Table Of ContentBe´zier and Splines in Image Processing
and Machine Vision
Sambhunath Biswas • Brian C. Lovell
Be´zier and Splines in Image
Processing and Machine Vision
SambhunathBiswas BrianC.Lovell
IndianStatisticalInstitute TheUniversityofQueensland
Kolkata,India Brisbane,Australia
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To my late parents, Kali Kinkar Biswas and Niharbala
Biswas, who were always inspiring
Sambhunath Biswas
To my wonderful and supportive wife, Vicki, and my beautiful
daughters, Adeleine, Quetta, and Tess, who were very
understanding during the many hours spent writing this
manuscript, both at home and abroad
Brian C. Lovell
Preface
Therapiddevelopmentofsplinetheoryinthelastfivedecades—anditswide-
spread applications in many diverse areas—has not only made the subject
richanddiverse,butalsomadeitimmenselypopularwithindifferentresearch
communities. It is well established that splines are a powerful tool and have
tremendousproblem-solvingcapability.Ofthelargenumberofsplinesdiscov-
ered so far, a few have established permanent homes in computer graphics,
image processing, and machine vision. In computer graphics, their significant
role is well documented. Unfortunately, this is not really the case in machine
vision, even though a great deal of spline-based research has already been
doneinthisarea.Thesituationissomewhatbetterforimageprocessing.One,
therefore, feels the need for something in the form of a report or book that
clearly spells out the importance of spline functions while teaching a course
on machine vision. It is unfortunate that despite considerable searching, not
even a single book in this area was found in the market. This singular fact
providesthemotivationforwritingthisbookonsplines,withspecialattention
to applications in image processing and machine vision.
The philosophy behind writing this book lies in the fact that splines are
effective, efficient, easy to implement, and have a strong and elegant mathe-
matical background as well. Its problem-solving capability is, therefore, un-
questionable. The remarkable spline era in computer science started when P.
E. B´ezier first published his work on UNISURF. The subject immediately
caught the attention of many researchers. The same situation was repeated
withthediscoveryofIngridDaubechi’swavelets.Differentwaveletsplinesare
now well known and extensively found in the literature. As splines are rich
inproperties,theyprovideadvantagesindesigningnewalgorithmsandhence
theyhavewide-scaleapplicationsinmanyimportantareas.B´ezierandwavelet
splines, can, therefore, be regarded as two different landmarks in spline the-
ory with wide application in image processing and machine vision, and this
justifies the title of the book.
In writing this book, therefore, we introduce the Bernstein polynomial
at the very beginning, since its importance and dominance in B´ezier spline
VIII Preface
models for curve and surface design and drawing are difficult to ignore. We
omittedthedesignproblemsofcurvesandsurfacesbecausetheyaredealtwith
inalmostallbooksoncomputergraphics.Someapplicationsindifferentimage
processingareas,basedontheB´ezier-Bernsteinmodel,arediscussedindepth
in Chapters 1, 2, 3, and 4, so that researchers and students can get a fairly
goodideaaboutthemandcanapplythemindependently.Chapter1provides
a background for B´ezier-Bernstein (B-B) polynomial and how binary images
canbeviewed,approximated,andregeneratedthroughB´ezier-Bernsteinarcs.
Chapter 2 explains the underlying concept of graylevel image segmentation
and provides some implementation details, which can be successfully used for
image compression. In Chapter 3 of this book, we will show how one can
use one dimensional B-B function to segment as well as compress image data
points. Chapter 4 depicts image compression in a different way, using two
dimensional B-B function.
B-splines, discussed in Chapter 5, are useful to researchers and students
of many different streams including computer science and information tech-
nology, physics, and mathematics. We tried to provide a reasonably compre-
hensive coverage. Attention has been devoted to writing this chapter so that
students can independently design algorithms that are sometimes needed for
theirclasswork,projects,andresearch.Wehavealsoincludedapplicationsof
B-splines in machine vision because we believe it also has strong potential in
research.ThebetasplinesdiscussedinChapter6arerelativelynewandmuch
work remains to be done in this area. However, we tried to discuss them as
much as possible and indicated possible directions of further work.
In Chapter 7, discrete splines are discussed, along with the feasibility of
their use in machine vision. The application is appropriate and informative.
It shows how the problem of recovering surface orientations can be solved
throughasystemofnonlinearequations.Splinesinvisionisanopenareaand
much attention needs to be paid for further research work. Wavelet splines
are relatively new, so we took special care to write the theory in a clear,
straightforwardwayinChapter8.Toaidinunderstanding,weusedexamples
whenever necessary.
Snakes and active contours are explained in Chapter 9, and we discuss
their intimate relationship with mathematical splines. Minimizing snake en-
ergy using both the original calculus of variations method and the dynamic
programmingapproacharediscussed.Thischapteralsoincludesproblemsand
pitfalls drawn from several applications to provide a better understanding of
the subject. Chapter 10, on the other hand, discusses powerful globally opti-
malenergyminimizationtechniques,keepinginmindtheneedofstudentsand
researchers in this new and promising area of image processing and machine
vision.
Finally, we believe that this book would help readers from many diverse
areas,asitprovidesareasonablygoodcoverageofthesubject.Webelievethis
book can be used in many different areas of image processing and machine
vision. It is our hope that this book differs from many other books, as we
Preface IX
made a considerable effort to make these techniques as easy to understand
and implement as possible. We do hope the reader will agree with us.
Sambhunath Biswas Brian C. Lovell
Indian Statistical Institute The University of Queensland
Kolkata, India Brisbane, Australia
March 2007 March 2007
Acknowledgments
We have freely consulted different books, articles from reputed journals and
conference proceedings, and Ph.D theses. All of them are listed in the bibli-
ography. We gratefully acknowledge all the authors whose contributions we
have used in some minor forms. Among them, we express our sincere ac-
knowledgement to Roberto Cipolla and Andrew Blake for the application of
B-splineinmachinevision;BrianAndrewBarskyforbetasplines;Cohen,Ly-
che and Risenfeld, David Lee and B.K.P. Horn for some of the properties of
discrete splines and application, respectively. We believe these works are be-
fitting and informative. We extend our acknowledgments to Charles K. Chui
and S. Mallat for inclusion of a few articles on wavelet splines. Chapter 10
outlinesanumberofresearchthemescurrentlybeingpursuedwithintheIntel-
ligentReal-TimeImagingandSensingGroupandNationalICTAustralia.We
would like to acknowledge the contributions of Terry Caelli, Hugues Talbot,
Peter Kootsookos, and Brian’s current and former students Pascal Bamford,
Ben Appleton, Carlos Leung, David McKinnon, Christian Walder, Stephen
Franklin, and Daniel Walford. We would also like to acknowledge the ANU
Centre for Mental Health for providing the labeled brain images.
Contents
Part I Early Background
1 Bernstein Polynomial and B´ezier-Bernstein Spline......... 3
1.1 Introduction ............................................ 3
1.2 Significance of Bernstein Polynomial in Splines .............. 3
1.3 Bernstein Polynomial .................................... 5
1.3.1 Determination of the Order of the Polynomial......... 6
1.3.2 B´ezier-Bernstein Polynomial ........................ 8
1.4 Use in Computer Graphics and Image Data Approximation ... 9
1.4.1 B´ezier-Bernstein Curves............................ 10
1.4.2 B´ezier-Bernstein Surfaces........................... 13
1.4.3 Curve and Surface Design .......................... 13
1.4.4 Approximation of Binary Images .................... 14
1.5 Key Pixels and Contour Approximation .................... 15
1.5.1 Key Pixels........................................ 15
1.5.2 Detection of Inflection Points ....................... 21
1.6 Regeneration Technique .................................. 23
1.6.1 Method 1 ........................................ 23
1.6.2 Method 2 ........................................ 24
1.6.3 Recursive Computation Algorithm................... 25
1.6.4 Implementation Strategies .......................... 26
1.7 Approximation Capability and Effectiveness ................ 28
1.8 Concluding Remarks..................................... 31
2 Image Segmentation ....................................... 33
2.1 Introduction ............................................ 33
2.2 Two Different Concepts of Segmentation ................... 33
2.2.1 Contour-based Segmentation........................ 34
2.2.2 Region-based Segmentation......................... 35
2.3 Segmentation for Compression ............................ 35
2.4 Extraction of Compact Homogeneous Regions............... 36
