Mathematical Modeling for Image Segmentation and Restoration

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Mathematical Modeling for Image Segmentation and Restoration
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Zhang, Haili
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Chen, Yunmei
Committee Members:
Zhang, Lei
Rao, Murali
Mccullough, Scott A
White, Keith D

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algorithm -- deblurring -- denoising -- nonparametric -- restoration -- segmentation
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This thesis is devoted to the formulation, implementation, and improvement of mathematical models and efficient algorithms for image segmentation and image restoration using tools based on partial differential equation (PDE), calculus of variations, numerical optimizations, and statistical methods.  Image segmentation or pattern classification is to partition  image domain into disjointed subregions such that each region corresponds to a single subject. This problem is of fundamental importance in digital image processing and has wide applications. During the last few decades, a considerable amount of approaches have emerged to tackle this issue. However, the difficulties caused by intensity inhomogeneity, higher level of noise,  and unevenly distributed illumination still need to be addressed. In this thesis, we propose two main results: a nonparametric image segmentation model based Renyi's statistical dependence measure and a fast algorithm for multiphase image segmentation with bias correction. Our second focus is image restoration. Due to limitations of hardware or erroneous transmission, images we obtain in real world may contain certain kind of distortion and noise.  The goal of image restoration  is to recover the latent clear image from observed contaminated data. The fields of image restoration is very broad, and it is impossible to cover all the related fields. In this thesis, we only deal with  joint image deblurring and denoising. This is in general an ill-posed problem, and certain regularization technique must be imposed. In this work, we choose to use sparse representation theory, and propose a sparseland model for deblurring images in the presence of impulse noise. The main results of this thesis are summarized as the following: 1. Nonparametric Image Segmentation Using R\'{e}nyi's Statistical Dependence Measure In this thesis, we present a novel  nonparametric active region model for image segmentation. This model partitions an image by maximizing the similarity between the image and a label image, which is generated by setting different constants as the intensities of partitioned subregions. The intensities of these two images cannot be directly compared  as they are of different modalities. In this work, we use R\'{e}nyi's statistical dependence measure, maximum cross correlation, as a criterion to measure their similarity. By using this  measure, the proposed model deals directly with independent samples and does not need to estimate the continuous joint probability density function. Moreover, the computation is further simplified by applying the theory of reproducing kernel Hilbert spaces. Experimental results based on medical and synthetic images are provided to demonstrate the effectiveness of the proposed method. 2. An Efficient Algorithm for Multiphase Image Segmentation with Intensity Bias Correction In addition, we propose a variational model for simultaneous multiphase segmentation and intensity bias estimation for images corrupted by strong noise and intensity inhomogeneity. Since the pixel intensities are not reliable samples for region statistics due to the presence of noise and intensity bias, we use local information based on the joint density within image patches to perform image partition. Hence, the pixel intensity has a multiplicative distribution structure. Then, the maximum-a-posteriori principle with those pixel density functions generates the model. To tackle the computational problem of the resultant nonsmooth nonconvex minimization, we relax the constraint on the characteristic functions of partition regions, and apply primal-dual alternating gradient projections to construct a very efficient numerical algorithm. We show that all the variables have closed-form solutions in each iteration, and the computation complexity is very low. In particular, the algorithm involves only regular convolutions and pointwise projections onto the unit ball and canonical simplex. Numerical tests on a variety of images demonstrate that the proposed algorithm is robust, stable, and attains significant improvements on accuracy and efficiency over other state-of-the-art image segmentation  technology . 3. Sparse Image Deblurring in the Presence of Impulse Noise Joint image deblurring and denoising has long been an interesting problem in the field of image processing and computer vision. Traditional deconvolution methods (like the ROF model) only deals with Gaussian noise. Median-based approaches are generally concerned with the removal of impulse noise, which are more likely to hamper the deblurring process. In this thesis, we propose a novel approach for deblurring images corrupted by impulse noise. The key point is to approximate the residual  by Gaussian mixtures. We also use the sparse representation theory to regularize the image. Experimental results are provided at the end of this paper to demonstrate the effectiveness of the proposed method.
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In the series University of Florida Digital Collections.
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Statement of Responsibility:
by Haili Zhang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Chen, Yunmei.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-02-28

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MATHEMATICALMODELINGFORIMAGESEGMENTATIONANDRESTORATIONByHAILIZHANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013HailiZhang 2

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Tomyadvisorandmyparents 3

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ACKNOWLEDGMENTS Firstofall,Iwouldliketoexpressmysincereappreciationtomyadvisor,ProfessorYunmeiChen.Iamverygratefulforherprofessionalguidance,invaluableadviceandconsistentsupportthroughoutmylifeatUF.Withoutthese,thisthesiswouldnothavebeenpossible.SheisoneofthegreatestprofessorsIhaveevermet,veryknowledgeableandfriendly.IhavelearnedalotfromherandIreallyenjoytheexperienceofworkingwithher.Next,Iwanttothankmyotherexcellentcommitteemembers,ProfessorMuraliRao,ProfessorScottMcCullough,ProfessorLeiZhangandProfessorKeithWhite.Thankthemforbeingmycommitteemembers,attendingmyoralqualifyingexamandthenaldefense,providingconstructiveadviceandsuggestionstomyresearch.Inaddition,Dr.Chen,Dr.Rao,Dr.McCulloughandDr.Zhanghavepreparedrecommendationlettersformyjobhunting,andIreallyappreciatetheirkindnessandhelp.Iwouldalsoliketothankmyparents,mybrother,mysister-in-lawandmylittletwinnephewsinChinafortheircontinuousloveandunconditionalsupportthroughoutmylife.Theyarethemostimportantpeopleinmylifeandtheyhavegivenmemuchjoyandencouragement.Finally,Ithankallmygroupmembers,FuhuaChen,XiaojingYe,OuyangYuyuan,JiangliShi,fortheirexcellentcommentsanddiscussionsaboutmyresearch.Iwouldalsoliketoexpressmygratefulnesstoallthestaffmembersinourdepartment,MargaretSomers,SandyGagnon,KristenCason,andConieDobyfortheirpatienceandvaluableassistancetomygraduatestudiesatUF. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1Introduction ...................................... 11 1.1ImageSegmentation .............................. 11 1.1.1MethodsOverview ........................... 11 1.1.2WhyNonparametricImageSegmentation? ............. 13 1.1.3SegmentationwithBiasCorrection .................. 15 1.2ImageRestoration ............................... 18 2NonparametricImageSegmentationUsingRenyi'sStatisticalDependenceMeasure ........................................ 21 2.1ProblemStatementandRelatedWorks ................... 21 2.2Renyi'sStatisticalMeasure .......................... 23 2.3ProposedModelandNumericalMethod ................... 24 2.3.1Levelsetformulationandnumericalmethod ............. 25 2.3.2Asoftformulationandnumericalmethod ............... 28 2.4ExperimentalResults ............................. 30 2.4.1Comparisonwithtwoparametricmodels ............... 32 2.4.2Testonrealmedicalimages ...................... 34 2.5Conclusion ................................... 35 3AnEfcientAlgorithmforMultiphaseImageSegmentationwithIntensityBiasCorrection ....................................... 38 3.1ModelFormulation ............................... 38 3.1.1MultiphaseSegmentationandMAPApproach ............ 38 3.1.2ModelingtheIntensityInhomogeneity ................ 39 3.2NumericalAlgorithm .............................. 42 3.2.1FirstVariationsofb,c,and ..................... 43 3.2.2Solutiontou ............................... 44 3.2.3Algorithm ................................ 46 3.3ExperimentalResults ............................. 46 3.3.1ExperimentSettings .......................... 46 3.3.2QuantitativeEvaluationandComparisonwithExistingMethods .. 48 3.3.2.1MethodsOverview ...................... 48 5

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3.3.2.2Experiment1 ......................... 49 3.3.2.3Experiment2 ......................... 51 3.3.3FurtherEvaluationsoftheProposedAlgorithm ........... 52 3.3.3.1Robusttoinitialization .................... 52 3.3.3.2Robusttointensityinhomogeneity ............. 53 3.3.3.3Robusttonoiselevel .................... 53 3.3.3.4Differentparametersettings ................. 53 3.4ConcludingRemarks .............................. 54 4SparseImageDeblurringinthePresenceofImpulseNoise ........... 62 4.1Background ................................... 62 4.2ModelFormulation ............................... 64 4.3SpareRepresentationTheory ......................... 66 4.4ProposedModelandNumericalSchemes .................. 67 4.5ExperimentalResults ............................. 70 4.5.1Testwithseverblurand30%impulsenoise ............. 71 4.5.2Testwithseverbluranddifferentlevelsofimpulsenoise ...... 72 4.6Conclusion ................................... 75 APPENDIX AReproducingKernelHilbertSpace ......................... 77 BSimplexProjectionAlgorithm ............................ 79 REFERENCES ....................................... 80 BIOGRAPHICALSKETCH ................................ 87 6

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LISTOFTABLES Table page 2-1NumberofiterationsandCPUtime(s)fortheexperiments(Figure 2-1 2-2 ,and 2-3 ). ....................................... 31 3-1QuantitativeevaluationofFigure 3-2 ........................ 52 3-2QuantitativeevaluationofFigure 3-3 ........................ 52 3-3QuantitativeevaluationofFigure 3-4 ........................ 53 3-4QuantitativeevaluationofFigure 3-5 ........................ 54 3-5QuantitativeevaluationofFigure 3-6 ........................ 54 4-1PSNRvaluesforFigure 4-4 4-5 .......................... 75 4-2PSNRvaluesforFigure 4-6 ............................. 75 7

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LISTOFFIGURES Figure page 1-1Imagesegmentationwithintensitybiascorrection. ................ 15 2-1Segmentationresultsofthreesyntheticimages. ................. 32 2-2Segmentationresultsofacleanbrainimage(rstrow)anditsnoisyversion(secondrow). ..................................... 36 2-3Segmentationresultsofacleanlungimage(rstrow)andanoisyone(secondrow). .......................................... 37 3-1ComparisonoftheproposedmodelwithWKVLS,SVMLSandCLIConanMRbrainimage ................................... 56 3-2Comparisonoftheproposedmodel,WKVLS,SVMLSandCLIConanMRbrainimagewithseverenoiseandintensityinhomogeneity. ........... 57 3-3Robusttodifferentinitializations. .......................... 58 3-4Robustnesstestoftheproposedalgorithmondifferentintensityinhomogeneity 59 3-5Robustnesstestoftheproposedalgorithmondifferentimagenoiselevels ... 60 3-6EfciencyoflocalintensityestimationinFastSEGwhenappliedtoimageswithstrongnoise. ................................... 61 4-1Truelogarithmichistogramofnanditscorrespondingapproximation. ...... 63 4-2Samplesoftrainingimages. ............................. 71 4-3Partofthecleantestimagesandtheircorrespondingblurryones. ....... 72 4-4Testonsalt-and-peppernoise. ........................... 73 4-5Testonrandom-valuedimpulsenoise. ....................... 74 4-6Testondifferentlevelofnoise. ........................... 76 8

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMATHEMATICALMODELINGFORIMAGESEGMENTATIONANDRESTORATIONByHailiZhangAugust2013Chair:YunmeiChenMajor:MathematicsThisthesisisdevotedtotheformulation,implementation,andimprovementofmathematicalmodelsandefcientalgorithmsforimagesegmentationandimagerestorationusingtoolsbasedonpartialdifferentialequation(PDE),calculusofvariations,numericaloptimizations,andstatisticalmethods.Imagesegmentationorpatternclassicationistopartitionanimagedomainintodisjointedsubregionssuchthateachregioncorrespondstoasinglesubject.Thisproblemisoffundamentalimportanceindigitalimageprocessingandhaswideapplications.Duringthelastfewdecades,aconsiderableamountofapproacheshaveemergedtotacklethisissue.However,thedifcultiescausedbyintensityinhomogeneity,higherlevelofnoise,andunevenlydistributedilluminationstillneedtobeaddressed.Wehavetwomainresultsforimagesegmentation.Therstoneisanonparametricimagesegmentationmodel,whichpartitionsanimagebymaximizingthesimilaritybetweentheimageandalabelimage,generatedbysettingdifferentconstantsastheimageintensitiesofpartitionedsubregions.Renyi'sstatisticaldependencemeasure,isselectedasacriteriontomeasurethesimilarity,andthecomputationisfurthersimpliedbyapplyingthetheoryofreproducingkernelHilbertspaces.Anotheroneisafastalgorithmformultiphaseimagesegmentationwithbiascorrection,whichisdesignedtosegmentimagescorruptedbystrongnoiseandintensity 9

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inhomogeneity.Inouralgorithm,allthevariableshaveclosed-formsolutions.Moreover,thealgorithmonlyinvolvesregularconvolutionsandpoint-wiseprojectionsontotheunitballandcanonicalsimplex.Therefore,thecomputationcomplexityisverylow.Oursecondfocusisimagerestoration.Duetolimitationsofhardwareorerroneoustransmission,realworldimagesmaycontaincertainkindofdistortionandnoise.Thegoalofimagerestorationistorecoverthelatentclearimagefromobservedcontaminateddata.Theeldsofimagerestorationareverybroad,anditisimpossibletocoveralltherelatedelds.Inthisthesis,weonlydealwithjointimagedeblurringanddenoising.Thisisingeneralanill-posedproblem,andcertainregularizationtechniquemustbeimposed.Inthiswork,wechoosesparserepresentationtheory,andproposeasparselandmodelfordeblurringimagesinthepresenceofimpulsenoise. 10

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CHAPTER1INTRODUCTIONChapter 1 containsthebackgroundinformationregardingimagesegmentationandimagerestoration. 1.1ImageSegmentationImagesegmentationorpatternclassicationistopartitiontheimagedomainintodisjointedsubregionssuchthateachregioncorrespondstoasinglesubject.Mathematicallyspeaking,givenanimageI:!R,whereRDisaclosedandboundedregionthatrepresentsthedomainofI,thepurposeofimagesegmentationistopartitionitsdomainintoseveral(sayM)regions,suchthateachregiondelineatesanimagepatterndistinctfromthosebyotherregions.Namely,weneedtosolveforasetofregionsfigMi=1suchthat=[Mi=1i,andfigMi=1aredisjoint,whereiindicatesthesupportofthei-thpatterninimageI.Intheabovenotation,Disthedimensionoftheimage(usually2or3).Foreaseofpresentation,weonlyconsiderrectangulargray-valuedimages.Imagesegmentationhaslongbeenachallengingandfundamentalprobleminimageprocessingandcomputervisionwithawiderangeofapplications.Inparticular,theemergingdevelopmentsinmedicalimagingdemandmoreeffectiveandrobustalgorithmsforimagesegmentation.Approachestoimagesegmentationcanberoughlyclassiedintotwocategories:edge-basedmodels(e.g.[ 1 5 ])andregion-basedmodels(e.g.[ 6 11 ]).Edge-basedmodelsrelyonedgeinformationtolocatetheboundariesofregions.Regionbasedmodelspartitiontheimagedomainintoseveraldisjointregionssuchthateachregionexhibitsdistinctstatisticalpropertiesfromthosebyothers. 1.1.1MethodsOverviewTheproblemofimagesegmentationcanbeformulatedasanenergyminimizationproblem.MostoftheknownmodelsinthiseldarecloselyrelatedtotheMumford-Shah 11

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model[ 10 ]proposedbyMumfordandShah.Mumford-Shahmodelcanbewrittenas minEMS(J,\=Z(J(x))]TJ /F3 11.955 Tf 11.96 0 Td[(I(x))2dx+Zn)]TJ /F2 11.955 Tf 7.31 10.79 Td[(jrJ(x)j2dx+j)]TJ /F2 11.955 Tf 6.77 0 Td[(j,(1)whereIistheoriginalimagetobesegmented,JisasmoothapproximationofI,j)]TJ /F2 11.955 Tf 6.78 0 Td[(jrepresentsthelengththesegmentingcurve)]TJ /F1 11.955 Tf 6.78 0 Td[(.Thersttermin( 1 )ensurestheclosenessoftheoriginalimageIanditsapproximationJ,thesecondtermguaranteesthesmoothnessoftheJ,andthelasttermminimizestheedgelength.,>0areparametersusedtobalancethestrengthofthesethreeterms.Mumford-Shahmodelisawell-knownmodelforsimultaneoussmoothingandsegmentation,however,thewell-posednesshasnotbeenestablishedyetanditisnotcommonlyusedinmostreal-worldsegmentationapplications.Mostreal-worldsegmentationproblemsseekforobjectswithcloseboundaries,whilemodel( 1 )mayresultinopenboundaries.Inreal-worldsegmentationproblems,avariantofMumford-Shahmodelismoreapplicable.Thisvariantistheso-calledthepiecewiseconstantMumford-ShahmodelortheChan-VeseModel,whichisproposedin[ 6 9 ].Insteadofseekingforsmoothapproximations,Chan-Vesemodellooksforapiecewiseconstantapproximation.Morespecically,thismodelassumesJisaconstantciineachi,andinthiscase,model( 1 )becomes minECV(i,ci)=MXi=1Zi(I(x))]TJ /F3 11.955 Tf 11.96 0 Td[(ci)2dx+j@ij,(1)whereMreferstothenumberofpartitionedsubregions.Notethatthesecondtermin( 1 )disappearsbecauseJisconstantoni.TheChan-Vesemodel( 1 )isshowntobewell-posed,anditcanseparatetworelativelyhomogeneousregionswithoutusinganyedgeinformation.However,thehomogeneityassumptionlimitsitsapplications.Amoregeneralapproachisparametricregionbasedactivecontourmethod.Itassumethatateachx2i,theimageintensityI(x)isanindependentrandom 12

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variabledrawnfromtheprobabilitydensityfunction(p.d.f.)P(Ii(x)ji),whereIiistherestrictionofItoiandiisaparametervectorwhichneedstobeestimated.Thenthesegmentationisobtainedbyminimizingthenegativelog-likelihoodfunctionaltogetherwiththelengthterm,i.e., minC,i0s)]TJ /F8 7.97 Tf 15.89 14.94 Td[(MXiZilogP(Ii(x)ji)dx+jCj.(1)Model( 1 )ismoregeneralandhasvariousspecicformulationsdependinghowtheprobabilitydensityfunction,P(Ii(x)ji),isselected.TheregioncompetitionmodelbyZhuetal.[ 11 ]andgeodesicactiveregionmodelsbyRoussonetal.[ 12 ]andParagiosetal.[ 13 ]assumeP(Ii(x)ji)isaGaussiandistribution: P(Ii(x)jci,i)=1 p 2iexp )]TJ /F5 11.955 Tf 11.3 -.17 Td[((Ii(x))]TJ /F3 11.955 Tf 11.95 0 Td[(ci)2 22i!.(1)Ifallthei'sarethesameandprexed,model( 1 )( 1 )reducestomodel( 1 ).Wewanttomentionthatmodel( 1 )( 1 )isaglobalGaussianmodelasitassumesallrandomvariablesI(x)intheregionisharethesamemeanciandvariancei. 1.1.2WhyNonparametricImageSegmentation?Notethatmostoftheabovementionedmethodsareparametriconesinthattheyexplicitlyorimplicitlyassumeimagessharecertainkindofspecialcharacteristics.Somemayassumeimagesarehomogeneous,whileothersmayassumeimageintensitiesobeyspecicdistribution.However,aspecicassumptionoftheintensitydistributioncanbeasignicantrestrictioninrealapplications,especiallywhentheimagehasheavynoiseorisofmulti-modalintensitydistributions.Toovercomethisproblem,nonparametricmodels[ 14 ]havebeendevelopedtoincreasetherobustnessandsuccessfullyappliedtoimagesegmentationandregistration.Thesemethodsarefeaturedbyusingnonparametricdensityestimationtoreplacetheparametricdensityestimation.Forinstance,thenonparametricactivecontourmodel[ 15 ]isdrivenbythedisparityoftheforegroundandbackgroundp.d.f.'s, 13

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whichareapproximatedbyParzenwindowdensitymethod.In[ 16 ],thedynamicsegmentationofvideoimagesequencesisobtainedbyminimizingthedisparityofthep.d.f.ofthecurrentframewiththepreviousoneandthep.d.f.'sarealsoestimatedusingParzenwindowmethod.Thevariationalsegmentationmodelin[ 17 ]incorporatesboundaryinformationwithregioninformation,wheretheboundaryinformationisobtainedfromtheedgemapimageandtheinteriorregioninformationisrepresentedbytheintensityp.d.f.capturedusingParzenwindowdensityestimation.Moryetal.[ 18 ]considertheforegroundandbackgroundp.d.f.'stobeunknown,whichareintegratedintheregioncomputationmodel.Theproposedmodelcouldsimultaneouslyperformsegmentationandnonparametricdensityestimation,whichareupdatedusingtheParzenwindowdensitymethod.[ 19 ]regardstheforegroundandbackgroundcumulativedistributionfunction(c.d.f.)asunknownandutilizestheWassersteindistancetomeasurethedisparityoflocalc.d.f.withtheestimatedc.d.f.'s.Aworkcloselyrelatedtooursis[ 20 ],inwhichKimetal.segmentimagesthroughmaximizingmutualinformationbetweentheimagetobesegmentedanditscorrespondinglabelimagedenedbysettingdifferentconstantsasimageintensitiesofpartitionedsubregions(ref.Chapter 2 ),whichturnsouttobeminimizingthedisplacementoftheLogarithmicoftheforegroundandbackgroundp.d.f.'sandthep.d.f.'sareagainestimatedusingParzenwindowmethod.Borrowingtheideafrom[ 20 ],weproposeanewapproachofnonparametricimagesegmentationthatusesRenyi'sstatisticaldependencemeasure,maximumcorrelationcoefcient,asasimilaritymeasureoftwoimagesindifferentmodalities.Byusingthismeasureasanalternativechoiceofdependencemeasuretomutualinformation,wedonotneedtoestimatethecontinuousjointprobabilitydensityfunctionoftwoimages,whichissensitivetoimagequantizationandalsomakestheoptimizationprocesscomplicated.Moreover,thecomputationisfurthersimpliedbyapplyingthetheoryofreproducingkernelHilbertspaces. 14

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(a)Input (b)Chan-Vese (c)ProposedFigure1-1. Imagesegmentationwithintensitybiascorrection.(a):inputimagewithinitialpartitioncontourinred.(b):segmentationresultbythecommonusedpiecewiseChan-Vesemodel.(c):Correctsegmentationbytakingintensityinhomogeneityintoaccount. 1.1.3SegmentationwithBiasCorrectionInmanyrealworldapplications,imagesmayencountersignicantintensityinhomogeneityduetospatialvariationsinilluminationsandphysicalconstraintsinacquisitionsensitivities.Aparticularexampleismagneticresonance(MR)imaging,wheretheinhomogeneityispresentedasbiaseldmainlycausedbynonuniformmagneticelds[ 21 ].Inthesecases,thesameobjectinagivenimagemayexhibitvariouscontrastsatdifferentlocationsoftheimagedomain.Theselargevariationsinimageintensitiescancausefalseidenticationofregionsasaconsequenceofambiguousstatisticspresentedbypixelintensities.Forexample,thewidelyusedChan-Vesemodel[ 9 ]failedtogeneratecorrectsegmentationduetointensityinhomogeneityasshowninFigure 1-1(b) whereasthedesiredsegmentationshouldbetheoneshowninFigure 1-1(c) .Therefore,segmentationforimageswithinhomogeneousintensitiesisachallengingproblem,andusuallyrequiresacombinationofsegmentationandintensitybiascorrection.Therehavebeenaseriesofworkproposedtotacklethesegmentationproblemwithbiaseldcorrection(e.g.[ 21 30 ]andreferencestherein).Duetothespacelimitation, 15

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wehereonlyreviewseveralveryrecentmodelsthatarecloselyrelatedtothepresentwork.In[ 29 ],theauthorsproposedavariationalmodelfortissueclassicationofMRimages.Intheirmodel,localintensitiesofdifferenttissueswithinaneighborhoodareusedtoformseparableclusters,wherethecentersareapproximatedbytheproductofthebiaswithintheneighborhoodandatissue-dependentconstant.Thislocalclusteringcriterioniscombinedwithmembershipfunctionstoformanenergyfunctional.Thenthetissueclassicationandbiaseldestimationaresimultaneouslyachievedbyestimatingthemembershipfunctions,biaseld,andtheparametersthatapproximatethetruesignalsineachregionviaminimizingtheenergyfunctional.In[ 31 ],aminimizationframeworkwasdevelopedformultiphasesegmentationandbiascorrection.Thismodelusedthesamelocalclusteringcriterionasthatin[ 29 ]todeneanenergyfunctionalinlevelsetformulation.Later,Lietal.extendedtheirmodelin[ 30 ]tosimultaneousmulti-phasesegmentationandbiascorrection.In[ 30 ],multiplelevelsetfunctionswereusedtorepresentthesubregions.Minimizationoftheenergywasachievedbyaninterleavedprocessoflevelsetevolution[ 32 ]andtheestimationofthebiaseld.Recently,Zhangetal.in[ 33 ]alsoproposedalevelsetapproachforsimultaneoustissuesegmentationandbiascorrectionforMRimages.Butdifferentfromtheworkin[ 29 31 ],wheretheintensitiesineachclusterwereapproximatedbyitsmeaninL2sense,intheirmodellocalintensitiesofdifferenttissueswereassumedtobedistributedasGaussianswiththemeansasthecentersoftheclusterandvariancestobeoptimized.Weaimtoproposeajointimagesegmentationandbiaseldcorrectionframeworkthatuniestheserecentmodels,anddevelopanefcientnumericalalgorithmtosolvethemodel.Weconsideramultiplicativestructureofintensitydensityfunctionforcenterpointsofimagepatches,thenweutilizethemaximum-a-posteriori(MAP)principletoconstructageneralizedmodelforjointimagesegmentationandbiaseldestimation.Besidesthemodelingaspect,computationisalsoacriticalissueofsegmentationinrealapplications.Manyvariationalsegmentationmodelsuselevelsetformulation[ 34 ] 16

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todealwiththeproblemoftopologychangeduringcontourevolution,andhaveshowedpromisingresults.However,thecomputationalcostoflevelsetbasedapproachescanbehighinthecommonlyusedsemi-implicitimplementation.Also,ithasbeendemonstratedthatthemethodissensitivetotheinitialconditionandmayrequireperiodicreinitializationofthelevelsetfunction[ 9 ].Asanalternative,wedirectlyworkonthecharacteristicfunctionsofthepartitionregions.Thentheregularizationtermthatpenalizeslengthsofpartitioncurvesisequaltothetotalvariations(TV)ofthecharacteristicfunctions,andthenewobjectivefunctionisconvexwithrespecttothesefunctions.However,theminimizationisstilldifculttocarryoutduetothenon-convexityofthesolutionset,whichislatershowntobethetensorproductofverticesofacanonicalsimplex.Toresolvethisproblem,werelaxtheconstrainttotheentiresimplexbeforesolvingtheminimizationproblem.Thenwetruncatethenalresultandobtainacharacteristicfunctionasoursegmentationresult.Thisisanextensionoftheideaproposedin[ 35 36 ]fortwo-phasesegmentation.Inthatcase,thereisasinglecharacteristicfunctionwithbinaryconstraintappearedintheminimization.Afterrelaxation,theconstraintbecomestheunitintervalandhencecanbehandledrelativelyeasily,see,e.g.[ 35 37 ].Moreover,theoreticalresultsontheequivalencebetweentheoriginalandrelaxedproblemscanbeestablished.However,thesituationbecomesmuchmorecomplicatedwhentheproblemlevelsuptomultiphasesegmentation,mainlyduetothesimplexconstraintsandmultiplenon-smoothTVtermsinvolvedinoptimization.InChapter 3 ,weproposeaneffectivenumericalalgorithmthatutilizestheprimal-dualformulationofTVnormsandspecialpropertiesofcanonicalsimplextoquicklyapproximateasolution.Theprimal-dualformulationhasbeensuccessfullyappliedtoTVbasedimagereconstructiontoachieveverypromisingefciency[ 38 39 ].Inthisthesis,weutilizethesimilarideatoderiveafastsegmentationalgorithmwhichinvolvesonlyconvolutionsusingkernelfunctionwithsmallsupport,andpointwiseprojectionsontotheunitballandcanonicalsimplex.Todemonstratetheeffectivenessof 17

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ourmethod,weshowthenumericalexperimentsonvariouskindsofimages,andmakecomparisonswiththestate-of-the-artalgorithms. 1.2ImageRestorationImagerestorationisanotherfundamentalprobleminimageprocessingandithasbeenextensivelystudiedinthepastseveralyears.Basically,itistorecoverthelatentclearimagefromdegradeddata.Thedegradedprocesscanbemathematicallymodeledasg=kf+n,whereg,fandnrepresenttheobservednoisyandblurredimage,theoriginalclearimageaswellastheadditivenoiserespectively,kstandsforaknownblurkernelwhichisspaceinvariantandthesymbolreferstotheconvolutionoperator.Theproblemofimagerecoverydealswithhowtogettheclearimagefbasedontheobserveddatag.Thisisgenerallyanill-posedinverseproblem,andweneedtoemploysomesortofregularizationtomakeitwell-posed.OneofthefamousoneistheROFmodel[ 40 ],proposedbyRudin,Osher,andFatemi,whichusestotalvariationregularizationandrecoversfbyminimizingthefunctional f=argminfE(f)=Zjrfjdx+1 2kkf)]TJ /F3 11.955 Tf 11.95 0 Td[(gk22.(1)In( 1 ),thersttermreferstothetotalvariationoff,whichisaveryfamousregularizationapproachduetoitsexcellentabilityinpreservingedgeswhilestillkeepingthesmoothnessoftherecoveredimage.Thesecondtermisadata-delityterm,and>0isaparameterwhichbalancesthestrengthofthesetwoterms.FortheGaussianwhitenoisecase,theROFmodelcouldbequiteefcientbecausethesecondtermturnsouttobethelog-likelihoodofthenoiseifweassumethevarianceisxed.However,inrealapplications,thenoisemaynotobeyGaussiandistributionandtheROFmodelwouldfail.Atypicalexampleisimpulsenoise,whichisoftengeneratedbymalfunctioningpixelsincamerasensors,faultymemorylocationsinhardware,or 18

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erroneoustransmission.Impulsenoisehastwocommontypes,salt-and-peppernoiseandrandom-valuedimpulsenoise.PleaserefertoChapter 4 foradetaileddenition.Classicalimpulsenoiseremovaltechniquesaregenerallybasedonmedian-basedlters[ 41 ]anditsvariousmodications[ 42 47 ].Theproblemwiththeselter-baseddenoisingmethodsisthattheytendtomodifypixelsthatarenotaffectedbynoiseandthusbringblurtotherecoveredimagesespeciallywhenthenoiseratioishigh.Toimproveperformance,variousdecisionbasedmethodshavebeenproposed,likethemedianlterbasedonhomogeneityinformation[ 48 ],theboundarydiscriminativenoisedetection[ 49 ],directionalweightedmedianlter[ 50 ],modiedadaptivecenter-weightedmedianlter[ 51 ]andregularizeddata-preservingmethods[ 52 ].Thesedecisionbasedmethodsgenerallyincludetwosteps:theyrstidentifypossiblenoisypixelsandthenreplacethemusingcertaintypeofltersorsomeregularizationmethodwhileleavingallotherpixelsunchanged.Alloftheabovementionedmethodsonlyconsiderthenoiseremovalproblem,andthustheycouldnothandlethecasewhentheimageiscontaminatedbyblurandimpulsivenoisesimultaneously.Therearealsosomeinterestingworkonimagedeblurringinthepresenceofimpulsenoise.In[ 53 55 ],theauthorsproposedamodelinvolvingaL1datadelityandtheMumford-Shahregularizationterm.Thetwo-phaseapproachisemployedin[ 56 58 ].Afteridentifyingthepossiblenoisypixelsusingmedian-typelters,theauthorsin[ 56 57 ]useMumford-Shahregularizationwhile[ 58 ]useTVbasedregularizationonlyonthedatasamplesthatarenotoutliercandidates.In[ 59 ],Yangetal.proposedanefcientalgorithm,FTVd(fasttotalvariationdeblurring),andtheauthorsin[ 60 ]usedaugumentedlagrangemethodtoforTVL1modeltodeblurimageswithimpulsenoise.Liuetal.[ 61 62 ]developedaTVregularizedandadaptiveL2-basedweightedmodeltodeblurimagesinthepresenceofimpulsenoise.Wealsowanttomentionsomenovelimagerecoveredmodelsbasedonsparserepresentationtheory.Therstaredecisionbasedmethods[ 63 64 ]aimingatremoving 19

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impulsenoise.Thenoveltyofthesetwomethodsisthattheyrstdetectthenoisypixelsusingtheknownmethods([ 63 ]uses[ 49 ],and[ 64 ]employs[ 51 ]),andthenapplythesparserepresentationtheorytorecoverthenoisypixels.In[ 65 ],Louetal.proposedamethodthatdirectlydealswiththesparserepresentationofblurredimages.Theirintuitionisthatthesparsecoefcientsofalatentimagewithrespecttoanover-completebasisarethesameasthosethatencodetheblurredversionoftheimagewithrespecttotheblurredversionoftheover-completebasis.However,itisonlyusedforGaussiannoiseandnotapplicableforimpulsenoise.InChapter 4 ,weproposeanovelapproachfordeblurringimagescorruptedbyimpulsenoise.ThekeypointistoapproximatetheprobabilitydensityfunctionofimpulsenoisebymixedGaussiandistributions.Wealsousethesparserepresentationtheorytoregularizetheimage.Experimentalresultsareprovidedtodemonstratetheeffectivenessoftheproposedmethod. 20

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CHAPTER2NONPARAMETRICIMAGESEGMENTATIONUSINGRENYI'SSTATISTICALDEPENDENCEMEASURE 2.1ProblemStatementandRelatedWorksLetbeaboundedLipschitzdomain,I:!RbeagivenimageandCbeanarbitrarycurveinthedomain.ThesegmentationproblemcanbeformulatedasmoveCsuchthatitseparatestheforegroundfromthebackground.Todoso,wegenerateabinaryimageLcorrespondingtothepositionofCinthefollowingway L(x)=8><>:Fifx2R;Bifx2Rc.(2)RandRcdenotetheregioninsideandoutsideCrespectively.NotethatthelabelimageLinthissettingchangesasthecurveCevolves.WhenCreachesthepositionoftherightsegmentation,theintensitiesinsideCandoutsideChavedifferentstatistics,forinstance,differentmeansor(and)variances.Obviously,theintensitiesinsideCandoutsideCforthelabelimageLarealwaystwodifferentconstants.Therefore,whenCprovidesagoodsegmentation,theimageIandlabelimageLshouldbebettermatchedstatistically.ThisisthebasicideaofusingmatchingIandLtoassistsegmentation.However,itisusuallynoteasytomatchthesetwoimagesastheyareofdifferentmodalitiesanditdoesnotmakesensetodirectlycomparetheirintensities.Tocopewiththisdifculty,anumberofsimilaritymeasuresbasedonstatisticaldependencehavebeenproposed.Forinstance,[ 20 ]choosestomaximizethemutualinformationbetweenoriginalimageIanditslabelimageL,togetherwithaconstraintofalengthterm,i.e., E(C)=)]TJ /F3 11.955 Tf 9.3 0 Td[(MI(I,L)+ICds.(2)In( 2 ),IandLareviewedasrandomvariables.Ateachpointx2,theimageintensityI(x)(orL(x))isasampledrawnfromtherandomvariableI(orL)andallthe 21

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samplesfI(x)jx2g(orfL(x)jx2g)areassumedtobeindependent.ThemutualinformationMI(I,L)isdenedasfollows:MI(I,L)=h(I))]TJ /F3 11.955 Tf 11.95 0 Td[(h(IjL)=h(I))]TJ /F3 11.955 Tf 11.95 0 Td[(Pr(L=F)h(IjL=F))]TJ /F3 11.955 Tf 11.96 0 Td[(Pr(L=B)h(IjL=B),whereh()referstothetheentropy.ForacontinuousrandomvariableZ,theentropyofZisdenedas:h(Z)=)]TJ /F10 11.955 Tf 11.3 16.27 Td[(ZRNpZ(z)logpZ(z)dz.Sinceh(I)isindependentofthecurveC,weonlyneedtoestimateh(IjL=F)andh(IjL=B),whichareestimatedbyusingthenonparametricParzenwindowdensitystrategy,i.e.,h(IjL=F))]TJ /F5 11.955 Tf 27.8 8.09 Td[(1 jRjZRlog1 jRjZRK(I(x))]TJ /F3 11.955 Tf 11.96 0 Td[(I(^x))d^xdx,h(IjL=B))]TJ /F5 11.955 Tf 30.37 8.09 Td[(1 jRcjZRclog1 jRcjZRcK(I(x))]TJ /F3 11.955 Tf 11.96 0 Td[(I(^x))d^xdx.Intheaboveequations,K()istheso-calledwindowfunctionorkernel,whichispositive,symmetric,vanishingatinnityandsatisfyiesZR2K(s)ds=1.Forinstance,wecanchooseKtobetheGaussianp.d.f.,i.e.,K(s)=1 p 2exp)]TJ /F3 11.955 Tf 14.27 8.09 Td[(s2 22,whichistheParzen-windowdensityestimationkernel.Notethatthiskernelisinnitelydifferentiableandthusleadingtothesamepropertyfortheestimatedp.d.f.Themutualinformationcouldbeeffectivelyusedasasimilaritymeasuretomatchtheimagetobesegmentedanditslabelimage.However,itrequirestoestimatethejointp.d.f.ofIandL,whichissensitivetoimagequantizationandalsoincreasesthecomplexityofcomputation.Inthiswork,wechoosetouseRenyi'sstatisticalmeasure, 22

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maximumcrosscorrelation,asasimilaritymeasure.Thismeasuredealsdirectlywithsamplesanddoesnotneedtoestimatethecontinuousjointp.d.f..Abriefreviewofthismeasureisprovidedinthenextsection. 2.2Renyi'sStatisticalMeasureIn[ 66 ],RenyiproposedasetofpostulatesforasuitabledependencemeasureQoftworandomvariables/vectorsXandY,whichhasdrawnmuchattentioneversince.Thesepostulatesinclude 1. Q(PX,Y)iswell-dened; 2. 0Q(PX,Y)1; 3. Q(PX,Y)=0ifandonlyifX,Yareindependent; 4. Q(PX,Y)=1ifY=f(X)orX=g(Y),wherefandgareBorelmeasurablefunctions.PX,YreferstothejointdensityfunctionofXandY.IfQ(PX,Y)satisestheabovepostulations,thenitcanbeusedasadependencemeasure.RenyialsoshowedthatonefunctionsatisfyingtheseconditionsisQ(PX,Y)=supf,g2VCC(f(X),g(Y)),whereVisthespaceofallBorelmeasurablefunctionswithnitepositivevariance,andCC(f(X),g(Y))isthecorrelationcoefcientoff(X)andg(Y),i.e., CC(f(X),g(Y))=cov(f(X),g(Y)) p var(f(X))p var(g(Y)).(2)In( 2 ),thecovariancebetweenf(X)andg(Y)isdenedascov(f(X),g(Y))=E[(f(X))]TJ /F3 11.955 Tf 11.96 0 Td[(E[f(X)])(g(Y))]TJ /F3 11.955 Tf 11.96 0 Td[(E[g(Y)])],whereE[f(X)]istheexpectationvalueoff(X).ThedifcultyofusingRenyi'smeasureliesinthefactthatweneedtondtheoptimalfandginthespaceV,whichisthesetofallBorelmeasurablefunctionswith 23

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nitepositivevariance.Itisextremelydifculttosearchfandginsuchahugespace.Fortunately,wehaveshownin[ 67 ]thatthesupremuminVcouldbeattainedinamuchsmallerspace,whichisareproducingkernelHilbertspace(RKHS)associatedwithareproducingkernelthatiscontinuous,symmetric,positivedeniteandvanishingatinnity.Foreaseofpresentation,weincludetheRKHStheoryintheAppendix.Inthiswork,wechoosetheGaussianfunctiontobethereproducingkernel,i.e.,K(x,y)=1 p 2exp()]TJ /F5 11.955 Tf 10.49 7.92 Td[((x)]TJ /F3 11.955 Tf 11.96 0 Td[(y)2 22).AccordingtothetheoryofRKHS(see[ 67 ]orAppendix),anytwofunctionsfandgintheRKHSassociatedwiththeGaussiankernelcanbeapproximatedbyfunctionspandqoftheform,p(x)=nXi=1i p 2exp()]TJ /F5 11.955 Tf 10.5 7.92 Td[((x)]TJ /F3 11.955 Tf 11.95 0 Td[(yi)2 22),andq(x)=mXj=1j p 2exp()]TJ /F5 11.955 Tf 10.49 7.92 Td[((x)]TJ /F3 11.955 Tf 11.95 0 Td[(zj)2 22),forsomeparameters,yi,i,zj,j,i=1,2,,n,j=1,2,,m.Inpractice,wecanchoosefandgtobeoftheaboveformandx,yi,zj.Thusweonlyneedtoestimatethecoefcientsiandj,whichcouldsignicantlysimplifythecomputation. 2.3ProposedModelandNumericalMethodInsection 2.3 ,weproposeourmodelandcorrespondingnumericalschemes.OuraimistondacurveCsuchthattheresultedlabelimageLdenedin( 2 )matchesthebestwiththeoriginalimageI.InthisworkweusetheRenyi'sstatisticaldependencemeasure,maximumcorrelationcoefcient,asasimilaritymeasuretoalignIandL.Byusingthismeasurewedon'tneedtoestimatethecontinuousjointp.d.f.ofthetwoimagesasinthemodelsbasedonmutualinformation.Asshowninthepreviouspostulates,whentheimageIanditslabelimageLarefunctionsofeachother,i.e.,L=f(I)orI=g(L)forsomefunctionforg,themaximum 24

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crosscorrelationattainsitsmaximumvalue1.NotethatLispiecewiseconstant,soistheresultingimageg(L).Itdoesnotmakeabigdifferencebymaximizingthecrosscorrelationbetweenf(I)andLorg(L),sointhefollowingwechoosetomaximizethecrosscorrelationbetweenf(I)andthelabelimageL.Theobjectiveenergyfunctionalisobtainedbycombiningthecrosscorrelationoff(I)andLandthelengthofC,i.e., E(C,a1,,an)=ICds+ 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(CC(f(I),L))2,(2)wheref(I(x))=nXiaiK(I,yi),andthecorrespondinglabelimageL(x)isdenedbyL(x)=8><>:c1ifx2R;c2ifx2Rc. 2.3.1LevelsetformulationandnumericalmethodEnergyfunctional( 2 )canbeminimizedusingthelevelsetapproach[ 6 34 68 69 ].ThecurveCisrepresentedbythezerolevelofaLipschitzfunction:!Randtheresultingenergyfunctionalbecomes E(,a1,,an)=ZjrH((x))jdx+ 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(CC(f(I),L))2,(2)whereHistheHeavisidefunctionandL(x)=c1H((x))+c2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(H((x))).Thealternateminimization(AM)approach[ 70 ]isemployedtosolvethisproblem.First,wekeepa1,,anxedandsolveforusingthegradientdescentapproach,i.e., @ @t=()divr jrj+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(CC(f(I),L))F,(2) 25

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whereistheregularizedDiracfunctionand F=(f(I))]TJ ET q .478 w 152.41 -29.09 m 173.42 -29.09 l S Q BT /F3 11.955 Tf 152.41 -39.73 Td[(f(I))var(L))]TJ /F5 11.955 Tf 11.95 0 Td[(cov(f(I),L)(L)]TJ ET q .478 w 311.58 -29.75 m 318.35 -29.75 l S Q BT /F3 11.955 Tf 311.58 -39.73 Td[(L) var(f(I))1 2var(L)3 2(c1)]TJ /F3 11.955 Tf 11.96 0 Td[(c2).(2)In( 2 ),()divr jrjistherstvariationofthesmoothingterm(TV-term)in( 2 )withrespectto.Inthesameway,()(1)]TJ /F3 11.955 Tf 12.39 0 Td[(CC(f(I),L))Fistherstvariationofthedelitytermin( 2 )withrespectto.Notethat()FistherstvariationofCC(f(I),L)withrespecttoandthisderivationcanbeobtainedbyapplyingthechainrule,thatis,therstvariationofCC(f(I),L)withrespecttoLtimestherstvariationofLwithrespectto,i.e., ()F=dCC(f(I),L) d=dCC(f(I),L) dLdL d.(2)NotethatCC(f(I),L)=cov(f(I),L) p var(f(I))p var(L)=E[(f(I))]TJ ET q .478 w 321.05 -298.68 m 342.06 -298.68 l S Q BT /F3 11.955 Tf 321.05 -309.32 Td[(f(I)))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(L)]TJ ET q .478 w 375.8 -299.35 m 382.58 -299.35 l S Q BT /F3 11.955 Tf 375.8 -309.32 Td[(L] q E[(f(I))]TJ ET q .478 w 310.46 -322.34 m 331.47 -322.34 l S Q BT /F3 11.955 Tf 310.46 -332.98 Td[(f(I))2]q E[(L)]TJ ET q .478 w 394.02 -323 m 400.79 -323 l S Q BT /F3 11.955 Tf 394.02 -332.98 Td[(L)2],thus dCC(f(I),L) dL=(f(I))]TJ ET q .478 w 209.59 -365.95 m 230.61 -365.95 l S Q BT /F3 11.955 Tf 209.59 -376.59 Td[(f(I))var(L))]TJ /F5 11.955 Tf 11.96 0 Td[(cov(f(I),L)(L)]TJ ET q .478 w 368.76 -366.62 m 375.54 -366.62 l S Q BT /F3 11.955 Tf 368.76 -376.59 Td[(L) var(f(I))1 2var(L)3 2.(2)Ontheotherhand,dL d=()(c1)]TJ /F3 11.955 Tf 12.24 0 Td[(c2).Therefore,wecanget( 2 )through( 2 )and( 2 ).Thenwekeepxedandminimize( 2 )withrespecttoai's.Inthisthesis,wepresenttwoalgorithmstoupdateai's.Therstoneistousethegradientdescentmethod@ai @t=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(CC(f(I),L))E.Intheaboveequation,EistherstvariationofCC(f(I),L)withrespecttoai,i.e.,E=cov(pi,L)var(f(I)))]TJ /F5 11.955 Tf 11.95 0 Td[(cov(f(I),L)cov(f(I),pi) var(f(I))3 2var(L)1 2, 26

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wherepi=K(I,yi).Thesecondmethodisbasedontheformulationoftheenergyfunctional.Tominimize( 2 )withrespecttoai'sisequivalenttomaximizingCC(f(I),L)withrespecttoai'sasthelengthtermisindependentofai's.Forsimplicity,werstintroducesomenotations.TheimageIandLareviewedasvectorsoflengthN,whereNisthenumberofpixels.Foreachi=1,2,,n,setpi=K(I,yi)beavectoroflengthN,anddeneP=[p1,p2,,pn],a=[a1,a2,,an].Foranyvectorx,wedenotethemeanofxtobe x.Thusf(I)=nXiaiK(I,yi)=nXi=1aipi=Pa,andCC(f(I),L)=hPa)]TJ ET q .478 w 229.1 -288.29 m 242.71 -288.29 l S Q BT /F3 11.955 Tf 229.1 -298.26 Td[(Pa,L)]TJ ET q .478 w 269.58 -288.29 m 276.35 -288.29 l S Q BT /F3 11.955 Tf 269.58 -298.26 Td[(Li jPa)]TJ ET q .478 w 228.52 -305.75 m 242.13 -305.75 l S Q BT /F3 11.955 Tf 228.52 -315.73 Td[(PajjL)]TJ ET q .478 w 270.16 -305.75 m 276.93 -305.75 l S Q BT /F3 11.955 Tf 270.16 -315.73 Td[(Lj=hP0a,L0i jP0ajjL0j,whereP0ij=Pij)]TJ /F9 7.97 Tf 14.23 4.71 Td[(1 NPNk=1Pk,jandL0=L)]TJ ET q .478 w 233.86 -326.76 m 240.64 -326.76 l S Q BT /F3 11.955 Tf 233.86 -336.74 Td[(L.Let^PbeamatrixwithorthonormalcolumnvectorswhichspansthecolumnspaceofP0,thenwehaveP0a=^Pforsomevector.Undertheseformulations,weget CC(f(I),L)=hP0a,L0i jP0ajjL0j=h^P,L0i j^PjjL0j=T^PTL0 jjjL0j=TUVTL0 jjjL0j=(UT)T(VTL0) jUTjjVTL0j.(2)In( 2 ),UVTisthesingularvaluedecompositionofthematrix^PT,whereUisannnunitarymatrix,VTisanNNunitarymatrixandisannNdiagonalmatrixwithnonnegativediagonalentries1,2,,n,whicharelistedindecreasingorder.Nowletx=UT jUTj,y=VTL0 jVTL0j,thenx2Rn,y2RN,jxj=jyj=1,thusCC(f(I),L)=xTy=xTz=hx,zi, 27

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wherez=[y1,y2,,yn]andisannndiagonalmatrixwithdiagonalentriesp 1,p 2,,p n.Therefore,CC(f(I),L)ismaximizedwhenxandzhavethesamedirection.Notethatjxj=1,wegetx=+(z) j+(z)j,where+referstothepseudo-inverseof.Fromthedenitionofx,wecanpickaparticular=UxandthenwecanupdateafromtherelationP0a=^P.Inthiswork,andai'sarealternativelyupdateduntilwereachasatisfactoryresult.Toincreasetherateofconvergence,thesemi-implicitdifferenceschemeisappliedin( 2 ),i.e.,n+1)]TJ /F4 11.955 Tf 11.95 0 Td[(n t=(n)divrn+1 jrnj+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(CC(f(I)n,Ln))Fn.Thisequationcouldbeeffectivelysolvedbyusingtheadditiveoperatorsplitting(AOS)method.Notethatthecurvaturetermin( 2 )isapproximatedby@ @x x p 2x+2y+2!+@ @y y p 2x+2y+2!,whereisasmallpositivenumberincasethatthedenominatorsbecomezero.However,itmaystillcausestabilityissuesandlimittheconvergencerate. 2.3.2AsoftformulationandnumericalmethodToavoidlocalminimumproblemweproposeasoftformulationoftheenergyfunctional( 2 )byusingthesamestrategyin[ 36 ],anduseChambolle'sdualmethod[ 35 71 ]tosolveit.Letu:![0,1]beafuzzymembershipfunctionandrewritetheenergyfunctionalasE(u,a1,,an)=Zjru(x)jdx+ 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(CC(f(I),L))2,whereL(x)=c1u(x)+c2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(u(x)). 28

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Inrealapplications,wewantLbeabinaryimagesothatitcouldgiveareasonablesegmentation.Soduringeachiteration,weresetL(x)=c1(u>0.5)+c2(1)]TJ /F4 11.955 Tf 11.95 0 Td[((u>0.5)),andreferstothecharacteristicfunction.Followingthestrategyin[ 35 71 ],weintroduceanauxiliaryvariablev:![0,1]andconsiderthefollowingapproximatedenergyfunctionalE(u,v,a1,a2,,an)=Zjru(x)jdx+1 2ku)]TJ /F3 11.955 Tf 11.96 0 Td[(vk2+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(CC(f(I),L))2,whereL(x)=c1v(x)+c2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(v(x)),andischosentobesmallenoughsuchthattheminimizersu?andv?areclosetoeachother.Westillemploythealternateminimization(AM)approachtosolvethisminimizationproblem,i.e.,wegoontoalternativelysolvethefollowingtwoproblems: minuZjru(x)jdx+1 2ku)]TJ /F3 11.955 Tf 11.96 0 Td[(vk2(2)and min0v1,ai1 2ku)]TJ /F3 11.955 Tf 11.95 0 Td[(vk2+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(CC(f(I),L))2.(2)Theminimizationproblem( 2 )couldbeeffectivelysolvedbyapplyingChambolle'smethod[ 71 ]andthesolutionisu(x)=v(x))]TJ /F4 11.955 Tf 11.95 0 Td[(divp(x),wherep=(p1,p2)isgivenby )-222(r(divp)]TJ /F3 11.955 Tf 11.96 0 Td[(v)+jr(divp)]TJ /F3 11.955 Tf 11.96 0 Td[(v)jp=0.(2) 29

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Equation( 2 )couldbesolvedbyaxedpointmethod,i.e.,pn+1=pn+r(divpn)]TJ /F3 11.955 Tf 11.95 0 Td[(v=) 1+rjdivpn)]TJ /F3 11.955 Tf 11.95 0 Td[(v=j.Followingthesamestrategyin[ 35 ],thesolutionvof( 2 )isgivenbyv=min(max(u+G,0),1),whereG=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(CC(f(I),L))(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(c2)(f(I))]TJ ET q .478 w 256.86 -172.54 m 277.87 -172.54 l S Q BT /F3 11.955 Tf 256.86 -183.18 Td[(f(I))var(L))]TJ /F5 11.955 Tf 11.95 0 Td[(cov(f(I),L)(L)]TJ /F3 11.955 Tf 11.95 0 Td[(L) var(f(I))1 2var(L)3 2.ai'sareonlyincludedinthecrosscorrelationterm,sotheoptimizationschemeisexactlythesameasthelevelsetapproach.Wewanttomentionthatbothofthesetwomethods(levelsetmethodandthesoftsegmentationbyusingChambolle'smethod)arequiteeffective.However,itisstillpossiblethatwemayresultinalocalminimum.Thisisduetothenon-convexityofthecrosscorrelationtermintheenergyfunctional( 2 ).Therefore,( 2 )mayhavemorethanoneminimizerandwecannotguaranteethisapproachconvergestoaglobalminimizer. 2.4ExperimentalResultsInsection 2.4 ,weshowourexperimentalresultsonvariousimagestodemonstratetheperformanceoftheproposedmodelforsegmentation.AllthesimulationsarepreformedinMatlab7.9(R2009b)onaPCwithanIntelCore2DuoCPUat2.4GHzand3GBRAM.Wecomparetheproposednonparametricmodelwithtwoparametricmodels,namely,theChan-Vesemodel( 1 )andalsotheparametricGaussianmodel( 1 )( 1 ).Forcompleteness,werewritethesemodelsasthefollowing minC,c1,c2Z1(c1)]TJ /F3 11.955 Tf 11.95 0 Td[(I)2dx+Z2(c2)]TJ /F3 11.955 Tf 11.96 0 Td[(I)2dx+jCj,(2) 30

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Table2-1. NumberofiterationsandCPUtime(s)fortheexperiments(Figure 2-1 2-2 ,and 2-3 ). Model( 2 )Model( 2 )Proposed ImagesSizeIterationCPUIterationCPUIterationCPUFig.1(a)100100502.93632.61352.31Fig.1(e)1001001764.481827.21833.81Fig.1(i)256256867.07909.16585.90Fig.2(a)210180--984.63201.40Fig.2(f)210180--1007.14503.24Fig.3(a)336406----205.89Fig.3(e)336406----337.34 and minC,c1,c2,1,22Xi=1Zi(I)]TJ /F3 11.955 Tf 11.96 0 Td[(ci)2 22i+logidx+jCj.(2)Weapplythesamesoftformulationtotheabovetwomodelsandcomparetheresultswiththosegotfromtheproposedmodel.Thenumericalalgorithmwouldbeterminatedoncekuk+1)]TJ /F3 11.955 Tf 11.96 0 Td[(ukk=kukk<10)]TJ /F9 7.97 Tf 6.58 0 Td[(5,whereukreferstothelevelsetfunctionorthemembershipfunctionduringthekthiteration.ThecorrespondingnumberofiterationsandCPUtimeforallthefollowingexperimentsaresummarizedinTable.Fromthistable,wecanseethatcomparedtotheparametricmodels,theproposednonparametricmodelneedslesstimeanditerationstoobtaingoodresults. Forthelevelsetapproach,theinitialissettobethesigneddistancefunctionoftheinitialcirclesintheimage.However,Chambolle'sapproachismorerobusttotheinitialization.Theinitialuandvcanbegeneratedasrandomeldsintherange[0,1].Allofthetestimagesarerescaledtotheinterval[0,1]andtheparametersc1,c2inthelabelimagearexedtobe1and2.Unlessotherwisestated,ineachgure,weincludethetestimageI,thenaltransformedimagef(I),thenallabelimageLandthesegmentationresult,i.e.,thecontours(=0oru=0.5)superimposedontheoriginalimage. 31

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2.4.1Comparisonwithtwoparametricmodels (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o)Figure2-1. Segmentationresultsofthreesyntheticimages.(a)(f)(k)TestimagesI;(b)(g)(l)SegmentationresultsofthepiecewiseconstantMumford-Shahmodel;(c)(h)(m)ResultsoftheparametricGaussianmodel;(d)(i)(n)Resultsoftheproposedmodel;(e)(j)(o)Distributionsoftheforeground(blue)andbackground(red). ThepurposeofExperimentI(Figure 2-1 )istotesttheabilityoftheproposedmodelonthreesyntheticimages,wheretheforegroundandforegroundaregeneratedbytwodifferentdistributions.Meanwhile,wealsocomparethesegmentationresultswiththeChan-Vesemodel( 2 )andtheparametricGaussianmodel( 2 ).Ineachrow,fromlefttoright,wepresenttheoriginalsynthetictestimage,correspondingboundaryoverlaidontheimageobtainedfromthetheChan-Vesemodel( 2 ),theparametricGaussianmodel( 2 ),theproposedmodel,thedistributionsoftheforeground(blue)andbackground(red). 32

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Inthersttestimage(Figure 2-1(a) ),theintensitiesofforegroundandbackgroundaredrawnfromtwoGaussiandistributionswithdifferentmeanandthesamevariance(Figure 2-1(e) ).Whiletheintensitiesoftheforegroundandbackgroundinthesecondtestimage(Figure 2-1(f) )sharethesamemeananddifferentvariance(Figure 2-1(j) ).Thelasttestimage(Figure 2-1(k) )showsauni-modalGaussianforegroundoverabi-modalGaussianbackground(Figure 2-1(o) ).Wecaneasilyseethatalltheabove-mentionedthreemodelsworkeffectivelyforthersttestimage.However,theChan-Vesemodelfailsforthesecondandthirdtestimagesasthepiecewiseconstantassumptiondoesnotholdforthesetwocases.AsfortheparametricGaussianmodel,itcouldautomaticallyestimatethemeanandvariancebetweenandoutsidethecurveandthenusetheseinformationtoaidthesegmentation.Therefore,theresultforthesecondtestimage(Figure 2-1(f) )issatisfactoryeveniftheimageintensitiesoverlaps.However,itfailsforthethirdtestimage(Figure 2-1(k) )asthebackgroundisbi-modalGaussiandistributed,whichmakestheassumptionfortheparametricGaussianmodeldonothold.Ourproposednonparametricmodelsuccessfullyseparatestheforegroundfromthebackgroundinallthesethreecases.Theimageintensitiesofobjectandbackgroundinthersttestimagevaryalotinthersttestimage(Figure 2-1(a) )anditwouldbeeasilysegmented.Regardingthelasttwoimages,thebackgroundortheforegroundiscomparativelyhomogeneous.Hence,wecouldselectyi'sfromtheintervalswheretheintensitiesofthebackgroundorthebackgroundlie.Thusafterapplyingthefunctionf,linearcombinationofaseriesGaussianfunctionscenteredatyi's,totheimageI,thevaluesoff(I)intheinhomogeneousregionbecomeextremelysmallandthetransformedimagesf(I)aremorehomogenous.Therefore,bycarefullychoosingyi'sinthefunctionfandmaximizingthecorrelationcoefcientofthetransformedimagesf(I)withthelabelimageL,wegetthedesirableresults. 33

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Thecomparisonwiththesetwomodelsindicatesthattheparametricmodelsfailswhentheimageintensitydistributionismorecomplicated.Thatiswhyweneedtoexploreformoregeneralnonparametricmodels. 2.4.2TestonrealmedicalimagesWetesttheproposedmodelontworealmedicalimages,anMRbrainimageandalungimage.InExperiment2(Figure 2-2 ),thersttestimage(Figure 2-2(a) )isacleanbrainimageandthesecondtestimage(Figure 2-2(f) )isgeneratedbyaddingGaussiannoisewithzeromeanandvariance0.1tothecleanimage(Figure 2-2(a) ).Meanwhile,wealsocomparetheresultswiththoseobtainedfromtheparametricGaussianmodel( 2 ).ThesegmentationresultsoftheparametricGaussianmodel( 2 )areplacedattheendofeachrow(Figure 2-2(e) 2-2(j) ).Wecanseethatitsuccessfullyseparatesbackground,cerebrospinaluid(csf)fromwhitematterandgraymatter.However,thisisquitedifferentfromtheresults(Figure 2-2(d) 2-2(i) )obtainedfromtheproposedmodel.From(Figure 2-2(d) 2-2(i) ),wecanseethatgraymatterisseparatedfromtherestandthewholeimageisactuallysegmentedintothreepartsevenifweonlydothetwophasesegmentation.Thisisreasonablebecauseweutilizethehistograminformation(Figure 2-2(k) )ofthetestimage.Asindicatedinthegraph(Figure 2-2(k) ),therearethreepeaks,whichfromlefttorightstandforbackgroundandcsf,graymatter,whitematter.Notethatbackgroundandcsfareconsideredasawholeandtheintensitiesofthegraymatterlieintheinterval[0.4,0.6].Weuniformlyselectyi'sfromtheinterval[0.4,0.6],thenafterapplyingthefunctionftothetestimagesI,intensitieswhicharenotinthisinterval(whitematter,backgroundandcsf),wouldbecomealmostzerowhileintensitiesofthegraymatterareenlarged.Thatiswhywhitematter,csfandbackgroundlookdarkwhilethegraymatterlooksbrightinthetransformedimagesf(I)(Figure 2-2(b) 2-2(g) ).In 34

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otherwords,bydoingthistransformation,wecanviewwhitematter,csfandbackgroundasawholeandseparatethemfromthegraymatter.Thelastimage(Figure 2-2(l) )showsthatthecrosscorrelationbetweentheimagef(I)andthelabelimageL,i.e.,CC(f(I),L),keepsincreasingastheiterationprocessgoes.AndthistrendcoincideswiththemutualinformationbetweenthetestimageIandthelabelimageL.Sowecanconcludethattheproposedmethodisconsistentwiththemutualinformationbasednonparametricimagesegmentationmethod[ 20 ].Figure 2-1 and 2-2 indicatethatbychoosingspecicyi's,theproposedmodelworkswellforimageswithinhomogeneity,unevenlydistributedilluminationanditcangetmultiphasesegmentationresultswhileonlyusingtwophases.Inthefollowingexperiment,wedonotpaytoomuchattentionontheselectionofyi'sandletthemtobeequallyspacedintheinterval[0,0.5].Experiment3(Figure 2-3 )aimstotestwhethertheproposedmodelworksforimageswithnestructures.Wechoosethetestimagetobealungimagewithlotsofnedetails.Thisrsttestimage(Figure 2-3(a) )isacleanimageandthesecondone(Figure 2-3(e) )ismoreinhomogeneous,whichisgeneratedbyaddingGaussiannoisewithzeromeanandvariance0.1tothecleanimage.AfterapplyingthefunctionftotheoriginaltestimagesI(Figure 2-3(a) 2-3(e) ),theresultedimagesf(I)(Figure 2-3(b) 2-3(f) )havemorestrongcontrastbetweendifferentfeatureswhilestillpreservingthedetailedstructures.Thesameparametersareappliedforthesetwotests.Thenalresults(Figure 2-3(d) 2-3(h) )showthatmostofthenestructuresarecapturedandthenoiseinhomogeneitydoesnotexertabigdifference. 2.5ConclusionInChapter 2 ,weproposeanovelimagesegmentationframeworkbasedonRenyi'sstatisticaldependencemeasure.ThecomputationisgreatlysimpliedbyapplyingthetheoryofreproducingkernelHilbertspace.Twonumericalapproaches,thelevelsetmethodandChambolle'sdualapproach,areemployedduringtheimplementation. 35

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(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)Figure2-2. Segmentationresultsofacleanbrainimage(rstrow)anditsnoisyversion(secondrow).Fromlefttoright:TestimageI(a)(f);Finaltransformedimagesf(I)(b)(g);FinallabelimagesL(c)(h);Segmentationresultsoftheproposedmodel(d)(i);SegmentationresultsobtainedfromtheparametricGaussianmodel(e)(j).Thirdrow:(k)Histogramofthetestimage(a);(l)theinformationofCC(f(I),L)andMI(I,L)duringeachiteration. 36

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(a) (b) (c) (d) (e) (f) (g) (h)Figure2-3. Segmentationresultsofacleanlungimage(rstrow)andanoisyone(secondrow).Fromlefttoright:TestimagesI(a)(e),transformedimagesf(I)(b)(f),labelimagesL(c)(g)andthenalcontour(u=0.5)superimposedonthetestimages(d)(h). Finally,theproposedmodelisappliedtodifferentkindsofimagesandgetssatisfactoryresults. 37

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CHAPTER3ANEFFICIENTALGORITHMFORMULTIPHASEIMAGESEGMENTATIONWITHINTENSITYBIASCORRECTION 3.1ModelFormulationInSection 3.1 ,weestablishageneralizedmultiphaseimagesegmentationframeworkforjointregionpartitioningandintensitybiascorrection. 3.1.1MultiphaseSegmentationandMAPApproachLetI:!Rbetheinputimagetobesegmented,theimagesegmentationproblemistondasetofregionsfigMi=1suchthat=[Mi=1i,figMi=1aredisjoint.Thisisequivalenttosolvingforthecollectionofcharacteristicfunctionsi(x)ofi,where i(x)=8><>:1ifx2i,0otherwise,(3)fori=1,,M,andPii(x)=1,8x2.Asaddressedintheintroductionsection,theinputimageIcanbecorruptedbynoiseandunknownintensitybiaseldb:!R.ThisprocessisusuallymodelledasI(x)=b(x)I0(x)+n(x),whereI0istheidealcleanimage.Considerthesimplecasewheretheidealimageisconstantciineachregioni,andthenoisen(x)isnormallydistributedandindependentofthoseatotherlocations.Moreprecisely,ifx2i,then I(x)=b(x)ci+ni(x),(3)whereni(x)isnormallydistributedwithmeanzeroandunknownvariance2i.Itisworthnotingthatdifferentapplicationsmayyieldchangesinthemodelingof( 3 )[ 21 ].Nevertheless,thederivationandresultingalgorithmsgivenbelowstillworkwithappropriatemodicationsaccordingly. 38

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Tothisend,wecanseethatacompletesolutionpackagetoanimagesegmentationwithbiaseldestimationproblemisf,b,c,g,where=(1,,M)T:!f0,1gM,c=(c1,...,cM)T2RMand=(1,...,M)T2RM+.Herei,thei-thcomponentof,isthecharacteristicfunctionofiasdenedin( 3 ),ciand2irepresenttheoriginalmeanintensityandnoisevarianceinregioni,respectively.Finally,bisunknownbiaseldthatcausesintensityinhomogeneityintheimage.Werstndtheposteriorprobabilitydistributionp(f,b,c,gjI)off,b,c,ggivenimageI,andthenobtainanoptimalsegmentationandbiaseldestimationbythemaximum-a-posteriori(MAP).NotethattheBayes'ruleimpliesthat p(f,b,c,gjI)/p(Ijf,b,c,g)p(f,b,c,g).(3)Therefore,weneedtodeterminep(f,b,c,g),thepriorinformationimposedtof,b,c,g,andp(Ijf,b,c,g),thejointdistributionofpixelintensitiesgivenf,b,c,g. 3.1.2ModelingtheIntensityInhomogeneityBasedonmodel( 3 ),onecanreadilyseethatI(x)isnormallydistributedasN(b(x)ci,2i)ifx2i(orequivalentlyi(x)=1)giventhesegmentationf,b,c,g.However,theobservedintensityI(x)ismerelyonerealizationanditisusuallynotreliabletorecover,b,candsimultaneously.Toovercomethisdifculty,weestimateintensitydensityfunctionbasedonpixelintensitiesinalocalimagepatch.Moreprecisely,weassumeamultiplicativedensitystructureofI(x)asfollows, p(I(x)jf,b,c,g)/Yy2Wx(p(I(y)jf,b,c,g)x(y),(3)whereWx=fy2:jy)]TJ /F3 11.955 Tf 12.59 0 Td[(xjgisaroundimageneighborhoodwithprescribedradiusandcenteredatx.Ontherighthandsideof( 3 ),weconsiderthatI(y)closelyfollowsthemodel( 3 )andcontributestothedensityfunctionofI(x)viaaweightedproductasin( 3 ).Notethat( 3 )isanadhocmodicationofthelog-likelihoodfunctionandthusthemethodisnotreallyaMAPmethodintherigoroussense. 39

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In( 3 ),x:Wx![0,1]givestheweightsofintensitydistributionsofthepointsinWxsuchthatPy2Wxx(y)=1.Onecansimplychoosex(y)=1=jWxjforally2WxiftheintensitiesofneighborpointsinWxmakeequalcontributionstotheprobabilitydistributionp(I(x)jf,b,c,g).Inthisthesis,weusemoreadaptiveweightsx(y)accordingtothedistancefromytothecenterxviax(y)=Ks(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x),whereKsa(truncated)Gaussiankernelfunctiondenedby Ks(z)=8><>:Cexp)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jzj2=2s2,ifjzj0,otherwise.(3)forsomes>0.In( 3 ),CisanormalizingconstantthatmakesRjzjKs(z)dz=1.Weobservethattheintensitybiaseldinpracticalapplicationsusuallyvariesgraduallyacrosstheimagedomain.Inotherwords,thevalueb(y)isnearlyconstantforpointsyinanimagepatchWxprovidedthatisnottoolarge.Therefore,weapproximateb(y)byb(x),thebiasatthecenterpointx,andobtainthatI(y)N(b(x)ci,2i)fory2Wx\i.Hence,thejointdistributionp(Ijf,b,c,g)in( 3 )canbewrittenas p(Ijf,b,c,g)=Yx2Yy2Wxp(I(y)jf,b,c,g)Ks(y)]TJ /F8 7.97 Tf 6.59 0 Td[(x)(3)wherep(I(y)jf,b,c,g)isGaussian-type1 p 22iexp)]TJ 10.49 8.08 Td[(jI(y))]TJ /F3 11.955 Tf 11.96 0 Td[(b(x)cij2 22iforthosepointsythati(y)=1.Bynow,wehaveestablishedtheconditionalprobabilitydensityp(Ijf,b,c,g)in( 3 ).Ontheotherhand,wesetthepriorofaccordingtothedescriptivelengthoftheboundaries@itoexponentialdistributionwithparameter,whichimplicitlypenalizes 40

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undesiredirregularandzigzagpartitioncurves.Thepriorsofb,candareimposednon-informatively.Inaddition,termsinf,b,c,gareassumedtobeindependent.Consequently,thepriorp(f,b,c,g)canbesimpliedto p(f,b,c,g)/MYi=1exp()]TJ /F4 11.955 Tf 9.3 0 Td[(j@ij).(3)Basedon( 3 )and( 3 ),theMAPof( 3 )isequivalenttothefollowingminimizationafterweapplynegativelogarithmtobothsidesof( 3 ), min,b,c,(MXi=1j@ij+L(f,b,c,g)).(3)HereL(f,b,c,g)isthenegativelog-likelihoodfunction L(f,b,c,g)=)]TJ /F5 11.955 Tf 11.95 0 Td[(logp(Ijf,b,c,g)=ZMXi=1ZWx\iKs(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)li(y;x)dydx=ZMXi=1Zi(y)Ks(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)li(y;x)dydx(3)andli(y;x)isdenedfory2i\Wxby li(y;x):=jI(y))]TJ /F3 11.955 Tf 11.95 0 Td[(b(x)cij2 22i+1 2log(22i).(3)In( 3 ),wesubstitutethesummationbyintegraltoaccommodatethecontinuoussettingofourderivation,andomitWxandiinthelastequalityaccordingtothedenitionsofKsandiin( 3 )and( 3 ),respectively.Towritethersttermin( 3 )usingthecharacteristicfunctionsi,werecallthatthetotalvariationofafunctionf:!Risdenedby TV(f)=supp2Y)]TJ /F10 11.955 Tf 11.29 16.27 Td[(Zfdivpdx(3) 41

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wheretheadmissiblesetYis Y:=fp2C10(;Rd):jp(x)j1,8x2g.(3)Sinceiisthecharacteristicfunctionofi,thetotalvariationofiisthenthedescriptivelengthof@i,namely, TV(i)=j@ij.(3)Plug( 3 )and( 3 )into( 3 ),weobtainageneralizedmultiphasesegmentationmodelasfollows, min,b,c,MXi=1TV(i)+Zi(x)hi(x)dx(3)subjecttotheconstraintthatonlyonecomponentin(x)=(1(x),...,M(x))Tisoneandtherestarezerosateachx2.In( 3 ),thefunctionhiisdenedby hi(x)=ZKs(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)li(x;y)dy.(3)Notethatthesecondtermin( 3 )isobtainedbyexchangingthesymbolsxandy,followedbyswitchingtheorderofintegrationsin( 3 ). 3.2NumericalAlgorithmAlthoughthesegmentationproblemhasbeenuniedto( 3 ),thisminimizationproblemcannotbesolvedefcientlyingeneralduetothenondifferentiabilityoftheTVterm,andthenonconvexityoftheobjectivefunctionwithrespecttof,b,c,g.Conventionalapproachesbasedonlevelsetformulationrequireextensivecomputationsandsufferthelocalminimumsseverely.Inthisthesis,wedevelopanumericalalgorithmtotackleproblem( 3 )effectively.Werstrelaxtheconstraintonthecharacteristicfunction=(1,,M)Tin( 3 )toXdenedby X:=fu:![0,1]Mju(x)2M,8x2g(3) 42

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andthecanonicalsimplexMisdenedbyM=f(z1,,zM)T2RM+:z1++zM=1gTherelaxedmodelof( 3 ),usingnotationuinsteadofconventionalbinaryfunction,becomes minu,b,c,MXi=1TV(ui)+Zui(x)hi(x)dx(3)subjecttou2X.Onecanreadilyseethattheoriginalconstraintin( 3 )furtherrequiresu(x)tobeoneofthevertexesofM.Thisrelaxationsubstitutesextendsthesolutionsetofutoacontinuousandconvexset.Notethatthisrelaxationissimilartotheideaproposedin[ 35 37 ]andpleaserefertothecorrespondingpapersformoredetailedinformation.Intherestpartofthissection,weusealternatingminimizationstoconstructaniterativealgorithm.Namely,weneedtominimizetheobjectivefunctionwithrespecttooneofthevariablesinfu,b,c,gwithothersbeingxed. 3.2.1FirstVariationsofb,c,andFirstofall,weobservethatthevariablesb,c,andonlyappearinthesecondtermoftheobjectivefunctionin( 3 ),andtheirsolutionscanbeobtainedbyrstvariations.Fixu,cand,wecomputetheEuler-Lagrangian(E-L)equationforbandobtain b(x)=PMi=1(ci=2i)[Ks(uiI)](x) PMi=1(c2i=2i)[Ksui](x),x2,(3)whereistheconvolutionoperator.Next,wexu,b,andiandobtaintheE-Lequationofciforeachi=1,,Mas ci=R[Ks(uibI)](x)dx R[Ks(uib2)](x)dx.(3) 43

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FinallywehavetheE-Lequationofias 2i=R)]TJ /F5 11.955 Tf 5.48 -9.69 Td[([Ks(uiI2)])]TJ /F5 11.955 Tf 11.95 0 Td[(2cib[KsI]+cib2dx R[Ksui]dx.(3)Therefore,theupdatesofb,candhaveclosedformsandthemaincomputationsareregularconvolutionsusingkernelfunctionKsdenedin( 3 ). 3.2.2SolutiontouNowweturntotheminimizationoftheobjectivefunction( 3 )withrespecttou.Forxedb,cand,theminimizationcanbewrittenas minu2XMXi=1TV(ui)+Zui(x)hi(x)dx,(3)wherehidoesnotdependonuaccordingtoitsdenitionin( 3 ).Weremarkthat( 3 )isaconstrainednonsmoothoptimizationproblemduetotheconstraintonu(x)2Mforeachx2andthenondifferentiableTVtermintheobjectivefunction.Soweneedtondaneffectivewaytotacklethesetwoissues.Foreachui,weintroducethedualvariablepi2Yaccordingtothedenitionin( 3 ),andreformulatetheminimizationproblem( 3 )asamin-maxproblem minu2Xmaxpi2YMXi=1)]TJ /F4 11.955 Tf 9.3 0 Td[(Zuidivpidx+Zuihidx,(3)whereXisdenedin( 3 )andYistheadmissiblesetofpi'sdenedin( 3 ).InthediscretesettingwheretheimageIconsistsofNpixels,wecanvectorizeeachuiintoacolumnvectorinRN,thenitsdualvariablepiisamatrixinRND,whereDisthedimensionoftheimage(e.g.2or3).Hence,theoptimizationproblem( 3 )canbewrittenas minu2Xmaxpi2YF(u,p):=MXi=1hui,rTpi+hii,(3)wherer:RN!RNDisthediscretizedgradientoperator,thesuperscriptTistheconjugateoperator,andh,irepresentstheregularinnerproductinRN. 44

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NotethatbothofXandYareclosedandconvexsets.Henceasolutiontothemin-maxproblem( 3 )canbeobtainedbyalternatelysolvingfortheprimalvariableuanddualvariablepuk+1=X([uki)]TJ /F4 11.955 Tf 11.96 0 Td[(kruiF(uk,pk)]Mi=1) (3)pk+1i=Y(pki+krpiF(uk+1,pk)),8i (3)wherekandkactasthestepsizesoftheprimalanddualvariablesuandpinthek-thiteration,respectively,and[ui]Mi=1denotesthematrix[u1,,uM]thathasuiascolumns.HereX:RNM!XandY:RNd!YareprojectionoperatorsontothesetsXandY,respectively.Moreprecisely,Xmapseachrowofitsargument,sayz2RM,tothesimplexMusingthealgorithmshownin[ 72 ](ref.Appendix B ),andYprojectseachrowofitsargument,sayz2Rd,totheunitballBd:=fz2Rd:kzk2=1gviaz7!z maxfkzk2,1g.AccordingtothedenitionofF(u,p)in( 3 ),weknowthat( 3 )and( 3 )haveclosedformsasuk+1=X([uki)]TJ /F4 11.955 Tf 11.95 0 Td[(k(DTpk+1i+hi)]Mi=1), (3)pk+1i=Y)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(pki+kDuki,i=1,,M. (3)WenotethattheprojectionsXandYin( 3 )( 3 )havecomplexityMlogMandD,respectively.Therefore,themaincomputationalcostisNM(logM+D)ineachiteration.NotethatMisthenumberofphasesintheimageandisusuallylessthan10,andDisthedimensionoftheimagesuchas2or3.Moreover,theseprojectionsareappliedtoeachoftheNpixelsandhencethecomputationsinbothofXandYcanbecarriedoutinparallel.Onthecontrary,levelsetfunctionbasedsegmentationwithcommonlyusedsemi-implicitgradientdescentschemeusuallyrequiresGauss 45

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eliminationstosolvetridiagonallinearsystems,andhencethecomputationcannotbeparallelizedeasily. 3.2.3AlgorithmInconventionalsettingsofalternatingminimizations,weneedtoiterate( 3 ),( 3 )untilconvergencetogetubeforeupdatingtheothervariablesb,ciandi,i=1,,M.However,wefoundthatempiricallyitismoreefcienttosimplysolveforuandponlyonceandimmediatelyupdatetheremainingvariables.Thestoppingcriterionoftheproposedalgorithmissettokuk)]TJ /F3 11.955 Tf 12.15 0 Td[(uk)]TJ /F9 7.97 Tf 6.58 0 Td[(1k2=kukk2><>>:1ifui(x)=max1jMfuj(x)g0otherwise(3)foreachx2.Ifthereareseveralequallymaximalvaluesinu(x),wejustpickonerandomly.Tosumup,weproposeafastsegmentation(FastSEG)algorithminAlgorithm 1 below. 3.3ExperimentalResultsInSection 3.3 ,wetesttheAlgorithm 1 onavarietyofimagesandcomparewithseveralrecentlyproposedmethodsforimagesegmentationinthepresenceofnoiseandintensitybias. 3.3.1ExperimentSettingsTheproposedalgorithmisimplementedandallthetestsareperformedinMATLABr7.9(R2009b)computingenvironmentonaPCwithIntelDualCore2Duo 46

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Algorithm1Algorithm 1 :FastMultiphaseSegmentation(FastSEG) Input>0andtol.Initializeu0andp0,andsetb0=1,k=0. repeat Updateckusing( 3 )withukandbk; Updatekusing( 3 )withuk,bkandck; Computehkusing( 3 )withbk,ckandk; Computeukusing( 3 )withhk; Computepkusing( 3 )withhk; Updatebkusing( 3 )withuk,ckandk; k k+1. untilkuk)]TJ /F3 11.955 Tf 11.95 0 Td[(uk)]TJ /F9 7.97 Tf 6.59 0 Td[(1k2=kukk2
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Weusetestimagesanddefaultparametersettingsintheoriginalcodeofcomparisonalgorithmsifavailable.Fortheproposedalgorithm,thestoppingcriteriontolissetto10)]TJ /F9 7.97 Tf 6.58 0 Td[(3throughouttheexperiments.Theparameters,kandkaresettobe10)]TJ /F9 7.97 Tf 6.59 0 Td[(4,0.2and0.5respectively.Thepatchradiusissetto8,andthevariancesis4in( 3 ).Forallimageswetested(whoseintensitiesarescaledto[0,1]),theyseemtoprovidegoodcompromisebetweensmoothnessandaccuracyaswellasspeedandstableness.Wealsofoundthatmoderatechangesintheseparametersdonotyieldsignicantdifferenceinsegmentationresults. 3.3.2QuantitativeEvaluationandComparisonwithExistingMethodsWeuseJaccardsimilaritycoefcientasaquantitativemeasuretoevaluatethesegmentationresults.Letibethei-thregionobtainedbythealgorithmandibeitscorrespondingregioninthegroundtruthimage,thentheJSCbetweeniandiisdenedasJ(i,i)=ji\ij ji[ij,wherejjrepresentstheareaofaregion.Generallyspeaking,Jaccardsimilaritycoefcientsisboundedin[0,1]andlargervaluesimplymoreaccuratesegmentation.Todemonstratetheeffectivenessoftheproposedmodel,wecompareitwiththreerecentlydevelopedmethodsinthiseld.Forcompleteness,wegiveabriefsummaryregardingthesemethodsinthefollowing. 3.3.2.1MethodsOverviewTherstmethodwearegoingtocompareistheWeightedK-meansVariationalLevelSet(WKVLS)method[ 31 ].Forthetwo-phasecase,theWKVLSmodelcanbewrittenas EW(,b,c1,c2)=ZjrH()jdx+Z(jrj)]TJ /F5 11.955 Tf 17.94 0 Td[(1)2dx+ZZH()Ks(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)jI(y))]TJ /F3 11.955 Tf 11.95 0 Td[(b(x)c1j2dydx+ZZ(1)]TJ /F3 11.955 Tf 11.95 0 Td[(H())Ks(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)jI(y))]TJ /F3 11.955 Tf 11.96 0 Td[(b(x)c2j2dydx,(3) 48

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whereisthelevelsetfunctionwhosezerolevelsetrepresentsthepartitioncontour,andHistheHeavisidefunctiondenedbyH(z)=1ifz0and0otherwise.Thersttwotermsin( 3 )penalizethelengthofpartitioncontourandforcethelevelsetfunctiontobeupstraight(hasslope1)duringevolutions,respectively.Thelasttwotermsin( 3 )arefordatattingasintheproposedalgorithm,butlackthevariabilityofnoiseleveli.ThenextoneistheStatisticalandVariationalMultiphaseLevelSet(SVMLS)method[ 33 ],whichalsoutilizeslevelsetformulationandminimizesthefollowingenergyfunctionalES(,b,c,)=4Xi=1ZZMi((y))Ks(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x)li(y;x)dydx,whereli(y;x)isthesameasthatin( 3 ),=(1,2),andMi()isdenedasfollows:8>>>>>>><>>>>>>>:M1()=H(1)H(2),M2()=H(1)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(H(2)),M3()=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(H(1))H(2),M4()=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(H(1))(1)]TJ /F3 11.955 Tf 11.95 0 Td[(H(2)).ThelastmethodwewouldcompareistheCoherentLocalIntensityClustering(CLIC)method[ 29 ].CLICpartitionsanimagebysolvingaconstrainedminimizationproblem,EC(b,u,c)=MXi=1ZZui(y)Ks(y)]TJ /F3 11.955 Tf 11.96 0 Td[(x)jI(y))]TJ /F3 11.955 Tf 11.96 0 Td[(b(x)cij2dydx,subjecttoMXi=1ui(x)=1,8x2. 3.3.2.2Experiment1Intherstexperiment,wecomparetheproposedmodelwiththeaforementionedthreemethodsonanMRbrainimagewithstrongintensityinhomogeneityandnoise.Weusethedefaulttestimage(showninFigure 3-1(a) )fromthesourcecodepackage 49

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ofSVMLSpublishedonline1.Theinitialconditions(showninFigure 3-1(b) )isalsothedefaultsettingfortheSVMLScodeandisusedforallthetestedalgorithms.Forthisexperiment,weonlyprovidevisualresultsinFigure 3-1 sinceagroundtruthsegmentationisnotavailableforthisdataset.TheinputimageshowninFigure 3-1(a) containsstrongintensityinhomogeneityandhenceitisdifculttodistinguishdifferenttissueintensitiesfromitshistogramasshowninFigure 3-1(c) .Therefore,conventionalapproachesbasedonintensityclusterscannotyieldcorrectsegmentationsinthiscase.Ontheotherhand,allthefourtestedalgorithmscangeneratereasonableresultsbytakingtheintensityinhomogeneityintoaccount.Astheintensitybiaseldbisalsoestimatedbybothalgorithms,weplotI=b,theimagesafterbiascorrection,inthesecondrowofFigure 3-1 .Here=representspointwisedivision.Itcanbeseenthatthecorrectedimageshavelessintensitybiasescomparedto 3-1(a) .ThiscanalsobeobservedintheirhistogramsshowninthethirdrowofFigure 3-1 .Thehistogramsofthecorrectedimageshaveclearintensitypeaksandhencedifferenttissuescanbedistinguishedmoreeasily.Whenwelookintothedetailsofthesegmentationresults,wecanobservethatthoseobtainedbyCLICandtheproposedalgorithmaremoreaccuratethanthosebyWKVLSandSVMLS:theformertwocanbetterseparatethegrayandwhitemattersasindicatedbytheredarrowsinFigure 3-1(l) 3-1(o) .OneofthepossiblereasonsisthatWKVLSandSVMLSareformulatedinthatlevelsetframeworkandhencecanbeeasilytrappedintolocalminimum. 1 http://www4.comp.polyu.edu.hk/~cslzhang/code.htm 50

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3.3.2.3Experiment2ThetestimageinExperiment2isanMRimageobtainedfromBrainWeb2.Asgroundtruthsegmentationisavailableinthiscase,weuseJaccardsimilaritycoefcienttoevaluatetheperformanceofthetestalgorithmsquantitatively.InFigure 3-2 ,weusethesameinitializationforalltheabovementionedsegmentationmodelsasshownbytheredandbluerectanglesinFigure 3-2(a) .ThegroundtruthobtainedfromBrainWebarepresentedinFigure 3-2(b) ,whichconsistsoffourparts:background,whitematter,graymatterandcerebrospinaluid(CSF).Thesegmentationresultsobtainedbytheproposedalgorithm,WKVLS,SVMLSandCLICareshowninFigure 3-2(c) 3-2(d) 3-2(e) 3-2(f) ,respectively.TheJaccardsimilaritycoefcientsandCPUtimearesummarizedinTable 3-1 .WKVLSandSVMLScannotreturncorrectsegmentationasevolutionofthelevelsetfunctionscanbeeasilystuckatalocalminimum.TheresultofCLIClooksbetterthanthosegotfromWKVLSandSVMLS,butitcontainstoomanysuperuouspointsduetothelackofproperregularizationinsuchnoisycase.Ingeneral,theproposedmodelismoreaccurateandefcientthanallthethreemethods,whichcanbeaddressedinthefollowingaspects.Firstly,insteadofusinglevelsetformulation,wedirectlychoosecharacteristicfunctions,whichcanavoidthepossiblelocalminimumphenomenaandreinitializationprocess.Next,weusetotalvariationtoregularizethecharacteristicfunctions,whichismoreaccurateasitturnsouttothelengthofthesubregions.Finally,byapplyingtheprimal-dualformulationofTVnormsandspecialpropertiesofcanonicalsimplex,theproposedalgorithminvolvesonlyconvolutionswithGaussiankernelsandprojectionstotheunitballandsimplex.Therefore,theproposedmodelismoreaccurateandefcient. 2 http://www.bic.mni.mcgill.ca/brainweb/ 51

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Table3-1. QuantitativeevaluationofFigure 3-2 .JaccardsimilaritycoefcientsJSCofthefourregions,background(b),CSF(c),greymatter(g)andwhitematter(w),obtainedbythetestalgorithmsandtheirCPUtimesinseconds. Method JSCb JSCc JSCg JSCw CPU(s) WKVLS 48.87% 5.51% 25.63% 6.68% 160.82SVMLS 69.71% 18.79% 42.84% 61.68% 21.90CLIC 98.75% 70.76% 60.57% 65.96% 41.07Proposed 99.13% 80.47% 80.69% 82.54% 33.75 3.3.3FurtherEvaluationsoftheProposedAlgorithmItisimportantthatanautomatedsegmentationprocedureisrobustwithrespecttodifferentinitialsegmentations,intensitybiasstatus,noiselevel,andparametersettings.InSubsection 3.3.3 ,wefurtherevaluatetheperformanceofAlgorithm 1 ontheseaspects. 3.3.3.1RobusttoinitializationAstheobjectivefunctionsappearedinsegmentationproblemsareusuallynonconvex,mostalgorithmsespeciallythoseformulatedusinglevelsetfunctions,sufferlocalminimumsandhenceareverysensitivetoinitializations.Onthecontrary,theproposedalgorithm 1 appearstoberobust:wetestAlgorithm 1 onanMRimageusingvedifferentinitializationsasshowninFigures 3-3(a) (generatedbyK-means), 3-3(b) 3-3(c) 3-3(d) ,and 3-3(e) (generatedbysomeseedsshowninred,greenandbluesquares).ThenalcharacteristicfunctionsiobtainedbytheproposedalgorithmareshowninFigures 3-3(f) 3-3(g) 3-3(h) 3-3(i) and 3-3(j) ,respectively.Theresultsimplythattheproposedalgorithmisquiterobustwithrespectdifferentinitialconditions. Table3-2. QuantitativeevaluationofFigure 3-3 .JaccardsimilaritycoefcientsJSCofthefourregions,background(b),CSF(c),greymatter(g)andwhitematter(w),obtainedbythetestalgorithmsandtheirCPUtimesinseconds. JSCb JSCc JSCg JSCw CPU(s) (f) 99.98% 99.58% 99.83% 99.91% 14.63(g) 99.98% 99.46% 99.88% 99.97% 11.85(h) 99.99% 99.58% 99.90% 99.97% 12.18(i) 99.98% 99.46% 99.88% 99.96% 12.02(j) 99.83% 96.75% 99.00% 99.46% 13.75 52

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3.3.3.2RobusttointensityinhomogeneityWeconductmoreexperimentsonAlgorithm 1 onMRimagestotestitscapabilityondifferentintensityinhomogeneities.TheresultsareshowninFigure 3-4 .TheoriginalMRimagesareobtainedfromBrainWeb.Weaddsyntheticintensitybiasestotheimage(concentratedatthemiddle,topandbottomoftheimagedomain,respectively),asshownintherstrowofFigure 3-4 .WeshowthecorrectedimagesI=b,therecoveredbiaseldsb,segmentationresults,histogramsofthetestimagesandcorrectedimagesundereachofthesethreeimagesinFigure 3-4 .WecanseethatAlgorithm 1 successfullydetectstheintensitybiasesandobtainsdesiredsegmentationsregardlessofbiasstatus. Table3-3. QuantitativeevaluationofFigure 3-4 .JaccardsimilaritycoefcientsJSCofthethreeregions,backgroundandCSF(bc),greymatter(g)andwhitematter(w)obtainedbythetestalgorithmsandtheirCPUtimesinseconds. JSCbc JSCg JSCw CPU(s) Left 98.96% 91.65% 94.76% 8.34Middle 98.82% 86.86% 91.78% 20.40Right 99.48% 95.34% 95.87% 18.01 3.3.3.3RobusttonoiselevelThepurposeofthisexperimentistotestAlgorithm 1 onMRimageswithintensityinhomogeneityanddifferentlevelsofnoise.ThetestimagesaregeneratedbyrstmultiplyingasimulatedbiaseldtothecleanMRimageandthenaddinglow,medium,andstrongGaussiannoise.ThesegmentationresultsarepresentedinFigure 3-5 .FromtheresultsshowninthesecondrowofFigure 3-5 ,wecanseethattheproposedalgorithmconsistentlyreturnsreasonablepartitionsoftheimage,buttheaccuracycanbeslightlyaffectedbythenoiselevel. 3.3.3.4DifferentparametersettingsAsshowninSection 3.2 ,Algorithm 1 involvesthepenaltyparameter,andstepsizesk,kfortheprimalanddualvariables.Wefoundthattheproposedalgorithmperformswellforavarietyofimagesunderthesamesettingoftheseparameters. 53

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Table3-4. QuantitativeevaluationofFigure 3-5 .JaccardsimilaritycoefcientsJSCofthethreeregions,backgroundandCSF(bc),greymatter(g)andwhitematter(w)obtainedbythetestalgorithmsandtheirCPUtimesinseconds. JSCbc JSCg JSCw CPU(s) (e) 98.04% 76.47% 81.13% 2.68(f) 96.89% 74.53% 83.73% 3.08(g) 95.75% 67.38% 78.37% 3.51 However,thepatchsizeusedforlocaldensityweightcalculationin( 3 )canimpacttheresultsunderdifferentlevelofnoise.Inthefollowingexperiments,wetestAlgorithm 1 with=0,,8.Inparticular,weshowtheresultswith=0,2,8inFigures 3-6(e) 3-6(g) ,respectively.NotethatinFigure 3-6(e) ,therearemanysuperuouspointsinthecaseof=0,wherelocalpatchinformationisnotutilizedtoestimateI(x).Ontheotherhand,theresultslookmuchbetterwhenweusetheneighboringinformation.Thissuggeststheimportanceofusinglocalintensitydensityestimation( 3 ),especiallyinthepresenceofstrongnoise. Table3-5. QuantitativeevaluationofFigure 3-6 .JaccardsimilaritycoefcientsJSCofthethreeregions,backgroundandCSF(bc),greymatter(g)andwhitematter(w)obtainedbythetestalgorithmsandtheirCPUtimesinseconds. Method JSCbc JSCg JSCw CPU(s) =0 89.06% 65.52% 82.07% 10.99=2 86.83% 67.14% 84.59% 12.30=8 89.61% 66.81% 86.15% 12.10 3.4ConcludingRemarksInChapter 3 ,wepresentageneralmultiphasesoftsegmentationframeworkwhichcandealwithsevereintensityinhomogeneityandnoiseininputimages.Ourmodelestimatestheintensitydistributionataparticularpixelusingamultiplicativestructureofdistributionsofallpixelsinaneighborhood,andderivetheminimizationproblemusingMAP.Totacklethecomputationaldifcultyduetothehighlynonsmoothandconstrainedformulationofthesegmentationmodel,weapplyprimal-dualgradientprojectionstodevelopafastnumericalalgorithm.Numericalresultsonvariousimagesshowthat 54

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ourmethodismoreefcientandaccurateincomparisonwithotherrecentlyproposedalgorithms. 55

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(a)InputI (b)Initialcontours (c)Histogramof(a)Correctedimages (d)WKVLS (e)SVMLS (f)CLIC (g)ProposedHistogramsofthecorrectedimages (h)WKVLS (i)SVMLS (j)CLIC (k)ProposedSegmentationresults (l)WKVLS (m)SVMLS (n)CLIC (o)ProposedFigure3-1. ComparisonoftheproposedmodelwithWKVLS,SVMLSandCLIConanMRbrainimage(a)withstrongintensityinhomogeneity.Thecolorsquaresin(a)depictinitialcontoursusedbyallmethods. 56

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(a)Input (b)GroundTruth (c)Proposed (d)WKVLS (e)SVMLS (f)CLICFigure3-2. Comparisonoftheproposedmodel,WKVLS,SVMLSandCLIConanMRbrainimagewithseverenoiseandintensityinhomogeneity. 57

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(a)Initial1(byK-means) (b)Initial2 (c)Initial3 (d)Initial4 (e)Initial5 (f)Finalcharacteristicfunctionsusing(a)asinitial (g)Finalcharacteristicfunctionsusing(b)asinitial (h)Finalcharacteristicfunctionsusing(c)asinitial (i)Finalcharacteristicfunctionsusing(d)asinitial (j)Finalcharacteristicfunctionsusing(e)asinitialFigure3-3. Robusttodifferentinitializations. 58

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Figure3-4. Robustnesstestoftheproposedalgorithmondifferentintensityinhomogeneity.Left,middle,andrightcolumnsshowsegmentationresultsonimageswithintensitybiasinthemiddle,top,andbottomofimagedomain,respectively.Fromtoptobottom:inputimages,segmentationresults,estimatedbiaselds,correctedimages,histogramsofinputimages,andhistogramsofcorrectedimages,respectively. 59

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Figure3-5. Robustnesstestoftheproposedalgorithmondifferentimagenoiselevels.Left,middle,andrightcolumnscorrespondtosmall,mediumandstrongnoiselevels,respectively.Topandbottomrowsshowtheinputimagesandsegmentationresults,respectively. 60

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(a)Inputimage (b)=0 (c)=2 (d)=8 (e)Finalcharacteristicfunctionfor=0 (f)Finalcharacteristicfunctionfor=2 (g)Finalcharacteristicfunctionfor=8Figure3-6. EfciencyoflocalintensityestimationinFastSEGwhenappliedtoimageswithstrongnoise. 61

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CHAPTER4SPARSEIMAGEDEBLURRINGINTHEPRESENCEOFIMPULSENOISE 4.1BackgroundTwocommontypesofimpulsenoisearesalt-and-peppernoiseandrandom-valuedimpulsenoise.Ifthelevelofimpulsenoiseisp,thentheimpulsenoisedegradationprocesscanbesummarizedas:theintensityofeverypixelstaysthesamewithprobability1)]TJ /F3 11.955 Tf 12.89 0 Td[(pandchangestosomenewvaluewithprobabilityp.Iftheintensitychanges,itwillbecometheminimumormaximumvalueoftheimageintensityforsalt-and-peppernoiseanditwillbecomeasampledrawnfromauniformdistributionontheimageintensityintervalforrandom-valuedimpulsenoise.Inthefollowing,weassumethecleanimagefisrescaledtotheinterval[0,1]andweconsiderthecasewhenfisrstdegradedbyblurandthentheblurryimageisfurthercontaminatedbyimpulsenoise.Therefore,theobservationimagecanbemodeledbyg=N(kf),whereNreferstotheimpulsenoiseoperator.Mathematicallyspeaking,theprocesscanbemodeledas Salt-and-peppernoisegi,j=(N(kf))i,j=8<:0withprobablilityp=2,1withprobablilityp=2,(kf)i,jwithprobablility1)]TJ /F3 11.955 Tf 11.95 0 Td[(p. Random-valuedimpulsenoisegi,j=(N(kf))i,j=awithprobablilityp,(kf)i,jwithprobablility1)]TJ /F3 11.955 Tf 11.95 0 Td[(p,whereaisasampledrawnfromauniformdistributionon[0,1].Itisnotdifculttoseethatimpulsenoiseisnotadditive,butwecanviewitasadditivebysetting n=g)]TJ /F3 11.955 Tf 11.95 0 Td[(kf=N(kf))]TJ /F3 11.955 Tf 11.95 0 Td[(kf,(4)andthemodelg=kf+nstillholds.Anexperimentiscarriedoutonasyntheticimagetostudythebehaviorofimpulsenoise.Forbetterunderstanding,weplotthelogarithmic 62

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histogramofninFigure 4-1 (red).Thehistogramshowsthattheintensitiesofimpulse Figure4-1. Truelogarithmichistogramofnanditscorrespondingapproximation. noiseaccumulateat0,sharplygodownanductuateawayfrom0.However,thereisnospecicformregardingtheprobabilitydistributionofimpulsenoise.In[ 73 ],theauthorsproposedanefcientimagerestorationmodelbasedonEMalgorithm.Later,in[ 62 ],theysimpliedthemodelas E(f,u,21,22)=1 2Zuln21dx+1 2Z(1)]TJ /F3 11.955 Tf 11.96 0 Td[(u)ln22dx+1 2Zu 21+1)]TJ /F3 11.955 Tf 11.96 0 Td[(u 22(g)]TJ /F3 11.955 Tf 11.95 0 Td[(kf)2dx+1Zjru(x)jdx+2Zjrf(x)jdx.(4)Therstfourtermsin( 4 )turnouttobethenegativelog-likelihoodoftwoGaussianmixturewithdifferentvariances.Inthisthesis,wechoosetousetworandomlymixedGaussianwhitenoisetoapproximatetheresidualn.ThisisplausibleasindicatedinFigure 4-1 .ThegreenlinereferstothelogarithmichistogramoftwodifferentrandomlymixedGaussianwhitenoisewithvariance0.5and0.0001.Thisapproximation 63

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looksquitegoodevenifthevariancesarechosenbyourselves.Inthefollowing,weiterativelyupdatethevariancestogetabetterapproximation. 4.2ModelFormulationInSection 4.2 ,wegiveadifferentformulationaboutthemodelproposedin( 4 ),whichiscloselyrelatedtoourownwork.Afterthat,weproposeanewmodelusingsparserepresentationtechnique.Asdiscussedabove,itisappropriatetoapproximatethedistributionofnusingGaussianmixture.Letu:!f0,1gdenotewhetherapixelisnoisyornot.Morespecically,foreachx2,u(x)=1indicatesthepixelisfreeofnoiseandu(x)=0impliesxisanoisypixel.Thisimmediatelyleadstoapartitionoftheimagedomain=1\2,with1=fx2ju(x)=0gand2=fx2ju(x)=1g.Ifxisacleanpixel,thenwehaven(x)=0by( 4 )andthusitisplausibletoapproximatelyviewitasasampledrawnfromthedistributionN(0,21)withasufcientlysmall21.Nevertheless,ifitisanoisypixel,thenweregarditasasampleofN(0,22)foraproper22.Ourpurposeistondthetheposteriorprobabilitydistributionp(ff,u,21,22gjg)offf,u,21,22ggiventheobservedimageg,andobtainanoptimalrecoverybythemaximum-a-posteriori(MAP).NotethattheBayes'ruleimpliesthat p(ff,u,21,22gjg)/p(gjff,u,21,22g)p(ff,u,21,22g).(4)Therefore,weneedtodeterminep(ff,u,21,22g),thepriorinformationimposedtothesegmentationff,u,21,22g,andp(Ijff,u,21,22g),whichisthejointdistributionofpixelintensitiesinIgiventhesegmentationff,u,21,22g.Basedontheabovediscussion,onecanreadilyseethattherandomvariablesfn(x)ju(x)=0garenormallydistributedasN(n(x),21)andtherandomvariablesfn(x)ju(x)=1garedistributedasN(n(x),22)giventhevaluesff,u,21,22g.Ifwefurtherassumealltherandomvariablesfn(x)jx2gareindependent,thenthejoint 64

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distributionp(njff,u,21,22g)canbewrittenas p(njff,u,21,22g)=2Yi=1Yx2i1 p 22iexp)]TJ 10.49 8.09 Td[(jn(x)j2 22i(4)Bynow,wehaveestablishedtheconditionalprobabilitydensityp(njff,u,21,22g)in( 4 ).Ontheotherhand,wecansetthepriorsof21and22to(non-informative)uniformdistributions.Thepriorofuissettobethedescriptivelengthoftheboundaries@1(or@2)totheexponentialdistributionwithparameter.Moreover,termsinff,u,21,22gareassumedtobeindependent.Regardingthepriorforf,letisignoreitforamomentorwecantemporallysetittobeuniformdistribution.Therefore,thepriorp(ff,u,21,22g)canbesimpliedto p(ff,u,21,22g)/exp()]TJ /F4 11.955 Tf 9.3 0 Td[(j@1j).(4)Basedon( 4 ),( 4 ),theMAPof( 4 )isequivalenttothefollowingminimizationwhereweappliednegativelogarithmtobothsidesof( 4 ), minf,u,21,22j@1j+L(ff,u,21,22g).(4)HereL(ff,u,21,22g)isthenegativelog-likelihoodfunction L(ff,u,21,22g)=)]TJ /F5 11.955 Tf 11.29 0 Td[(logp(Ijff,u,21,22g)=2Xi=1Zijn(x)j2 22i+1 2log(22i)dx=1 2Zuln21dx+1 2Z(1)]TJ /F3 11.955 Tf 11.95 0 Td[(u)ln22dx+1 2Zu 21+1)]TJ /F3 11.955 Tf 11.95 0 Td[(u 22(g)]TJ /F3 11.955 Tf 11.95 0 Td[(kf)2dx+Const.(4)Sinceuisthecharacteristicfunctionof1,thetotalvariationofu1isthenthedescriptivelengthof1,namely, j@1j=TV(u),Zjru(x)jdx.(4) 65

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Plug( 4 ),( 4 )into( 4 )andignoretheconstantterm,weobtaintheimagerecoverymodelasfollows,minf,u,21,22TV(u)+L(ff,u,21,22g),whereL(ff,u,21,22g)isdenedasthatin( 4 )subjecttotheconstraintuonlytakestwovalues(1and0).Inthiscase,thesolutionsetforuisnotconvex,soweittotheunitinterval.Finally,letusconsiderthepriorfortherecoveredimagef.Thecleanimagesareusuallysmoothandallowedtohavelargejumpsacrosstheimageboundaries.Ifweemploythetotalvariationtoregularizetherecoveredimagef,thenwegetthemodelin[ 62 ].Inthisthesis,weproposeanewmodelforimagereconstructionbyusingthesparerepresentationtheorytoregularizetherecoveredimagef. 4.3SpareRepresentationTheoryThetheoryofsparserepresentationhasgainedagreatdealofattentionrecentlyandhasbeensuccessfullyappliedtotheeldofimageprocessing,suchasdenoising[ 74 75 ],deblurring[ 65 76 78 ],super-resolution[ 79 80 ],etc.Tothebestofourknowledge,thereislittleworkonimagedeblurringinthepresenceofimpulsenoiseviasparserepresentationtheory,whichisthemainpurposeofthispaper.Sparserepresentationtheoryisfoundedontheassumptionthatanimagecouldberepresentedasasparselinearcombinationofaseriesofatomimages.Thesetofatomsiscalledadictionary,whichcouldeithermanuallychosenorlearnedfromtrainingdataset.ThedictionaryDisaredundantmatrixinRstfort>s(s=81andt=324inourpaper),aimingtohandleimagepatchesofsizep sp s,whicharetreatedascolumnvectorsinRsunlessotherwisestated.Supposetheimagef2Rp Sp S(Ss),wedecomposeitintoJsmallpatchesofthesizep sp ssuchthatallthepatchescovertheentireimagefandmayoverlap.LetRj2RsSbeamaskwiththepropertythatRjfextractsthej-thpatchoff. 66

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Forthemoment,letusassumethedictionaryDisknown.ThesparelandmodelassumeseachpatchRjfcanbesparselyrepresentedbythedictionaryD.Morespecically,thereexisti2Rtsuchthatkjk0
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convenience,wedenoteH(f,u,21,22)=1 2Zuln21dx+1 2Z(1)]TJ /F3 11.955 Tf 11.31 0 Td[(u)ln22dx+1 2Zu 21+1)]TJ /F3 11.955 Tf 11.96 0 Td[(u 22(g)]TJ /F3 11.955 Tf 11.95 0 Td[(kf)2dx.Firstofall,forxedf,u,D,,weconsiderthevariables2i,i=1,2,whichcanbeobtainedthroughtheirEuler-Lagrangian(E-L)equations 21=Ru(g)]TJ /F3 11.955 Tf 11.95 0 Td[(kf)2dx Rudx,22=R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(u)(g)]TJ /F3 11.955 Tf 11.96 0 Td[(kf)2dx R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(u)dx.(4)Theusubproblemmin0u1H(f,u,21,22)+1Zjru(x)jdxisaconstrainedTVproblem,whichissolvedbythesplitBregmanalgorithm. 8><>:(un+1,dn+1)=argmin0u1H(f,u,21,22)+1Zjdnjdx+ 2Zjdn)-222(ru)]TJ /F25 11.955 Tf 11.96 0 Td[(bnj2dx,bn+1=bn+run+1)]TJ /F25 11.955 Tf 11.96 0 Td[(dn+1.(4)Thealternatingminimizationschemeisagainemployedtosolvetheproblem( 4 ).TheEuler-Lagrangeequationof( 4 )withrespecttoucanbewrittenas )]TJ /F5 11.955 Tf 11.96 0 Td[(u=div(bni)]TJ /F25 11.955 Tf 11.96 0 Td[(dni))]TJ /F10 11.955 Tf 11.95 16.86 Td[(ln21)]TJ /F5 11.955 Tf 11.96 0 Td[(ln22+1 1 21)]TJ /F3 11.955 Tf 15.78 8.09 Td[(u 22(g)]TJ /F3 11.955 Tf 11.95 0 Td[(kf)2,(4)whichcouldbesolvedbyGauss-Seidel(GS)method.Thedsubproblemin( 4 )isaL1)]TJ /F3 11.955 Tf 12.43 0 Td[(L2minimizationproblem,whichcouldbeexplicitlysolvedbythetwodimensionalshrinkagealgorithm,dn+1=shrink(run+1+bn,1=),wheretheoperatorshrinkisdenedasshrink(y,)=y jyjmaxfjyj)]TJ /F4 11.955 Tf 17.93 0 Td[(,0g. 68

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Let!=u 21+1)]TJ /F3 11.955 Tf 11.95 0 Td[(u 22,thentheEulerLagrangeequationof( 4 )associatedwithfcanbewrittenas ^k[!(kf)]+2 XjRTjRj!f=^k(!g)+2JXj=1RTj(Dj),(4)where^kreferstotheconjugatedofk.NotethatXjRTjRjisadiagonalmatrixwithpositivediagonalentries.So( 4 )couldalsobetransformedtoalinearmatrixequationAf=bwithAbeingpositivesymmetricdeniteandinthispaperwechoosetheconjugategraduate(CG)methodtosolveit.Finally,letusturntothe(D,)subproblem.Thejsubproblemn+1j=minjjkjk0+1 2kDj)]TJ /F3 11.955 Tf 11.95 0 Td[(Rjfnk22,isasparse-codingproblem,whichisequivalentto j=argminjkjk0,s.t.kDj)]TJ /F3 11.955 Tf 11.96 0 Td[(Rjfnk<.(4)Severalalgorithmcouldbeappliedtosolvethisproblem,suchasthresholding,matchingpursuit(MP),orthogonalmatchingpursuit(OMP),etc.Inthispaper,wechoosetouseOMP.Giventhecoefcientsj,thedictionaryDisupdatedthroughtheK-SVDalgorithm[ 81 ].AccordingtotheK-SVDalgorithm,thedictionaryisupdatedonecolumnatatime.Foraxedcolumn,theupdatecanbedonebyperformingasingularvaluedecomposition(SVD)operationonitsresidualdatamatrix,computedonlyontheimagepatchesthatusethisatom.Inthisway,thepenaltytermisguaranteedtodropandatthesametime,therepresentationcoefcientschangeaswell,whilekeepingtheirsparsitystructure.Tosumup,weproposeanalgorithmforsparseimagerestorationinAlgorithm 2 below. 69

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Algorithm2Algorithm 2 :SparseImageDeblurringandDenoising InputdictionaryD,f0=g,b0=0,d0=0,u0=0,(2i)0,i=1,2. repeat Update(2i)n+1through( 4 ),i=1,2; Sparsecodingstage:solve( 4 )foreachj; Dictionarylearningstage:updateDifnecessary; Updatefn+1through( 4 )usingCGmethod; Updateun+1through( 4 )usingGSmethod; dn+1=shrink(run+1+bn,1=); bn+1=bn+rfn+1)]TJ /F25 11.955 Tf 11.96 0 Td[(dn+1; n n+1. untilkfn)]TJ /F3 11.955 Tf 11.95 0 Td[(fn)]TJ /F9 7.97 Tf 6.59 0 Td[(1k1
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people,building,animalandower,takenfromtheBerkeleySegmentationDatabase[ 82 ].PartofthetrainingimagesareshowninFigure 4-2 Figure4-2. Samplesoftrainingimages. Letforibetheoriginalcleanimageandfbetherestoredimage.Weusepeaksignaltonoiseratio(PSNR)tomeasurethegoodnessoftherestoredimage,PSNR=10log10m1m2 kf)]TJ /F3 11.955 Tf 11.96 0 Td[(forik22,where[m1,m2]referstothesizeoftheimage.Allthetestimagesarerescaledtotheunitinterval[0,1].Thecleantestimagesarecorruptedby99Gaussiankernelwith=15andfurthercontaminatedbyimpulsivenoise.PartoftheoriginalimagesandblurryonesareshowninFigure 4-3 .Wecompareourresultswithtwodifferentimagerestorationmodels.TherstoneistheTVL1model,minfZjrfj+kkf)]TJ /F3 11.955 Tf 11.96 0 Td[(gk1,proposedin[ 59 ],whichissolvedbytheFTVdalgorithmandthecodeispublishedonline1.Thesecondoneismodel( 4 )proposedin[ 62 ].Foreachalgorithm,wetestdifferentparametersandchoosetheoneswiththebestrecoveryresults. 4.5.1Testwithseverblurand30%impulsenoiseInSection 4.5.1 ,wecomparetheproposedmodelwithTVL1andthemodel( 4 )inrecoveringimagesfromblurryimagescontaminatedby30%impulsenoise.Theresults 1 http://www.caam.rice.edu/~optimization/L1/ftvd/ 71

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cleanimages blurryimagesFigure4-3. Partofthecleantestimagesandtheircorrespondingblurryones. forsalt-and-peppernoiseispresentedinFigure 4-4 andtheresultsforrandom-valuedimpulseareshowninFigure 4-5 .ThecorrespondingPSNRvaluesarepresentedinTable 4-1 .ItiseasytoseethattheresultsgotfromtheFTVdalgorithm(secondrow)sufferstrongstair-casingeffect,whichhasbeenefcientlysuppressedintheresultsbythemodel( 4 )(thirdrow)andtheproposedmodel(forthrow).Moreover,theproposedmodeliscapabletorecovermoredetailedinformationasshowninthetextureoftheimageBarbaraandalsothecoatofthebear.Inaddition,highPSNRvaluesoftheproposedmodelalsorevealsthattheresultsoftheproposedmodelarethebestamongthethree. 4.5.2TestwithseverbluranddifferentlevelsofimpulsenoiseThepurposeofthesecondexperiment(Figure 4-6 )istotesttherobustnessoftheproposedinrecoveringimageswithdifferentlevelsofnoise.ThetestimagesareshownintherstrowofFigure 4-6 ,whicharegeneratedbyrstconvolvingtheoriginalcleanimagewitha99Gaussiankernelwithvariance=15.Theresultedimagesare 72

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noisedblurryimages(salt-and-peppernoise) FTVdL1 Model( 4 ) ProposedModel( 4 )Figure4-4. Testonsalt-and-peppernoise. 73

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blurrynoisyimages(impulsenoise) FTVd1 Model( 4 ) ProposedModel( 4 )Figure4-5. Testonrandom-valuedimpulsenoise. 74

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furthercontaminatedby10%,50%salt-and-peppernoiseand10%,50%random-valuedimpulsenoise,respectively.TheresultsgotfromFTVdL1,model( 4 )andtheproposedmodelarepresentedinthesecond,thirdandforthrow.ThecorrespondingPSNRvaluesareshowninTable 4-2 .FromFigure 4-6 ,wecanseethatwhentheresultsgotfromtheFTVdL1aresatisfactoryforthelownoiselevelcase,butthestair-casingeffectbecomesverystrongwhenwefurtherincreasethenoiselevel.Ontheotherhand,bothmodel( 4 )andtheproposedonearenotaffectedbythehighnoiselevel.Inaddition,theproposedmodelisveryeffectiveinrecoveringtinydetailedinformationasshowninthehairofLenaanditsPSNRvaluesarealwaysthelargestamongallthethreemodels. Table4-1. PSNRvaluesforFigure 4-4 4-5 Figure 4-4 :Salt-and-pepperFigure 4-5 :Random-valued FiguresFTVdL1Model( 4 )ProposedFTVdL1Model( 4 )ProposedBarbara27.527428.932432.374627.589329.093231.2386Boat26.699328.825928.840526.790928.151428.7988Bear33.341935.773737.642833.082035.853035.9038Flower32.075233.439933.455532.082331.363233.6063 Table4-2. PSNRvaluesforFigure 4-6 Salt-and-pepperRandom-valued FiguresFTVdL1Model( 4 )ProposedFTVdL1Model( 4 )ProposedLena(10%)28.127629.006829.100928.045629.098529.1901Lena(50%)25.433227.229728.271526.589527.858028.2207 4.6ConclusionInChapter 4 ,weproposeasparelandmodelfordeblurringimagesinthepresenceofimpulsenoise.Impulsenoiseisnon-additive.Ourmodelusesmixturegaussiantoapproximatetheresidualinformation.Moreover,toavoidthestair-casingeffectsbroughtbytotalvariation,weutilizethetheoryofsparserepresentationtoregularizetherecoveredimage.Experimentalresultsondifferentimageshavedemonstratetheaccuracyoftheproposedmodel. 75

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(a)10%saltandpeppernoise (b)50%saltandpeppernoise (c)10%random-valuedimpulsenoise (d)50%random-valuedimpulsenoise FTVdL1 Model( 4 ) Model( 4 )Figure4-6. Testondifferentlevelofnoise. 76

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APPENDIXAREPRODUCINGKERNELHILBERTSPACELetEbeanarbitrarysetandHbeaHilbertspaceofrealfunctionsonE.WesaythatHisaRKHSifthelinearmapFx:f!f(x)isaboundedfunctionalforanyx2E.Bythisdenition,Fx2H?,whichisthedualspaceofH.Therefore,ReiszrepresentationtheoremshowsthatthereexistsauniqueKx2H,suchthatf(x)=hFx,fi=hKx,fi,8f2H.DeneK:EE!RasK(x,y)=hKx(),Ky()i.ItiseasytoseethatKhasthefollowingproperties: 1. Kissymmetric:K(x,y)=K(y,x). 2. Reproducingproperty:f(x)=hK(x,),f()i. 3. Kispositivedenite:Pni,jaiajK(xi,xj)0holdsforallx1,x2,,xn2E,a1,a2,,an2Randtheequalityholdsifandonlyifai=0,i=1,2,,n.WecallsuchanKthereproducingkernelfortheHilbertspaceH.Ontheotherhand,supposeK:EE!Rissymmetricandpositivedenite,thenaccordingtotheMoore-Aronszajntheorem([ 83 ]),thereisauniqueHilbertspaceoffunctionsonEforwhichKisareproducingkernel.Infact,letH0(E)bethelinearspanofthefunctionsfK(x,)jx2EganddenetheinnerproductinH0(E)tobehnXi=1aiK(xi,),mXj=1bjK(yi,)i=nXi=1mXj=1aibjK(xi,xj).LetH(E)bethecompletionofH0(E)withrespecttothisinnerproduct.ItisnotdifculttocheckthatH(E)istheuniqueRKHSwithreproducingkernelK.FortheparticularcaseE=R.LetC0(R)bethespaceofrealvaluedcontinuousfunctionsvanishingatinnitywiththesupremumnorm.Thenwehavethefollowingresult supf,g2V(R)CC(f(X),g(Y))=supf,g2H0(R)CC(f(X),g(Y)),(A) 77

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whereV(R)isthespaceofallrealBorelmeasurablefunctionswithnitepositivevariance.Thisisthemainresultwehaveusedinthispaperand(??)(??)followsdirectlyfromthisresult.Theproofofthisresultcanbeobtainedthroughthefollowingthreesteps(Weomitthedetailshere). 1. H0(R)isdenseinC0(R). 2. LetV(B)bethespaceofallrealboundedBorealmeasurablefunctions,thensupf,g2V(B)CC(f(X),g(Y))=supf,g2C0(R)CC(f(X),g(Y)). 3. supf,g2V(R)CC(f(X),g(Y))=supf,g2V(B)CC(f(X),g(Y)). 78

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APPENDIXBSIMPLEXPROJECTIONALGORITHMTheprojectionontosimplexMcanbeaccomplishedeasilyasshowninAlgorithm 3 [ 84 ].Ascanbeseen,themaincomputationisinthesortingstep,andhencecomplexityisMlogM.Aninspiringderivationwithdetailedanalysisofcomputationalcomplexitycanbefoundin[ 72 ]. Algorithm3ProjectionontothesimplexM Inputz=(z1,,zM)T2RM; Sortzintheascendingorderasz(1)z(M),andseti=M)]TJ /F5 11.955 Tf 11.95 0 Td[(1; repeat Computeti=(PMj=i+1z(j))]TJ /F5 11.955 Tf 11.95 0 Td[(1)=(M)]TJ /F3 11.955 Tf 11.95 0 Td[(i). iftiz(i)then Set^t=tiandbreak else Seti i)]TJ /F5 11.955 Tf 11.95 0 Td[(1 endif untili=0 ifi=0then Set^t=(PMj=1zj)]TJ /F5 11.955 Tf 11.95 0 Td[(1)=M endif Return(z)]TJ /F5 11.955 Tf 11.55 1.62 Td[(^t)+astheprojectionofzontoM. 79

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BIOGRAPHICALSKETCH HailiZhangwasborninLuoyang,HenanProvince,P.R.China.In2005,shereceivedherBachelorofScienceinmathematicsfromHenanNormalUniversity,P.R.China.Afterthat,shewasadmittedtoBeijingNormalUniversitygraduateprogramundertheguidanceofDr.ZhongdanHuanandcompletedherMasterofScienceinappliedmathematicsin2008.Infall2008,shejoinedthemathgraduateprogramattheUniversityofFloridaandstartedtoworkwithDr.YunmeiChen.ShereceivedherPh.D.inappliedmathematicsfromtheUniversityofFloridainthesummerof2013.Herresearchinterestsaremathematicalmodeling,numericalanalysisandoptimizationtheorieswithapplicationstocomputervisionandimagingscienceusingtoolsbasedonpartialdifferentialequations,calculusofvariationsandstatisticalmethods. 87