients for low-pass and high-pass filters, respectively.
)1(*)1()(1 khkh o
k −−= (3.6)
The earliest such features involved calculating the energy, or a similar measure, present
in each of the subbands resulting from the wavelet decomposition of an image [40, 41]. It
has been shown that such features perform reasonably well in classification and
segmentation tasks. Using multiple analyzing, wavelets have also been shown to improve
)2(2)( 2/ ntt mm −= −− ψψ
30 3.3 Feature Extraction
the overall classification accuracy, as each can detect different characteristic features of
textures [42].
Other first order statistics are computed by constructing a histogram of the wavelet
coefficients at each level [34]. To obtain such a histogram, uniform quantization of the
coefficients is used. The wavelet packet transform has been extensively used in texture
analysis as it allows greater resolution in the frequency domain. Lee and Pun [43] use the
wavelet packet transform to select only the dominant energy band for use as texture
features, allowing for good classification accuracy at reduced computational expense.
3.3.4 Gabor Features
Gabor functions were introduced by Dennis Gabor in 1946 in his ‘Theory of
Communication’. The purpose was to represent a signal in both the time and frequency
domains simultaneously. The work of Gabor was extended to 2-D by Daugman [44]. The
Gabor filter takes the form of a 2-D Gaussian modulation complex sinusoidal grating in
the spatial domain [45], which is given by:
)(2exp),(),( VyUxjyxgyxh +−′′= π (3.7)
where ),( VU defines the position of the filter in the Fourier domain with a centre
frequency of 22 VUf += and an orientation of )/arctan( UV=θ . The term
),( yxg ′′ represents a Gaussian function orientated at an angle φ , where ),( yx ′′ are the
3.3 Feature Extraction 31 rotated co-ordinates given by φφ sincos yxx +=′ and φφ cossin yxy +−=′ . The general
form of the Gaussian function is:
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
22
5.0exp2
1),(yxyx
yxyxgσσσπσ
(3.8)
where yx σσ / is known as the aspect ratio, which determines the eccentricity of the
Gaussian envelope. The product of xfσ determines the spatial frequency bandwidth b.
In recent times, Gabor filters have emerged as one of the most commonly used techniques
in the field of texture analysis. They are useful because they capture frequency and
orientation, which are important properties. By applying a rotation-invariant transform to
these features, it is then possible to create a set of texture descriptors that is invariant to
rotation. In this thesis, Gabor features will be compared with higher order spectral
features in virus recognition.
3.3.5 Higher Order Spectra 3.3.5.1 Time Domain Definition and Properties
Higher order spectra [46] consist of moment and cumulant spectra and can be defined for
both deterministic signals and random processes. If {X (k)}, k = 0, ±1, ±2, ±3…. is a real,
stationary, discrete-time signal and its moments up to order p exist, then
( ) ( ) ( ) ( ){ }11121 ...,...,, −− ++= pp
xp kXkXkXEm τττττ (3.9)
32 3.3 Feature Extraction represents the pth-order moment function of the signal, where E is the expectation
operation. The expectation operation can be performed over the ensemble (ensemble
averaging) or along the time dimension (time averaging). For ergodic processes, the two
expectations are the same. For deterministic signals, there is no ensemble averaging
possible. The first order moment xm1 is the mean value and second order moment )(2 τxm
is the auto-correlation function. The third-order moment ),(3 kjxm ττ is often called
bicorrelation function, and fourth order moment function ),,(4 srqxm τττ is often called
tricorrelation function, and so on.
There exists a general relationship between moment and cumulant functions. If we
consider a zero-mean stationary random process )(tx , then the relationship between these
two functions (or sequences) becomes quite simple. Specifically, the second and third
order moment are the same as the second and third order cumulant, respectively [47]
given by:
( ) ( ){ }τττ +== txtxEmc xx )()( 22 (3.10)
( ) ( ) ( ) ( ) ( ){ }2133 ,, ττττττ ++== txtxtxEmc kjx
kjx (3.11)
The relationship between the fourth order cumulant and moment is as follows:
)()()()()()(),,(),,( 22222244 qr
xs
xqs
xr
xrs
xq
xsrq
xsrq
x mmmmmmmc τττττττττττττττ −−−−−−= (3.12)
3.3 Feature Extraction 33 By putting 0=== srq τττ into the equations above, we obtain;
( ){ } )0(222
xckXE ==γ (variance) (3.13)
( ){ } )0,0(333xckXE ==γ (skewness) (3.14)
( ){ } ( ) )0,0,0(3 4
2244
xckXE =−= γγ (kurtosis) (3.15)
The following are important properties that any pth-order cumulants satisfy [48]
(i) Scaled quantities: The cumulants of scaled quantities equal the product of all
the scale factors times the cumulant of the unscaled quantities, i.e., if iλ , i = 1,2, …, p
are constants and ix , i = 1,2, …, p are random variables, then:
),...,,(),....,,( 21
p
1i2211 Π ppp xxxcumxxxcum
i⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
==
λλλλ (3.16)
(ii) Symmetry: Cumulants are symmetric in their arguments, i.e.,
),...,,(),....,,( 2121 ipiip xxxcumxxxcum = (3.17)
where ( 1i ,..., pi ) is a permutation of (1,…, p); interchanging the arguments of the
cumulant in any way does not change its value, e.g.:
),,(),,(),,( 132412343214 τττττττττ CCC ==
34 3.3 Feature Extraction
(iii) Additivity: Cumulants are additive in their arguments, that is the cumulants of
sums equal sums of cumulants. For example, even if ox and oy are not statistically
independent, it is true that
),...,,(),...,,()....,,( 111 popopoo zzycumzzxcumzzyxcum +=+ (3.18)
(iv) Additive constants: Cumulants are insensitive to additive constants, that is, for
α constant:
),...,(),....,( 11 pp zzcumzzcum =+α (3.19)
(v) Sums: The cumulants of a sum of statistically independent quantities equals the
sum of the cumulants of the individual quantities, i.e., if the random variable [xi] are
independent of the random variables [ iy ] for i = 1,2, …, p then:
),...,,(),...,,(),...,,( 21212211 pppp yyycumxxxcumyxyxyxcum +=+++ (3.20)
Note that if ii yx ... were not independent, then from equation 3.18 there would be 2p terms
on the right hand side. Statistical independence reduces these terms to just 2.
(vi) Independent subsets: If a subset of the random variables is independent from the
rest, then
0)....,,( 21 =pxxxcum (3.21)
3.3 Feature Extraction 35 3.3.5.2 Frequency Domain Definition and Properties The Weiner-Khintchine relation [49] indicates that the spectral density function
)(wM xp and the correlation function )(τx
pm constitute of Fourier transform pair, that is
)()( wMm x
pxp ↔τ (3.22)
where w denotes frequency, and ↔ denotes a Fourier transform pair. The second order
moment spectrum is the classical power spectrum, and the third order moment spectrum
is the bispectrum and fourth-order is frequently referred to as the trispectrum. However,
the words bispectrum and trispectrum also are used to describe third-order and fourth
order cumulant spectra. The second-, third-, and fourth-order cumulant spectra also are
known as the power spectrum, bispectrum, and trispectrum, respectively as shown below:
General formula
⎥⎦
⎤⎢⎣
⎡−= ∑∑∑
−
=
∞
−∞=−
∞
−∞=−
−
1
1121121
1
exp),...,(...),...,,(p
iiipppp wjCwwwS
p
ττττττ
(3.23)
Power Spectrum: p =2
[ ]∑∞
−∞=
−=1
111212 exp)()(τ
ττ jwCwS (3.24)
In the above definitions, it is assumed that the moment or cumulant functions satisfy the
conditions necessary for a Fourier (spectral) representation. This implies that they decay
36 3.3 Feature Extraction and are at least square integrable. For discussion on existence of polyspectra for random
processes, refer to [50]. For a deterministic signal x(n), the power spectrum can be
expressed in terms of the Fourier transform of the underlying signals as:
)()(*)(2 wXwXwS =
Bispectrum: p =3
The bispectrum is the 2D-Fourier transform of the third cumulant function:
( ) ( ) ( )[ ]2211213213 exp,,1 2
τττττ τ
wwjCwwS +−= ∑ ∑∞
−∞=
∞
−∞=
(3.25)
for πππ ≤+≤≤ 2121 and,, wwww . For a deterministic, zero-DC signal the bispectrum
may be expressed in terms of the Fourier transform of the underlying signal since:
( ) ( ) ( ) ( ) ( )[ ]221121213 exp,1 2
τττττ τ
wwjnxnxnxwwSn
+−++= ∑ ∑ ∑∞
−∞=
∞
−∞=
∞
−∞=
(3.26)
setting knmn =+=+ 21 and ττ and splitting the exponent yields:
( ) ( ) [ ] ( ) [ ] ( ) ( )[ ]⎭⎬⎫
⎩⎨⎧
+⎭⎬⎫
⎩⎨⎧
−⎭⎬⎫
⎩⎨⎧
−= ∑∑∑∞
−∞=
∞
−∞=
∞
−∞= nkm
nwwjnxkjwkxmjwmxwwS 2121213 expexpexp,
)(*)()( 2121 wwXwXwX += (3.27)
Note that in the expressions above for the power spectrum and the bispectrum of a
deterministic signal, these spectra are Fourier transforms of the time averaged (or sample)
3.3 Feature Extraction 37 correlations of order 2 and 3 respectively, and there is no ensemble averaging. The
sample correlations are themselves deterministic functions of the lag variables in these
cases, and must satisfy conditions for existence of their Fourier transforms. If x(n) is a
finite duration sequence (as is the case for all cases in this thesis and for finite 1-D
patterns or finite length projections of 2-D patterns) the existence is guaranteed.
The symmetry conditions of ),( 213 wwS follow from those of the third cumulant, namely:
),(),( 123213 wwSwwS =
),(*),(* 213123 wwSwwS −−=−−=
),(*),(* 21132213 wwwSwwwS −−=−−=
),(*),(* 21231213 wwwSwwwS −−=−−=
Thus, knowledge of the bispectrum in the triangular region π≤+≥≥ 21122 ,,0 wwwww is
sufficient to describe the rest (Figure 3.2). This region (labeled 1) is often termed the
principal region of the bispectrum.
38 3.3 Feature Extraction
Figure 3.2: The symmetry regions of the bispectrum (labeled 1). The region labeled 1 contains a unique set of values and those in the other labeled regions can be mapped to these.
3.3.5.3 Motivations for Using Higher Order Spectra in Feature Extraction A wide range of applications has developed due to these unique properties of higher order
spectra such as image reconstruction, pattern recognition, image restoration and edge
detection. As mentioned by Chandran et. al. [11], the motivations for using higher order
spectra in feature extraction are as below:
(i) Higher order spectra preserve both amplitude and phase information from the
Fourier transform of a signal, unlike the power spectrum. The phase of the Fourier
transform contains important shape information [51].
1
2
34
5
6
7
8
9 10 11
12
w2
w1
w2=w1
w2+w1= π
-(w2+w1)= π
w1= - π
w2= π
w2= - π
w1= π
3.3 Feature Extraction 39 (ii) The Fourier phase of a one-dimensional (1-D) pattern is a shape-dependent
nonlinear function of the frequency and higher order spectra can extract this
information.
(iii) Finite-length patterns that are symmetric about their centres of support have a
Fourier phase that is a linear function of frequency. Therefore, the shape
information resides in the Fourier magnitudes. However, the Fourier magnitude
for positive frequencies is an asymmetric function whose shape is related to the
shape of the original input. Therefore, higher order spectra can still be used to
extract features indirectly from Fourier magnitudes.
(iv) Higher order spectra are translation invariant because linear phase terms are
cancelled in the products of Fourier coefficients that define them. Functions that
can serve as features for pattern recognition can be defined from higher order
spectra that satisfy other desirable invariance properties such as scaling,
amplification and rotation invariance.
(v) Higher order spectra are zero for Gaussian noise and, thus provide high noise
immunity to features.
(vi) Multidimensional signals can be decomposed into 1-D projections.
Transformations such as shift, scaling, or rotation of the multidimensional signal
can be related to shift or scaling of the projections. Higher order spectral features
40 3.3 Feature Extraction
derived from the projections can be used to derive invariant features for the
multidimensional signal.
3.3.5.4 Rotation, Translation and Scaling (RTS) Invariants Generation The bispectrum of a 1-D deterministic sequence, x(n) may be defined as in (3.27). It is
assumed that the sequence is oversampled for all scales of interest such that X(f) = 0 and
for f ≤ 1/2 , with the frequency f normalized by the Nyquist frequency.
The bispectrum (3.27) is a triple product of Fourier coefficients, and is a complex-valued
function of two frequencies, similar to the power spectrum, )(*)()( fXfXfP = , a
function of only one frequency. As mentioned earlier, the bispectrum retains information
about the phase of the Fourier transform of a sequence. For a symmetric sequence of
finite extent, the phase of the Fourier transform is a linear function of frequency, thus the
biphase is zero. As for an asymmetric sequence, the phase of the Fourier transform is a
nonlinear function of frequency and this nonlinearlity is extracted by the biphase.
Based on these properties, Chandran and Elgar [52] proposed a type of bispectral
projection that yields DC level, amplification, scale and translation invariant features
from the input.
These features are defined as;
))(/)(arctan()( aIaIaP ri= (3.28)
where
3.3 Feature Extraction 41
∫+
+==+= )a1/(10f 111ir 1
df)af,f(B)a(jI)a(I)a(I (3.29)
for 0< a ≤ 1, and j = √-1. The bispectral values are integrated along straight lines with
slope a passing through the origin in the bifrequency space are shown in Figure 3.3. Refer
to [52] for the discrete-time version of )a(I . In practice, the fast Fourier transform (FFT)
is used to obtain )K(X where NKf = , 12N,,....1,0K −= and the integral in (3.29) is
computed as a summation.
Figure 3.4 shows the flow chart of computation of these invariant parameters. P(a) is
invariant to translation and scaling. The direct and indirect methods can be used to
compute these invariant parameters. It has been shown [52] that the indirect method that
discards all the phase information from the original sequence achieves better scale
invariance. For different sequences that have the same DFT magnitude, the direct
procedure can be use to classify them.
42 3.3 Feature Extraction
Figure 3.3: Owing to symmetries of the bispectrum in equation (3.27), the bispectrum possesses redundancy and needs to be only computed for the triangular region shown above. Features are extracted by integrating the bispectrum along a radial line as shown and taking the phase of the complex-valued integral. 1f and 2f are frequencies normalized by one half of the sampling normalized frequency.
Invariance properties of the bispectral features that have been proved in [11] is shown
below:
Claim: P(a) are translation invariant
Proof: Translation produces linear phase shifts of sequence x(n) that cancel in (3.27).
Integrating the bispectrum along lines passing through the origin in bifrequency space
preserves the translation because the integral is translation invariant if the integrand is.
The phase of this complex entity I(a) must also be translation invariant because its real
and imaginary parts are. Thus, P(a) are translation invariant.
3.3 Feature Extraction 43
Claim: P(a) are scale invariant
Proof: Scaling the sequence x(n) results in an expansion or contraction of the Fourier
transform that is identical along the f1 and f2 directions. The real and imaginary parts of
the integrated bispectrum along a radial line are multiplied by identical real-valued
constants upon scaling and therefore the phase, P(a) of the integrated bispectrum is
unchanged.
Rotation invariance is achieved by deriving invariants from the Radon transform of the
image and using the cyclic-shift invariance property of the discrete Fourier transform
magnitude [11, 52].
3.3.5.4.1 Feature Extraction from 2-D Images The 1-D bispectral invariant features can be applied to 2-D images by taking Radon
transform projections and computing features from these projections [11]. Let g(u,v) be
an N x N image and {xθ(n)} be the Radon transform of the image, that is, a set of parallel
beam projections of the image at angles θ with respect to the horizontal axis as shown in
Figure 3.5. The 2-D image thus is reduced to a set of 1-D projections by computing
projections at equal increments of angle θ, or θi = iπ/ Nθ for i = 0 to Nθ – 1.
Features P(a) that are invariant to shift, scaling, or amplification of the 1-D function, x(n)
will therefore provide a set {P(a)(θ), a=1, Na } of 1-D functions of θ, which are cyclically
shifted when the image is rotated. The radial spectra of the bispectral invariants, P(a,ωθ)
44 3.3 Feature Extraction
where ωθ is a frequency in cycles per 180 degrees is a set of features that are invariant to
rotation, translation and scaling.
Figure 3.4: The direct and indirect methods for computation of invariant parameters for a 1-D sequence.
Input n
x(n)
DIRECT INDIRECT
FFT
Triple products X(f1)X(f2)X*(f1+f2)
Integrate along line of slope a
Parameter P(a)
DFT Magnitude For positive frequencies
3.4 Classifiers 45
Figure 3.5: The Radon transform of a 2-D image yields 1-D parallel beam projections, )n(x at various angles, θ .
3.4 Classifiers
An important consideration in the problem of texture classification is that of classifier
design. Most evaluations of texture features rely on simple classifiers, such as minimum
distance and nearest neighbour classifiers, while other researchers have used artificial
neural networks to good effort.
Classifiers for pattern recognition tasks have been widely studied. In theory, the Bayes
classifier is optimal for any problem, as it minimizes the probability of error. The true a
posteriori probability that an observation x belongs to a class iw is given by Bayes rule
∑ =
= Nn nn
iii
wxpwP
wxpwPxwPr
1)|()(
)|()()|( (3.30)
46 3.4 Classifiers
where N is the total number of classes, )( iwP is the a priori probability of being in class
iw and )|( iwxp is the true conditional density function for the class iw .
In practical applications, this true a posteriori probability can never be achieved, because
the true conditional density functions )|( nwxp can never be known with finite training
data. An estimate of this probability is given, such that
)()|()|( xxwPrxwPr ii ∈+=∧
(3.31)
where )(x∈ is an error term due to the limitations stated above. Thus the aim of a
classifier system is to provide an estimate of )|( iwxp such that this error is minimized.
The following section discusses some of the classifiers that are commonly used in texture
analysis.
3.4.1 Parametric and Non-parametric Classifiers
Classifiers are categorized into two types, parametric and non-parametric classifiers
which depend on the method used to find this decision function. Parametric classifiers
use the statistics of the data to implement the best discriminant function. The error of
classification in both supervised and unsupervised learning can be minimized with the
parametric approach, because this becomes a deterministic optimization problem. Non-
parametric classifiers do not use such parameters, but classify the features in feature
space using statistical optimality criteria.
3.4 Classifiers 47
3.4.2 Gaussian Mixture Models
A Gaussian Mixture Model (GMM) is a parametric model used to estimate a continuous
probability density function from a set of multi-dimensional feature observations. A
GMM probability density is described by the additive contribution of N multi-
dimensional Gaussian components. The Gaussian mixture model is described by mixture
component weights iw , means iµ and covariances iΣ . For a single observation, x, the
probability density given a GMM described by λ given by:
( ) )Σ,|(| iiµxx gwpN
ii∑
=
=1
λ (3.32)
The probability density of a single Gaussian component of D dimensions is given by:
( )( )
( ) ( )⎟⎠⎞
⎜⎝⎛ −−−= −
iiii µxµxµx 1i
iD 2
1exp2
1g Σ'|Σ|
Σ,|π
(3.33)
The vector or matrix transpose is represented by (').
The solution for determining the parameters of the GMM is by the Maximum Likelihood
(ML) parameter estimation criterion. The joint likelihood of T independent and
identically distributed feature vector observations, { }T21 xxxX ....,,= , may be specified
according to Equation 3.34,
( ) ( )∏=
=T
t
pp1
λλ |x|X t (3.34)
48 3.4 Classifiers This may conveniently be represented in log form as
( ) ( ) ( )∑=
==T
t
ppL1
|log|log λλλ txX (3.35)
In terms of the mixture component densities, the log-likelihood function to be maximized
is given by:
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛= ∑∑
==
N
iii
T
t
gwL11
log Σ,| it µxλ (3.36)
The model parameters are estimated such that they maximize the likelihood of the
observations. A method for maximizing the log-likehood of the observations is by the
general form of the Expectation-Maximization (E-M) algorithm. The E-M algorithm,
given the parameters of an initial estimate ⎭⎬⎫
⎩⎨⎧
⎭⎬⎫
⎩⎨⎧
⎭⎬⎫
⎩⎨⎧
⎭⎬⎫
⎩⎨⎧=
∧∧∧∧∧∧∧
NNww Σ,...,Σ,,...,,..., 11 N1 µµλ ,
will determine new estimates of the parameters, { } { } { }{ }NNww Σ,...,Σ,,...,,,..., 111 Nµµ=λ
such that ( ) ⎟⎠⎞
⎜⎝⎛≥
∧
λλ || XX pp . On each EM iteration, the parameters are updated using
the following equations:
Mixture Component Weights:
∑=
∧
⎟⎠⎞
⎜⎝⎛=
T
ti iPr
Tw
1
1 λ,| tx (3.37)
3.4 Classifiers 49 Mixture Component Means:
∑
∑
=
∧
=
∧
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
=T
t
T
t
iPr
iPr
1
1
λ
λ
,|
,|
t
tt
i
x
xxµ (3.38)
Mixture Component Covariances:
( )( )
∑
∑
=
∧
=
∧
⎟⎠⎞
⎜⎝⎛
′−−⎟⎠⎞
⎜⎝⎛
=T
t
T
t
i
iPr
iPr
1
1
λ
λ
,|
,|Σ
t
ititt
x
µxµxx (3.39)
The a posteriori probability for the Gaussian mixture component class i is given by
∑ =
∧∧∧∧
∧∧∧∧
∧
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
=⎟⎠⎞
⎜⎝⎛
N
j jjj
ii
ggw
ggwiPr
1Σ|
Σ|,|
µx
µxx
t
it
t λ (3.40)
One of the important attributes of the GMM is its ability to form smooth approximations
for any arbitrarily-shaped densities. As ‘real world’ data has multi-modal distributions,
GMM provide an excellent tool to model the characteristics of the data. Another
extremely useful property of GMM is the possibility of employing a diagonal covariance
matrix instead of the full covariance matrix [53]. Thus, the amount of computational time
and complexity can be reduced significantly. GMMs have been used widely in many
areas of pattern recognition and classification, and with great success in the area of
speaker identification and verification [54, 55].
50 3.4 Classifiers 3.4.3 Support Vector Machines
In the recent years, Support Vector Machines (SVMs) classifiers have demonstrated
excellent performance in a variety of pattern recognition problems [56-58]. Support
vector machines are particularly suited for binary classification tasks. In SVMs, patterns
from each class are categorized based on their location with respect to a hyperplane. The
hyperplane is obtained from the training data, in such a way that the best discriminance is
achieved even to unseen data. The advantage of SVMs is that complexity of the classifier
is defined not by the dimensionality of the transformed space, but rather by the number of
support vectors used to define the boundary hyperplane.
An SVM binary classifier selects an indicator function { }1,1: −aNRf from a class of
functions Λ∈αα ),,(xf , such that the function f correctly classify the test
examples ( )y,x . Like all learning machines, an SVM seeks to minimize the expected
error of classification. Since the underlying probability distribution of the data in each
class ),( yxP is unknown, the observed training data ( ) ( ) ( ){ }nn yxyxyx ,,...,,,, 2211 is used
to approximate it. This, implies approximating the actual risk by the empirical risk
(empirical in the sense that it is obtained from the data and not the distribution). Hence,
minimizing the empirical risk somehow contributes to the minimization of the actual risk.
A better estimation of the actual risk can be obtained in addition to the empirical data, the
confidence interval is considered. Hence the actual risk is bounded by the empirical risk
and the confidence interval as the following equation shows:
3.4 Classifiers 51
),(][][ hnfRfR emp Φ+≤ (3.41)
where R[ f ] is the actual risk, Remp[ f ] is the empirical risk and Φ(n,h) is the confidence
interval and is explicitly given by
( )
nnnh
hn4ln12ln
),(η−⎟
⎠⎞
⎜⎝⎛ +
=Φ (3.42)
which is a function of the training sample count n and the VC-dimension h of the
function class we seek. The probability that the bound in equation (3.41) holds is η−1 .
The VC-dimension h of a function is defined as the largest number of points that it can
correctly classify, when the training examples are drawn from an underlying distribution
( ) { }1, ±×∈ Nii Ryx hi ,....,2,1= in all the possible 2h ways.
The minimization of the right hand side of (3.41) is known as Structural Risk
Minimization (SRM), where we seek to minimize the upper bound on the actual risk, by
minimizing either Remp[ f ] or Φ(n,h). This principle of SRM is implemented by
expressing the concept of VC dimension in practical terms, such as margin of the optimal
separating hyperplane (OSH). The margin is defined as the minimum distance of a
sample to the decision surface as in Figure 3.6.
52 3.4 Classifiers The OSH f(x) is a linear function expressed as:
f(x) = (w.x) + b (3.43)
where the length of weight vector w can be used as a measure of the margin length, and b
is an offset parameter. The weight vector may not be easily found in the input space. The
coefficients w and b are found by solving the quadratic programming problem [58, 59].
The solution (or hyperplane) is generally sought in a feature space that is higher than the
input space. This feature space of higher dimension is a result of a non-linear transform
of the input space. It is easier to obtain a linear hyperplane in the higher dimensional
space. The solution is obtained using optimization theory and involves the inner product,
which appears as we proceed in the quest of obtaining w. This ubiquitous inner product
expression is called the kernel. There is a freedom of choice of kernels depending on the
nature of the data being handled. Some of the common kernels are polynomial kernels,
Gaussian RBF's and the Sigmoidal [60]. Table 3.1 shows the descriptions of these
kernels.
3.4 Classifiers 53
-5
0
10
5
5
0 10
*1
*6
*8
*4
*5
o
2
o
7 o
9
o10o
3
Margin
-5
0
10
5
5
0 10
*1
*6
*8
*4
*5
o
2
o
7 o
9
o10o
3
Margin
Figure 3.6: SVM Margin
Kernel Description k(x,y) Polynomial dyx )).(( θ+
Gaussian RBF ⎟⎟⎠
⎞⎜⎜⎝
⎛ −−β
2||||exp yx
Sigmoidal )).(tanh( θ+yxk Inverse Multiquadric
22||||1
β+− yx
Table 3.1: Some commonly used kernels. The method to determine these kernels parameters are illustrated in [60].
54 3.5 Summary 3.5 Summary
In this chapter, the process of creating an automated recognition system was discussed
with emphasis on feature extraction and classification methods. Of particular interest is
the Gabor features which will be compared with higher order spectral features in virus
recognition and Support Vector Machine (SVM) and Gaussian Mixture Model (GMM) as
classifiers. The remaining sections of the chapter outlined a number of popular feature
extraction techniques which have been used over the years such as co-occurrence
matrices, autocorrelation and wavelet. Only a brief summary of each technique is
presented due to the large body of material on the subject in the literature and research
activity currently in progress.
This chapter provides a brief background of HOS and its properties. This chapter also
presents an algorithm for invariant feature extraction from 1-D sequence and extends the
algorithm to 2-D images. This algorithm used radial spectra of higher order spectral
features to provide desirable invariance properties such as rotation, translation and
scaling and to provide noise immunity. The virus recognition system developed for this
thesis incorporates the method used to extract these features.
Chapter 4 A Study of Texture and Contour
4.1 Introduction
This chapter follows the review of virus morphology, the role of electron microscopy in
virology in Chapter 2 and the steps of creating an automated virus recognition system that
includes the approach to texture analysis in Chapter 3. Chapter 3 also discussed a brief
background on Higher Order Spectra (HOS), the motivations for using HOS in feature
extraction and generation of the bispectral invariant features. In this chapter, the goal is
to involve knowledge of feature extraction using HOS, the direct and indirect methods to
a set of synthetic images with different textures and contours. The knowledge gained will
then be applied to virus recognition.
56 4.1 Introduction
Texture is an important term used to characterize the surface of an object and is a
significant feature used in many applications of image processing and pattern recognition.
The evaluation of texture and contour features is important for virus recognition. This
forms a basis for creating an automated virus recognition system because different
viruses appear differently on electron micrographs. The morphological (texture and
contour) differences appearing on the micrograph are useful in distinguishing one virus
group from another. As mentioned in the earlier chapter, the morphology differences
arise due to the arrangement of capsomeres that are evidenced through negative staining
of the viral particle. Some of the negative-stained virus particles are shown in Figure 4.1.
When stained, adenoviruses appear as hexagonal shaped with distinct, closely packed
capsomers. Astroviruses appear circular in outline and a 5- or 6-pointed white star
configuration in certain orientations; Caliciviruses giving a ‘Star of David’ appearance
(refer to Figure 4.1). These differences might not be visible to the naked eye, thus
making virus recognition a challenge.
In this chapter, the analysis of texture and contour was carried out using a set of synthetic
texture images from the Brodatz album. The first segment of work is to investigate the
relative importance of phase and magnitude in texture. In the subsequent experiments,
noise was added to the images to stimulate EM images that are normally noisy due to the
high magnification used to capture the viruses. These experiments will enhance our
understanding of texture and contour classification of noisy images using higher order
spectral features and also will help to determine the effectiveness of using processing
4.2 The Relative Importance of Phase and Magnitude in Texture Analysis 57
techniques such as averaging of features and input decision fusion for improvement of
classification results.
(a) (b) (c)
Figure 4.1: A single virus particle of (a) Adenovirus (b) Astrovirus (c) Calicivirus. Note that although there are small differences in texture and contour, it is difficult to tell them apart by visual examination.
4.2 The Relative Importance of Phase and Magnitude in
Texture Analysis
Phase and magnitude-only have been studied in acoustical and optical holograms [51].
Research has shown that reconstructed objects from the magnitude-only holograms are
not of much value in representing the original object, whereas reconstructions from
phase-only holograms have many important features in common with the original objects.
Similar observations also have been made in the context of speech signals, X-ray
crystallography and images. As with holograms, only phase-only images have Fourier
transform phase equal to that of the original image. These studies suggest very strongly
58 4.2 The Relative Importance of Phase and Magnitude in Texture Analysis
that in many contexts the phase contains much of the essential information in a signal
[51, 61, 62] .
Currently no research has been done in the area of investigating the relative importance
of Fourier phase and magnitude in texture analysis. Thus, the aim of this section is to
compare the best representation of texture when magnitude and phase information are
used in isolation and to compare the importance of phase and magnitude in different
types of textures.
4.2.1 Experimental Procedure
Two images or textures, known as Texture 1 and Texture 2 were selected from the
Brodatz album. Each of the 512 X 512 textures was reduced to 256 X 256, and a circular
mask was applied to the image. A Fourier transform was computed to obtain the
magnitude and phase of the textures. The magnitude of Texture 1 was combined with
phase of Texture 2 and the magnitude of Texture 2 with phase of Texture 1. Contrast
stretching was applied to the images after inverse fast Fourier transform (FFT) was
performed on these images. These steps are shown in the flow chart in Figure 4.2.
The textures used in this experiment can be divided into 2 categories, spatially
homogeneous (pseudo periodic and periodic) and inhomogeneous. Tests were conducted
with 2 randomly chosen images of (a) homogeneous and homogeneous textures, (b)
inhomogeneous and inhomogeneous textures, and (c) homogeneous and inhomogeneous
4.2 The Relative Importance of Phase and Magnitude in Texture Analysis 59
textures. Examples of homogeneous and inhomogeneous textures used are shown in
Figure 4.3. This analysis was evaluated with 12 sets of two different types of textures
using the combinations stated above. In each set, two of these textures were subjected to
steps shown in Figure 4.2. If magnitude information is more important, the inverse FFT
of the magnitude of Texture 1 and phase of Texture 2 will represent Texture 1. This is
opposite when phase information is more important.
Figure 4.2: Flow chart of analysis of two different textures to evaluate the importance of phase or magnitude in representing the texture.
INPUT Texture 1
Fourier-transform
Combine Mag1 with Phase2
INPUT Texture 2
Fourier-transform
Combine Mag2 with Phase1
Inverse FFT Inverse FFT
Contrast stretching Contrast stretching
Does it represent Texture1 or Texture 2?
Does it represent Texture1 or Texture 2?
OUTPUT If Texture 1,magnitude is important
If Texture 2,phase is important
OUTPUT If Texture 1,phase is important
If Texture 2,magnitude is important
60 4.2 The Relative Importance of Phase and Magnitude in Texture Analysis
(a)
(b)
Figure 4.3: Examples of (a) homogeneous textures (b) inhomogeneous textures used in the experiment. Two of the images above are randomly chosen and subjected to steps in Figure 4.2.
4.2.2 Results and Discussion
Some of the results presented in terms of input and output textures are shown in Figure
4.4. The rest can be found in Appendix A.1. In all cases, the magnitude information
seems to be more important when both input textures are homogeneous. As shown in
Figure 4.4(b), combining the magnitude of a homogenous texture (Texture 1) with the
phase of another homogenous texture (Texture 2) shows an output texture of Texture 1.
4.2 The Relative Importance of Phase and Magnitude in Texture Analysis 61
This is opposite when testing with 2 inhomogeneous textures, where phase information is
more important (Figure 4.4(a)). Whereas, combining the magnitude of a homogeneous
texture with the phase of an inhomogeneous texture shows a clearer representation of the
homogeneous texture at the output (Figure 4.4(c)), it contains more features of the
homogeneous pattern than the inhomogeneous pattern. In contrast, combining the phase
of a homogeneous texture with the magnitude of an inhomogeneous texture represents
neither of the images at the output (Figure 4.4(c)).
This analysis shows that the importance of phase and magnitude dependents largely on
the pattern of the textures and does not necessarily reside on phase-only. The results
illustrate that magnitude information will better represent the texture when both the input
textures are homogeneous, and that if the magnitude information is eliminated, many of
the important characteristics of texture will not be retained. As outlined in Chapter 3,
although the direct method uses both magnitude and phase information, it emphasizes
phase information. Most of the magnitude information is lost in the integration over the
bifrequency plane and the division to obtain phase parameters. Relative performance of
the direct features for phase-only and magnitude-only speech data can be found in [63].
The indirect method discards all phase information because it uses the magnitude of the
DFT for positive frequencies of the input. However, it extracts the magnitude information
in a manner superior to mere extraction of energy in spectral bands. Virus images show
homogeneous textures, and thus the indirect method is used in this thesis and in the
evaluation of the following sections.
62 4.2 The Relative Importance of Phase and Magnitude in Texture Analysis
Figure 4.4: Results when images of (a) 2 inhomogeneous textures (b) 2 homogeneous textures (c) a homogenous and inhomogeneous texture were used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
(b) (a)
(c)
4.3 Texture and Contour Analysis using HOS 63 4.3 Texture and Contour Analysis using Indirect Method for
HOS Feature Extraction
4.3.1 Introduction The performance of using the indirect method to obtain bispectral features is evaluated
experimentally using a selection of images from the Brodatz album, which can be
downloaded from http://www.ux.his.no/~tranden/brodatz.html. These images were
chosen on the basis of being relatively uniform and regular in appearance, or in other
words, they are homogeneous textures.
In this section, four experiments were carried out. In the first experiment, 40 different
texture images with different contours were used to perform classification to determine
the ability of higher order spectral features to discriminate a large number of images with
different textures and contours. In the second experiment, 4 texture images were chosen
and 4 different contours were applied to each image to distinguish images with the same
texture but different contours. In the third experiment, images with different levels of
signal and noise ratio were used to investigate the robustness of higher order spectral
features to noise. The final experiment seeks to improve the classification results by
averaging the features and the performance of input fusion.
In the following experiments, each of the 512 X 512 textures was tiled into 64 X 64 sized
images, resulting in 64 sample images for each texture. Each of the 64 X 64 images was
rotated at 3 random angles to produce a total of 192 images for each texture. Noise was
64 4.3 Texture and Contour Analysis using HOS
added to the images. 150 bispectral features were then extracted from each image.
Section 3.3.5.4 outlined the procedure of extracting the features using the indirect
method. These features extracted are invariant to rotation, translation, scaling and robust
to noise.
The features used for training and testing were randomly chosen, and 15 realizations of
the experiments were undertaken for each binary classification task, and the results were
averaged. Forty were used for training and the remaining for testing. The training and test
images were extracted from separate parts of the image such that no overlap between the
two sets is possible. In all cases, a RBF kernel with width = 0.1 in the Support Vector
Machine (SVM) was used for classification.
4.3.2 Experiment with a Large Database of Different Textures and
Contours
The experiment was carried out on 40 texture images chosen from the Brodatz album; 10
of these textures has a circular contour, 10 of 5-fold-symmetry, 10 of 6-fold-symmetry
and 10 of 7-fold-symmetry contour. Figure 4.5 shows some of the different texture
images used with different contours. Each image is labeled using the standard notation in
this texture database. White Gaussian noise was added to the image with signal to noise
ratio (SNR) equal to 5dB. Each trial used average features from 5 images.
4.3 Texture and Contour Analysis using HOS 65
(a) (b)
(c) (d)
Figure 4.5: Four different texture images with four different contours (a) D5 with 5 fold symmetry contour (b) D17 with 6 fold symmetry contour (c) D21 with 7 fold symmetry contour (d) D33 with circular contour chosen out of the forty Brodatz images that were used for classification.
66 4.3 Texture and Contour Analysis using HOS
The misclassification results of the first 10 textures are tabulated as shown in Table 4.1.
The rest are in Appendix A.2. The overall performance over 40 textures shows an
average misclassification of 1.37%, demonstrating that low misclassification can be
achieved in a large database of different textures and contours. Thus, bispectral features
are able to distinguish images of different textures and contours.
D1 D2 D3 D5 D6 D8 D9 D11 D16 D17
D1
D2 1.88
D3 0.35 0.27
D5 0.79 3.54 0.08
D6 0.02 0.06 0.33 0
D8 0 0.98 0 0.69 0
D9 1.00 0.54 1.94 0.04 1.06 0
D11 6.23 1.69 0.94 0.27 0 0.21 1.25
D16 0 0 0.04 0 1.02 0.04 2.25 0.04
D17 0 0 0 0 1.48 0.02 0.42 0 0.33
Table 4.1: Misclassification rate (%) of the first 10 textures out of the 40 that were chosen from the Brodatz album. The overall performance over 40 textures shows an average misclassification of 1.37%.
4.3 Texture and Contour Analysis using HOS 67 4.3.3 Experiment with Images of the Same Texture of Different
Contours
Four texture images from the Brodatz album, D5, D8, D17, D21 were chosen as shown in
Figure 4.6. Four different contours (circular, 5-,6-,7-fold symmetry) were applied to each
texture image. Similar to the previous section, white Gaussian noise of SNR equal to 5dB
was added to the images and feature averaging of 5 images were used for each trial.
(a) (b)
(c) (d)
Figure 4.6: Four texture images from the Brodatz album (a) D5 (b) D8 (c) D17 (d) D21 used in the experiment. 4 different contours were applied to each of these texture images.
68 4.3 Texture and Contour Analysis using HOS
The classification results of D5 and D21 using higher order spectral features are
presented in Table 4.2 and the rest are in Appendix A.3. It shows the percentage of
misclassification. The misclassification among the 5-6-7 fold symmetry is slightly higher
than the misclassification of circular against the 5-6-7 fold symmetry, which is as
expected. It also is observed that misclassification of textures D5 and D8 is slightly
higher than D17 and D21, because in low spatial frequency textures (both D5 and D8)
and in texture images that are homogeneously random (D5), each of the extracted tiles
contains only part of the texture and thus does not represent the underlying pattern that
spans the entire source image (Figure 4.7).
4.3 Texture and Contour Analysis using HOS 69 Circular 5 fold sym 6 fold sym 7 fold sym
Circular
5 fold sym 8.06
6 fold sym 7.64 12.58
7 fold sym 4.47 7.94 9.94
(a)
Circular 5 fold sym 6 fold sym 7 fold sym
Circular
5 fold sym 2.92
6 fold sym 0.92 4.77
7 fold sym 1.60 5.98 4.48
(b)
Table 4.2: Misclassification rate (%) of four different contours of (a) D5 and (b) D21 texture images using higher order spectral features.
70 4.3 Texture and Contour Analysis using HOS
Figure 4.7: The 512 X 512 texture is tiled into 64 X 64 sized images. The texture inhomogeneity between the tiles causes higher classification inaccuracy. 4.3.4 Experiment with Various Level of Noise The robustness against noise of the methodology is studied in this section. The textures
are degraded with noise at different levels. The amount of noise added is measured using
signal to noise ratio (SNR), which is the average signal to noise ratio. White Gaussian
noise was added to four different textures with SNR equal to -80dB, -50dB, -40dB,
-30dB, -20dB, -10dB and 0dB. The texture images chosen were D5, D8, D17 and D21
from the Brodatz album (Figure 4.6). Each trial consists of feature averaging of 5
different images. Figure 4.8 shows the texture of D5 with added noise of SNR equal to
4.3 Texture and Contour Analysis using HOS 71
-50dB, -40dB and -30dB. The results in percentage of misclassification of these images
using higher order spectral features is presented in Table 4.3. The rest is in Appendix A.4.
(a) (b)
(c) Figure 4.8: Gaussian noise added to texture D5 with SNR equal to (a) -50 dB (b) -30 dB (c) -20 dB
72 4.3 Texture and Contour Analysis using HOS
The misclassification in percentage when SNR equal to -50dB is fairly high. This is
expected because the original image is barely visible when noise is added (Figure 4.8(a)).
The misclassification improves with decreasing noise as shown in Figure 4.9. An average
misclassification of textures D5, D8, D17 and D21 was used for this plot, and shows that
the method is robust down to a SNR of -30dB.
D5 D8 D17 D5 D8 27.1 D17 22.5 21.0 D21 29.2 22.1 23.7
(a)
D5 D8 D17 D5 D8 6.10 D17 0.42 0.44 D21 0.46 0.56 14.4
(b)
D5 D8 D17 D5 D8 5.90 D17 0.23 0.33 D21 0.31 0.77 12.6
(c)
Table 4.3: Misclassification rate (%) of four textures with added white Gaussian noise of SNR equal to (a) -50 dB (b) -30 dB (c) -20 dB using higher order spectral features.
4.3 Texture and Contour Analysis using HOS 73 4.3.5 Experiment with Averaging of Features and Input Fusion of Noisy Images
In the previous sections, feature averaging over 5 images was used in all testing. In this
section, we increased the test ensemble for feature averaging from 5 images to 10 and 20
images. White Gaussian noise was added to the images (D5, D8, D17, D21) with SNR
equal to -30dB.
The results are presented in Table 4.4, which shows the percentage of misclassification of
feature averaging of 10 images and 20 images with SNR = -30dB using higher order
spectral features. As clearly seen (Table 4.4), as the number of test ensemble increases,
the misclassification rate drops.
The result of feature averaging of 5 images (Table 4.3(b)) produces an average
misclassification of 3.6%. Feature averaging of 10 images gives an average
misclassification of 1.8% and averaging of 20 images has a 0.8% misclassification rate
(Table 4.4). This shows that averaging of features improve the classification
performance.
74 4.3 Texture and Contour Analysis using HOS
-80 -70 -60 -50 -40 -30 -20 -10 0 100
5
10
15
20
25
30
35
Signal to Noise Ratio (dB)
Mis
clas
sific
atio
n Ra
te (%
)
Figure 4.9: Noise performance of bispectral features. Misclassification rate shown is an average misclassification of textures D5, D8, D17 and D21.
D5 D8 D17 D5 D8 1.69 D17 0 0.5 D21 0 0.52 8.14
(a)
D5 D8 D17 D5 D8 1.48 D17 0 0.44 D21 0 0.27 2.4
(b) Table 4.4: Misclassification rate (%) when feature averaging of (a) 10 images (b) 20 images were used for classification using higher order spectral features.
4.3 Texture and Contour Analysis using HOS 75 Another method to reduce the percentage of misclassification is by input fusion where the
features are combined from N images, and soft decisions (likelihoods) are averaged from
M such sets of N images. In the following experiments, different numbers of M were
chosen while N was fixed at 5. Table 4.5 presents the misclassification percentage when
M = 4 and 8 and N = 5 using higher order spectral features. The rest of the results, where
M = 2, 5 and 10 is in Appendix A.5. From the results, it can be seen that the average
misclassification of textures D5, D8, D17 and D21 with added noise (SNR = -30dB)
decreases as M increases. Figure 4.10 shows the misclassification in percentage with
different M.
D5 D8 D17
D5 D8 0.5 D17 0.083 0 D21 0.083 0.25 4.42
(a)
D5 D8 D17
D5 D8 0 D17 0 0 D21 0 0 1.2
(b)
Table 4.5: Misclassification rate (%) when M = (a) 4 (b) 8 sets and N=5 images were used for classification using higher order spectral features. Gaussian noise of SNR = -30 dB was added to the image.
76 4.4 Summary
4.4 Summary This chapter begins by investigating the relative importance of Fourier phase and
magnitude. Experimental evaluations were carried out using a set of texture images from
the Brodatz album. This analysis shows that the importance of phase and magnitude
dependents largely on the pattern of the textures. Through this investigation, the indirect
method of obtaining bispectral features was used for the subsequent experiments.
A large number of experiments were conducted to illustrate the effectiveness of the
proposed bispectral features for classification of texture and contour. The experimental
results show the ability of bispectral features in classifying a large database
of different texture images with different contours, as well as images with the same
texture, but different contours. Since it is important that any texture classification scheme
can operate successfully in a noisy environment, particularly in the application to EM
virus images, the robustness of this feature against noise was evaluated. The method
shows robustness to noise down to a SNR of -30dB.
This chapter has presented two approaches for improving the classification accuracy
particularly the noisy images. Experimental evaluations were carried out by increasing
the feature averaging of N number of images and input fusion, where the features are
combined by averaging of M sets where each set consists of N number of images. Both
methods have shown the ability to reduce the percentage of misclassification. The
analytical and experimental study of texture and contour in this chapter has provided an
important foundational knowledge that can be applied to virus recognition.
4.4 Summary 77
1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
4
Mis
clas
sific
atio
n ra
te (%
)
M sets
Figure 4.10: Average misclassification rate in percentage by input fusion where the features are combined by averaging of M sets where each set consists of 5 images.
Chapter 5 Virus Recognition using Higher
Order Spectral Features
5.1 Introduction Viruses can be viewed only by using electron microscopes, and although large
populations can be imaged, the individual images often are noisy and viral particles can
vary in orientation and size. Additional processing such as averaging an ensemble of
particles is necessary to get a better result. Higher order spectral features, that are
invariant to translation, rotation and scaling and robust to noise, have been defined and
applied in previous pattern recognition work, but not in virus recognition.
80 5.1 Introduction
This segment of work examines the use of radial spectra of higher order spectral invariant
features to capture information about symmetries and asymmetries of different types of
viruses in electron microscope images. Biological reproduction processes often impart
symmetries to the forms of organisms, and viruses are even seen assembled in crystal-like
structures. Radial spectra of invariant higher order spectral features can be averaged over
an ensemble of particles regardless of the orientation and size changes. Shape information
can thus be extracted and used for classification even when an individual segmented virus
image is difficult to classify on its own.
In the chapter, experiments were carried out to identify gastroenteritis virus population
from digitised EM images of four types of viruses; Rotavirus, Adenovirus, Astrovirus and
Calicivirus, whose morphologies are quite similar. The performance of the method on
viruses of similar size was separately evaluated using Astrovirus, Hepatitis A virus
(HAV) and Poliovirus. The viral particles from one or more images are segmented and
analyzed to verify whether they belong to a particular class (such as Adenovirus,
Rotavirus etc) or not. Two experiments were conducted - depending on the populations
from which virus particle images were collected for training and testing, respectively. In
the first, disjoint subsets from a pooled population of virus particles obtained from
several images were used. In the second, separate populations from separate images were
used.
5.2 Higher Order Spectra 81
A Gaussian Mixture Model (GMM) and Support Vector Machine (SVM) were used
separately to train the viruses and results were compared. A threshold on the log
likelihood is varied to study false alarm and false rejection trade-off. Features from many
particles and/or likelihoods from independent tests are averaged to yield better
performance. Results are presented in detection error trade-off (DET) curves.
5.2 Higher Order Spectra (HOS) As outlined in Chapter 3, the bispectrum is a function of two frequencies and in contrast
to the power spectrum this function is complex-valued in general and thus retains some of
the phase information in the Fourier transform. Figure 5.1 presents an illustration of two
different signals which that identical power spectra but different bispectra. A one-
dimensional input sequence that is finite in length and has a z-transform with zeros inside
and outside the unit circle (non-minimum phase) is used in this illustration (Figures 5.1(a)
and 5.1(b)). All the zeros that are outside the z-plane are inversed conjugate to produce a
sequence that is of minimum phase. The z-transform of the minimum phase is shown in
Figure 5.1(e). The power spectrum of the minimum and non-minimum phase sequence is
identical (Figures 5.1(c) and 5.1(f)). A distinct difference can be seen in the plot of the
real and imaginary parts of the bispectrum (Figure 5.2) showing that the information of
phase only be captured only by higher order spectra.
Especially for asymmetric sequences the phase is nonlinear and higher order spectra
retain the nonlinear phase information. Higher order spectra also are unaffected by a
82 5.2 Higher Order Spectra
translation of the input. These unique properties of higher order spectra are useful in
pattern recognition. If an input is even (or odd) symmetric, the phase of the Fourier
transform is zero (or pi) or a linear function of frequency if the input is shifted. In either
case, the phase of the bispectrum will be zero. This is expected because all the
information resides in the Fourier magnitude for such inputs. The magnitude of the DFT
of the input for positive frequencies may then be used to compute higher order spectral
invariants (referred to as indirect HOS invariants) [11]. Because virus images can exhibit
symmetry in projections, the indirect method is used in this work. An added advantage of
using the indirect method is that the DFT magnitude sequence is band-limited, and scale
invariance is better satisfied for these indirect features. Preliminary experimental and
analytical evaluations of texture and contour in Chapter 4 also justify and support the use
of the indirect method in obtaining the bispectral features in the application to viruses.
5.2 Higher Order Spectra 83
0 2 4 6 8 10 12 140
1
2
3
4
5
6
7
8
Time Index, k
h(k)
0 2 4 6 8 10 12 14
0
0.2
0.4
0.6
0.8
1
Time Index, k
G(k
)
(a) (d)
-2 -1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Real Part
Imag
inar
y Pa
rt
Z plane
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Real Part
Imag
inar
y Pa
rt
Z plane
(b) (e)
0 2 4 6 8 10 12 140
10
20
30
40
50
Frequency, w
| H(w
) |
0 2 4 6 8 10 12 140
1
2
3
4
5
Frequency, w
| G(w
) |
(c) (f)
Figure 5.1: Illustration of the inability of the power spectrum to retain phase information. Figure 5.1(a) Non minimum phase of input sequence, h(k), Figure 5.1(b) Zeros of the non-minimum phase, Figure 5.1(c) Power spectrum of the non-minimum phase sequence. The zeros outside the unit circle in Figure 5.1(b) are inversed conjugate to produce a minimum phase sequence. Figure 5.1(d) 1-D plot of the minimum phase sequence, Figure 5.1(e) Zeros of the minimum phase sequence, Figure 5.1(f) Power spectrum of the minimum phase sequence.
84 5.2 Higher Order Spectra
20 40 60
5
10
15
20
25
30
20 40 60
5
10
15
20
25
30
20 40 60
5
10
15
20
25
30
20 40 60
5
10
15
20
25
30
0 0.5 1f1
0 0.5 1f1
0 0.5 1f1
0 0.5 1f1
0.5
f2
0
0.5
f2
0
0.5
f2
0
0.5
f2
0
Figure 5.2: The left top and bottom figures show the imaginary and real parts of the bispectrum of Figure 5.1(a) and the right top and bottom figures show the imaginary and real parts of the bispectrum of Figure 5.1(d). The differences observed in the plots show the ability of bispectrum to retain the phase information.
5.2.1 Illustration of the Indirect Method in obtaining Bispectral
Features using Synthetic Images
The indirect method in obtaining the bispectral invariant parameters is outlined in
Chapter 3. The steps can be summarized using the flow chart in Figure 5.3. Using Radon
transform projections, the 2-D image is reduced to 1-D image [11], rotation invariance is
achieved by taking radial spectra of these features. The radial spectra of the bispectral
invariants, ),( θωaP where θω is a frequency in cycles per 180 degrees is a set of features
that are invariant to rotation, translation and scaling.
5.2 Higher Order Spectra 85
The steps of computing these invariant features are clearly illustrated using synthetic
images of n-fold symmetries as shown in Figures 5.4, 5.5 and 5.6. Figures 5.4(c) and
5.4(d) show the Radon transform projections at 45 degree angle of images with a 5 fold
symmetry and 7 fold symmetry binary object, respectively. Figure 5.5 shows the real and
imaginary parts of the bispectrum of these projections. Figure 5.6 shows the radial
spectrum of the bispectral features, )2/1(P . Note that the 5 fold symmetry image
produces a peak showing a dominant symmetry at 5 cycles per 180 degree, whereas the 7
fold symmetry image produces a peak at 7 cycles per 180 degrees.
To illustrate how robustness can be achieved by averaging these features, the 7 fold
symmetry image is used and white Gaussian noise is added to the image with SNR equal
to 0dB. With only one image used, the plot of the radial spectrum of the bispectral
features, as shown in Figure 5.7(a) does not reveal a dominant symmetry at 7 cycles per
180 degrees. In Figure 5.7(b), the individual spectra are accumulated over 75 of the 7 fold
symmetry images with similar signal to noise ratio (SNR = 0dB). Ensemble averaging
images produce spectra (see top line in Figure 5.7(b)) that eventually converge to a form
revealing a visible peak at 7 cycles per 180 degrees. This is applicable to EM images
where due to the low signal to noise ratio, individual viral particles are difficult to discern
visually and present a challenge to feature extraction. Averaging of these features
improves the classification performance.
86 5.2 Higher Order Spectra
Figure 5.3: Flow chart of computation of invariant parameters. ),( θaP is invariant to scaling and translation and ),( θωaP is invariant to scaling, translation and rotation. The algorithm was tested on a 5 fold symmetry and a 7 fold symmetry image and results are presented in Figures 5.4, 5.5 and 5.6.
n
x(n)(θ)
Input
Zero pad and FFT
Triple Product X(f1)X(f2)X*(f1+f2)
Integrate along line of slope a
Parameter P(a,θ)
| DFT|
Parameter P(a,ωθ)
| DFT|
Obtain the DFT magnitude, retain positive frequencies spectrum only
5.2 Higher Order Spectra 87
Figure 5.4: Figure 5.4(c) and 5.4(d) show the Radon transform projection at 45 degree angle of the 7 fold symmetry image, Figure 5.4(a) and the 5 fold symmetry image, Figure 5.4(b).
7 fold symmetry 5 fold symmetry 5.4(a) 5.4(b)
5.4(c) 5.4(d)
5.5(a) 5.5(b)
88 5.2 Higher Order Spectra
Figure 5.5: Figure 5.5(a) and Figure 5.5(c) show the real and imaginary parts of the bispectrum of the Radon transform projection at 45 degree angle of Figure 5.4(a). Figure 5.5(b) and Figure 5.5(d) show the real and imaginary parts of the bispectrum of Figure 5.4(b). The bispectrum is a triple product of Fourier coefficients and is a complex valued function of two frequencies, f1 and f2, where f1 and f2 are frequencies normalized by one half of the sampling frequency. Different shaped projections result in different bispectra. Invariant features are extracted by integrating along radial lines and taking the phase. The scale shown at the colorbar above is the log of the absolute value of the real and imaginary parts of the bispectrum. The above plots show that features, )a(P close to
21a = may capture differences as well.
Figure 5.6: Figure 5.6(a) and 5.6(b) show plot of ))(2/1( θωP as a function of θω , (where θω is a frequency in cycles per 180 degrees) of Figure 5.4(a) and Figure 5.4(b).
))(2/1( θωP is invariant to scaling, translation and rotation. Note that Figure 5.6(a) shows a dominant symmetry at 7 cycles per 180 degrees whereas Figure 5.6(b) shows a dominant symmetry at 5 cycles per 180 degrees.
5.6(b) 5.6(a)
5.5(c) 5.5(d)
5.2 Higher Order Spectra 89
Figure 5.7: Figure 5.7(a) shows the radial spectrum of the bispectral features,
))(2/1( θωP as a function of θω , (where θω is a frequency in cycles per 180 degrees) of a 7 fold symmetry. White Gaussian noise has been added and SNR = 0dB to the image. In Figure 5.7(b), the individual spectra are accumulated over 75 such images. Note that Figure 5.7(a) does not demonstrate a dominant symmetry at 7 cycles per 180 degrees due to the low signal to noise ratio. As an ensemble of images is taken and the spectra are accumulated, it eventually converges to a shape. A peak at 7 cycles per 180 degrees can be seen in Figure 5.7(b). Robustness to noise can thus be achieved by averaging these features.
Next, a 3-D reconstructed Adenovirus displayed on greyscale image in 2-D is compared
with a 5 fold symmetry image. In Figure 5.8, plots of ))(( θωaP are compared, where
a =1/2 and θω = 1, 2……. 16 cycles per 180 degrees. ))(( θωaP which is invariant to
translation, rotation and scaling captures information from the contour and texture
properties of the virus image that is useful for verification.
3-D reconstructions of virus particles are normally taken from a large set of electron
cryomicroscopy (cryo-EM) images. Comparing with conventional EM, where the
specimens are metal stained and dried for observation, in cryo-EM, the unstained
particles are preserved in a flash-frozen aqueous environment. The drying process in
5.7(a) 5.7(b)
90 5.2 Higher Order Spectra
conventional EM (example negative staining) tends to flatten the structure onto the
support plane, causing distortions to the 3D structure. In other words, what is seen in the
electron micrographs might not be a faithful representation of the virus. Consequently,
the 3D reconstruction image is not used here to build the reference feature vectors for
classification. A large set of negative stained EM images was used, which will produce a
more accurate classification result when the testing set is of a similar preparation method.
Figure 5.8: Comparison between a 5 fold symmetry image and a 3D reconstructed virus image using plots of radial spectra of the bispectral features, P(1/2). These features which are invariant to translation, rotation and scaling contain information from the contour and texture that are useful for verification.
5 fold symmetry 3D reconstructed Adenovirus
5.3 Gastroenteric Viruses 91 5.3 Gastroenteric Viruses Acute gastroenteritis is one of the most common diseases that affect humans, and is a
significant cause of morbidity and mortality worldwide [64], causing 3-5 million deaths
per year. The majority of these occur in developing countries [65] and children under 5
years of age [64]. Viruses, apart from bacteria and other parasites, have been known to be
the important causes of gastroenteritis. Four major categories of viruses are now
recognized as clinically important to this problem. These are Rotavirus, Astrovirus,
Adenovirus and Calicivirus [64, 66]. Many studies have been conducted to determine
which gastroenteric viruses are more prevalent with respect to geography, sex, seasonal
pattern and age distribution [67-72]. Research shows that Rotavirus is responsible for the
majority of these deaths and 20-52% of all acute gastroenteric episodes [73-75]. Accurate
understanding of the relative prevalence of these agents would help design strategies to
control the disease.
Generally there are 3 diagnostic methods for gastroenteritis viruses in stool samples;
(immune) transmission electron microscopy (TEM), antigen ELISA and polymerase
chain reaction (PCR) [76]. The diagnostic significance, advantages and disadvantages of
these methods have been compared in Section 1.1 and 2.4.
92 5.4 Image Analysis 5.4 Image Analysis Gastroenteric viruses are normally shed in high concentrations, often reaching particle
concentrations of 1011ml-1 which makes it suitable to diagnose these viruses using EM.
The difficulty arises in distinguishing these viruses with one another because they are all
nearly circular in shape with little visual differences (Figure 5.9 and Figure 5.10), and
produce the same pathological symptoms in the patients. This can be overcome by
having an automated verification system. We considered the problem of detecting
Rotavirus against the other gastroenteric viruses such as Calicivirus, Adenovirus and
Astrovirus. Rotavirus was chosen as the target virus due to prevalence of this virus in
causing acute gastroenteritis. Although size alone can distinguish these viruses, this study
is conducted under the assumption that the magnification level and true particle size is
unknown. Classification of these viruses are based upon contour and texture. In Section
5.8, another experiment is conducted with viruses of similar size. The viruses chosen are
Astrovirus, Hepatitis A virus (HAV) and Poliovirus.
The following steps comprise the image analysis:
Segmentation: Segmentation is performed to separate individual viral particles from the
image. Viral particles in the images are segmented out into subimages (64 by 64 pixel),
and the images are aligned using the centroid of each subimage. This alignment need not
be perfect because the features extracted are translation invariant. Then a circular mask is
applied to each particle to eliminate the peripheral region that may contain neighbouring
virus cells. A circular mask will not corrupt the type of periodicity of the bispectral
5.4 Image Analysis 93
features as exhibited by the virus image within it. The features extract asymmetry
information from the virus image and a perfectly circular mask will not introduce any
asymmetry as a result of masking. Examples of segmented image of each type of virus
are shown in Figure 5.10.
(a) (b)
(c) (d)
Figure 5.9: A sample image of each type of virus used for testing. These images are different in magnification and resolution. (a) Adenovirus (b) Astrovirus (c) Rotavirus (d) Calicivirus
94 5.4 Image Analysis
(a) (b) (c) (d)
Figure 5.10: A single virus of each type. (a) Adenovirus (b) Astrovirus (c) Rotavirus (d) Calicivirus. These subimages are extracted from portions shown by square boxes in figure 5.9 and a circular mask is applied to each. Note that although there are small differences in texture, it is difficult to tell them apart by visual examination. Pseudo colouring could be used to emphasize the differences in texture but the difficulty arises when there is some variation in texture within the same virus type, on images that are obtained from various sources of different background, scale, contrast and noise.
Extraction of cell features: Each subimage (Figure 5.10) is subjected to the steps shown
in Figure 5.3. Radon transform projections at 32 angles are computed from each
subimage to yield one-dimensional functions. Ten bispectral features, ))(( θaP (where
1,.....,102,101a = and ),.....,322,32)rad( πππθ = are computed from each projection.
A total of 320 features can thus be extracted from each subimage. These features are not
rotation invariant. A discrete Fourier transform (DFT) is then computed on the features
considered as sequences with the angle of rotation as the index, to yield a new set of
features, ))(( θωaP , where θω is a frequency in cycles per 180 degrees. The resulting
features are invariant to rotation, translation and scaling, and are therefore robust to small
changes in the sizes of viral particles and their orientation. They are sensitive to
asymmetries in the shape of the virus, and because of their robustness to orientation and
size they can be combined for a population of virus particles, to pick up useful shape
5.4 Image analysis 95
differences that are not visually evident from electron micrograph images of single
particles.
Feature Selection: In general, the dimension of the feature vector (i.e., the number of
features) may be very large at the feature generation stage. Given a number of available
features, the main task of the feature selection (or reduction) process is to select
information rich features providing a large interclass distance and a small intraclass
variance in the feature vector space. In this work, the dimensionality is reduced by
computing the largest distance separation, F between the target virus and the background
virus, given by
)/()( 2
221
221 σσµµ +−=F (5.1)
where 1µ and 2µ are the mean of the target virus and the background virus and 1σ and
2σ are the standard deviation of the target virus and background virus computed over
some subset of the training set. These 150 features are then arranged according to the
ones that produced the largest distance separation between the target and background
virus. Then the dimensionality of the features used for training are varied in the interval
of 50 to 150 at an increment of 10 to produce an optimal result.
The steps of creating an automated recognition system outlined in Chapter 3 did not
include the process of reducing the dimensionality of features. Numerous techniques are
found in the literature and will be reviewed briefly here. Two means of reducing the
dimensionality are through selection of a subset of features and by extracting the
96 5.5 GMM Modeling principal components of variation from the vector. One of the commonly used feature
transformation techniques, the principal component analysis (PCA) [77], did not perform
well on these images. The ineffectiveness of using PCA might be due to the various
clusters or modes that exist from the data (images) obtained from multiple sources that
might not be linearly separable when projected on their first principal component.
Although, PCA chooses the components that best represent the data, it does not
necessarily choose the components that would be best for discriminating one class of data
from another.
The feature selection approach, described above was used here because it has several
advantages over the feature transformation approach. Feature selection retains the
integrity of each of the features within the set, allowing easier examination and
interpretation of classification results. Another advantage of the feature selection
approach is that features that are not selected need not be computed at all for future
unknown observations, which in some cases can save considerable computation time.
5.5 GMM Modeling Gaussian Mixture Models (GMMs) were trained for feature distributions from the viral
particles. GMM was chosen to model the feature density functions from the target virus
and background virus because of the range of modes or clusters produced by the images
with different scale, background and contrast that were used in training each virus type.
Figure 5.11 shows the cluster plot of 2 randomly chosen bispectral features of three
5.5 GMM Modeling 97
different set of images of Rotavirus. They differ in scale, background and contrast. Each
feature is an average from a subpopulation of 10 viral particles. As can be seen, the plot
that represents the probability distribution of the features shows quite compact and
isolated clusters in feature space.
As determined in Section 3.4.2, each type of virus is represented by a GMM describing
its features, with its mean vectors, covariance matrix and the mixture weights as
parameters. One of the critical factors in training a GMM is selecting the order of the
mixture. There are no good theoretical means to guide these selections, so they are best
experimentally determined for a given task. An experimental examination of these factors
is discussed in the following section.
5.5.1 Experimental Evaluation of Model Order and Dimensionality of
Training Observations
Determining the number of components in a mixture needed to model the features
adequately is an important, but challenging problem. There is no theoretical way to
estimate the number of mixture components a priori. Choosing too few mixture
components can produce a model that does not accurately model the distinguishing
characteristics of the feature distribution. Choosing too many components can reduce
performance when there are a large number of model parameters relative to the available
training data, and also can result in excessive computational complexity both in training
and classification. The following experiments examine the performance of the GMM to
98 5.5 GMM Modeling model feature distribution from the virus particles for different model orders using a fixed
and variable amount of training data.
To investigate the virus recognition performance of the GMM, a different combination of
number of mixtures, M and dimensionality of training observations, D were used in the
experiments. First the number of training observations is fixed while varying the number
of mixtures. As mentioned earlier, the number of training observations is incremented at
the interval of 10 starting from 50 to 150 features.
In all cases, a diagonal covariance form of the GMM was used with a covariance matrix
for each component. The input data were pre-processed with a k-means algorithm that
performs an unsupervised learning to find centres of clusters that reflect the distribution
of the data, followed by iterations of the Expectation Maximization (E-M) algorithm. The
iteration was stopped when the change in log likelihood of the error function at the
solution between two steps of the E-M algorithm of the target and background virus was
below a preset threshold and considered insignificant.
Using different combinations of training sets, 5000 trials where each trial consists of
feature averaging of 15 particles were used for training. The features used for training and
testing were randomly chosen and there was no overlapping between the training and test
set. Table 5.1 presents the test results in terms of percentage of efficiency, which is
calculated from miss and false alarm probabilities of Rotavirus against the other viruses.
The result is an average over 1000 trials. It is found that at a higher dimension of features,
the GMM model becomes unstable and sensitive to the selection of mixture components.
5.5 GMM Modeling 99 Consequently, the results of more than 130 features are not presented in Table 5.1. The
highest efficiency, 97.4% is achieved when D=110 and M=10. This combination will be
used in the following section to train the features.
(Features, Mixture)
(D,M)
Miss (%) False alarm (%), FA Efficiency (%)
((100-Miss) + (100-FA))/2
(130, 10) 1.1 3.2 93.1
(130, 20) 2.1 3.4 94.1
(130, 30) 0.6 4.8 95.2
(120, 10) 1.0 5.3 92.1
(120, 20) 1.2 3.8 92.5
(120, 30) 5.1 3.5 95.7
(120, 40) 3.0 7.6 94.7
(110, 10) 0.9 4.3 97.4
(110, 20) 2.6 5.4 96.0
(110, 30) 1.3 6.7 96.0
(110, 40) 2.2 6.9 95.5
(110, 50) 4.3 6.4 94.7
(100, 10) 1.4 8.5 95.1
(100, 20) 2.3 8.3 94.7
(100, 30) 1.9 7.1 95.5
(100, 40) 1.2 9.0 94.9
(100, 50) 1.2 11.8 93.5
(90, 10) 1.5 9.0 94.8
(90, 20) 1.1 10.1 94.4
(90, 30) 2.4 11.4 93.1
(90, 40) 2.2 8.0 94.9
(90, 50) 1.8 12.7 92.8
(80, 10) 2.0 7.9 95.1
(80, 20) 2.6 13.6 91.9
(80, 30) 1.5 24.1 87.2
(80, 40) 1.7 18.9 89.7
100 5.6 GMM Verification (60, 5) 1.4 13.9 92.4
(60, 10) 1.3 15.7 91.5
(60, 15) 2.1 14.6 91.7
(60, 20) 1.6 19.5 89.5
(60, 25) 1.5 22.4 88.1
(60, 30) 1.8 31.0 83.6
(50, 5) 3.2 24.7 86.1
(50, 10) 2.2 24.9 86.5
(50, 15) 2.6 26.9 85.3
(50, 20) 1.2 29.5 84.7
Table 5.1: Efficiency (%) of different combinations of number of mixtures, M and dimensionality of training observations, D used to examine the performance of GMM.
5.6 GMM Verification The test set is scored against both the target model (Rotavirus) and the background model
(Calicivirus, Astrovirus and Adenovirus). In the experiment of viruses of similar size, the
target model is Astrovirus and the background models are HAV and Poliovirus. In the
first verification test, the decision score ),( NXS λ is a log likelihood ratio of a
subpopulation of N particles, where N is the number of particles used to compute a
feature vector by averaging, given by
)|('log)|('log),( NiNiN XfXfXST βλλ −== (5.2)
5.7 Experiments 101 In the second verification test, each decision score, ),(' NXS λ is an average of M sets
where each set consist of a subpopulation of N particles, as can be shown in the formula:
)|('log)|('log),('1
NiNi
M
iN XfXfXST βλλ −== ∑
=
(5.3)
where Nλ is the target model, iX is the test set comprised of a subpopulation of N
particles, )|(' NiXf λ is the likelihood of the target virus, and )|(' NiXf β is the average
likelihood of the three background virus, with each of the background virus is weighted
equally.
If T
5.7 Experiments Two types of experiments were conducted. In the first experiment, the training and
testing were carried out on a population of virus images pooled from all the EM images
of that type obtained from various sources (referred to as a pooled population). A pooled
population is useful when a single image does not provide enough viral particle
subimages for statistically reliable training or testing. Although the population is pooled,
viral particle subimages used for training are different from those used for testing.
> 0, then the virus is identified as the target virus < 0, then the virus is identified as non-target virus
102 5.7 Experiments
In the second experiment, the testing and training is done on populations derived from
separate images. The images used in the training set were different from those in the test
set. All viral particles in a given population are selected from the same image (referred to
as a single image population). The feature vector in each case is obtained by averaging
features from a set of N number of particles as shown in Figure 5.7. The first verification
test was performed on both the single image population and the pooled population, while
the second verification test was only performed on the pooled population. Figure 5.12
illustrates the selection of viral particles from EM images used for testing and training in
the pooled population and the single image population cases.
Figure 5.11: Cluster plot of features from three different sets of Rotavirus images with different backgrounds, contrast and scale. Each point is an average feature from a subpopulation of 10 viral particles. The plot shows quite compact and isolated clusters or modes in feature space.
5.7 Experiments 103
Figure 5.12: Illustration of selection of viral particles from different sets of EM images used for testing and training in pooled population and single image population. In the single image population, the testing and training is done on populations derived from separate images. In a pooled population, the virus images pooled from all the EM images of that type obtained from various sources.
Single image population Pooled population Training Testing Training Testing
U
U
U
EM image 1 EM image 3 EM image 4 EM image 2
104 5.7 Experiments 5.7.1 Experiment on a Pooled Population Different sets of digitised electron microscope images obtained from various sources of
these gastroenteric viruses were used for testing and training. Ten images of Rotavirus of
different scale, background, contrast and appearance of contour were chosen. A total of
12 images of Adenovirus, Astrovirus and Calicivirus were used as the background virus.
For each virus type, the training set consists of 5000 trials where each trial is a feature
average of N particles. The remaining particles were used to select the test set. Twenty
particles of each of the three background viruses were chosen randomly to form 60
particles for a test set of the background virus.
A subpopulation of N particles was chosen from the test sets (a pooled population) of the
target virus and the background virus. In the first verification test of the pooled
population, each decision score is produced by a subpopulation of N particles, where N is
the number of particles used to compute a feature vector by averaging (referred to as a
subpopulation test). In the second verification test, likelihood scores of M sets where each
set consists of a subpopulation of N particles, were averaged to produce a decision score
(referred to as the averaged score subpopulation test).
The verification tests were performed on subpopulations of N viral particles where N = 5,
8, 10, 13 and 15. For each subpopulation size, 100 tests (scores) with randomly selected
subpopulations from the test set were conducted and the results are presented in Detection
Error Tradeoff (DET) curves. In the second verification test, M was set to 2.
5.7 Experiments 105
Results of the subpopulation test are presented in Figures 5.13 and 5.14. From Figure
5.13, it can be seen that as the test ensemble for feature averaging is increased from 5 to
15 particles, the equal error rate (EER) drops quite significantly from 15% to 2.5%
(Figure 5.14). The test was stopped at 15 particles because there was no improvement in
the EER as the subpopulation size for feature averaging was increased, from 13 particles
to 15 particles.
Legend
Figure 5.13: DET curve using the bispectral features from subpopulations of 5, 8, 10, 13 and 15 viral particles on a pooled population. As the test ensemble size for feature averaging is increased, the EER drops. EER is the point on the DET curve where the false alarm probability is equal to the miss probability. Refer to Figure 5.14 for EER values of each subpopulation size.
106 5.7 Experiments
Figure 5.14: Plot of EER versus ensemble size shows that as the test ensemble for feature averaging increases, the EER drops for N= 15 to 2.5%.
The EER drops even lower when the number of sets of subpopulations used for
comparing probability densities modeled by GMMs increases. Results of an averaged
score subpopulation test with M=2 and N=15 particles is presented in Figure 5.15. 5000
tests (scores) with randomly selected subpopulations from the test set were conducted.
From the DET curve, the EER drops from 2% to less than 0.2% as M goes from 1 to 2.
If two test populations are thrown at the GMM, and log-likelihood scores are obtained,
they could be used in different ways. In output fusion, decisions are combined. Each
score could be used to obtain a decision and the decisions may be combined. For
example, a decision to accept may be made only if both individual decisions are to
accept. In this case, false acceptance rate will be the product of individual false
acceptance rates. However, the false rejection rate will be the sum of the individual rates.
02468
10121416
5 8 10 13 15Ensemble size
EER
(%)
, N
5.7 Experiments 107
The false acceptance will go from 2% to 0.04%, but at the expense of false rejection
which goes from 2% to approximately 4% at the same threshold (assuming an equal error
rate of 2% at that threshold).
Alternately, a decision to reject may be made only if both individual decisions are to
reject [78]. In this case, the false rejection rate will go to approximately 0.04%, but false
acceptance to 4% at that threshold (Figure 5.15).
By contrast, in input fusion, features or scores are combined. Individual scores may be
weighted and combined depending upon the confidence one has in each score. There will
exist some optimal weighting of the scores for which the performance is best. When there
is no prior knowledge of confidence or the two tests are equal in all respects, the scores
may simply be averaged (50% weight for each of the two scores). This is done here. It
turns out that averaged scores yield better performance than output fusion in this case.
5.7.2 Experiment on a Single Image Population In the second experiment, only images that have more than 20 viral particles per image
were chosen because the testing and training were conducted on subpopulations of
particles that are drawn from the same image. Nine images of rotavirus of different scale,
background, contrast and appearance of contours were chosen, and 5 of these were used
for training and the rest for testing. For the background virus, a total of 11 images were
used; 5 images of Adenovirus, 4 images of Astrovirus and 3 images of Calicivirus.
108 5.7 Experiments
Out of these images, 3 Adenovirus images were used for training and two each of
Astrovirus and Calicivirus; and the rest were used for testing. Similar to experiment on a
pooled population, 5000 trials where each trial is an average feature of N viral particles
were used for training. There is no overlap in training and test set.
Legend
Figure 5.15: DET curve using the bispectral features from subpopulation of N=15 viral particles on a pooled population. The solid curve shows an average of 2 sets of subpopulation of 15 viral particles, while the dotted curve shows the case for M = 1 (M, N, refer to equations 5.2, 5.3). The EER drops to less than 0.2%. The darker solid curve shows an output fusion of two test populations. In this case, false acceptance rate will be the product of individual false acceptance rates, and the false rejection rate will be the sum of the individual ones. This shows that the averaged scores yield better performance than output fusion in this case.
______ ---------- M = 1 ______
M = 2
Output fusion
5.7 Experiments 109 The verification test was performed on subpopulations of viral particles of sizes N = 5, 8,
10, 13, 15 and 18. One hundred tests (scores) were performed for each subpopulation size
(Figures 5.16 and 5.17). The figures show that as the test ensemble for feature averaging
is increased from 5 to 18 particles, the equal error rate (EER) drops from 20% to 2%. The
test was stopped at 18 particles (Figure 5.17) because there was no improvement in EER
as the subpopulation size increased from 15 to 18 particles.
Legend
Figure 5.16: DET curve using the bispectral features from subpopulations of 5, 8, 10, 13, 15 and 18 viral particles on a single image population. The EER drops as the feature averaging of the subpopulation size increases. Refer to Figure 5.17 for EER values of each subpopulation size.
110 5.8 Virus of Similar Size
Figure 5.17: Plot of EER versus ensemble size shows that as the test ensemble for feature averaging increases, the EER drops for 18 particles to 2%
5.8 Virus of Similar Size
An experiment was conducted to determine the ability to distinguish particles similar true
size. Two types of viruses with similar size; Astrovirus of diameter 28-30nm and viruses
of Parvoviridae family, Hepatitis A virus (HAV) and Poliovirus of diameter 22-30nm
were chosen. Six images of Astrovirus and 7 images of HAV and Poliovirus were used
for training and testing. Figure 5.18 shows the electron micrograph of Astrovirus and
HAV of the same magnification level. Astrovirus was used as the target virus and HAV
and Poliovirus were used as the background virus.
0%
5%
10%
15%
20%
25%
5 8 10 13 15 18Ensemble size
EER
(%)
, N
5.8 Virus of Similar Size 111
(a) (b)
Figure 5.18: A sample image of (a) Astrovirus and (b) Hepatitis A virus. The magnifications of these images are the same. The verification test of a pooled population was performed on subpopulations of viral
particles, where N= 5, 10, 15, 20. The results in subpopulations of 500 tests show that as
the subpopulations is increased to 20 viral particles, the EER drops to 5% (Figure 5.19).
The EER drops further as M is increased from 1 to 2 of a subpopulation of 20 particles to
less than 2%.
112 5.9 Evaluation with SVM Classifier
Legend
M=1, N=5
M=1, N=10
M=1, N=15
M=1, N=20
M=2, N=20
M=1, N=5
M=1, N=10
M=1, N=15
M=1, N=20
M=2, N=20
Figure 5.19: DET curve using the bispectral features from subpopulation of N = 5, 10, 15, 20 particles, M =1 and subpopulation of N = 20, M=2 on a pooled population. The EER drops to less than 2% when features of 2 sets of subpopulation of 20 particles were averaged. 5.9 Evaluation with SVM Classifier 5.9.1 Introduction
The higher order spectral features resulting from the virus particles also were trained
using Support Vector Machines (SVMs). SVM was chosen and compared with the GMM
classifier. Among other classifiers such as Bayes to neural networks, SVMs appear to be
a good candidate because of their ability to generalize in high-dimensional spaces, such
as spaces spanned by texture patterns. The appeal of SVMs is based on their strong
5.9 Evaluation with SVM Classifier 113 connection to the underlying statistical learning theory. That is, as outlined in Section
3.4.3, a SVM is an approximate implementation of the structural risk minimization
(SRM) method [56]. For several pattern classification applications, SVMs have been
shown to provide better generalization performance than traditional techniques, such as
neural networks [79, 80].
5.9.2 Experimental Procedure
Similar to GMM verification, the test set is scored against both the target virus
(Rotavirus) and the background virus (Calicivirus, Astrovirus and Adenovirus). For
SVM, which is a binary classification, the decision score, ),( NXS λ in equation 5.2 is an
average of three scores. Each score is obtained by subtracting the log likelihood of the
target virus with one of the background virus when tested with a test set comprised of a
subpopulation of N particles (equation 5.2).
Bispectral features from the resulting virus particles are extracted using techniques
outlined in Section 5.4. The virus particles used for testing and training are chosen using
the pooled population method (first paragraph of Section 5.7.1). For definition of pooled
population, please refer to Section 5.7. In all cases, the Gaussian RBF kernel was used,
which is given by:
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
β
2||exp, yxyxk (5.4)
114 5.9 Evaluation with SVM Classifier
___ N=5, GMM ___ N=10, GMM ___ N=15, GMM * N=5, SVM * N=10, SVM * N=15, SVM
A β of 0.1 was used, which was found to perform better in verification of rotavirus for the
bispectral features. The number of features used for training and testing also were varied
to obtain the optimal result (see Chapter 6).
5.9.3 Results and Discussion
The verification tests were performed on subpopulations of N viral particles where N = 5,
10, and 15. For each subpopulation size, 100 tests (scores) with randomly selected
subpopulations from the test set were conducted and the results were presented in
Detection Error Tradeoff (DET) curves (Figure 5.20). It can be seen that there is not
much difference in the equal error rate (EER) in all the subpopulations.
Legend
Figure 5.20 DET curve using the bispectral features from subpopulations of 5, 10 and 15 viral particles on a pooled population using SVM and GMM classifier.
5.10 3D virus reconstruction 115 5.10 3D Virus Reconstruction
In the last decade, there has been a burst of activity in the use of EM for the elucidation
of virus activity. This has resulted from two advances in techniques; cryo-EM has
allowed the preservation of fragile specimens in the EM, and the development of efficient
algorithms for processing micrographs to produce 3D structures of icosahedral particles.
The 3D structures are normally reconstructed from its 2D projections. One of the most
commonly used and well-known algorithms to reconstruct 3D virus is the weighted back
projection method (WBP) [81], which is based on superposing 3D functions ("back-
projection bodies") obtained by translating the 2D projections along the directions of
projection.
The 2D projections are obtained through several thousand experimental particles using
the untilted and tilted images (for random conical tilt) of the same micrograph. In the
random conical tilt reconstruction method [82], the actual reconstruction is done from the
tilted particles, but the alignment and classification comes from the untitled images. In
this method is it necessary that the untitled images with the same orientation (the particles
are showing the same face to the viewer, but the only difference between them is that
they can be rotated by some angle in the plane of the image) are aligned and classified to
obtain their class averages. With these clearer class images, the 3-D density is then
reconstructed by using the back projection method as mentioned earlier. The benefit of
using higher order spectral features is that alignment is not necessary because of the
invariant properties.
116 5.11 Summary However, the differences between 2D projections of the same virus when the angle is
different (random conical tilt) can affect the higher order spectral method. In this case, it
is necessary to divide these images into clusters, and build the class averages from there.
Thus, the ‘extra’ step of alignment of the images before averaging can be eliminated by
using higher order spectra due to the rotation and shift invariant properties, allowing
computational savings. Experimentally this has not been carried out, but is one of the
possible avenues for future research.
5.11 Summary The chapter shows that higher order spectral features that are invariant to translation,
rotation and scaling are effective in identifying viruses from digitized electron
micrographs. As mentioned in Chapter 3, the system can be made fully automated by
automatically segmenting the individual virus particles.
Verification tests were conducted on 4 major types of viruses that cause gastroenteritis;
Adenovirus, Astrovirus, Rotavirus and Calicivirus, where Rotavirus was chosen as the
target virus and the rest as the background virus. Results are presented for tests with
various subpopulation sizes, N, used for averaging feature values and varying number of
subpopulations, M, used for averaging likelihoods. EER of around 2% is achieved for
N=15, M=1. EER drops to less than 0.2% for M=2.
Tests also were conducted on viruses of similar true size. Astrovirus was scored against
HAV and Poliovirus. Results show that the EER drops to less than 2% for N=20, M=2.
5.11 Summary 117
Support Vector Machine (SVM) was compared with Gaussian Mixture Model (GMM) to
classify the viruses. Experimental evaluation shows that there are no substantial
differences in the EER when trained and tested with subpopulations of 5, 10 and 15
particles using these classifiers separately to verify Rotavirus. In the next chapter,
Rotavirus, Adenovirus and Calicivirus are used separately as the target virus in a
comparative study of higher order spectral features and Gabor features.
Chapter 6 Relative Performance Evaluation
6.1 Introduction
The previous chapter investigates the use of radial spectra of higher order spectral
features in virus recognition. The aim of this chapter is to provide a comparative study of
higher order spectral and Gabor features in virus recognition. Gabor filtering is one
prominent filtering method, and has been shown to provide an excellent basis for
identifying textured images, and thus has been used in many applications. A few
comparisons between Gabor filtering with other filtering methods, as well as with other
feature extraction techniques have been presented.
120 6.1 Introduction Randen and Husoy [83] did a comparative study of the filtering methods, reviewing
Wavelets, Gabor filters and Wedge filters, and concluded that no single approach
performs best or very close to the best for all images because different images yield
different results. Pichler [84] compares wavelet transforms with adaptive Gabor filtering
feature extraction, and reported superior results using the Gabor technique. However, the
computational requirements are much larger for the wavelet transform, and in certain
applications accuracy may be compromised for a faster algorithm. Filtering features also
have been compared to the co-occurrence features in some other studies, with different
conclusions. Strand and Taxt [85] concluded that the co-occurrence features were
performing best, while Laws [86] and Clausi and Jernigan [87] drew the opposite
conclusion. Different setups, different test images, and different filtering methods may be
the reasons for the contradicting results.
Comparison of Gabor features with higher order spectral features has been investigated
using a set of images from the Brodatz album by Elunai [88]. In this study, experiments
show that higher order spectral invariant features produce better classification results for
more texture pairs than the Gabor features.
6.2 Gabor Filter 121 6.2 Gabor Filter
As outlined in Chapter 3, a 2D Gabor function, ),( yxg can be expressed as
)(2exp),(),( VyUxjyxgyxh +−′′= π (6.1)
where
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
22
5.0exp2
1),(yxyx
yxyxgσσσπσ
(6.2)
and ))cos()sin(),sin()cos((),( θθθθ yxyxyx +−+=′′ are the coordinates viewed by
system of rotated axis. The axis is rotated by a specified angle θ anticlockwise from the
positive x-axis. The frequencies in the orthogonal directions, U and V, can be combined
into the resultant frequency )( 22 VUf += cycles per image. For a square image of
dimension N x N, the response frequencies θf used in the experiment for each orientation
θ are obtained from octave frequencies given by:
]sin*max[cos0
θθθf
f = (6.3)
Thus, using (6.1) and (6.2) it is possible to design Gabor filters with arbitary
frequency, 0f , orientation, θ and bandwidth, b which is the product of xfσ . In this
experiment, 30 different orientations from 0° to 360° in steps of 12°, and five frequencies
per degree using (6.3) for 0f = 1, 2, 3, 4 and 5 cycles per image were used. b was adopted
122 6.3 Discriminant Analysis Testing and Results as 0.56, which is a commonly used bandwidth in feature extraction. These features are
rotation variant, so a DFT coding step is required to obtain rotation invariance. This
method of extracting rotation invariant features is used by [89].
6.3 Discriminant Analysis Testing and Results In Chapter 5, two types of experiments were carried out; pooled population and single
image population (refer to Section 5.7 for definition). In this segment of work, only an
experiment of the pooled population will be conducted. The experimental evaluation is
carried out using a set of negative stained EM images of Rotavirus, Adenovirus,
Calicivirus and Astrovirus. Rotavirus, Adenovirus and Calicivirus are used separately as
the target virus. Similar to the experimental procedure outlined in Section 5.4, the images
are segmented into individual viral particles and Gabor features were extracted from each
viral particle. The Gabor features that were extracted using the method and parameters
outlined in Section 6.2 are invariant to translation and rotation.
The experimental procedure outlined above was repeated using bispectral features. 150
features were extracted using the steps outlined in Figure 5.3. Radon transform
projections at 30 angles were computed to yield a one dimensional function. Five
bispectral features, ),( θaP (where 1,.....,5/2,5/1=a and ),.....,30/2,30/)( πππθ =rad
were computed from each such projection. The training and test samples used were
similar to the ones used in the Gabor features.
6.3 Discriminant Analysis Testing and Results 123
To evaluate and compare the quality of both the feature sets, discriminant testing is
performed. The Fisher linear discriminant (FLD) is used to measure and compare the
separability of Gabor and HOS features. The FLD is used since it is a recognized
nonparametric method to analyze the class separation in the feature space. The FLD is
determined by calculating the Fisher criterion
( ) ωωωωτωτ w
TB
T SS== (6.4)
where BS and wS are the between-class and within-class scatter matrices.
The average feature sets of Gabor and HOS and the ratio of this average are presented in
Table 6.1. The average feature sets of Gabor of Rotavirus, Adenovirus and Calicivirus
used separately as target virus is less than that obtained by the HOS feature sets. Clearly,
the higher order spectra feature sets have stronger separability than the Gabor feature
sets.
Features Target virus
Average feature of
Gabor ( )gabavg _τ
Average feature of HOS ( )hosavg _τ
Ratio
( )hosavggabavg __ ττ Rotavirus
2753
5318
0.52
Adenovirus
664
1310 0.51
Calicivirus
671
1208
0.55
Table 6.1 Comparison of the feature space separability of Gabor and HOS features using FLD.
124 6.4 Training and Classification using SVM 6.4 Training and Classification using SVM
Support Vector Machine (SVM) was used to train and classify populations of viral
particles. The features used are obtained using the experimental procedure outlined in
Section 6.3. When designing and training a SVM, selection of the kernel function and the
constraint parameter C are important. Traditionally, this has been done by trial-and-error
for kernel selection and exhaustive search for C. More recently, an algorithm for
automatically selecting the kernel parameters to minimize an upper bound on the
generalization error was given [90]. In either case, the SVM must be optimized for a large
number of different choices of parameters.
To make the comparison of Gabor and higher order spectral features in virus recognition
more effective, the constraint parameter C and number of features N that were used for
training and testing were varied. In this work, a trial-and-error method was used focusing
on SVMs with the RBF kernel. RBF widths, C of 0.001, 0.1 and 10 were considered. For
determining the optimal values of C and N in all cases, the same number of training sets
was used, with samples for training up to 40%, i.e. 400 samples, and the remaining 60%
for testing. The features used for training and testing were randomly chosen and there
was no overlapping between them. The recognition performance will be affected by the
selection of training data, and thus each experiment is conducted 1000 times and the
reported results given are an average. Each trial consists of feature averaging of 10
different particles.
6.4 Training and Classification using SVM 125 6.4.1 Results and Discussion
The comparison classification results of Gabor and higher order spectral features are
shown in Figure 6.1. The result presented in percentage of efficiency is calculated from
the miss and false alarm probabilities. The highest efficiency, 83% is obtained by using
higher order spectral features with kernel width of 0.1 and 75 features when Rotavirus is
the target virus. The experimental evaluation also is carried out using Adenovirus and
Calicivirus as the target virus. In both cases, higher order spectral features perform much
better than Gabor features, with 100% efficiency in the verification of adenovirus using
higher order spectral features compared with 69% using Gabor features. In the
verification of Calicivirus, higher order spectral features achieved an efficiency of 87%
while Gabor features only produced 62%.
126 6.5 Summary
30 40 50 60 70
50
60
70
80
90
100
No of features
Effic
ienc
y (%
)gabor,kernel = 0.001
gabor,kernel = 0.1
gabor,kernel = 10
HOS, kernel = 0.001
HOS, kernel = 0.1
HOS, kernel = 10
Figure 6.1: The plot of efficiency versus number of features for a subpopulation of 10 viral particles trained using SVM classifier. A choice of 3 kernel widths were used, 0.001, 0.1 and 10 to train the features. The efficiency is calculated from the miss and false alarm probabilities.
6.5 Summary
In this chapter, Gabor features were compared with higher order spectral features in virus
recognition. Experimental evaluation of both the feature sets were performed using
negative stained EM images of four types of viruses; rotavirus, adenovirus, astrovirus and
calicivirus, whose morphologies are quite similar. To evaluate and compare the quality
of both the feature sets, Fisher linear discriminant (FLD) was used to measure and
compare the feature space separability of Gabor and HOS features. The HOS feature sets
demonstrated to produce higher feature space separation compared to Gabor feature sets.
6.5 Summary 127
A SVM was used to train the features. The optimal result is gained by trying different
combinations of kernel widths and number of features in training and testing
observations.The result shows that the highest efficiency, 83% is achieved by using
higher order spectral features of the RBF kernel width of 0.1 in the verification of
Rotavirus. In the verification of Calicivirus and Adenovirus, higher order spectral
features are superior to Gabor features (Table 6.2).
Rotavirus
Calicivirus
Adenovirus
HOS features
83%
87%
100%
Gabor features
79%
62%
69%
Table 6.2: Efficiency (%) of higher order spectral features and Gabor features in the
verification of Rotavirus, Calicivirus and Adenovirus.
Chapter 7 Conclusion and Further Work
7.1 Conclusion
This thesis has made an original and substantial contribution to science. The methodology
developed is able to detect and classify different types of viruses, that are difficult to
distinguish visually from two-dimensional images obtained from an electron microscope.
It is based on radial spectra of higher order spectral invariant features that capture
information about variation in texture and contour and differences in symmetries of
different types of viruses. This method takes advantage of the large numbers of particles
available in these images using invariant properties of the higher order spectral features
and statistical techniques of feature averaging and soft decision fusion to improve
classification accuracy.
130 7.1 Conclusion
Experimental evaluations were carried out on negative stained EM images of
gastroenteritis viruses. A high statistical reliability and low misclassification rates were
achieved using this methodology. An Equal Error Rate of less than 0.2% was achieved in
verifying Rotavirus by input fusion, where the features are combined from 15 images,
and soft decisions (likelihoods) are averaged from 2 such sets of 15 images. Even though
only 6 types of viruses were used in this scope of work, preliminary studies of texture and
contour (Chapter 4) have shown the ability of higher order spectral features in
distinguishing a large number of images of different textures and contours. An average
misclassification of 1.37% was obtained over forty different textures chosen from the
Brodatz album.
Gabor features were compared with higher order spectral features to verify Rotavirus,
Calicivirus and Adenovirus. Overall results show a superiority of higher order spectral
features over Gabor features in this application. An efficiency of 83% was achieved in the
verification of Rotavirus using HOS versus 79% using Gabor. In the verification of
Calicivirus and Adenovirus, efficiencies of 87% and 100% were achieved using HOS
compared with only 62% and 69% using Gabor. The results presented used a feature
averaging of 10 particles and SVM as the classifier. Higher efficiency can be achieved by
increasing the number of particles for feature averaging or averaging the likelihood
scores (as shown in Chapter 5).
This methodology can be incorporated with the existing automated virus segmentation
methods to make it fully automated. With the simple and fast negative stain preparation
7.2 Further Work 131 and automated pattern recognition, electron microscopy permits a rapid detection and
identification of infectious agents. This will further encourage a broad application of EM
such as investigation of potential bioterrorist events [91-93] and identification of an
outbreak.
7.2 Further Work
Continuing on from the research presented in this thesis, a number of possible avenues
for further research have been identified, including:
(i) Chapter 3 discussed some of the segmentation methods of negative stained EM
virus images. These segmentation methods can be applied and incorporated with our
system to have a fully automated virus recognition system.
(ii) Chapter 5 showed the comparison of higher order spectral (bispectral) and Gabor
features in virus recognition, with results showing a higher degree of accuracy using
bispectral features in this application. There are a great number of other techniques that
remain untested. Thus, there exists much potential to improve on the results presented in
this thesis by testing other texture-based approaches. In this scope of work, input fusion,
where the likelihood scores were averaged from M sets, each set consists of feature
averaging of N images were used. There are different kinds of fusion strategies that are
available. Thus, the combination of these features in an intelligent manner also is a
132 7.2 Further Work topic worthy of future consideration although this investigation can be computationally
quite demanding.
(iv) A multiple classifier system that is based on a combination of outputs of a set of
different classifiers has been shown as a method to develop high performance
classification systems. The dimensionality of the data that can be reduced either through
selection of a subset of features, or by extracting from the vector the principal
components of variation also leads to an increase in classification accuracies. Numerous
examples of these techniques are found in the literature. Thus, further investigation on
combining classifiers and feature selection methods is another avenue for possible future
research.
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Appendix A A Study of Texture and Contour A.1 Experimental Results of the Relative Importance of Phase and Magnitude in Texture
D101 D92D101 D92
D78 D101D78 D101
(a) (b)
Figure A.1(a): Subfigures of (a)-(b) Results when images of 2 homogeneous textures used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
144 Appendix A
D101 D56D101 D56 D46 D34D46 D34
(c) (d)
D34D20 D34D20 D20 D6D20 D6
(e) (g) Figure A.1(a): Subfigures of (c)-(g) Results when images of 2 homogeneous textures used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
Appendix A 145
D6D3 D6D3 D6 D11D6 D11
(g) (h)
D11 D82D11 D82 D11 D101D11 D101
(i) (j) Figure A.1(a): Subfigures of (g)-(j) Results when images of 2 homogeneous textures used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
146 Appendix A
D34 D78D34 D78 D11 D78D11 D78
(k) (l) Figure A.1(a): Subfigures of (k)-(l) Results when images of 2 homogeneous textures used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
D42D31 D42D31 D97 D107D97 D107
(a) (b) Figure A.1(b): Subfigures (a)-(b) Results when images of 2 inhomogeneous textures used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
Appendix A 147
D62 D42D62 D42 D62 D91D62 D91
(c) (d)
D88 D91D88 D91 D13 D45D13 D45
(e) (f) Figure A.1(b): Subfigures (c)-(f) Results when images of 2 inhomogeneous textures used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
148 Appendix A
D88 D7D88 D7 D44 D40D44 D40
(g) (h)
D42D90 D42D90 D42 D45D42 D45
(i) (j) Figure A.1(b): Subfigures (g)-(j) Results when images of 2 inhomogeneous textures used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
Appendix A 149
D30 D88D30 D88 D41 D62D41 D62
(k) (l) Figure A.1(b): Subfigures (k)-(l) Results when images of 2 inhomogeneous textures used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
D62D55 D62D55 D42D46 D42D46
(a) (b) Figure A.1(c): Subfigures (a)-(b) Results when images of a homogenous and inhomogeneous texture used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
150 Appendix A
D20 D61D20 D61
D45D101 D45D101
(c) (d)
D43 D56D43 D56 D46 D45D46 D45
(e) (f) Figure A.1(c): Subfigures (c)-(f) Results when images of a homogenous and inhomogeneous texture used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
Appendix A 151
D11 D42D11 D42
D11 D40D11 D40
(g) (h)
D56 D62D56 D62 D16 D108D16 D108
(i) (j) Figure A.1(c): Subfigures (g)-(j) Results when images of a homogenous and inhomogeneous texture used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
152 Appendix A
D21 D30D21 D30 D61 D52D61 D52
(k) (l) Figure A.1(c): Subfigures (k)-(l) Results when images of a homogenous and inhomogeneous texture used as input textures after subjected to steps in Figure 4.2. The top images of each subfigure are the input and the bottom images are the output.
Appendix A 153 A2. Classification Results of a Large Database of Different Textures and Contours using Higher Order Spectral Features
D1 D2 D3 D5 D6 D8 D9 D11 D16 D17 D19 D20 D21 D22 D24 D28
D2 1.87
D3 0.35 0.27
D5 0.79 3.54 0.08
D6 0.02 0.06 0.33 0
D8 0 0.97 0 0.68 0
D9 1.00 0.54 1.93 0.04 1.06 0
D11 6.22 1.68 0.93 0.27 0 0.20 1.25
D16 0 0 0.04 0 1.02 0.04 2.25 0.04
D17 0 0 0 0 1.47 0.02 0.41 0 0.33
D19 4.60 3.27 1.70 0.64 0.22 0.12 1.25 3.79 0.77 0.20
D20 0.68 2.35 0.16 2.00 0.02 0.10 0.66 0.25 0 0 1.97
D21 0.06 0 0.02 0 0 0 0.79 0 0.04 0.68 0.02 0
D22 0.81 0.37 2.33 0.33 0.08 0 0.93 1.66 0.12 0 2.72 1.29 0
D24 0.02 0.16 0.56 0 1.64 0.04 2.52 0.18 2.95 1.06 0.29 0 2.39 0.47
D28 3.64 7.06 0.31 2.31 0 0.41 0.29 2.10 0 0 7.72 2.50 0 0.56 0
D29 0.6 0.52 1.91 0 2.02 0.04 4.06 0.52 2.02 0.81 1.83 0.02 1.41 0.85 3.64 0.02
D32 0.27 0.02 2.6 0 3.20 0.02 3.58 0.16 1.62 0.39 0.70 0.04 0.52 1.18 5.25 0
D34 1..31 0.54 1.14 0.02 0.43 0 5.10 2.10 0.85 0.08 3.95 0.16 0.22 2.70 2.20 0.50
D36 0.29 0.66 5.75 0.12 0.27 0 1.45 0.43 0.08 0.02 2.27 0.10 0 3.62 0.10 0.27
D38 0.93 0.14 0.41 0.06 0.18 0.25 2.68 0.93 1.31 0.45 0.47 0 0.10 0.04 1.33 0
D52 0.20 0.25 3.77 0.02 0.58 0 1.56 0.66 0.41 0.16 0.47 0.29 0.16 2.45 3.16 0.12
D53 0.04 0 0 0 0 0.02 1.14 0 0.25 0.93 0.14 0 3.16 0.02 1.70 0
D55 0.35 0.22 0.41 0 2.29 0 2.64 0.25 1.64 0.52 0.39 0 0.27 0.18 3.29 0
D57 0 0.29 1.06 0 1.77 0 2.10 0.08 1.54 0.47 0.39 0.02 0.02 0.47 4.93 0.06
D68 2.8 0.52 0.81 0.06 0.22 0.08 2.68 2.75 0.54 0.16 2..35 0.14 0.02 0.43 1.83 0.06
D77 0 0 0 0 0.04 0.02 0.02 0 0.02 0.37 0 0 1.41 0 0.39 0
D78 0.27 0.04 0.37 0 1.66 0.10 2.16 0.02 1.91 1.00 0.06 0.08 1.43 0.08 4.04 0
D80 0.06 0.18 0.41 0 1.70 0.02 4.25 0.02 2.12 0.47 0.31 0.04 1.39 0.58 7.31 0
D82 0.14 0.08 0.37 0 2.45 0 3.97 0.14 2.00 2.33 0.27 0.20 0.75 0.45 6.68 0
D83 0.41 0.29 0.66 0 0.68 0.10 3.5 0.12 1.06 0.31 1.14 0.60 0.62 0.37 1.50 0.10
D84 1.00 0.85 6.27 0.12 1.62 0 4.70 1.83 1.25 0.25 1.12 0.95 0.16 3.54 3.16 0.62
D85 0.06 0.02 0.12 0 0 0.04 1.58 0.02 0.83 0.02 0.16 0 0.12 0.66 4.87 0
D92 2.70 1..35 2.58 0.41 0.29 0 3.66 2.60 0.43 0 4.54 1.66 0.33 3.25 1.18 2.00
D93 0.04 0.18 0.66 0 1.83 0 2.89 0.22 3.33 1.14 0.29 0.02 1.43 0.43 6.41 0
D101 0.79 5.12 0.35 3.58 0.06 0.45 0.39 1.79 0.14 0 2.06 1.52 0.10 0.33 0.06 6.08
D103 0.08 0.18 3.08 0 1.54 0 3.02 0.52 0.62 0.16 0.68 0.16 0.14 1.33 3.08 0.18
D104 0.14 0.12 6.43 0.02 0.72 0 1.72 0.81 1.08 0.08 0.43 0.02 0.10 2.02 2.87 0.04
D110 2.39 0.45 3.10 0.02 0.18 0.10 6.16 1.77 0.29 0 3.50 0.37 0.10 3.06 1.33 0.22
D111 2.12 3.60 0.22 1.54 0 0.16 0.14 2.12 0 0 7.83 2.31 0 0.97 0.02 6.06
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154 Appendix A
D32 D34 D52 D53 D55 D57 D68 D77 D78 D80 D82 D83 D85 D92 D93 D101
D34 3.79
D52 0.54 2.31
D53 1.65 0.77
D55 2.08 2.50 0.02
D57 1.31 0.18 1.37 0.22
D68 1.79 0.75 2.25 0.22 3.54
D77 1.29 2.10 0.93 0.10 0.50 0.68
D78 1.93 3.31 0.02 1.08 0.16 0.45 0
D80 0.41 0.04 1.08 0.54 1.95 0.68 1.72 0.06
D82 2.27 0.68 0.85 1.50 4.33 2.77 0.93 0.20 3.37
D83 4.87 0.83 2.91 0.64 4.87 2.29 1.37 0.04 1.91 4.45
D84 3.37 2.18 0.72 0.29 0.75 0.37 2.43 0.16 0.97 4.47 2.06
D85 3.33 1.08 4.58 0.25 2.20 2.56 2.10 0.25 1.16 2.35 4.43 1.56
D92 4.62 6.89 0.18 0.75 2.18 2.18 0.10 0.06 1.45 3.93 2.10 2.14 0.68
D93 2.64 0.35 2.27 0.31 0.60 0.87 3.75 0 0.97 0.93 0.85 2.54 5.45 0.25
D101 3.22 9.35 1.54 1.00 4.91 3.20 1.00 0 3.33 7.83 3.89 2.00 2.75 5.20 0.66
D103 5.35 1.33 0.12 0.04 0.12 0.06 0.27 0.04 0.18 0.14 0.31 0.20 0.33 0.08 0.72 0.08
D104 0.25 0.64 7.72 0.27 1.18 5.00 0.70 0.02 1.00 2.12 3.27 0.77 4.27 0.50 2.02 0.31
D110 2.93 3.56 4.18 0.04 1.83 4.29 0.66 0.08 1.06 1.66 1.06 1.41 5.75 0.58 1.12 0.41
D111 1.45 2.43 3.27 0.08 0.58 1.72 4.39 0 0.39 0.72 1.35 2.54 6.52 0.16 5.72 4.00
Table A.2: Misclassification in percentage of a large database of different textures and contours images that were chosen from the Brodatz album using higher order spectral features. The overall performance over forty textures shows an average misclassification of 1.37%.
Appendix A 155 A.3 Classification Results of Images of the Same Texture but Different Contours using Higher Order Spectral Features
Circular 5 fold sym 6 fold sym Circular 5 fold sym 6.19 6 fold sym 4.25 9.35 7 fold sym 3.44 8.04 7.63
(a)
Circular 5 fold sym 6 fold sym Circular 5 fold sym 5.21 6 fold sym 2.54 5.42 7 fold sym 5.75 4.96 3.73
(b)
Table A.3: Misclassification rate (%) of four different contours of (a) D8 and (b) D17 texture images using higher order spectral features.
156 Appendix A A.4 Classification Results of Images with Various Level of Noise using Higher Order Spectral Features
D5 D8 D17 D5 D8 31.0 D17 28.8 25.6 D21 32.9 31.5 30.4
(a)
D5 D8 D17 D5 D8 8.58 D17 0.16 1.2 D21 0.21 1.2 10.2
(b)
D5 D8 D17 D5 D8 7.8 D17 0.21 1.25 D21 0.17 1.25 11.2
(c)
D5 D8 D17 D5 D8 5.5 D17 0.39 1.0 D21 0.48 0.98 13.2
(d)
Table A.4: Misclassification rate (%) of four textures with added white Gaussian noise of SNR equal to (a) -80 dB (b) -10 dB (c) 0 dB (d) 10dB using higher order spectral features.
Appendix A 157 A.5 Classification Results with Averaging of Features and Input Fusion of Noisy Images using Higher Order Spectral Features
D5 D8 D17 D5 D8 5.33 D17 1.04 0.83 D21 1.25 0.5 12.8
(a)
D5 D8 D17 D5 D8 0.104 D17 0 0 D21 0 0 1.98
(b)
D5 D8 D17 D5 D8 0 D17 0 0 D21 0 0 0
(c)
Table A.5: Misclassification rate (%) when M = (a) 2 (b) 5 and (c) 10 sets and N=5 images were used for classification using higher order spectral features. Gaussian noise of SNR = -30 dB was added to the image.
Glossary Antigen: Any agent that initiates antibody formation and/or induces a state of active
immununological hypersensitivity and that can react with the immunoglobulins that are
formed.
Capsid: A protein shell comprising the main structural unit of a virus particle.
Capsomers: The individual protein units that form the capsid of a virus
Envelope: A lipid membrane enveloping a virus particle.
Nucleocapsid: The core of a virus particle consisting of the genome plus a complex of
proteins.
Virions: Structurally mature, extracellular virus particles.
Protomer: Set of DNA sequences necessary for initiation of transcription by a DNA-
dependent RNA polymerase.
Pleomorphic: The variation in size