Table Of ContentDIGITAL IMAGE PROCESSING
Mathematical and Computational Methods
"You British asses, who expect to hear ever some new thing, I have nothing to
tell, but what I fear may be a true thing. For Tait1 comes with his plummet
and his line, quick to detect your old stuff, now dressed in what you call a fine
popular lecture."
James Clerk Maxwell, 1868
"Talkingofeducation, peoplehave now a-days" (he said) "got astrangeopinion
that every thing should be taught by lectures. Now, I cannot see that lectures
can do so much good as reading the books from which the lectures are taken.
I know nothing that can be best taught by lectures, except where experiments
are to be shown. You may teach chymestry by lectures - You might teach
making ofshoes by lectures!"
Samuel Johnson, 1766
DEDICATION
To all those students with whom I had the good fortune
to work and, in using the material herein, taught me how
to teach it
HORWOOD
1An experimentalist and closecolleagueofMaxwell
ABOUT THE AUTHOR
Jonathan Blackledge graduated in physics from Imperial College and music
from the Royal College of Music, London, in 1 ")0 and obtained a Doctorate
in theoretical physics from the same university in 1983. He was appointed
as Research Fellow of Physics at Kings College, London from 1983 to 1988
specializingin inverseproblemsin electromagnetismand acoustics. During this
period, he worked on a number of industrial research contracts undertaking
theoretical and computational work on the applications of inverse scattering
theory for the analysis of signals and images.
In 1988, hejoinedthe AppliedMathematicsandComputingGroupat Cran
field University as Lecturer and later, as Senior Lecturer and Head of Group
where he promoted postgraduate teaching and research in applied, engineering
and industrial mathematics in areas which included computer aided engineer
ing, digitalsignalprocessingandcomputergraphics. In 1994, he was appointed
ProfessorofAppliedMathematicsandComputingandHeadofthe Department
of Mathematical Sciences at De Montfort University where he established the
Institute of Simulation Sciences. He is currently Professor of Digital Signal
Processing and Communications Technology in the Department of Electronics
and ElectricalEngineeringat LoughboroughUniversity, EnglandandProfessor
ofComputer Science in the Department of Computer Science at the University
of the Western Cape, South Africa. He is also a co-founder and Director of
a group of companies specializing in communic, lions technology and financial
analysis based in London and New York.
ProfessorBlackledgehaspublishedover one hundredscientific andengineer
ing research papers and technical reports for industry, six industrial software
systems, fifteen patents, ten books and has been supervisor to sixty research
(PhD) graduates. He lectures widely to a variety of audiences composed of
mathematicians, computer scientists, engineers and technologists in areas that
include cryptology, communications technology and the use of artificial intelli
gence in process engineering, financial analysis and risk management. His cur
rent researchinterests includecomputationalgeometryandcomputergraphics,
image analysis, nonlinear dynamical systems modelling and computer network
security, working in both an academic and commercial context. He holds Fel
lowships with England's leading scientific and engineering Institutes and Soci
eties including the Institute of Physics, the Institute of Mathematics and its
Applications, theInstitutionofElectricalEngineers, the InstitutionofMechan
ical Engineers, the British Computer Society, the Royal Statistical Society and
the Institue of Directors. He is a Chartered Physicist, Chartered Mathemati
cian, CharteredElectricalEngineer, CharteredMechanicalEngineer, Chartered
Statistician and a Chartered Information Technology Professional. He has an
additionalinterestin music for whichhe holds aFellowshipoftheRoyalSchools
of Music, London.
DIGITAL IMAGE PROCESSING
Mathematical and Computational Methods
JONATHAN M. BLACKLEDGEt
Professor of Digital Signal Processing and Communications Technology,
Department of Electronic and Electrical Engineering,
Loughborough University, England
Horwood Publishing
Chichester, West Sussex
tProfessor of Computer Science, Department of Computer Science,
University ofthe Western Cape, Republic ofSouth Africa.
HORWOOD PUBLISHING LIMITED
CoIl House, Westergate, Chichester, West Sussex, P020 3QL, England.
First published in 2005.
@J. M. Blackledge, 2005
All Rights Reserved. No part of this publication may be reproduced, stored
in a retrieval system, or transmitted, in any foriu or by any means, electronic,
mechanical, photocopy, recording, or otherwise, without the permissionofHor
wood Publishing Limited, CoIl House, Westergate, Chichester, West Sussex,
P020 3QL, England.
ISBN 1-898563-49-7
British Library Cataloguing in Publishing Data
A catalogue record of this book is available from the British Library.
Typeset in LaTeX, the TeXnicCenter graphical user interface and the stylefile
of the Institute of Mathematics and its Applications.
Printedand bound byAntony Rowe Ltd, Eastbourne
---.
v
Foreword
Newspapers and the popular scientific press today publish many examples of
highlyimpressiveimages. Theseimagesrange, forexample, from thoseshowing
regions of star birth in the distant Universe to the extent of the stratospheric
ozone depletion over Antarctica in springtime, and to those regions of the
human brain affected by Alzheimer's disease. Processed digitally to generate
spectacular images, often in false colour, they all make an immediate and deep
impact on the viewer's imagination and understanding.
Professor Jonathan Blackledge's erudite but very useful new treatise Digi
tal Image Processing: Mathematical and Computational Methods explains both
the underlying theory and the techniques used to produce such images in con
siderable detail. It also provides many valuable example problems - and their
solutions -so that the reader cantest his/her graspofthe physical, mathemat
ical and numerical aspects of the particular topics and methods discussed. As
such, this magnum opus complements the author's earlier work Digital Signal
Processing. Bothbooksare a wonderful resourcefor studentswho wish tomake
their careers in this fascinating and rapidly developing field which has an ever
increasing number of areas of application.
The strengths of this large book lie in:
• excellent explanatory introduction to the subject;
• thoroughtreatmentofthetheoreticalfoundations, dealingwithbothelec
tromagnetic and acoustic wave scattering and allied techniques;
• comprehensive discussion of all the basic principles, the mathemati
cal transforms (e.g. the Fourier and Radon transforms), their inter
relationshipsand, in particular, Bornscatteringtheoryandits application
to imaging systems modelling;
• discussion in detail - including the assumptions and limitations - of opti
cal imaging, seismic imaging, medical imaging (using ultrasound), X-ray
computer aided tomography, tomography when the wavelength of the
probing radiation is of the same order as the dimensions ofthe scatterer,
SyntheticApertureRadar (airborneor spaceborne), digitalwatermarking
and holography;
• detaildevotedtothemethodsofimplementationoftheanalyticalschemes
in various case studies and also as numerical packages (especially in
C/C++);
• coverage of deconvolution, de-blurring (or sharpening) an image, maxi
mum entropy techniques, Bayesian estimators, techniques for enhancing
the dynamic range of an image, methods of filtering images and tech
niques for noise reduction;
• discussionofthresholding, techniquesfor detectingedgesin an imageand
for contrast stretching, stochastic scattering (random walk models) and
models for characterizing an image statistically;
vi
• investigation of fractal images, fractal dimension segmentation, image
texture, the coding and storing of large quantities of data, and image
compression such as JPEG;
• valuablesummaryoftheimportantresultsobtainedineach Chaptergiven
at its end;
• suggestions for further reading at the end of each Chapter.
I warmly commend this text to all readers, and trust that they will find it
to be invaluable.
ProfessorMichaelJRycroft
Visiting Professor at the International Space University, Strasbourq, France,
and at Cranfield University, England.
In 2003 Jonathan Blackledge published Digital Signal Processing, a book
based on material developed by him for the first semester of the MSc course
in Digital Systems Engineering offered by the Department of Electronic and
ElectricalEngineering at Loughborough University. Thecontentofthe present
text forms the basis of the second semester of that course, and it completes an
authoritative and comprehensive account of the subject. The requisite mathe
matical and computational techniques are coveredin satisfying detail, but the
really significant feature is the way in which tlu fundamental physics underly
ing the generationofdata is consistently and thoroughlyexplored. This is not
simply a transcript of a course oflectures aiming to describe the methods used
to process images but a painstaking study of the principles involved, together
with a generous supply of wide-ranging examples and tutorial problems (all
provided with detailed model answers). The aim, in the author's own words,
has been to 'encourage the reader to design some example software solutions
for digital image processing' and to 'develop a small digital image processing
library that can be developed further and tailored to his/her learning and/or
researchinterests'. That aim has beenmost satisfactorilyachieved. Digitalim
age processing is, of course, a most rapidly changing and developing field, but
this book promisesto remaina standard andessentialguide toits fundamental
ideas and techniques for a considerable time to come.
ProfessorRoy FHoskins
Visiting Professor, Loughborough University, England
vii
Preface
Digital Image Processing complements Digital Signal Processing (Horwood
Publishing 2003) which was based on teaching material developed for the MSc
programme in Digital Systems Engineering at Loughborough University. Digi
tal Image Processingextends this material further by exploring the character
istics of imaging systems, the computational techniques used to process digi
tal images and the Interpretation of the information which an image conveys
through an understanding of the physical processes that occur.
Many excellent image processing systems, software libraries and packages
are currently available for low-level general applications whereas others have
been designed for specific applications. Users can process images using either
a command line language (e.g. the MATLAB2 image processing toolbox) or a
graphicaluser interface (e.g. Adobe Photoshop) to improvethe generalquality
and fidelity of a digital image and/or to achieve results conveying specific as
pects ofitsinformationcontent (featureextraction). This can be accomplished
without the user having a thorough understandingof the computational meth
ods involved or how and why such methods have evolved, e.g. the application
of a particular filter. For those who are only interested in using a particular
processing system to 'get the job done' working in a commercial environment
for example, application of a specific commercial package or packages with
an appropriate selection of image processing options is all that is required.
However, for those who wish to contribute to the future development of such
systems and/or develop their own 'home-spun' versions for research purposes,
a deeper understanding of the mathematical and computational techniques is,
by necessity, required.
This work provides a study of the computational methods that are used to
process images, but in such a way that there is a direct link (where possible)
between the process that is used, the data to which it is applied and, most
of all, the 'physics' that underpins the generation of the data. In order to
do this, it is necessary to spend some time discussing the principles of how
waves and wavefields propagate and interact with objects whose images are
required. Depending on the wavelengthof the field, the interactionsthat occur
are usually in the form of some scattered wavefield. Hence, after a review of
the mathematical and computationalbackgroundto thesubject given in Part I
(which includes materialon vector fields, the 2D Fourier transform and the 2D
FIR filter), weprovidean introductiontothefieldequationsandwaveequations
used to model different types of wavefields and the scattering theory needed
to develop appropriate models for the images that these wavefields produce in
terms ofthe information on the imagedobject that they convey. We formulate
some of the analytical methods and results that are required to compute a
scattered wavefield and provide details on the equations that are used in later
chapters. Some of this material is based on a previous work published by the
author, namely, Quantitative Coherent Imaging (Academic Press, 1989), which
was concerned with the principles of interpreting the structure and material
properties of objects by the way in which they scatter electromagnetic and
2High-leveltechnical computing language by MathWorks Inc.
viii
acoustic radiation with the aim of exploring tlu theory, methods and some of
the applications of incoherent and coherent imaging systems.
Having established the principal theoretical background to modelling an
imaging system, we look at a range of imaging techniques which are classified
into two main types, namely, incoherent and coherent imaging. In Part II, in
coherent optical systems are studied and an introduction given to the method
ofprojectiontomographywhereit is assumedthat the probe (i.e. the radiation
field) used to interrogate an object can be described in terms of a sequence of
raystraceablethroughtheobject and'back-projected'. PartII includesastudy
ofcoherent imagingmethods and investigates the principles ofcoherent optics,
the imaging of layered media, diffraction tomography and synthetic aperture
imaging. Both electromagnetic and acoustic imaging systems are discussed.
In the case of diffraction tomography for example, the aim is to interpret the
internalstructure and compositionof an object by the way in whichit diffracts
electromagneticor acoustic radiation. Two types ofdiffraction tomographyare
discussed where the object isilluminated/insonifiedwith a wavefield oscillating
at a fixed frequency (Continuous Wave or CW case) or with a short pulse of
radiation. In the material on synthetic aperture imaging, attention is focused
on the use ofRadar for imagingthesurfaceofthe Earthand a model presented
to describe the scattering of a pulse of frequencv modulated microwave radia
tion by the ground. This material also includes a case study which develops a
solution to the so called 'sea spikes' problem.
In the 'light' of the preceding material, Part III introduces the basis of
digital image processing including the problem of image restoration, image
reconstruction and image enhancement. The methods discussed are all related
in one form or another to the physical principles presented in Parts I and II
and forms the basis for Part IV of this work which studies the principles of
pattern recognition and computer vision. This includes an introduction to
statistical modelling and analysis, an extended chapter on fractal images and
fractal image processing, and a chapteron datacoding andimagecompression,
including fractal image compression.
The author has attempted to provide the reader with the mathematical
methods requiredfor imageanalysiswhicharethen usedtodevelopmodels and
algorithms for processing digital images and, finally, to encourage the reader
to design some example softwaresolutions for Digital Image Processing (DIP).
In this way, the reader is invited to develop a small DIP library that can then
be developedfurther and tailored to his/herlearning and/or research interests.
This is accomplished by the inclusion of a series of tutorial problems which are
given at the end of each Part with model ans- ers pro' ided in Appendix A.
These problems include theoretical, computational and programming exercises
in the C programming language.
Theemphasisthroughout ison the mathematicalfoundations ofthesubject
whicharecommonto avarietyofimagingsystems and methods. In somecases,
examples have been provided to illustrate the conversion of a computational
algorithm into a computer program. Either pseudo code, C or MATLAB code
is used for this purpose. The book has been designed to serve the reader with
enough formal detail for him/her to acquire a firm foundation on which to
ix
build. References to other important texts and/or key scientific papers are
included at the end of each chapter or within the text for this purpose.
The material presented in this book is based on the lecture notes and sup
plementary material developed by the author as part of an advanced taught
MSc programmein 'Digital SignalProcessing'. This programmewas originally
established at Cranfield University in 1990 and modified at De Montfort Uni
versity in 1994. The programmes are still operating at these universities and
the material has been used by more than 500 graduates since its creation and
development in the early 1990s. The material was enhanced and developed
further when the author moved to the Department ofElectronic and Electrical
Engineering at Loughborough University in 2003, and now forms part of the
department's post-graduateteaching and learning activities. Theoriginal MSc
programmewas basedon taught componentscoveringa periodofeight months
and consisting of two semesters, each semester, being composed of four mod
ules; the third semester focused on a minor research project. The material in
this work covers the second semester and is 'index-linked' through this teach
ing programme to the publication Digital Signal Processing (Horwood, 2003)
which covers the first semester. The classification of this work into four parts
reflects the four modules given in the second semester. It has been necessary
to include some of the material published previously with the view of revising
some of the principal themes such as those concerned with the properties and
computational methods associated with the Fourier transform. This has been
done for reasons of completeness and to provide the reader with an account
of the field that does not necessarily require significant reference to previous
publications (by the author or otherwise).
An attempt has been made to cut through much of the jargon characteriz
ing different fields of research in imaging science presenting an account of the
fundamental physicalprinciples commonto nearly all imagingsystems. This is
done by illustrating the similarity ofthe underlying mathematical models used
to process data on a wavefield in a variety of applications. The approach has
beento unifythe principlesofdifferent imagingsystems andtoprovideacourse
text covering the theoretical foundations of imaging science in an integrated
and complete form.
Finally, while every effort has been made by the author and publisher to
provide a work that is free from error, it is inevitable that in a first edition,
typing errorsor 'typos' and 'bugs' will occur. Ifthe readerstartstosuffer from
a lack of comprehension over certain aspects of the material (due to errors or
otherwise) then he/she should not assume that there is something wrong with
him/herselfas the fault may lie with the author and his imaging system!
Professor Jonathan M Blackledge
Department of Electronic and Electrical Engineering, Loughborough Univer
sity.
