Fuzzy clustering algorithms and their application to medical image analysis

169 Pages·3.245 MB·English

by  Shihab Ahmed Ismail.

F C A
UZZY LUSTERING LGORITHMS AND THEIR
A M I A
PPLICATION TO EDICAL MAGE NALYSIS
A dissertationsubmittedin partialfulfilmentoftherequirements
forthedegreeofDoctorofPhilosophyoftheUniversity ofLondon.
A I S
HMED SMAIL HIHAB
DepartmentofComputing
ImperialCollegeofScience, Technologyand Medicine
UniversityofLondon,LondonSW7 2AZ.
DECEMBER 2000.
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Abstract
Thegeneralproblemofdataclusteringisconcernedwiththediscoveryofagroup-
ing structure within a finite number of data points. Fuzzy Clustering algorithms pro-
vide a fuzzy description of the discovered structure. The main advantage of this de-
scriptionisthatitcapturestheimprecisionencounteredwhendescribingreal-lifedata.
Thus, the user is provided with more information about the structure in the data com-
pared toacrisp, non-fuzzyscheme.
During the early part of our research, we investigated the popular Fuzzy c-Means
(FCM) algorithm and in particular its problem of being unable to correctly identify
clusters with grossly different populations. We devised a suite of benchmark data
sets to investigate the reasons for this shortcoming. We found that the shortcoming
originatesfromtheformulationoftheobjectivefunctionofFCMwhichallowsclusters
with relatively large population and extent to dominate the solution. This led to a
search for a new objectivefunction, which we have indeed formulated. Subsequently,
we derived a new so-called Population Diameter Independent (PDI) algorithm. PDI
was tested on the samebenchmark data used to study FCM and was found to perform
better than FCM. We have also analysed PDI’s behaviourand identified how it can be
furtherimproved.
Sinceimagesegmentationisfundamentallyaclusteringproblem,thenextstepwas
to investigatetheuse offuzzy clustering techniquesfor imagesegmentation. We have
identified the main decision points in this process. Furthermore, we have used fuzzy
clusteringtodetecttheleftventricularbloodpoolincardiaccineimages. Specifically,
theimageswereoftheMagneticResonance(MR)modality,containingbloodvelocity
dataaswellastissuedensitydata. Wehaveanalysedtherelativeimpactofthevelocity
data in the goal of achieving better accuracy. Our work would be typically used for
qualitative analysis of anatomical structures and quantitative analysis of anatomical
measures.
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DEDICATION
To my parents and sisters, with much love, appreciation, and affection.
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Acknowledgments
While a thesis has a single author by definition, many people are responsible for
its existence. Dr Peter Burger, my supervisor, is perhaps the most important of these
people. Peter provided me with regular weekly meetings and many ideas. I wish to
thank him sincerely for being very supportive and friendly throughout the whole of
my PhD. I would also like to thank my examiners: Professor Michael Fairhurst of
the ElectronicEngineeringDepartment, UniversityofKent, Canterbury and Professor
XiaohuiLiuoftheDepartmentofComputerScience, Brunel University. Iam grateful
to Dr. Daniel Ru¨ckert for commenting extensively on an earlier draft of this thesis. I
wouldalso liketoacknowledgeDrGuang-ZhongYang forhishelp inmymock viva.
During my PhD journey I met a number of excellent people with whom I have
become good friends and therefore made the journey particularly enjoyable. I would
hope that weremain friends after we haveall gone separate ways. MoustafaGhanem:
thank you for your wise and light-hearted chats. Daniel Ru¨ckert and Gerardo Ivar
Sanchez-Ortiz: thankyouforbeingspecialfriendswithwhomboundariesfaded. Ioan-
nisAkrotirianakis: thankyouforourmanysharedmagicalmoments. TarkanTahseen:
thank you for your friendship, inspiration, and all those netmaze sessions. Khurrum
Sair: thankyouforputtingupwithallsortsofinconveniencesfrommeandforournew
friendship. OutsideofCollege,IwouldliketothankAtifSharafandWalidZgallaifor
being mygood(half-Egyptian!) Arab friends withwhomIshared manyagoodtime.
On a more personal level, I would like to thank my parents, Amaal and Ismail —
their hard work gave me the opportunity to choose the path that led here — and my
sisters,Fatimaand Iman, fortheircontinuoussupportand encouragement.
Lastbutnotleast,ImustthanktheDepartmentofComputing,ImperialCollegefor
kindly allowingme to use its facilities even after the expiry of my registration period.
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Contents
1 Introduction 17
1.1 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.1.1 ClusteringApplications . . . . . . . . . . . . . . . . . . . . 19
1.1.2 ClusteringParadigms . . . . . . . . . . . . . . . . . . . . . . 21
1.1.3 Fuzzy Clustering . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2 ImageAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 General Framework and Motivation . . . . . . . . . . . . . . . . . . 24
1.4 Research Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5 MainResearch Contributions . . . . . . . . . . . . . . . . . . . . . . 25
1.6 OutlineofthisDissertation . . . . . . . . . . . . . . . . . . . . . . . 26
2 The BasicsofData Clustering 28
2.1 NotationandTerminology . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 ExamplesofDataSets . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Hierarchical and PartitionalClustering . . . . . . . . . . . . . . . . . 36
2.3.1 Hierarchical Clustering . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 Example: Singlelinkalgorithm . . . . . . . . . . . . . . . . 38
2.3.3 PartitionalClustering . . . . . . . . . . . . . . . . . . . . . . 39
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2.3.4 Example: Hard -Means(HCM) . . . . . . . . . . . . . . . . 40

2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Fuzzy Clustering 44
3.1 Fuzzy Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 TheFuzzy -Means Algorithm . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 FCM OptimisationModel . . . . . . . . . . . . . . . . . . . 47
3.2.2 ConditionsforOptimality . . . . . . . . . . . . . . . . . . . 48
3.2.3 TheAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.4 AnExample . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.5 AnalysisofFCM Model . . . . . . . . . . . . . . . . . . . . 51
3.2.6 Noteson UsingFCM . . . . . . . . . . . . . . . . . . . . . . 53
3.2.7 Strengthsand Weaknesses . . . . . . . . . . . . . . . . . . . 54
3.3 ExtensionsofFCM . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1 Fuzzy CovarianceClustering . . . . . . . . . . . . . . . . . . 56
3.3.2 Fuzzy -ElliptotypesClustering . . . . . . . . . . . . . . . . 58

3.3.3 ShellClustering . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 Modificationsto theFCM Model . . . . . . . . . . . . . . . . . . . . 60
3.4.1 Possibilistic -Means (PCM) Clustering . . . . . . . . . . . 60
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3.4.2 HighContrast . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.3 CompetitiveAgglomeration . . . . . . . . . . . . . . . . . . 63
3.4.4 CredibilisticClustering . . . . . . . . . . . . . . . . . . . . . 64
3.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 A New AlgorithmforFuzzy Clustering 67
4.1 TheExperimentalFramework . . . . . . . . . . . . . . . . . . . . . 68
4.1.1 TestsforClusteringAlgorithms . . . . . . . . . . . . . . . . 70
4.1.2 DesigningtheSyntheticData. . . . . . . . . . . . . . . . . . 71
4.2 TheBehaviourofFCM . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 FCM’sResults ontheSyntheticData . . . . . . . . . . . . . 75
4.2.2 DiscussionofFCM’s Results . . . . . . . . . . . . . . . . . . 82
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