Multiphase Image Segmentation Based on Intensity Statistics

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Multiphase Image Segmentation Based on Intensity Statistics Modeling and Applications
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english
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Chen, Fuhua
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Chen, Yunmei
Committee Members:
Hager, William W
Groisser, David J
Rao, Murali
Song, Sihong

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Subjects / Keywords:
image -- multiphase -- processing -- segmentation -- supervised
Mathematics -- Dissertations, Academic -- UF
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Abstract:
Image segmentation is  the first stage of image processing and analysis. The result of image segmentation is used for image post-processing. In computer vision, segmentation refers to the process of partitioning a digital image into multiple segments, called phases (or classes). The goal of segmentation is to simplify and/or change the representation of an image into something that is more meaningful and easier to analyze. An important task of image segmentation is to distinguish objects from background without (or least) affected by noise, intensity-inhomogeneity and artifacts. Based on the classification policy that each pixel is classified to only one phase exclusively, or each pixel can partly belong to more than one phase, image segmentation methods are divided into two categories: hard segmentations and soft segmentations. Variational methods are powerful in hard segmentation. Roughly speaking, variational models for hard segmentations contain region-based methods and edge-based methods. The level-set technique is extensively used in the implementation of these models. However, the non-convexity of the energy functional in the level-set formulation is an inherent drawback of the level-set method. As a result, many level-set-based variational segmentation models are sensitive to initial values. This problem is more difficult to deal with for multiphase image segmentation. Another problem of hard segmentation is its applications to some real world problems. For example, in MRI brain image segmentation,  each voxel (3-D pixel) may contain more than one type of matter due to limited spatial resolution of imaging equipment. The effect is called partial volume. Instead of labeling each image voxel with a unique tissue type \cite {Zhang}, partial volume segmentation aims to estimate all the percentage of each voxel that belongs to each tissue-type. The second problem mentioned above produces the need directly to develop soft segmentation methods, while for the problem of non-convexity of variational hard segmentation models, it has been found that for some of them, the problems can be solved by relaxing the characteristic functions of interested regions to membership functions (also called ownership functions). In this research, after reviewing the existing image segmentation methods in Chapter $1$, we developed three models based mainly on stochastic theory. In all these models, we assume that the intensity of the image at each pixel is a random variable with Gaussian distribution (or mixed Gaussian distribution). Chapter $2$ - $5$ describe these models. In Chapter $2$, we extend the Sine-Sinc model to Gaussian-distribution-like image. Moreover, we choose a normalization of the original image as an initialization of the iterations so that it helps converge to the ``true segmentation''. Furthermore, we replaced the sinc function by the exponential function. With this change, the new model is more adaptable, and can still be implemented using convex-concave procedure (CCCP) which is guaranteed to converge to a local minimum or saddle point. In Chapter $3$, we define a piecewise function $h(x)\in C^1$ to replace the exponential function in the first model  and the Sinc function in Sine-Sinc model(discussed in Chapter $2$). The advantage of this change lies in the fact that the constructed function has a sum of $1$ at each point over all phases. This makes the set of composition functions $\{h_k(x)=h(z(x)-k)\}_{k=1^K}$ be essentially a set of membership functions. Another advantage of this function is that only the nearest neighbor branches can have an overlap of their supports. This property is similar to the partial volume effect in MRI partial volume segmentation where, approximately, different types of matter, called white matter, gray matter and CSF overlap only at their border. This similarity motivated us to apply our model to partial volume segmentation for MRI brain images. In Chapter $4$, we start from considering the piecewise constant Mumford-Shah model for images with intensity-inhomogeneity and develop a stochastic variational soft-segmentation model with mixed Gaussian distribution. The model is more robust to noise and robust to intensity-inhomogeneity too. The problem is formulated as a minimization problem to estimate the mixture coefficients, spatially varying means and variances in the Gaussian mixture. The optimized mixture coefficients lead to a desirable soft segmentation, as well as a hard segmentation. We apply the  primal-dual-hybrid-gradient (PDHG) algorithm to our model for iterations of membership functions and use a novel algorithm for explicitly computing the projection from $R^K$ to simplex $\Delta_{K-1}$ for any dimension $K$ using dual theory. Our algorithm is more efficient in both coding and implementation than existing projection methods. Unsupervised image segmentation models are usually efficient only for a specific kind of image. For example, intensity-based unsupervised models usually assumes images to be smooth. It usually fails to work on textured images. Another example is in medical images. When the interested part of some tissue in the image has the same or similar intensity as other tissues, the segmentation will lead to an incorrect result. On the other hand, supervised image-segmentation methods take a learning procedure with a labeled training set to form a classifier. Although supervised methods are likely to give a better result than unsupervised methods, marking the training set is very time-consuming. Semi-supervised segmentation can save the time of machine learning while still utilizing the advantage of unsupervised methods. So far, most of the semi-supervised segmentation methods are developed for two-phase case. Only a few papers have dressed this topic for multiphase segmentation. In Chapter $5$, we develop a framework for semi-supervised image segmentations based on the model in Chapter $4$. The frame work can be implemented interactively, and can actually be applied to many static image-segmentation models. By using semi-supervised and interactive image segmentation framework developed in this chapter, people can expected more meaningful segmentation results. All the three models and the semi-supervised frame work are demonstrated with experimental results. In Chapter $6$, we give a short prospectus of our anticipated future work related to the aforesaid models and methods.
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Fuhua Chen.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Chen, Yunmei.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-02-28

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Iamverygratefultoallmycommitteemembers.Iwanttothankthemforspendingtimetoattendmyoralqualifyingexamandnaldefense.Inparticular,IwanttothankDr.Rao,whohelpedmetobetterunderstandsomestochasticconceptsandprinciples.IwanttothankDr.Groisserwhogaveadetailedcheckofmyresearchproposalandhelpedtorevisemyresearchstatement.IwanttothankDr.Hagerwhowasnotmycommitteememberinthebeginning.WhenheknewthatoneofmypreviouscommitteemembersmovedoutfromUniversityofFlorida,hekindlyacceptedmyinvitationtolltheplace.IwanttothankDr.Song.Althoughheisverybusyinhisownteachingandresearch,hestillacceptedmyinvitationtobemycommitteememberforacademiccommitmentaswellasfriendship.IwanttoespeciallyexpressmygratitudetomyadvisorDr.YunmeiChenforherpatientguidance,franksuggestionsandsincerehelp.DuringmycurrentPhDstudy,shehasgivenmealotofhelpfromstudy,researchtoliving.Herinstructioninresearchmethodswillbenetmeformywholelife.Inpreparingforthisnaldefense,shespentlotsoftimehelpingme.So,Iwanttosaythanksoncemoretoher.Duringmystudy,Dr.LeiZhangalsogavemealotofacademichelp.Meanwhile,manystaffmemberssuchasConieDoby,MargaretSomers,SandyGagnonandKristenCasonalsoprovidedmewithmuchassistance.Iwanttoexpressmysincereappreciationtothemhere.Ialsowanttogivemythankstoallmygroupmembers.Inmypreviouswork,theyalsogavemealotofhelp. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTIONTOIMAGESEGMENTATION .................. 14 1.1SomeTopicsinImageSegmentation ..................... 15 1.2IntroductiontoExistingTechniques ...................... 18 1.2.1Level-SetBasedSegmentations ................... 18 1.2.2GraphBasedMethods ......................... 21 1.2.3SupervisedSegmentations ...................... 24 1.2.4SoftSegmentations ........................... 27 1.3Ourwork .................................... 33 2MULTI-PHASEIMAGESEGMENTATIONBASEDONPHASE-TRANSITIONTHEORY ....................................... 36 2.1ModelDevelopment .............................. 38 2.1.1ReviewonSine-SincModel ...................... 39 2.1.2NewModel ............................... 40 2.1.3ExistenceofSolution .......................... 42 2.2ImplementationandConsiderations ..................... 45 2.2.1Convex-ConcaveProcedure(CCCP) ................. 45 2.2.2IterationScheme ............................ 46 2.2.3Initialization ............................... 49 2.2.4SegmentationDecision ......................... 49 2.3ExperimentandDiscussions ......................... 49 2.4Conclusions ................................... 50 3MULTIPHASESOFTSEGMENTATIONUSINGCONSTRUCTEDMEMBERSHIPFUNCTIONS ..................................... 53 3.1FrameworkDevelopment ........................... 55 3.2ImplementationandConsiderations ..................... 58 3.2.1IterationSchemeforphasefunctionz(x) 60 3.2.2Constructapproximationfunctionh(x) 62 3.2.3SegmentationDecision ......................... 63 3.3AppliedtoPartialVolumeSegmentation ................... 64 3.3.1IntroductiontoPartialVolumeSegmentation ............. 64 5

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........ 65 3.4ExperimentsandDiscussion ......................... 66 3.5Conclusions ................................... 70 4ASTOCHASTICVARIATIONALMODELFORMULTI-PHASESOFTSEGMENTATIONINTHEPRESENCEOFINTENSITYINHOMOGENEITY ............ 71 4.1ModelDevelopment .............................. 73 4.2NumericalImplementation ........................... 75 4.2.1PrimalDualHybridGradientAlgorithm ................ 76 4.2.2OptimizemembershipfunctionsusingPDHG ............ 78 4.2.3ProjectiontosimplexK1 79 4.3Results ..................................... 80 5SEMI-SUPERVISEDMULTIPHASEIMAGESEGMENTATION .......... 87 5.1FromUnsupervisedSegmentationtoSemi-SupervisedSegmentation .. 89 5.2SolveSemi-SupervisedMultiphaseImageSegmentation ......... 91 5.3Experiments .................................. 93 6FUTUREWORK ................................... 95 REFERENCES ....................................... 97 BIOGRAPHICALSKETCH ................................ 105 6

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Table page 3-1ErrorComparison .................................. 68 7

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Figure page 1-1Level-setevolution. ................................. 19 1-2Comparisonbetween2-Devolvementand3-Devolvementwithlevel-set. ... 19 1-3Minimumcutsegmentation. ............................ 22 2-1Segmentationsforcleanimage. .......................... 51 2-2Segmentationsfornoisyimage. ........................... 51 2-3SegmentationsofrealMRIbrainimage. ..................... 51 2-4SegmentationsofrealMRIbrainimagewithnoise. ............... 51 3-1Differentbranchesoftheconstructedfunction. .................. 63 3-2Branchesoftheconstructionfunctionsandtheirsum. .............. 63 3-3FormulationofPartialvolume. ........................... 65 3-4Comparisonwithnoisyimage. ........................... 67 3-5Softsegmentation. ................................. 67 3-6Robustnesstobias:syntheticimage. ....................... 68 3-7ComparisonofMRIBrainimagesegmentation. ................. 68 3-8ComparisonofMRIBrainimagesegmentation. ................. 69 3-9Comparisonwithgroundtruthimage. ....................... 69 3-10Errorcomparison. .................................. 70 4-1Experiment1:Robustnesstonoise. ........................ 81 4-2Experiment2:Robustnesstobias. ........................ 82 4-3Comparisonbetweenvariancesxedandupdated. ............... 83 4-4MRIliversegmentation. ............................... 83 4-5MRIbrainimagesegmentations. .......................... 85 4-6Naturalimagesegmentationafterthresholding. ................. 85 4-7Naturalimagesegmentationafterthresholding. ................. 85 4-8Naturalimagesegmentationafterthresholding. ................. 86 8

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88 5-2Comparisonwithsyntheticimage. ......................... 93 5-3Comparisonwithower. .............................. 94 5-4ComparisonwithMRIbrainimage. ........................ 94 9

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31 ],partialvolumesegmentationaimstoestimateallthepercentageofeachvoxelthatbelongstoeachtissue-type.Thesecondproblemmentionedaboveproducestheneeddirectlytodevelopsoftsegmentationmethods,whilefortheproblemofnon-convexityofvariationalhardsegmentationmodels,ithasbeenfoundthatforsomeofthem,theproblemscanbesolvedbyrelaxingthecharacteristicfunctionsofinterestedregionstomembershipfunctions(alsocalledownershipfunctions).Inthisresearch,afterreviewingtheexistingimagesegmentationmethodsinChapter1,wedevelopedthreemodelsbasedmainlyonstochastictheory.Inallthesemodels,weassumethattheintensityoftheimageateachpixelisarandomvariablewithGaussiandistribution(ormixedGaussiandistribution).Chapter2-5describethesemodels.InChapter2,weextendtheSine-SincmodeltoGaussian-distribution-likeimage.Moreover,wechooseanormalizationoftheoriginalimageasaninitializationoftheiterationssothatithelpsconvergetothetruesegmentation.Furthermore,wereplacedthesincfunctionbytheexponentialfunction.Withthischange,thenewmodelismoreadaptable,andcanstillbeimplementedusingconvex-concaveprocedure(CCCP)whichisguaranteedtoconvergetoalocalminimumorsaddlepoint.InChapter3,wedeneapiecewisefunctionh(x)2C1toreplacetheexponentialfunctionintherstmodelandtheSincfunctioninSine-Sincmodel(discussedinChapter2).Theadvantageofthischangeliesinthefactthattheconstructedfunctionhasasumof1ateachpointoverallphases.Thismakesthesetofcompositionfunctionsfhk(x)=h(z(x)k)gk=1Kbeessentiallyasetofmembershipfunctions.Anotheradvantageofthisfunctionisthatonlythenearestneighborbranchescanhaveanoverlapoftheirsupports.ThispropertyissimilartothepartialvolumeeffectinMRI 11

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85 ].Imagesegmentationistypicallyusedtolocateobjectsandboundaries(lines,curves,etc.)inimages.Moreprecisely,imagesegmentationistheprocessofassigningalabeltoeverypixelinanimagesuchthatpixelswiththesamelabelsharecertainvisualcharacteristicsorsomereasonablemeaning.Inthiscaseeachpixelbelongsexclusivelytoexactlyonephase.Thesegmentationiscalledhardsegmentation.Insomeapplications,suchasmedicalimageprocessing,hardsegmentationisnotenoughforpost-processing.Forexample,inMagneticResonanceImaging(MRI),avoxel(3Dpixel)maycontainmorethanonetissues(whitematter,graymatterorcerebrospinaluid)duetolimitedresolution,i.e.,avoxelpartlybelongstomorethanonephases.Thegoalofsoftsegmentationistondalltheprobabilitiesthateachpixelbelongstoallphases.Thisprobabilityisalsocalledmembership(orownership)intheliteratures.Sincesoftsegmentationallowseachpixeltobelongtoseveralphaseswithcertainprobabilities,itprovidesamoreexiblemechanism,andtherebykeepsmoreoptionsavailableforpost-processingsteps.Forconvenienceofstatement,inthiswholework,wesupposethattheimageusedforsegmentationisalwaystwodimensionalone.However,mostofthemethodswedevelopedcanalsoappliedtothreedimensionalcaseorcanbeeasilyextendedtothreedimensionspaces.LetR2beanopenandboundeddomainwithLipschitzdomain.AdigitalimageonisaboundedfunctionI:!R.Eachpointx2iscalleda 14

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9 39 ],whichismuchmorerobusttonoisecomparedwiththeoriginalalgorithm.Undercontinuoussetting,noisescanbeeasilyrestrainedbyintroducingvariation[ 83 ].Onetypicalexampleisthewell-knownMumford-Shahmodel[ 74 ]whichassumesimagestobepiecewisesmooth.ROFmodelisarepresentativeofvariationaldenoisingmodel[ 83 ].Comparedwithdiscretesetting,variationalmodelsundercontinuoussettingaretheoreticallymorestrictandmorebeautiful. 2. 14 15 ],callednon-local(NL)segmentation.TheNLmodeliscapableofintegratingsemi-localandglobalimageinformationsimultaneouslythroughaspecicgraph,andthusimprovetheoriginalmodelthatdoesnotworkefcientlywithimageshavinglocalintensityinhomogeneities.Mostotherapproachestodealingwithintensityinhomogeneityfocusonbiascorrection[ 1 53 63 81 102 ].Suchmethodsareusuallyintegratedintosoftsegmentationschemes. 3. 29 77 ]undercontinuoussettingandgraph-cutmethods[ 13 14 48 56 ]underdiscretesetting(Recently,J.Yuanetal.developedaframeworkthatdiscussedgraph-cutmethodsundercontinuoussetting[ 6 ]).Thesemethodsalwaysassumethatthereexistsaboundaryseparatingtwospatiallyneighboringphases.Eachpointbelongsexclusivelytoonlyonephase.So,thesegmentationisequivalenttondtheboundarybetweendifferentphases.Differentfromhardsegmentation,thesoftsegmentation[ 34 87 92 ]istondalltheownerships(ormemberships)ofallpixels.Comparedwithhardsegmentation,softsegmentationhasmanyadvantages.Forexample,eachpixelinanimagemaynotbelongtoonlyoneclassduetolimitedresolution.AtypicalapplicationisthepartialvolumesegmentationforMRIbrainimages[ 42 60 ],whereneartheboundarybetweenwhitematterandgraymatter,eachpixelusuallycontainspartofwhitematterandpartofgraymatter.Simplyassigningapixeltoonepuremattermaycauseasignicantaccumulatederror.Inaddition,evenifinsomeapplications,softsegmentationhastobeconvertedtohardsegmentationatthenalstagebyusingthemaximummembershipclassication(Inthiscase,xbelongstoi-thphaseif 16

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4. 13 14 47 ].Undercontinuoussetting,itisusuallyhardtondglobalsolutionsforhard-segmentationmethods.Forinstance,itisimpossibletoobtainglobalsolutionsdirectlyfromalevel-setbasedmethodduetothenon-convexityofthemodel.However,byrelaxingahard-segmentationmodeltoasoft-segmentationmodel,itispossibletondaglobalsolutionsfortheoriginalhard-segmentationmodel.Forinstance,intwo-phasecase,theactivecontourmodelorsnakemodelcanbesolvedglobally[ 16 ].Formulti-phasecase,therehavebeenseveralpapersaddressingsuchtopic[ 5 17 18 52 ].However,comparedwithtwo-phasecase,theimplementationismuchmorecomplicatedandthecostismuchmoreexpensive. 5. 29 ],theinitialzero-level-setcanberandomlychosen.Whenthelevel-setmethodwasextendedtomultiphaseimagesegmentation,however,theinitialzero-level-setmustbechosenproperlyinorderfortheiterationstoconvergetoafeasiblesolution.Anotherexampleistherelaxationofhardsegmentation.Asmentionedabove,thetwo-phaseactivecontourmodelcanberelaxedtoasoftsegmentationmodelsothataglobalsolutionoforiginalhardsegmentationcanbeexactlyrecoveredfromaglobalsolutionofthesoftsegmentationmodel.However,whenthisprocedureisappliedtomultiphasesegmentation,itishardtoachievesuchasimilarresult.Thisisbecausetherearemorethanonemembershipfunctionsinthefunctional.Although 17

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6. 1.2.1Level-SetBasedSegmentationsThetaskofimagesegmentationiseithertondsubregionscorrespondingtodifferentobjectsortondtheboundariesofthoseobjects.Inrecenttwodecades,oneofthemostadvancedsegmentationmodelsiscalledactivecontourmodelorsnakemodel,whichisrstintroducedbyM.Kass,A.Witkin,andD.Terzopoulos[ 51 ].Themotivationofthismodelistondanoptimalcontouroftheobjectinanimagebyevolvinganinitiallygivencontour.Figure 1-1 showshowthismodelworks,where(A)istheoriginalobject,(B)showsthatthecontourmoves(shrinks)inwhenitisoutoftheobject,and(C)showsthatthecontourmovesout(expands)whenitisinsidetheobject.Theexistingactivecontourmodelscanbebroadlyclassiedaseitherparametricactivecontourmodelsorgeometricactivecontourmodelsaccordingtotheirrepresentationandimplementation.Inparticular,theparametricactivecontours[ 51 104 ]arerepresentedexplicitlyasparameterizedcurvesinaLagrangianframework,whilethegeometricactivecontours[ 22 23 66 ]arerepresentedimplicitlyaslevelsetsofatwo-dimensionalfunctionthatevolvesinanEulerianframework. 18

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22 ]andMalladietal.[ 66 ],respectively.Thesemodelsarebasedoncurveevolutiontheory[ 58 ]andlevel-setmethod[ 78 ].Geometricactivecontourspresentseveraladvantages[ 86 ]overthetraditionalparametricactivecontours.First,thecontoursrepresentedbythelevel-setfunctionmaybreakormergenaturallyduringtheevolution,andthetopologicalchangesarethusautomaticallyhandled.Second,thelevel-setfunctionalwaysremainsafunctiononaxedgrid,whichallowsefcientnumericalschemes.Earlygeometricactivecontourmodels[ 22 23 66 ]aretypicallyderivedusingaLagrangianformulationthatyieldsacertainevolutionPDEofaparametrizedcurve.ThisPDEisthenconvertedtoanevolutionPDEforalevel-setfunctionusingtherelatedEulerianformulationfromlevel-setmethods.Asanalternative,theevolutionPDEofthelevel-setfunctioncanbedirectlyderivedfromtheproblemofminimizingacertainenergyfunctionaldenedonthelevel-setfunction.Thistypeofvariationalmethodsareknownasvariationallevel-setmethods[ 29 97 106 ]. Level-setevolution. Figure1-2. Comparisonbetween2-Devolvementand3-Devolvementwithlevel-set. 19

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76 77 ]whichisusedtomodelcurveevolutionimplicitlybyintroducingahigherdimensionalfunctionandmadetheoriginalcontour(forexample,2-Dcontour)azero-level-setofthehigherdimensionalfunction(3-Dfunction).Theevolvementofthe2-Dcontouristhenimplementedbytheevolvementofthehigherdimensionalsurface.AsshowninFigure 1-2 ,theleftoneshowstheevolvementofthecurvein2-Dcasewhiletherightoneshowstheevolvementofthesurfaceaswellastheevolvementofzero-level-set.The2-Dtopologicalrestrictioncanbethereforesolvedbysuchadimensionlifting.ComparedwithpurePDEdrivenlevel-setmethods,thevariationallevelsetmethodsaremoreconvenientandnaturalforincorporatingadditionalinformation,suchasregion-basedinformation[ 29 ]andshape-priorinformation[ 97 ],intoenergyfunctionalsthataredirectlyformulatedinthelevel-setdomain,andthereforeproducemorerobustresults.Forexamples,ChanandVese[ 29 ]proposedanactivecontourmodelusingavariationallevel-setformulation.Byincorporatingregion-basedinformationintotheirenergyfunctionalasanadditionalconstraint,theirmodelhasmuchlargerconvergencerangeandexibleinitialization.VemuriandChen[ 97 ]proposedanothervariationallevel-setformulation.Byincorporatingshape-priorinformation,theirmodelisabletoperformjointimageregistrationandsegmentation.Inimplementingthetraditionallevel-setmethods,itisnumericallynecessarytokeeptheevolvinglevelsetfunctionclosetoasigneddistancefunction[ 78 80 ].Re-initialization,atechniqueforperiodicallyre-initializingthelevelsetfunctiontoasigneddistancefunctionduringtheevolution,wasextensivelyusedasanumericalremedyformaintainingstablecurveevolutionandensuringusableresults.C.Lietal.presentedanothervariationalformulationforgeometricactivecontoursthatforcesthelevel-setfunctiontobeclosetoasigneddistancefunction,andthereforecompletelyeliminatestheneedofthecostlyre-initializationprocedure[ 62 ]. 20

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98 ].Theproposedmethodisalsoageneralizationof[ 29 ]fortwo-phasesegmentation.Themethodusedlognlevelsetfunctionsfornphasesinthepiecewiseconstantcase.G.ChungandL.Veseappliedlevelsetmethodtomultiphaseimagesegmentationinanotherwayforpiecewiseconstantsegmentationofimages[ 36 ].Theyrepresentedthesetofboundariesofthesegmentationimplicitlyusingamultilayeroflevellinesofacontinuousfunction.Inthestandardapproachoffrontpropagation,onlyonelevellineisusedtorepresenttheboundary.Lateron,X.-C.Tai,etal.,proposedavariantofthelevelsetformulationforidentifyingcurvesseparatingregionsintodifferentphases.Comparedwithaforesaidmultiphaselevel-setmethods,thenoveltyinthismethodistointroduceapiecewise-constant-level-setfunctionanduseeachconstantvaluetorepresentanuniquephase.If2nphasesshouldbeidentied,thelevel-setfunctionmustapproach2npredeterminedconstants.Themethodalsoonlyneedsonelevelsetfunctiontorepresent2nuniquephases.However,there-initializationprocedurerequiredinclassicallevelsetmethodsissuperuoususingsuchapproach.Theminimizationfunctionalisconvexanddifferentiableandthusavoidsomeoftheproblemswiththenon-differentiabilityoftheheavisidefunctions. 21

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Minimumcutsegmentation. twopixelsare,thelargerthecostis.Moreover,inordertoapplygraphtheorytoimagesegmentation,twospecialnodesareaddedtothegraph,i.e.s(thesource:object)andt(thesink:background),calledterminalnodes,andanedgefromeachterminaltoeachnodeinVisalsoadded.Thenewformulatedgraphisdenotedbyatriple(G,s,t)andcalledans-tgraph.Ans-tcutisasubsetofedgesCEsuchthattheterminalsSandTbecomecompletelyseparatedontheinducedgraphG(C)=(V,EnC).Thenforagivencut,thetotalcostofthecutisdenedas 1-3 ,whereapixelhavingathickedgewithSmeansthatthepixelmostprobablybelongstotheobject,andapixelhavingathickedgewithTmeansthatthepixelmostprobablybelongstothebackground.ConsideragraphG=(V,E)withafunctioncdenedonVVsuchthatc(u,v)=0if(u,v)=2E.SuchafunctioniscalledacapacityfunctionofgraphG.Letthetriple(G,s,t)bethes-tgraph,stilldenotedby(V,E).Aowwithrespecttothes-tgraphisafunctionf:VV!Rsuchthat:

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103 ].ClusteringisachievedbyremovingedgesofGtoformmutuallyexclusivesubgraphssuchthatthelargestinter-subgraphmaximumowisminimized.Themethodwasimplementedusingcombinatorialoptimizationalgorithms.Agloballyoptimalsegmentationcanbecomputedefcientlyinlow-orderpolynomialtimeusingmax-ow/min-cutalgorithmsongraphs[ 37 44 46 ].Suchamethodwasthenextensivelystudiedbymanyotherpeople[ 43 55 96 ].J.ShiandJ.Malikusedadifferentwaycallednormalizedcutsforimagesegmentation[ 89 ].Thekeyideaofsuchmethodsisacompletelyautomatichigh-levelgroupingofimagepixels.Typically,thismeansthattheydivideanimageintoblobsorclustersusingonlygenericcuesofcoherenceorafnitybetweenpixels.Y.BoykovandM.-P.Jolly[ 12 ]arerstonesthatdemonstratedhowtousebinarygraphcutstobuildefcientobjectextractiontoolsforN-dimensionalapplicationsforanypositiveintegerNbasedonawiderangeofmodel-specic(boundaryandregion-based)visualcues,contextualinformation,andusefultopologicalconstraints.In2006,Boykovetal.showedaverystrongconnectionbetweengraph-cutsandlevel-sets[ 14 ].Inparticular,theydevelopedanovelintegralapproachtosolvingsurfacepropagationPDEsbasedoncombinatorialgraph-cutalgorithms.Themax-owmin-cuttheorem(byFordandFulkerson)statesthatamaximumowfromstotsaturatesasetofedgesinthegraphdividingthenodesintotwodisjointpartsfS,Tgcorrespondingtoaminimumcut.Thus,min-cutandmax-owproblemsareequivalent.Infact,themaximumowvalueisequaltothecostoftheminimumcut.In2003,H.Ishikawaextendedgraph-cutmethodstomultiphaseimagesegmentationusingMarkovRandomField(MRF)[ 56 ].HeintroducedamethodtosolveexactlyarstorderMarkovRandomFieldoptimizationproblemwiththeMRFhavingapriorterm 23

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56 ]byformulatingcontinuousmax-owandmin-cutmodelsoveraspeciallydesigneddomain.Thesemax-owandmin-cutmodelsareequivalentunderaprimal-dualperspective,whichcanbeseenasexactconvexrelaxationsoftheoriginalproblemandcanbeusedtocomputeglobalsolutions. 24

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59 ].Forbothcases,fewrobustcriteriaareavailableandtheusertypicallyturnstoheuristicsolutions.Moreover,theoutputsofclusteringmethodsareoftendifculttointerpretsincethereisnoexplicitlinkbetweentheclustersfoundbythemodelandtheclassesdesiredbytheuser.Theformerdescribesdatasimilaritiesandthelatteraresemanticinterpretationsoftheobjectsofinterest.Thus,theresultingclustersmayrepresentmixedsemanticpropertiesofthescene,harmingthelabelingofthenalclassicationmap.Mostofthemodelbasedsegmentationmethodsmentionedabovebelongtounsupervisedmethods.Semi-supervisedmethodstakeatrade-offbetweensupervisedmethodsandunsupervisedmethodsbyinferringtheclassicationfrompartiallylabeleddata.Thekeydifferencebetweensupervisedlearningandsemi-supervisedlearningisthatsemi-supervisedmethodsutilizethedatastructureinboththelabeledandunlabeleddatapoints[ 27 ].Hence,themainadvantageofsemi-supervisedimagesegmentationmethodsisthattheytakeadvantageoftheusermarkingstodirectthesegmentation,whileminimizingtheneedforuserlabeling.Thereareseveralgeneralapproachestowardssemi-supervisedlearning,butrecentdevelopmentshavefocusedongraph-basedmethods[ 27 ],probablybecausethegraph-basedrepresentationnaturallycopeswithnonlineardatamanifolds.Inthisformulation,dataarerepresentedbynodesinagraph,andtheedgeweightsaregivenbysomemeasureofdistanceorafnitybetweenthedata.Then,thelabelsfortheunlabeledpointsarefoundbypropagatingthelabelsoflabeledpointsthroughthegraph.Basedonthismethodology,anumberofmethodshavebeenproposed[ 7 50 100 107 110 111 ].However,these 25

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45 ]proposedasupervisedsegmentationmodelbasedonnon-localinformation.N.Houhouetal.proposedasemi-supervisedimagesegmentationmethodthatreliesonanon-localcontinuousversionofthemin-cutalgorithmandlabelsorseedsprovidedbyauser[ 50 ].Thesegmentationprocessisperformedviaenergyminimization.Theproposedenergyiscomposedofthreeterms.Thersttermdeneslabelsorseedpointsassignedtoobjectsthattheuserwantstoidentify.Thesecondtermcarriesoutthediffusionofobjectandbackgroundlabelsandstopsthediffusionwhentheinterfacebetweentheobjectandthebackgroundisreached.Thediffusionprocessisperformedonagraphdenedfromimageintensitypatches.Thegraphofintensitypatchesisknowntobetter 26

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90 ],BayesianMatting[ 35 ]andSpectralMatting[ 61 ].Moreprecisely,imagemattingisasemi-supervisedsoftimagesegmentation.Theopacityalphaisactuallythemembershipoftheforeground.Themethodassignssomepixelstobeforegroundandsomeotherpixelstobebackgroundbyheuristic.Theclassicationisthenpropagatedbysomealgorithm.Amongthesesemi-supervisedsegmentationsmentionedabove,allarefortwo-phaseimagesegmentationexceptfor[ 45 ]. 27

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74 ],geodesicactivecontour[ 23 ],geodesicactiveregion[ 79 ],andregioncompetition[ 109 ].Level-settechnique[ 76 ]hasbeenprovedtobepowerfulintheimplementationofvariationalmodels.Intwo-phasesegmentationthecompositionoftheheavisidefunctionwiththelevelsetfunctionisusedtorepresenttheregionsoftheobjectandbackground.In[ 98 106 ],theauthorsextendedthelevel-setmethodtomultiphasesegmentationbyusingmultiplelevel-setfunctions,whilein[ 36 65 ],theauthorsproposedanotherwaytoextendthelevel-setmethodbyusingmultiplelayersforeachlevel-setfunction.Withcarefullychoosingtheinitialvalues,thesemethodscanworkverywell.However,thenon-convexityoftheenergyfunctionalinthelevel-setformulationisaninherentdrawback.Asaresult,manylevel-setbasedvariationalsegmentationmodelsaresensitivetoinitialvaluesandmayconvergetoanundesirablelocalminimum.Thisproblemismoredifculttodealwithformultiphasesegmentation.Toovercomethenon-convexityproblemmentionedabove,oneapproachistoreplacethecompositionoftheheavisidefunctionwiththelevel-setfunctioninlevel-setformulationbyaweight/membershipfunction(ormoregenerallyreplacethecharacteristicfunctionswithmembershipfunctionsinregion-basedmodels).Thisrelaxationprovidesapossibilitytomaketheenergyconvexwithrespecttomembershipfunctionsandsoconvenienttondaglobalminimizer.Forexample,Chanetal[ 30 ]andBressonetal[ 10 ]restatedcertainnon-convexminimizationproblemsforimagesegmentationanddenosingasequivalentconvexminimizationproblemsbyusingmembershipfunctionstoreplacecharacteristicfunctions.Thesenewmodelsallowto 28

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10 ]efcientandfastnumericalschemestogloballyminimizethevariationalsegmentationmodelswereproposed.ThesealgorithmsarebasedonadualformulationoftheTV(totalvariation)normproposedanddevelopedin[ 4 21 25 26 28 ].Someapproachesofsoftsegmentationaredirectlyextendedfromthecorrespondinghardsegmentationmethods.Forexample,fuzzyregion-competitionmethod[ 71 72 ]isextendedfromregion-competitionmethod[ 109 ];fuzzyc-meanandadaptivefuzzyc-meanmethodaredevelopedfromc-meanclusteringmethod;softMumford-Shahmodel[ 87 ]isdevelopedfromMumford-Shahmodel[ 74 ],etal.Therearealsootherapproachesofsoftsegmentationthatarenotsimplyderivedfromhardsegmentationmethods,suchasmaximum-likelihood(ML)[ 38 ],maximum-a-posterioriprobabilitymethod(MAP)[ 64 102 ],Markov-random-eld(MRF)[ 64 ]methodandotherstochasticmethods[ 57 87 ].MoryandArdonextendedtheoriginalregion-competitionmodeltoafuzzyregion-competitionmethod[ 71 72 ].Thetechniquegeneralizessomeexistingsupervisedandunsupervisedregion-basedmodel.Theproposedfunctionalisconvex,whichguaranteesaglobalsolutioninthesupervisedcase.Unfortunately,thismethodonlyappliestotwo-phasesegmentationandishardtobeextendedtomultiphasesegmentation.FuzzyC-mean(FCM)[ 31 64 81 ]isamethoddevelopedforpatternclassicationandpatternrecognition.Itisalsoapplicabletoimagesegmentation.ThestandardFCMmodelpartitionsadatasetfxkgNk=1RdintoMclustersbythefollowingobjectivefunction[ 9 39 ] 81 ],wheretheconstantclustercentersvkusedintheFCMmodelaresubstitutedbyspatiallyvaryingfunctions. 29

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31 ]usedadifferentsimilaritymeasurefromthatin[ 81 ].Theirobjectivefunctionalreadsas 64 87 ].Intheseapproachestheintensityofpixelsareassumedtoberandomvariableswhichareeitherindependentorsomehowdependent.LetI(x)beanimagedenedonanopenboundeddomaincontainingNclasses.Letwbetheclasslabelvariable,w=1,,N.Ateachpixelx2,bothw(x)2f1,...,NgandI(x)areviewedasrandomvariablesindexedbyx.Theprobabilitythatxbelongstothei-thphaseisrepresentedbytheownershipfunctionspi(x),1iN.DenotebyProb(I(x)jw(x)=i)theprobabilitydensityfunction(pdf)oftherandomvariableI(x)belongingtothei-thpattern.ThenthepdfoftheimageI(x)ateachx2isamixeddistributiongivenby 38 ]andmaximum-a-posteriori(MAP)[ 64 102 ]principlebasedtechniqueshavebeenwidelyusedinsoftsegmentation.TheMLmethodsndtheoptimalparametersinthejointpdfsuchthatthelikelihoodfunctionismaximized[ 38 ].However,simplyusinglikelihoodto 30

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64 ],asegmentationframeworkbasedonMAPprinciplewasproposedforpartialvolume(PV)segmentationofMRIbrainimages.AmixtureoftheprobabilitydensityfunctionsisconsideredtoaddressthePVeffect.AMarkovRandomField(MRF)modelisusedtodenethepriordistributionofthemixturecoefcienteldimposingasmoothnessonthemixturecoefcients(ownerships).Thefuzzyc-meanmodelisextendedtodenethelikelihoodfunctionoftheobservedimage.J.Shen[ 87 ]proposedageneralmultiphasestochasticvariationalfuzzysegmentationmodelcombiningstochasticprinciple.TheintensityofimageswasmodeledasamixedGaussiandistribution.Theassumptioninthemodelthatmembershipfunctionsshouldbeeithercloseto1orcloseto0simpliedthemodelitselfbutlimiteditsapplication.Forexample,it'snotreasonabletoapplythemodeltopartialvolumesegmentationsinceinthatcasethemembershipfunctionsusuallyevaluatedneithercloseto1norcloseto0attheborderofdifferentmatters.AnotherfeatureofShen'spaperisthatitutilizedModica-Mortola'sphase-transitiontheory.Thesimilaritiesbetweenimagesegmentationandphasetransitiontheoryinmaterialsciencesanduidmechanicshaveinspiredpeopletoborrowsomeideasincontemporarymaterialsciences,e.g.,thediffuseinterfacemodelofCahn-Hilliard[ 20 ],anditsrigorousmathematicalanalysisintheframeworkof)]TJ/F1 11.95 Tf 6.77 0 TD[(-convergenceapproximationbyModicaandMortola[ 70 73 ].Theauthorsin[ 57 ]presentedamodelforimagesegmentationbaseduponthephasetransitiontheoryofModicaandMortolaanddiscusseditsconnectionstotheMumford-Shahsegmentationmodelandsomerelatedworks.Anotherwaytoimprovetheimplementationefciencyofvariationalmodelsistoreduceamodeltoapiecewiseconstantone[ 29 ].Underpiecewiseconstantassumption,theimplementationcanbesimpliedsincethetotalvariationofanobjectisexactlyitsperimeter.Piecewiseconstantassumptionisreasonableinmanycases.However,duetonon-uniformillumination,ornon-uniformimaging 31

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14 15 ],callednon-local(NL)segmentation.TheNLmodelisabletointegratesimultaneouslysemi-localandglobalimageinformationthroughaspecicgraph,andthusimprovetheoriginalmodelthatdoesnotworkwithimageswithlocalintensityinhomogeneities.Mostotherapproachesdealingwithintensityinhomogeneityfocusonbiascorrection[ 1 53 63 81 102 ].Thesemethodsareusuallyintegratedintosoftsegmentationschemes.Forexample,Wellsetal.proposedanexpectation-maximization(EM)algorithmtosolvethebiascorrectionproblemandthetissueclassicationproblem[ 102 ].TheEMalgorithmwasusedtoiterativelyestimatetheposteriortissueclassprobabilitieswhenthebiaseldisknown,andtoestimatetheMAPofthebiaseldwhentissueclassprobabilitiesareknown.Thedisadvantageofthismethodisthatthedirectlycomputedbiaseldisnotsmoothwhichwillleadtoapoorbiascorrectionandsegmentationresults.Phametal.intheirAFCMmethodreplacedtheconstantclustercentersbyspatiallyvaryingfunctions[ 81 ],whichareproductofbiaseldandtheconstantclusteringcenters.Smoothnessofthebiaseldisensuredbypenalizingitsrstandsecondorderderivatives,whichleadstoacomputationallyexpensiveprocedureforthesmoothingofthebiaseld.Ahemdetal.proposedtoaddaneighborhoodtermthatenabledtheclassmembershipofapixeltobeinuencedbyitsneighbors[ 1 ].Theneighborhoodeffectactsasaregularizerandforcesthesolutiontowardapiecewisehomogeneouslabeling.Thisapproachprovedtoleranttosaltandpeppernoise,resultinginsmoothersegmentation.Lietal.proposedavariationallevel-setbasedmethodformedicalimagesegmentationandbiascorrection[ 53 ],thesmoothnessofthebiaseldisintrinsicallyensuredbythedataterminthevariationalformulation,buttheschemeiscomputationallyexpensive.F.Lietal.proposedavariationalfuzzyMumford-Shahmodelformulti-phasesegmentation[ 63 ].Themodelisbasedonthe 32

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64 ],asegmentationframeworkbasedonMAPprinciplewasproposedforpartialvolume(PV)segmentationofMRIbrainimages.Themodelisdevelopedfromastochasticpointofview.ThenalenergyfunctionalisinfactequivalenttoPaper[ 63 ]withadiscreteform. 33

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34

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35

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20 ],anditsrigorousmathematicalanalysisintheframeworkof)]TJ/F1 11.95 Tf 6.77 0 TD[(-convergenceapproximationbyModicaandMortola[ 70 73 ].Thephase-eldrelaxationconsistsinapproximatingtheperimeterusingaCahn-Hilliardtypepenalizationfunctional[ 20 ],withtheform 11 19 101 ].In[ 84 ],theauthorsusedthephaseeldtoapproximatesharpedgesandavariationalphaseeldmodelisderivedtocomputeashapeaverageofagivennumberofshapes.In[ 19 ],theauthorsusedthephasetransitiontheoryinaCahn-Hilliardimpaintingmodel.Inpaper[ 87 ],J.ShenproposedageneralmultiphasestochasticvariationalfuzzysegmentationmodelcombiningstochasticprincipleandtheModica-Mortola'sphase-transitiontheory.Applyingphase-transitiontheoryintothemodelcanenhancepatternseparationandmakeboundariessmooth.Thestochasticvariablesareusedtorepresenttheownershipsofallclasses.Theregularizationismadeusingadoublewellpotentialborrowedfromthephase-transitiontheory.ByassumingthatallpatternsareGaussiandistributionswithmeaneldsui(i=1,...N),andaxedvariance2,thepdf 36

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minES(P,U)=NXi=1Z(uiI)2pi+NXi=1Zjruij2+NXi=1Z9jrpij2+(pi(1pi))2 57 ]presentedanothermodelforimagesegmentationbaseduponphase-transitiontheoryanddiscusseditsconnectionstotheMumford-Shahsegmentationmodelandsomerelatedworks.Themethodusesasine-sincmodelwhichcanalsobethoughttobemotivatedbytheMumford-Shahmodel.However,insteadofusingatargetimageu(x),thenewmodelusesasignaturefunctionz(x)thatissupposedtotakevalue0,1,...,K1(supposetherearetotallyKphasesintheimage).Withthetheoryof)]TJ/F1 11.95 Tf 6.77 0 TD[(-convergenceandtheconvex-concaveprocedure(CCCP)[ 8 20 105 ],theiterationschemecanguaranteeconvergetoalocalminimumorasaddlepoint. 37

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88 ].Onthesecondpoint,unlikeinpaper[ 87 ]wheretheauthorrecommendtouserandomvalueastheinitializationofthetargetfunction,wechoosetheinitializationofthesignaturefunctionz(x)basedontheoriginalimage(seeSection3).Theadvantageofourmethodisthatinmostcases,itcanpushtheiterationstoconvergeexactlytotherequirednumberofphases.Therestofthischapterisorganizedasfollows.InSection 2.1 ,werstreviewonthesine-sincmodel.Thendevelopourimprovedmodel.Section 2.2 istheimplementationandsomeconsiderations,wherewechooseanormalizationoftheoriginalimageastheinitializationofthesignaturefunctionz(x).InSection 2.3 ,weexhibitexperimentresultsfordifferentkindofimages,syntheticimageandhumanbrainimage.Weespeciallytakeacomparisonbetweenourmodelandthesine-sincmodel.Finally,thechapterisclosedwithashortconclusion. 38

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2 ),anduseG[zju0]todenotethesecondpartof( 2 ),i.e., 69 ],ModicaandMortolaestablishedthat 39

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2 )isanapproximationofthedifferenceoftheoriginalimageu0(x)andthepiecewiseconstantimageu(x).Sincetheenergyfunctionisnon-convexinz,theauthoradoptedtheconvex-concaveprocedure(CCCP)[ 105 ].WehaveashortreviewonthisprocedureinSection3. 40

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22k(k=0,1,...,K1).(2)Infurther,letc=(c0,c1,...,cK1)and=(0,1,...,K1).Wewanttomaximizethelikelihood,jointpdfoffu0(x),x2g 22k(2)orequivalently,tominimizethenegativelog-likelihood 2 )becomesthefollowingform 2 ).Inthispaper,wecansimplydenotethe 41

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2 )by 42

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43

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44

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2.1 88 ]. 105 ].

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99 ])thattheAMschemeismonotone:E[zn+1,cn+1,n+1ju0]E[zn,cn,nju0].

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2 )and( 2 ),onecansimplyhaveatpixellevel, 47

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sin2zn+1]+22 (sin2zni,j 2 ),( 2 )and( 2 ).Wecanusethealternatingminimizationscheme.Althoughonecantreatckandkasanindependentvariableinimagesegmentation[ 92 ],weupdateckandkineveryalternatingstep,asinanyusualAMscheme. 48

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MmKastheinitializationofz(x),whereeu0(x)isasmoothversionofu0(x),Kisthenumberofphases,andM=maximum(eu0(x))andm=minimum(eu0(x)),respectively.Wedosofortworeasons.First,wewanttosettherangeofz0(x)tobe[0,K]ifwewanttopartitiontheimageintoKregions,sincetheoptimalz(x)isasmoothversionofapiecewiseconstantfunctionrangingfromzerotoK.Thesmallerrangeofz0(x)canleadtoasegmentationwithlessnumbersofphasesasdesired.Second,notethateu0(x)m MmKisjustashiftandrescalingofthesmoothedimageu0,whichfullyreectsthefeatureoftheoriginalimageu0forsegmentation.Hence,ourchoiceofz0issomehowclosetotheexpectedoptimalsolution. 2z
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2-1 ,where(A)istheoriginalarticialsyntheticimage,(B)isthesegmentationusingModel1(=150),(C)isthesegmentationusingModel2(=150,=5),and(D)isthesegmentationusingModel3(=150,=5)).However,fornoisyimages,bothModel2andModel3arebetterthanModel1(SeeFig. 2-2 andFig. 2-4 ).InFig. 2-2 ,(A)istheoriginalArticialsyntheticimagewithGaussiannoise(m=0,=0.02),(B)isthesegmentationusingModel1(=150),(C)issegmentationusingModel2(=150,=5),and(D)isthesegmentationusingModel3(=150,=5).Fromtheresult,wecanseethattherearemorespuriousdotsinFig. 2-2 (B)thaninFig. 2-2 (C)andinFig. 2-2 (D).Meanwhile,comparingFig. 2-2 (C)andFig. 2-2 (D),theedgeofFig. 2-2 (D)isdamagedlessthanFig. 2-2 (C).Fig. 2-3 isthesegmentationforarealMRIbrainimage,where(A)istheoriginalhumanbrainimagewithoutnoise,(B)issegmentationusingModel1(=150),(C)isthesegmentationusingModel2(=150,=5),and(D)isthesegmentationusingModel3(=5000,=15).Fig. 2-4 showstheresultfortheimageinFig. 2-3 addedsomeGaussiannoise.InFig. 2-4 where=2,K=3,wecanalsoseethatModel3isbetterthanModel1andModel2inthatspuriousdotscanhardlybefoundfromthewhitemattersinFig. 2-4 (D),whileinFig. 2-4 (B)andFig. 2-4 (C),therearestillmanyspuriousdotsinthewhitematters. 50

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Segmentationsforcleanimage. Segmentationsfornoisyimage. SegmentationsofrealMRIbrainimage. SegmentationsofrealMRIbrainimagewithnoise. 51

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52

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14 ]andnon-localvariationbasedmethod(continuouscase)[ 15 ].Inthesemethods,whetherapointisanedgedependsnotonlyonthelocalintensitydifference,butalsoonndinghowoftenthesimilarfeaturesofthepointhavebeenrepeatedinthewholedomain.Bytakingnon-localinformation,theedgecanbewellpreservedwhilethenoiseissmoothed.Differentfromusingnon-localinformation,theframeworkinthispaperusesstochastictheorytorestrainnoiseandimprovesegmentation.ItcanbethoughtanextensionofpiecewiseconstantMumford-Shahkindmodelsmentionedabove.Moreprecisely,weassumethattheintensityofeachpointisaGaussiandistributedrandomsample.Ineachphasek,thepointsfollowasameGaussiandistributionwithmeanckandvariancek.Weassumethatthecleantrueimageu(x)isstillpiecewiseconstant(i.e.,insideeachphase,theintensitiesarealwaysaconstantequaltock)butcontaminatedbyaGaussiannoisen(x),i.e.,I(x)=u(x)+n(x).Asaresult,theintensitiesofpointsinasamephasewillnotbeaconstant,butafamilyofsamplesfrom 53

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32 33 36 40 65 87 88 98 106 ]wherevariantsarenotinvolved,forpointxwhere(ckI(x))2isrelativelylargerthan(ciI(x))2forsomei6=kduetonoise,themodelwithdelityterm( 3 )canstillclassifyittok-thphase(thecorrectphase)andthenoisecanthereforeberestrained.Thereasonisthattheeffectoflarge(ckI(x))2willbepartlycounteractedbythevariance.Ontheotherhand,basedonprobabilitytheory,weknowthattheprobabilitythatjIk(x)ckj>forsome>0andallx2SinaconnectedareaSkismuchsmallerthantheprobabilitythatjIk(x)ckj>foroneisolatedpointx.Thisfactcanguaranteethatthemodelbasedon( 3 )canpreservesmallstructurewhileremovingisolatednoise.Inthischapter,wedonotintroducemembershipfunctionsasanapproximationofcharacteristicfunction.Instead,weintroduceaconstructedfunctionh(x)sothatthecompositefunctionh(z(x)k)hasthepropertyofmembershipfunctionpk(x).Asaresult,themodelitselfisstillasoftsegmentation.Theadvantageofapplyingconstructedfunctioninthemodelliesinthefactthattherewillbelessvariablesintroducedinthemodelwhichmakesthediscussionandtheimplementationeasier.Forexample,aslongasweknowz(x),theprobabilitypk(x)isfollowedbypk(x)=h(z(x)k). 54

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3.1 ,wedeveloptheframeworkofmultiphasesegmentationbycombiningphasetransitiontheoryandGaussiandistribution.Section 3.2 istheimplementationandsomeconsiderations.TheninSection 3.3 ,weapplytheframeworktopartialvolumesegmentation.ExperimentsarecarriedoutinSection 3.4 .Weshowbyexamplestheadvantageoftheproposedmodelbycomparingwithothermultiphasesegmentationmodels.Finally,thechapterisclosedwithashortconclusion. (a) Ateachpointx2,theintensityI(x)isarandomvariable; (b) AlltherandomvariablesfI(x)jx2gareindependent; (c) IneachphaseIjk,0kK1,alltherandomvariablesfI(x):x2kgareidenticallydistributedasaGaussiandistributionwithsamemeanckandsamevariance2k(whicharetobedetermined).Wewanttomaximizethelikelihood,jointpdfoffI(x),x2g,whichisequivalenttominimizethefollowingenergy(thedetailcanbefoundin[ 34 ]): 55

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88 ]whichisbasedonphasetransitiontheory.LetIbetheimagedenedabove.Thesignaturefunctionz(x)isdenedby 3 ). 20 69 70 88 ]forfurtherunderstandingtophasetransitiontheory.Let~z(x)beasmoothedversionofthesignaturefunctionz(x),whichiscalledphaseelds.Tobesimple,westillusethesamenotationz(x)todenotethephaseeld. 56

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69 ]establishedthat 3 )by( 3 )whenissmallenough.Forthecharacteristicfunction1z=k,wecanuseanysmoothfunctionhk(x)asanapproximationifonly0hk(x)1andPK1k=0=1forallx2.Theidealsmoothfunctionhk(x)shouldhavethefollowingproperties. (a) Ateachpointx2wherez(x)=k,hk(x)iscloseto1; (b) Ateachpointx2wherejz(x)kj>0.5,hk(x)iscloseto0.Ifwecanchooseh(x)satisfyingthath(x)isclosetooneatsmallneighborhoodof0andclosetozeroelsewhere,thenhk(x)canbedenotedash(z(x)k)sincez(x)isalmostinteger.Then,thedelityterm( 3 )becomes 3 )andtherelaxeddelityterm( 3 ).Thenewenergyfunctionalis 57

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3 ),therearethreegroupsofparameterstobedetermined,themeansc=(c0,c1,...,cK1),thevariances2=(20,21,...,2K1),andthephaseeldz(x).WecomputeE[z,c,jI]regardingz,c,andasindependentvariables.Thisallowstheapplicationofthealternatingminimization(AM)scheme,i.e.,toalternatinglyoptimizethethreeconditionalenergies.E[zjc,,I],E[cj,z,I],andE[jc,z,I],undertheiterationsofz(n)!c(n)!(n)!z(n+1)givenby 99 ])thattheAMschemeismonotone:E[z(n+1),c(n+1),(n+1)jI]E[z(n),c(n),(n)jI].Forequation( 3 )and( 3 ),onecansimplyhaveatpixellevel, 58

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sin(2z(x))+K1Xk=0h(z(x)k)(logk+jckIj2 87 105 ].Wealsorecommendthereadertouseselectedinitialvaluetohelpconverge(butnotguaranteed).WerecallthefollowingtheoremonCCCP.

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60

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3.1 ,F1[z],F2[z],G1[zjI]andG2[zjI]areallconvex.InordertoapplyCCCP,westillneedthethefunctionaltobeboundedbelow.Todothat,wemustassumethattheimageisnotconstantforanyphase.Orindetail,wesupposeeachvariance2k6=0.Then 3.1 ,wecannowusetheCCCPiterationschemeviaFrechetderivative,i.e., 3 )),theaboveequationisequivalenttothefollowingPDE. sin2z(n+1)]+22 3 ),( 3 )and( 3 ).Wecanusethealternatingminimizationschemeasdiscussedatthebeginningofthissection. 61

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cos(x+1) 2,ifjx+1j1+x,if1+
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Differentbranchesoftheconstructedfunction. Figure3-2. Branchesoftheconstructionfunctionsandtheirsum. denedinanexplicitway,itsderivativecanbecalculatedeasily.SowecannallyusetheiterationschemeCCCPdevelopedinabovesection. 63

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3-3 )showstheprincipleofpartialvolumeformulation(whichisoriginallyusedby[ 60 ]).Theleftimagecontainstwophaseswithhigherresolution.Duetolowerresolution(halfoftheleftimageineachdimension),foursquaresintheleftimagecontributetoonesquareintherightimage.Asaresult,therightimagecontainsmorephases(forthisexample,itcontainsfourdifferentphases).Whenallfoursubsquareswithasamephaseintheleftimagecontributetoonesquareintherightimage,theresultedsquarewithlowerresolutionwillbestillthesamephaseasoriginalone,calledpurematter.However,whenthefoursubsquarescontaindifferentphases,theresultedsquarewillpresentaphaselookslikebetweentheoriginaltwophases.Inthiscase,theresultedsquareiscalledpartialvolume.Theintensityofthepartialvolumeisaweightedaverage 64

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FormulationofPartialvolume. oforiginalpurematterswithcombinationratiosdependingonthenumberofsubsquaresofeachpurematterintheoriginalimageformingthepartialvolume.Thiskindofpartialvolumemaycauseabigerrorintheestimationofpuretissuevolumes.Theerrorissometimesasbigas40-60percent[ 75 ].Thus,partialvolumesegmentationofMRIimageshasnowreceivedconsiderableattention.Theidealpartialvolumesegmentationshouldcontaintwoaspects:ndingpartialvolumeanddecidingitscombinationratioofdifferentpurematters.Mostrecentworkonpartialvolumesegmentationarebasedonstatisticalprincipal,e.g.,theexpectation-maximization(EM)method[ 42 60 ].Thesemethodsimprovedtheprecisionofpurematterestimation.However,theydonotcontainlengthterm,whichmakesthemsensitivetonoise.In[ 93 ],theauthorappliedareparameterizedlevelsetalgorithmtopartialvolumesegmentation.Themethoddoesincludethelengthterm.Ittakesthepartialvolumepartasseparatedclassesthatarecompositionofpuretissues.Thedrawbackofthepaperistousexedratios(e.g.,50%)ofcombinationsforpartialvolumes. 65

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88 ](wheretheauthorproposedamultiphasesegmentationmodelbasedonphase-transitiontheorybutitisnotasoftmodel),werstgiveacomparisonbetweenourmodelandtheSine-SincmodelinFig. 3-4 withsyntheticimageandMRIbrainimage,where(A)istheoriginalarticialsyntheticimagewithGaussiannoise(m=0,=0.02),(B)isthesegmentationusingtheSine-Sincmodel(=150),(C)issegmentationusingtheproposedmodel(=150,=0.2),(D)istheoriginalbrainMRimagewithGaussiannoise(m=0,=0.005),(E)isthesegmentationusingtheSine-Sincmodel(=150),and(F)isthesegmentationusingtheproposedmodel(=150,=0.2).TheninFig. 3-5 ,weshowthreemembershipfunctionsofanaturalimage,where(A)istheoriginalimageand(B)-(D)aremembershipfunctions,whosevalueisbetween0and1.Therestoftheexamplesgiveacomparisonbetweenadaptivefuzzyc-meanmethod[ 81 ]andtheproposedmodel.Fig. 3-6 isthesegmentationsforasyntheticbiasedimagewhichisrstusedbyX.BressonandT.Chan[ 15 ].Thecontourofsegmentationsinthethirdcolumnisobtainedbythresholdingthemembershipfunctions(samethingistruefornexttwogures).FirstlineisofAFCMmodel;secondlineisoftheproposedframework.Middlecolumnshowsthesoftsegmentations.Rightcolumnshowsthehardsegmentations.Fig. 3-7 andFig. 3-8 showdifferentresultsforMRIbrainimagesinasimilarwaytoFig.( 3-5 ).Finally,weapplyourmodeltopartialvolumesegmentationusingsimulatedbrainimages.Thencomparethegroundtruthofpuremattersandoursegmentationresults. 66

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(D)(E)(F)Figure3-4. Comparisonwithnoisyimage. Softsegmentation. Wecalculatetheerrorsofpartialvolumeestimationintwoways.Onewayisbasedonhardsegmentation.Anotherwayisbasedonsoftsegmentation.Fig. 3-9 isacomparisonbetweenthegroundtruthoftheoriginalsimulatedbrainMRimageandthemembershipfunctionsobtainedusingtheproposedframework.Fig. 3-9 (A)istheoriginalsimulatednoisedimage.Thecorrespondinggroundtruthofwhitematter,graymatter,andC.S.FareshowninFig. 3-9 (B),Fig. 3-9 (C),andFig. 3-9 (D),respectively.PhasemembershipfunctionsareshowninFig. 3-9 (E)-(G),Wecarriedouttheexperimentwith35consecutive2Dslicesofa3DsimulatedbrainMRimage.ThencomparetheerrorsbetweentheSine-Sincmodelandtheproposedmodel.Asanaverage,theerrorsareshowninTable1. 67

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(D)(E)(F)Figure3-6. Robustnesstobias:syntheticimage. (E)(F)(G)(H)Figure3-7. ComparisonofMRIBrainimagesegmentation. Table3-1. ErrorComparison methodswhitemattergraymatter .OldModel6.83%7.22%.NewModel4.68%2.73% Finally,weaddseriesofGaussiannoisestotheimageswithzeromeananddifferentvariances.ThencomparetheirerrorsamongtheAFCMmodel,theproposedmodelwithhardsegmentationbythresholding,andtheproposedmodelwithsoftsegmentation.TheerrorsareshowninFig. 3-10 (A).Fromthegraph,wecanseethatas 68

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(F)(G)(H)(I)(J)Figure3-8. ComparisonofMRIBrainimagesegmentation. (E)(F)(G)Figure3-9. Comparisonwithgroundtruthimage. thevarianceofthenoiserises,theerrorwillalsoriseforallthethreecases.However,comparedwiththeAFCMmodel,theerrorsusingtheproposedmodelrisesmuchmoreslowlyasthevarianceofthenoiserises.Wealsocomparedtheinuenceontheerrorsastheparameterinfunctionh(x)changes.ThisisshowninFig. 3-10 (B).Fromthegraph,purematterestimationbasedonphasefunctionz(x)looksalittlebetterthantheestimationbasedontheownershipfunctionh(z(x)k)whenisbigger,whilethepurematterestimationbasedonthe 69

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Errorcomparison. ownershipfunctionh(z(x)k)isalittlebetterthantheestimationbasedonthephasefunctionz(x)whenissmaller. 33 87 88 ].Thekeypointofthispaperistoconstructanapproximationfunctionsothatthemembershipfunctionscanbeobtainedbyitscompositionwithphasefunction.Withthisconstructedfunction,wecanavoidtoaddnewvariablesformembershipfunctionsandsoitsavesmemoryspaceandpromotesefciency.Moreover,sincethecompositionoftheconstructedfunctionandphasefunctionsformsmembershipfunctions,wealsoavoidthegeneralconstraintproblemforsoftsegmentationinimplementation.Theframeworkisthenappliedtopartialvolumesegmentation.Thefuturecontainschoosingbetterconstructedfunctionh(x),andbetterdiscretizationschemeanditerationscheme. 70

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31 64 71 72 81 87 ].MoryandArdonextendedtheoriginalregioncompetitionmodel[ 109 ]toafuzzyregioncompetitionmethod[ 71 72 ].Thetechniquegeneralizessomeexistingsupervisedandunsupervisedregion-basedmodels.Theproposedfunctionalisconvex,whichguaranteestheglobalsolutioninthesupervisedcase.Unfortunately,thismethodonlyappliestotwophasesegmentationandishardtobeextendedtomultiphasesegmentation.FuzzyC-mean(FCM)isamethoddevelopedforpatternclassicationandrecognition,andhasbeenappliedtoimagesegmentation[ 31 64 81 ].ThestandardFCMmodelpartitionsadatasetfxkgNk=1RdintoMclustersbythefollowingobjectivefunction 9 39 ].TheoriginalFCMmethodisverysensitivetonoise.Phamet.alproposedanadaptivefuzzyCmean(AFCM)model[ 81 ]whichismorerobusttonoisethanthestandardFCM,wheretheconstantclustercentersvkusedintheFCMmodel( 4 )aresubstitutedbyfunctionsthataresmoothenoughandclosetothecorrespondingclustercenters.Anotherclassofsoftsegmentationsarebasedonstochasticapproaches[ 34 64 87 ].Intheseapproaches,pixelintensitiesareconsideredassamplesofoneorseveralrandomvariables.Theadvantageofstochasticmethodisitsstrongerabilitytodealwithrandomnoise.Inmoststochasticsegmentationmodels,thelikelihoodfunctionsareusedtorepresentthettingterminanenergyfunctional.Itstartsfromtheassumptionthatreasonablesegmentationshouldmaximizethelikelihood.ThemethodiscalledMaximumLikelihood(ML)method[ 38 ].Anexpectation-maximization(EM)algorithmis 71

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64 ],asegmentationframeworkbasedonmaximumaposterioriprinciple(MAP)wasproposedforpartialvolume(PV)segmentationofMRIbrainimages,whichisaclassicapplicationofsoftsegmentation.J.ShenproposedageneralmultiphasestochasticvariationalfuzzysegmentationmodelcombiningstochasticprincipleandModica-Mortola'sphase-transitiontheory[ 87 ].TheintensityofimageswasmodeledasamixedGaussiandistribution.Themodelassumedthatmembershipfunctionsshouldbeeithercloseto1orcloseto0,whichsimpliedthemodelbutlimiteditsapplication.Forexample,it'snotreasonabletoapplythemodeltopartialvolumesegmentationsinceinthatcasethemembershipfunctionsareusuallyneithercloseto1norcloseto0attheboundaryofdifferentmatters.Biascorrectionisanimportantmeaninsoftsegmentationtodealwithintensityinhomogeneity[ 1 53 81 102 ].Forexample,Wellsetalproposedanexpectation-maximization(EM)algorithmtosolvethebiascorrectionproblemandthetissueclassicationproblem[ 102 ].TheEMalgorithmwasusedtoiterativelyestimatetheposteriortissueclassprobabilitieswhenthebiaseldisknown,andtoestimatetheMAPofthebiaseldwhentissueclassprobabilitiesareknown.Thedisadvantageofthismethodisthatthedirectlycomputedbiaseldmaynotbesmoothwhichwillleadtoapoorbiascorrectionandsegmentationresults.PhamandPrinceproposedanadaptivefuzzyC-meansalgorithmwhichisformulatedbymodifyingtheobjectivefunctioninthefuzzyC-meansalgorithmtoincludeamultiplicativebiaseld,whichallowsthecentroidsofeachclasstovaryacrosstheimage.Smoothnessofthebiaseldisensuredbypenalizingitsrstandsecondorderderivatives,whichleadstoacomputationallyexpensiveprocedureforthesmoothingofthebiaseld.Ahmedetalproposedtoaddaneighborhoodtermthatenabledtheclassmembershipofapixeltobeinuencedbyitsneighbors[ 1 ].Theneighborhoodtermactsasaregularizerandforcesthesolutiontowardapiecewise 72

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53 ],thesmoothnessofthebiaseldisintrinsicallyensuredbythedataterminthevariationalformulation.Inthischapter,weproposedastochasticvariationalmodelformulti-phasesoftsegmentationinthepresenceofnoiseandintensityinhomogeneity,wheretheimageintensityateachpointismodeledasamixedGaussiandistributionwithmeansandvariancestobeoptimized.DifferentfromJ.Shen'swork[ 87 ],ourmodeldoesnotsettheassumptionthatmembershipfunctionsmustbeclosetoeither1or0.So,ourmodelaremoresuitableforsoftsegmentationandapplicationtopartialvolumeanalysis.SinceourmodelisdevelopedbasedontheassumptionthattheimageintensityisamixedGaussiandistributionwithpossiblydifferentvariancesfordifferentphases,itisalsodifferentfrom[ 64 81 ]inthatourmodeladaptivelycorrectsbiasofintensitiesandremovesnoisebyndingoptimizedmeansandvariances.Itisdemonstratedbyexperimentsthatourmodelisnotonlyrobusttonoise,butalsorobusttobias.Therestofthechapterisorganizedasfollows.ThenewmodelisdevelopedinSection 4.1 .ThenumericalimplementationschemeispresentedinSection 4.2 .InSection 4.3 ,weshowsomeexperimentresultsandalsogivesomeexplanationandanalysis.Bothsyntheticimagesandauthenticimagesareused. 73

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(p 74

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14 ]andsoftMumford-Shahmodel[ 87 ].ThedifferencebetweentheproposedmodelandothermodelsliesinthefactthatallothermodelsassumedifferentGaussiandistributionssharingasamevariancewhichisusuallyxed.However,inourmodel,differentGaussiandistributionscanbedifferent,whichmakesthemodelmoreexible. 2iZfipi(Ici) 24iPKi=1fipidx#(n)(4)Thechallengeintheimplementationistheoptimizationofmembershipfunctionspi(x)becauseoftheconstraints 10 ],[ 30 ]and[ 95 ]).ThedrawbackofLagrangianmultipliermethodisitslowconvergencerate.Theso-calledexactpenaltytermisexactonlyundersomeconstraintandisnotdifferentiableatendpoints,andmustbereplacedbyasmoothedversionforapproximationwhichwillaffecttheexactness.AnotherwaytodealwiththesimplexconstraintistousetheEuler-LagrangianequationoftheunconstraintproblemforiterationsandthenprojecttheresulttosimplexK1ateach 75

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87 ].Thedrawbackofthismethodisthatnogeneralanalyticexpressioncanbewrittenforalldimensions.Fordifferentdimensions,theprojectionfunctionsaredifferent,andneedtobewritteninadifferentway.Especially,whenthedimensionisgreaterthanthree,theprojectionfunctionbecomescomplicated,whichleadstoalowefciencyinbothcodingandimplementation.Inthispaper,wegiveanovelwayofprojectionusingdualmethod.Theprojectioncanbeexpressedexplicitlyanduniformlyforalldimensions,andtheanalyticpropertyisguaranteedduetodualtheory.Dualmethodhasbeenextensivelystudiedtodealwithtotalvariationwhichisnotdifferentiableatpointswheretherstordervariationiszero.OneofthepopularexampleisChambolledualmethod[ 25 ].Recently,M.ZhuandT.F.Chandevelopedanewalgorithmcombiningthegradientdecentmethodanddualmethod,calledPrimal-Dual-Hybrid-Gradientmethod(PDHG)(see[ 108 ]fordetails).Themethodintegratestheadvantagesofbothgradientmethodanddualmethod,andthusfasterthanusingeithermethod.Itisprovedtobefasterthanusingdualmethodonlyanditsmodiediterationformisguaranteedtoconvergewhenstepsizesatisessomecondition(see[ 18 ],[ 41 ]and[ 82 ]).Inourapplication,weadoptedtheidealofPDHGandapplyittoourmodelwithconstraintonsimplexK1. 76

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4 )iscalledprimalROFmodel.Reorderingtheimagematrixu(resp.f)row-wiselyintoavectory(resp.z)andusingthematrixAlforthegradientoperatoratelementl,wegetthefollowingdiscreteformoftheprimalROFmodel 77

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4 )withrespecttomembershipfunctionspi(i=1,...,K)underconstraint( 4 ),itisequivalenttosolvethefollowingdiscretemin-maxproblem

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4 )withrespecttoqiisDpi,thedualstepis 47 )isgivenby 79

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2kxyk2+K1(x)=yargminz2RKK1(z)+1 2kzyk2(4)whereK1(z)istheLegendre-FencheltransformofK1.Bydenition,wehave 64 ],wepresentallexperimentresultscomparedwiththemodeldevelopedinpaper[ 64 ](WetemporarilycallthismodelMAP-AFCMmodel).Therstexperimentaimsattestingrobustnesstonoise.InFig. 4-1 ,theoriginalimagecontainsobviouslythreephases.WeaddedamixedGaussiannoisewithzeromeanandanoverallvariance0.03.First,weappliedMAP-AFCMmodel.Wechoose1=5,andstopiterationsusingcriterionmax1i3fjc(i)newc(i)oldjg<0.001,wherec(i)olddenotestheoldmeanbeforeeachiteration,andc(i)newdenotesthenewmeanaftereachiteration(thesamefortherestexperiments).Thenweappliedourmodel( 4 )totheimage.Obviously,theresultofthenewmodelismuchbetter,wheretherstlineshowsthesegmentationusingMAP-AFCM,thesecondlineshowsthesegmentationusingtheproposedmodel.Foreachline,fromlefttorightareoriginalimage,threemembershipfunctionsandhardsegmentationafterthresholding. 80

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Experiment1:Robustnesstonoise. 4 ),theeffectofisolatednoisetotheenergyfunctionalcanbecounteractedbythevariancesappearedinthedenominatorsofthettingterm.Sothenewmodelismorerobusttonoise.Thesecondexperimentaimsatcomparingrobustnesstobias.InFig. 4-2 ,therstlineistheoriginalbiasedimageanditsgroundtruthofallthreemembershipfunctions.ThesecondlineandthethirdlineshowthesoftsegmentationsobtainedusingMAP-AFCMmodelandtheproposedmodel,respectively.Obviously,theproposedmodelgivesmorepreciseresultcomparedwiththegroundtruthsincethereisnobiasinthesegmentation.Ourthirdexperimentaimstogiveacomparisonbetweenvariancesxedandvariancesupdatedinthenewmodel.Forallthevelines,fromlefttorightaretheoriginalimage,threemembershipfunctionsandhardsegmentation,respectively.Fromtherstlinetothefourthlinearetheresultswithvariancesxed.Forexample,weset2i=0.005forall(1i3)intherstline,andweset2i=0.010forall(1i3)inthesecondline,andsoon.However,thelastlineistheresultwherevariancesareupdated,andweobtainedthenalvariancesforthethreephases,whichare1=0.0069,2=0.0193,and3=0.0135,respectively.Obviously,thelastrow 81

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Experiment2:Robustnesstobias. givesthebestresult.Thisexperimentshowsthatupdatingvariancesisbetterthanxingvariancesandassumingallofthemareequal.SinceFCM(fuzzyc-mean)modelisaspecialcasewhenallvariancesarexedandthesame,thisexperimentshowsthattheproposedmodeloutperformsFCMmodel.Finally,wetestourmodelusingrealimages.InFig. 4-4 ,theliverisnotveryclearduetotheexistenceofbias.UsingMAP-AFCMleadstoawrongresultwhereabigpartoftheliverwasincorrectlyclassiedtobackgroundasshownintherstline.Thiscanbeeasilyseenfromthehardsegmentation.However,usingtheproposedmodelcangetmuchbetterresultasshowninthesecondline.Thisisbecausethettingterminthemodelcontainsbias,aswellasvariance.Bycalculatingthevariancesofthethreephases,theyare0.013,0.011and0.002,respectively.Thisfactalsoprovesthatitisreasonabletoassumethatdifferentphasesmayhavedifferentvariancesasinourmodel.Inthegure,therstlineshowsthesegmentationusingMAP-AFCMmodel,thesecondlineshowsthesegmentationusingtheproposedmodel.Fromlefttoright:originalimage,threemembershipfunctionsandhardsegmentations,respectively. 82

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Comparisonbetweenvariancesxedandupdated. Figure4-4. MRIliversegmentation. 83

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4-5 givesacomparisoninMRIbrainimagesoftsegmentation,wheretherstlineshowsthesegmentationusingMAP-AFCMmodel,thesecondlineshowsthesegmentationusingtheproposedmodel.Fromlefttoright:originalimage,threemembershipfunctions(whitematter,graymatterandCSF(cerebrospinaluid))andhardsegmentations,respectively..Thereisabigdifferencebetweenthesoftsegmentations(themembershipfunctions).ByusingMAP-AFCMmodel,mostpixelsareclassiedtobepartialvolume,i.e.,itsintensityisneithercloseto1,norcloseto0(Inthegure,brightnessofintensitymeanscloseto1,darknessmeanscloseto0,andintensitybetweenbrightnessanddarknessmeanspartialvolume).However,thisisnottruebecauseitiswellknownthatpartialvolumeofMRIbrainimageshouldappearmostlyoftenattheboundaryofdifferenttissues.Comparatively,usingtheproposedmodelcangetmorereasonableresults,wherethepartialvolumeonlyappearsattheboundaryofdifferenttissues.Wealsopresentsomenaturalimagesforcomparison.InFig. 4-6 ,theleftimageistheoriginalimage,andthemiddleoneandtherightonearehardsegmentationsafterthresholdingusingMAP-AFCMmodelandtheproposedmodel,respectively,wherefromlefttoright:originalimageandthreephasesofhardsegmentations.Line1isofMAP-AFCMmodelandLine2isoftheproposedmodel.InFig. 4-7 ,wepresentallthreephasesofhardsegmentationsafterthresholdingusingdifferentmodels.InFig. 4-8 ,therstcolumnistheoriginalimage.Wepresentallmembershipfunctionsandhardsegmentationsforreaderstocompare.Line1isthemembershipfunctionsusingMAP-AFCMmodel,Line2isthemembershipfunctionsusingtheproposedmodel,Line3isthehsrdsegmentationsusingMAP-AFCMmodel,andLine4isthehardsegmentationsusingtheproposedmodel.Forallthreeexamples,theresultsusingourmodelareallbetterthanusingAFCMmodel. 84

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MRIbrainimagesegmentations. Figure4-6. Naturalimagesegmentationafterthresholding. Figure4-7. Naturalimagesegmentationafterthresholding. 85

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Naturalimagesegmentationafterthresholding. 86

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87

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90 ],BayesianMatting[ 35 ]andSpectralMatting[ 61 ].Formally,imagemattingmethodstakeasinputanimageI,whichisassumedtobeacompositeofaforegroundimageFandabackgroundimageB.Thecolorofthei-thpixelisassumedtobealinearcombinationofthecorrespondingforegroundandbackgroundcolors, 5-1 .Thetrimapisarough(typicallyhand-drawn)segmentationoftheimageintothreeregions:foreground(showninwhite),background(showninblack)andunknown(showningray).Giventhetrimap,thesemethodstypicallysolveforF,B,andsimultaneously.Thisistypicallydonebyiterativenonlinearoptimization,alternatingtheestimationofFandBwiththatof.Inpractice,thismeansthatforgoodresultstheunknownregionsinthetrimapmustbeassmallaspossible.Justlikeimagematting,othersupervisedsemi-supervisedimagesegmentationsaremostlydevelopedfortwo-phaseimages.Inthischapter,wedevelopedanewsemi-supervisedmulti-phaseimagesegmentationframeworkbasedonthemodelstudiedinChapter4. 88

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2Z(I(x)ui(x))2pi(x)dx+NXi=1Zjrui(x)jdx+NXi=1Zjrpi(x)jdx(5)Theiterationsbasedonfastgradient-descentmethodare 5 )withrespecttouis 2Z(I(x)ui(x))2pi(x)dx+NXi=1Zui(x)divvidx(5)Theprimal-dualformof( 5 )withrespecttopis 2Z(I(x)ui(x))2pi(x)dx+NXi=1Zpi(x)divqidx(5)Theiterationonuandvis 89

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2ZU(I(x)ui(x))2pi(x)dx+NXi=1ZUjrui(x)jdx+NXi=1ZUjrpi(x)jdx(5)Ifwesolvethisproblemstillusingpreviousprocedures( 5 )and( 5 ),thenitisnotasupervisedsegmentationsincewedidn'tuseknowninformationtoinstructsegmentationforunknownarea.Thekeypointofourworkistoupdateeachpatternuibasedonthenearestpointprinciple,i.e., 90 ].Withinitiallygivenpatternsui(x)andundersmoothnessconstraintofpi(x),theobjectiveenergyfunctionalbecomes 2ZU(I(x)ui(x))2pi(x)dx+NXi=1ZUjrpi(x)jdx(5)whereeachui(x)isdeterminedby( 5 ). 90

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2ZU(I(x)ui(x))2pi(x)edi(x)dx+NXi=1ZUjrpi(x)jdx(5)whereisaparameteranddi(x)isthenearestdistanceobtainedfrom 5 .Thefactoredi(x)willforcetheinuencetobeignoredwhenthedistanceoftwopointsisfaraway.Correspondingly,werewritetheenergywithindicationfunctionsandreplacethetotalvariationbyweightedtotalvariation,wegetthenalenergyfunctional 1+x21+x22. 5 )is 91

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1. Initializeknownparts0iusingbrush; (b) Initializeunknownpartby0U=SNi=10i; (c) Initializememberships:Foreach1iNandx20i,setp0i(x)=1andp0j(x)=0forj6=i;forx20U,setpi(x)randomly. (d) Initializepatterns:Foreach1iNandx20i,setu0i(x)=I(x);Foranyx20U,setu0i(x)intermsofthenearestpointprincipleas( 5 ); 2. Updatemembershipspki(x)by( 5 ); (b) Updateknownareaskiby( 5 ); (c) UpdateunknownareakUby( 5 ); (d) Updatepatternsuki(x)by( 5 ); 3. 92

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Comparisonwithsyntheticimage. 5-2 ,weshowhowsemi-supervisedalgorithmworksdifferentlyfromgeneralunsupervisedsegmentation.TheoriginalimagecontainstwoobjectswithsameintensitiesasshowninFigure 5-2 (B).Supposethatonlytheleftoneistheobjectthatweneedtoseparate.Underunsupervisedsegmentation,thesegmentationoftheforegroundwillbeexactlythesameasshowninFigure 5-2 (B).However,ifwecircleagreenmaskastheseedforthebackgroundandcirclearedmaskastheseedfortheforegroundasshowninFigure 5-2 (A),thenthesegmentationoftheforegroundisshownasinFigure 5-2 (C),whichisthedesiredone.Figure 5-3 andFigure 5-4 showthesegmentationresultsofaowerandaMRImedicalimagerespectivelywithacomparisonbetweenusingunsupervisedsegmentationandusingsupervisedsegmentation.InFigure 5-3 ,(A1)istheoriginalimage,(A2)istheoriginalimagewithassignedclassmasks,(B1)and(C1)aretheunsupervisedsegmentation,and(B2)and(C2)aresupervisedsegmentation.Thedifferencesareobvious.InFigure 5-4 ,(A1)istheoriginalimage,(A2)istheoriginalimagewithassignedclassmasks,(B1)-(D1)areunsupervisedsegmentation,and(B2)-(D2)aresupervisedsegmentation. 93

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(A2)(B2)(C2)Figure5-3. Comparisonwithower. (A2)(B2)(C2)(D2)Figure5-4. ComparisonwithMRIbrainimage. 94

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FuhuaChenwasborninNanjing,Chinain1966.HewenttotheDepartmentofMathematicsin1984,andgothisbachelor'sdegreein1988andmaster'sdegreein1991bothfromNangjingUniversity.HethenworkedatHohaiUniversitytill2001asanassistantprofessorandalecturer.FromSeptember1999toJuly2003,hewasapart-timegraduatestudentattheNanjingUniversityofScience&Technologyreadingforhisdoctoratedegreeincomputersciencewithresearchinterestinpattern-recognitionandarticial-intelligence.InJuly2001,heimmigratedtoCanadaandbecameaCanadiancitizenin2006.FromSeptember2004,hewasagraduatestudentattheUniversityofWindsorandgothismaster'sdegreeinmathematicsandstatisticsin2006.FromJanuary2006toMay2006,hewasapostdoctoralfellowatYangmingUniversitywherehisresearchwasfocusedonpositron-emission-tomography.FromAugust2006,hewenttotheUniversityofFloridareadingforhisdoctoratedegreeinmathematicswithresearchinterestinimage-processing. 105