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Variational Models for Simultaneous Image Segmentation and Noise Removal

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Title:
Variational Models for Simultaneous Image Segmentation and Noise Removal
Creator:
Posirca, Iulia M
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (54 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Chen, Yunmei
Committee Members:
Groisser, David J
Rao, Murali
Mccullough, Scott
Samant, Sanjiv Singh
Graduation Date:
12/15/2012

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Subjects / Keywords:
Data smoothing ( jstor )
Image processing ( jstor )
Image reconstruction ( jstor )
Imaging ( jstor )
Mathematics ( jstor )
Modeling ( jstor )
Pixels ( jstor )
Statistical models ( jstor )
Supernova remnants ( jstor )
Ultrasonography ( jstor )
Mathematics -- Dissertations, Academic -- UF
image -- noise -- segmentation -- variational
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mathematics thesis, Ph.D.

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Abstract:
We present two projects for simultaneous image segmentation and noise removal. The first project concerns the images corrupted with Gaussian noise and the second one was developed for images contaminated with multiplicative noise. For both models we use soft segmentation, which allows each pixel to belong to each image pattern with some probability. Our work proposes also a functional with variable exponent, which provides a better noise removal with feature preserving. The diffusion resulting from the proposed models is a combination between the total variation (TV)-based and isotropic smoothing. To minimize the functional energy, we use the Euler-Lagrange equations on the (K-1)-simplex and the alternating minimization (AM) algorithm. The experimental and comparison results with some traditional models show the efficiency of our work, with improved denoising and segmentation of real and synthetic images. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Chen, Yunmei.
Statement of Responsibility:
by Iulia M Posirca.

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Copyright Posirca, Iulia M. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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LD1780 2012 ( lcc )

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VARIATIONALMODELSFORSIMULTANEOUSIMAGESEGMENTATIONANDNOISEREMOVALByIULIAMAGDALENAPOSIRCAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012IuliaMagdalenaPosirca 2

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ToDaniel TomyfatherIon InmemoryofmydearmotherMagdalena 3

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ACKNOWLEDGMENTS Iamdeeplygratefultomyadvisor,Dr.YunmeiChenforsharingherwisdom,forherconstantguidanceandsupportthroughmygraduatestudiesandforintroducingmetodigitalimagingresearchproblems.MyspecialthanksgoesalsotoDr.CeliaBarcelosforhercontributiontotheseprojects.Iamverygratefultoallmycommitteemembers,Dr.Groisser,Dr.McCullough,Dr.RaoandDr.Samantfortheirinputandadvice.Lastbutnotleast,Iwouldliketothanktomyfamilyfortheunderstandingandsupportgivenduringmydoctoralstudies. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 9 2THESEGMENTATIONPROBLEM ......................... 12 2.1DenitionandObjectives ........................... 12 2.2TheMumford-ShahModel .......................... 12 2.3GeodesicActiveContoursandTheLevelSetMethods ........... 13 3ASTOCHASTIC-VARIATIONALAPPROACHFORSOFTMUMFORD-SHAHSEGMENTATION ................................... 17 3.1SoftSegmentationvs.HardSegmentation .................. 17 3.2ExistingModels ................................. 17 3.3ProposedModel ................................ 21 3.4Algorithm .................................... 24 3.5Experiments .................................. 26 3.6Conclusions ................................... 29 4SEGMENTATIONANDDENOISINGOFIMAGESWITHMULTIPLICATIVENOISE ......................................... 33 4.1Introduction ................................... 33 4.2ExistingModels ................................. 34 4.3ProposedWork ................................. 37 4.4Algorithm .................................... 40 4.5Experiments .................................. 41 4.6Conclusions ................................... 45 REFERENCES ....................................... 50 BIOGRAPHICALSKETCH ................................ 54 5

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LISTOFTABLES Table page 3-1TheSNRandReErrforsyntheticimageinFigure3-3. .............. 28 3-2TheSNRandReErrforsyntheticimageinFigure3-4. .............. 29 4-1TheSNRandReErrforthesyntheticimageinFigure4-1. ............ 43 4-2TheSNRandReErrforsyntheticimageinFigure4-2. .............. 43 4-3ThecomputingtimeforultrasoundthyroidimageinFigure4-3. ......... 43 4-4ThenumberofpixelsperpartitionforsyntheticimageinFigure4-4. ...... 45 6

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LISTOFFIGURES Figure page 3-1Thesegmentationresultsformodel(3-16)andmodel(3-7)foranMRIbrainimage. ......................................... 29 3-2Comparisononvarianceforasyntheticimage. .................. 30 3-3Comparisononq(x)resultsforasyntheticimage. ................ 30 3-4Segmentationresultsforazebraimage. ..................... 31 4-1Noiseremovalresultsfortheproposedmodel(4-9)andtheRLOmodel. .... 44 4-2Comparisononthevariableexponentq(x)forasyntheticimage. ........ 45 4-3UltrasoundthyroidimagedenoisingresultsfortheproposedmodelandtheRLOmodel. ...................................... 46 4-4Segmentationcomparisonresultsforourmodel(4-9)andmodel(4-7). ..... 47 4-5Segmentationcomparisonresultsforourmodel(4-9)andmodel(4-7). ..... 48 4-6Thyroidimagesegmentationresultsfortheproposedmodel(4-9)andmodel(4-7). ......................................... 49 7

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyVARIATIONALMODELSFORSIMULTANEOUSIMAGESEGMENTATIONANDNOISEREMOVALByIuliaMagdalenaPosircaDecember2012Chair:YunmeiChenMajor:Mathematics Wepresenttwoprojectsforsimultaneousimagesegmentationandnoiseremoval.TherstprojectconcernstheimagescorruptedwithGaussiannoiseandthesecondonewasdevelopedforimagescontaminatedwithmultiplicativenoise.Forbothmodelsweusesoftsegmentation,whichallowseachpixeltobelongtoeachimagepatternwithsomeprobability. Ourworkproposesalsoafunctionalwithvariableexponent,whichprovidesabetternoiseremovalwithfeaturepreserving.Thediffusionresultingfromtheproposedmodelsisacombinationbetweenthetotalvariation(TV)-basedandisotropicsmoothing.Tominimizethefunctionalenergy,weusetheEuler-Lagrangeequationsonthe(K-1)-simplexandthealternatingminimization(AM)algorithm.Theexperimentalandcomparisonresultswithsometraditionalmodelsshowtheefciencyofourwork,withimproveddenoisingandsegmentationofrealandsyntheticimages. 8

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CHAPTER1INTRODUCTION Imagesegmentationandnoiseremovalaretwoessentialstepsinimageprocessingandanalysis.Incomputervision,segmentationreferstotheprocessofpartitioningadigitalimageintotwoormoreclassesorphases.Thetaskofsegmentationistopartitionanimageintodisjointconnectedcomponentsthatarehomogeneouswithrespecttosomeimagefeatures,suchasintensityandtexture.Accordingtothesegmentationclassicationrequirementthatapixelcanbelongexclusivelytoonlyonephase,orapixelcanbelongtomorethanonephase,imagesegmentationmethodscanbedividedintotwomajorcategories:hardsegmentationsandsoftsegmentations. Manyapproacheshavebeenproposedtotacklethehardsegmentationproblem.Amongthemwecanmentiontheedgebasedmodelsandregionbasedmodels.Edgebasedsegmentationmodelsusetheinformationoftheimagegradientas,forinstance,inthegeodesicactivecontourmodelproposedbyCaselles,KimmelandSapiro[7].Regionbasedsegmentationmodelsseparatesregionsstatistically,byusingthemaximumlikelihoodestimation(MLE)[14,20]andthemaximumaposterioriestimation(MAP)[27,34].Thesemodelsincluderegiondescriptorssuchasmeansandvariances.TheMumford-Shahmodel[35]isthemostpopularandwidelyusedregionbasedmodelforsimultaneoussegmentationandnoiseremovalforimagescorruptedwithadditivenoise.AlongwiththeMumford-Shahmodel[35],othervariationalmethodshavebeenusedforimagesegmentationsuchasregioncompetition[53]andgeodesicactiveregion[36].Invariationalformulation,imagesegmentationisachievedbysolvinganenergyminimizationproblem,whichincludessomeusefulinformationsuchaspriorshapeandconstraintsontheregularityofobjectboundaries.Levelsettechniques[11,18,29]havebeenusedaspowerfultoolstoimplementvariationalmodels.Theadvantageofusingclassicallevelsetmethodsistheabilitytoexpressgeometricalquantitiesandmotions.Thedrawbackofusinglevelsetbasedmethods 9

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isthattheenergyfunctionalisnotconvexwithrespecttothelevelsetfunction,factwhichcanleadtoundesirablesolutions.Toovercomethisproblem,someconvexmodels[6,9,10,12,22,39]wereproposed.Thesemodelsarenotsensitivetoinitializationanymore,and,inplus,theglobalminimumisattained. Anotherproblemofhardsegmentationisnotonlytoovercomethenon-convexityoftheenergyfunctional,butalsoconcernsitsapplicationstosomerealworldproblems. Softsegmentationderivedfromanalysisofnaturalimages,wherethepatternsoftendonothaveclearcutboundarieshardboundariesbetweendifferentobjectsorpatterns.Forexample,magneticresonanceimages(MRI)containwhitematter,graymatterandcerebrospinaluid(CSF).Accuratedetectionoftheboundariesofdifferentmattersishelpfulfordiagnosingsomebraindiseases,suchasbraintumors.Unliketheclassicalhardsegmentation,softsegmentationassumesthateachpointoftheimagemaybelongtomorethanonepatternwithsomeprobability,calledownershipormembership.Softsegmentationapproachoffersmoreexibilityinmodelingandcombinedwithothermethods,suchasthevariational-PDEmethodscanleadtomorepowerfulmodelsandefcientalgorithms[2,12,13,26,30,38,45,47,51]. Alongwiththesegmentationtask,thenoiseremovalisakeystepinimageprocessing.Theadditivenoiseremovalproblems,suchasthePDE-basedvariationalmethods,havebeenstudiedextensivelyoverthelastdecades.TheseincludetheRudin-Osher-Fatemi(ROF)model[42]andLysaker-Lundervold-Tai(LLT)model[32].Inpaper[42],theauthorsintroducedthetotalvariation(TV)-baseddenoisingmethod,whichpreserveswellthesharpedgesofimagescontaminatedwithadditivenoise.Variousmodicationsofthe(ROF)modelhavebeenintroduced[4,8,15,16,19,28,37,48,52]sothatanadaptivesmoothingisperformed,whichpreserveswellnotonlytheedges,butalsotheinsidefeaturesofdifferentpatternsofthenoisyimages. Thenoiseremovalismoredifcultforimagescorruptedwithmultiplicativenoise.Thistypeofnoisecanbefoundinmanyrealworldimages,suchasSyntheticAperture 10

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Radar(SAR)images,laserimagesandmedicalultrasoundimages.Incomparisontoadditivenoise,themultiplicativenoisehasnotyetbeenstudiedcompletely.Asweknowsofar,thevariationalapproachwasproposedrstlybyRudin,LionsandOsher[43].Animportantmodel,whichcanbeappliedtothenon-texturedSARimages,isthevariationalmodelproposedbyAubertandAujol(AAmodel)[3].Recently,morevariationalmodelsweredevelopedtoremovemultiplicativenoise[19,23-25,28,29]. Inourwork[40,41],wedevelopedtwomodelsforsimultaneousimagesegmentationandnoiseremovalforimagescontaminatedwithadditivenoiseand,respectivelyforimageswithmultiplicativenoise.Therstmodelusesthestochastic-variationalapproachtosegmenttheimagescontaminatedwithGaussianadditivenoise.Foreachpattern,weusethestatisticaldescriptorssuchasmeansandvariances,andforbothmodels,weperformanadaptivesmoothingbyusingafunctionalwithvariableexponent.Wehavemadeadetailedpresentationofourmodels,algorithmsandnumericalresultsinchapters3and4.Themodelsweretestedonrealandsyntheticimageswithgoodresults. 11

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CHAPTER2THESEGMENTATIONPROBLEM 2.1DenitionandObjectives Imagesegmentationisaveryimportantstepinvisionmodelingandanalysis.Itspurposeistopartitionanimageintoitsconstituentspartscalledphasesorpartitions.Moreprecisely,imagesegmentationistheprocessofassigningalabeltoeverypixelinanimagesuchthatpixelswiththesamelabelsharesomevisualcharacteristicssuchasintensityandtexture.Adjacentregionsaresignicantlydifferentwithrespecttothesamecharacteristics.Insegmentation,themaindifcultyisthatoneneedstomanipulateobjectsofdifferentkinds:twodimensionaldomains,functions,andcurves. 2.2TheMumford-ShahModel Mathematically,thesegmentationproblem,asformulatedbyMumfordandShah[35],canbedenedasfollows: GivenR2anopen,bounded,smoothdomainandI:!Ranobservedimage,ndadecompositioniofandapiecewisesmoothapproximationuofIsuchthatuvariessmoothlywithineachi,anddiscontinuouslyacrosstheboundariesofi. Tosolvethisproblem,D.MumfordandJ.Shah[35]proposedtheminimizationoftheenergyfunctional: E(u)=Z(u)]TJ /F3 11.955 Tf 11.95 0 Td[(I)2+Z)]TJ /F6 7.97 Tf 6.58 0 Td[()]TJ /F2 11.955 Tf 7.31 10.8 Td[(jruj2+j)]TJ /F2 11.955 Tf 6.77 0 Td[(j(2)]TJ /F4 11.955 Tf 11.95 0 Td[(1) whereuisthesmoothversionofI,and)]TJ /F1 11.955 Tf 6.78 0 Td[(=)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(uisthesetofthediscontinuitypointsofu.Wecalltherstterm,thesecondterm,andthethirdtermoftherighthandsideofequation(2-1)thettingterm,thesmoothingtermandrespectively,thelengthterm.Thepositiveparametersandarexedparameterstoweightthedifferenttermsinthefunctionalenergy.Areducedcaseofthemodel(2-1)isobtainedbyrestrictingthesegmentedimageutopiecewiseconstantfunctions,thatisu=ciinsideeach 12

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componenti.Thentheproblemiscalledtheminimalpartitionproblemand,theauthorsproposeasimpliedmodelof(2-1): E0(u)=Z(I)]TJ /F3 11.955 Tf 11.95 0 Td[(ci)2+j)]TJ /F2 11.955 Tf 6.77 0 Td[(j(2)]TJ /F4 11.955 Tf 11.95 0 Td[(2) Theoreticalresultsofexistenceandregularityofminimizersof(2-1)and(2-2)arediscussedin[39].Inpractice,itisnoteasytominimizethefunctionals(2-1)and(2-2)becauseoftheunknownsetofedges)]TJ /F1 11.955 Tf 10.1 0 Td[(oflowerdimension,andbecausetheseproblemsarenotconvex. 2.3GeodesicActiveContoursandTheLevelSetMethods AlongwiththeMumford-Shahmodel[35],othervariationalmethodshavebeensuccessfullyusedforimagesegmentationsuchasregioncompetition[53],geodesicactivecontour[7],geodesicactiveregion[36].Thegeodesicactivecontoursareedgebasedsegmentationmodels,whichusetheinformationofimagegradient.Forexample,themodelproposedbyCaselles,KimmelandSapiro[7]minimizesthefollowingenergyfunctional: E(C)=Zl(C)0h(jrI(C(s))j)dswherehisanedgedetectorfunction,meaningthattheevolvingactivecontourCstopswhenitarrivesatedges. Invariationalformulation,imagesegmentationisachievedbysolvinganenergyminimizationproblem,whichincludessomeusefulinformationsuchaspriorshapeandconstraintsontheregularityofobjectboundaries. Levelsettechniques[11,18,29,49]havebeenusedaspowerfultoolstoimplementvariationalmodels.Forthepiecewisesmoothcase,thelevelsetminimizationproblemforthetwo-phasecasecanbeformulatedasfollows: Letu0:!Rbetheoriginalimagefunctiondenedonatwodimensionalbounded,openandsmoothdomain.DenotebyCthecontourwhichseparatesinto 13

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tworegions1and2.LettheLipschitzfunctionbeonelevelsetfunction,i.e.C=f(x,y)j(x,y)=0g,andu+,u)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(areC1functionssuchthatu(x,y)=u+(x,y)(H((x,y))+u)]TJ /F4 11.955 Tf 7.09 -4.34 Td[((x,y)(1)]TJ /F3 11.955 Tf 12.07 0 Td[(H((x,y))whereHistheHeavisidefunction.Theenergytobeminimizedisgivenbytheformula[49]:F(u+,u)]TJ /F4 11.955 Tf 7.08 -4.93 Td[(,)=Zj(u+)]TJ /F3 11.955 Tf 11.95 0 Td[(u0)j2H()+Zj(u)]TJ /F2 11.955 Tf 9.74 -4.93 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(u0)j2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(H())+Zjru+j2H()+Zjru)]TJ /F2 11.955 Tf 7.08 -4.94 Td[(j2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(H())+ZjrH()j(2)]TJ /F4 11.955 Tf 11.95 0 Td[(3) Forthepiecewiseconstantcase,thereducedMumford-ShahmodelcanbewrittenasEquation(2-3)bydroppingthesmoothingterm[49].Formultiphasesegmentation,underthesameassumptionsforandu0,theminimizationproblemcanbeformulatedasfollows: LetCbeaclosedsubsetofconsistingofanitesetofsmoothcurveswhichpartitionthedomainoftheimageu0intomregionsfig,i=1,2,...m.Eachregionicanberepresentedbyalevelsetfunctioni,i=1,2,...,m,,i.e.i=f(x,y)ji(x,y)>0gwherem=log2nandnisthenumberofpartitions.Denotebycj,j=1,2,...,nthemeanoftheoriginalimageu0foreachphasej,j=1,2,...,n. Thentheenergyfunctionalcanbeexpressedbytheequality[49]:Fn(cj,i)=X1jn=2mZ(u0)]TJ /F3 11.955 Tf 11.95 0 Td[(cj)2j+X1imZjrH(i)j(2)]TJ /F4 11.955 Tf 11.95 0 Td[(4) wherejisthecharacteristicfunctionforeachclassorphasej,j=1,2,...,n. AnotherapproachtolevelsetmethodswasdonebyChungandVeseusingmultiplelayersofasinglelevelsetfunction[18].Themainideaistorepresentthediscontinuitysetofu0usingmorethanonelevel-lineoftheLipschitzcontinousfunction.Thesegmentationproblemcanbemodeledastheminimizationoftheenergyfunctional: F(c1,c2..cm+1,)=Zj(u0(x))]TJ /F3 11.955 Tf 11.96 0 Td[(c1)j2H(l1)]TJ /F9 11.955 Tf 11.96 0 Td[((x))dx 14

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+mXi=2Zj(u0(x))]TJ /F3 11.955 Tf 11.96 0 Td[(ci)j2H((x))]TJ /F3 11.955 Tf 11.96 0 Td[(li)]TJ /F6 7.97 Tf 6.59 0 Td[(1)H(li)]TJ /F9 11.955 Tf 11.95 0 Td[((x))dx+Zj(u0(x))]TJ /F3 11.955 Tf 11.95 0 Td[(cm+1)j2H((x))]TJ /F3 11.955 Tf 11.96 0 Td[(lm)dx+mXi=1ZjrH((x))]TJ /F3 11.955 Tf 11.96 0 Td[(li)jdx(2)]TJ /F4 11.955 Tf 11.95 0 Td[(5) wherel1
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regioninside)]TJ /F1 11.955 Tf 10.09 0 Td[(aremeasuredbyj)]TJ /F2 11.955 Tf 6.77 0 Td[(j=RjrH()jandjAj=RjH()jrespectively.TheunitnormalNandthecurvaturekto)]TJ /F1 11.955 Tf 10.1 0 Td[(aredenedby:N=r jrjandk=divN.Thedrawbackofusinglevelsetbasedmethodsisthattheenergyfunctionalisnotconvexwithrespecttothelevelsetfunctionandleadstoundesirablesolutions.Thisproblembecomesmoredifcultwhenwehavetohandlemultiphasesegmentation. Toovercometheissueofnonconvexityoftheenergyfunctional,severalvariationalmodelsweredeveloped[5,6,9,10,12,22].Thenewfunctionalsareconvexwithrespecttothemembershipfunctionandtherefore,theglobalminimumcanbeachieved.Inplus,thesemodelsarenotsensitivetoinitialization. 16

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CHAPTER3ASTOCHASTIC-VARIATIONALAPPROACHFORSOFTMUMFORD-SHAHSEGMENTATION 3.1SoftSegmentationvs.HardSegmentation Softsegmentationwasmotivatedbythenon-convexityofthehardsegmentationmodels,whichleadstoundesirablesolutions,andalsoderivedfromthepracticalanalysisofrealimages,wherethepatternsoftendonothaveclearcutboundaries.Unliketheclassicalhardsegmentation,softsegmentationassumesthateachpointoftheimagemaybelongtomorethanonepatternwithsomeprobability,calledownership(ormembership)intheliteratures.Softsegmentationapproachoffersmoreexibilityinmodelingandcombinedwithothermethods,suchasthevariational-PDEmethodscanleadtomorepowerfulmodelsandefcientalgorithms. 3.2ExistingModels Intherecentyears,thesoftsegmentationproblemhasbeenstudiedextensively[2,6,12,22,30,38,39,45]usingdifferentmethods.OneoftheearlydevelopedsoftsegmentationmethodiscalledfuzzyC-meansegmentation.ItusesfuzzyC-meansegmentationmethod(FCM)clustering[17,38].Mathematically,thestandardFCMobjectivefunctionofpartitioningadatasetfxjg,j=1,...,KintocclustersisgivenbycXi=1KXj=1umijjjxj)]TJ /F3 11.955 Tf 11.95 0 Td[(vijj(3)]TJ /F4 11.955 Tf 11.95 0 Td[(1) wherejj.jjistheEuclidiannormandvi,i=1,...,Kistheclustercenter.TheoriginalFCMmethodisverysensitivetonoise.AnadaptivefuzzyC-meanmethod(AFCM)wasproposedbyPhamandPrince[38],wheretheconstantclustercentersusedintheFCMmodelaresubstitutedbyspatiallyvaryingfunctionstoimposelocalspatialhomogeneity.In[38],theauthorsalsousedakernelversionofFCMtogetherwithanon-Euclideandistanceintheobjectivefunction. AnimportantworkwasdonebyBrown,ChanandBresson[6].TheirmethodreliesonthelevelsetframeworkproposedbyLieetal[31]andtheconvexicationapproach 17

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ofPocketal.[39].UsingthedualformulationoftheTVnormwritteninthepiecewiseconstantlevelsetframeworkallowstheirmethodtoobtainaconvexproblemfromwhichaglobalsolutionmaybecomputed.WebrieyreviewtheimportantmethodofPocketal.[39].TheauthorsconsiderthevariationalproblemminuE(u)=Zjru(x)jdx+Z(u(x),x)dx(3)]TJ /F4 11.955 Tf 11.95 0 Td[(2) where)-330(=[min,max],u:!)]TJ /F1 11.955 Tf 6.78 0 Td[(,andmaybeanonconvexfunction.Ifischosentobethedelityterm,forexample,(u,I)=(u)]TJ /F3 11.955 Tf 11.95 0 Td[(I)2 whereuisrecoveredimagefromthegivennoisyinputimageI,thenthemodel(3-2)canbeusedforsoftimagesegmentation.Denethesuperlevelsetsfunctionofu(x,)=1u>(x) Usingthecoareaformula[39],problem(3-2)canbeconvertedtoaconvexproblem:min2DE()=Zjrj+j@jd(3)]TJ /F4 11.955 Tf 11.95 0 Td[(3) where=x)]TJ /F1 11.955 Tf 10.09 0 Td[(andD=f:![0,1]j(x,min)=1,(x,max)=0g Usingtheapproachmentionedabove,Brownetal.[6]obtainedaglobalsolutiontothemultiphasesegmentationproblem,(alsoknownasthePottsmodel)inf1,...,K(KXi=1j@ij+Zigi(x)dx)(3)]TJ /F4 11.955 Tf 11.95 0 Td[(4) where=1[...[Kandgi(x)aretheregionsdescriptors,assumedtobeknownbeforehand.Forexample,ifgi(x)=(I(x))]TJ /F3 11.955 Tf 12.58 0 Td[(ci)2,whereciisthemeanoftheoriginalnoisyimageI:!Rforeachphasei,i=1,...,K,thenwehavethepiecewiseconstantMumford-Shahmodel. 18

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In[45],theauthorproposedastochastic-variationalmodelforsoftsegmentation.Wewillbrieyreviewthemainideaoftheworkpresentedin[45].LetI:!Rbeanimagedenedonanopen,bounded,smoothdomainR2.DenotebyithesupportofeachpatternIi=Iji,i=1,...,Kand)]TJ /F1 11.955 Tf 10.1 0 Td[(thesetofedges,i.e.n)-349(=[Ki=1i.Inthecaseofhardsegmentation,wehave:1(x)=PKi=11i(x). Denes:!f1,...,Kgthepatternlabelingandpi(x)=Prob(s(x)=i),i=1,...,Ktheprobabilitythatthepixelxfrombelongstotheregioni,withthesimplexconstraints:KXi=1pi(x)=1,0pi(x)1,i=1,...,K(3)]TJ /F4 11.955 Tf 11.95 0 Td[(5) Asoftsegmentationamountstoasofterpartitionoftheunit:1(x)=PKi=1pi(x)andthepatternsoftheimagearedenedbyi=fx2js(x)=igwheres(x)=argmaxs2f1,2,...,Kgps(x). Denotebyui,i=1,...,Kthemeaneldsofthepatterns,P(x)=(p1(x),.....pk(x))andU(x)=(u1(x),.....uk(x)).Theauthoralsoassumesthatthepatternssharethesamevariance2.Then,foreachpattern,thepixelintensityfI(x)jx2gisanindependentrandomvariable,indexedbyxdistributedwithGaussiandistributionofmeanui(x)andvariance2. Usingthebayesianformula[27,34],theposteriorprobabilitycanbeexpressedasaproductbetweentheconditionalprobabilityandthepriorprobabilitiesas:Prob(P,UjI)=Prob(IjP,U)Prob(P)Prob(U)=Prob(I) assumingthatthemixturepatternsUandPareindependent.BytakingthelogarithmiclikelihoodE[P,UjI]=)]TJ /F3 11.955 Tf 9.3 0 Td[(logProb(P,UjI)thesoftsegmentationproblemcanbewrittenas: minfE[P,UjI]=E[IjP,U]+E[U]+E[P]g(3)]TJ /F4 11.955 Tf 11.95 0 Td[(6) 19

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moduloaninsignicantconstant. Furthermore,theenergyfunctionaltobeminimizedcanbeexpressedas:E[P,UjI]=KXi=1Zjrui(x)j2+KXi=1Z(I(x))]TJ /F3 11.955 Tf 11.96 0 Td[(ui(x))2pi(x) 22+KXi=1Z9jrpi(x)j2+(pi(x)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pi(x))2 (3)]TJ /F4 11.955 Tf 11.95 0 Td[(7) ThethreetermsintherighthandsideoftheEquation(3-7)arethesmoothing,ttingandrespectively,thelengthterm(accordingtothe)]TJ /F1 11.955 Tf 10.1 0 Td[(-convergencetheory[33]).Thenalgoaloftheworkpresentedin[45]istominimizetheenergydenedbytherelation(3-7). Ourrstwork[40]isageneralizationofthemodel(3-7)inthefollowingaspects: 1)foreachpattern,thepixelintensityfI(x)jx2gisanindependentrandomvariable,withGaussiandistributionofmeanui(x)andvariance2i.Thevariancevariesfromonepatterntoanother,andthemeanisspatiallyvaryingdependingoneachpixeloftheimage.Inthisway,themodelbecomesmorerobusttonoise. 2)thedenoisingpartoftheproposedenergyfunctionalcontainsavariableexponentdenedasfollows: q(x)=8><>:1+1 1+jrG~I(x)j2ifjru(x)j1ifjru(x)j>(3)]TJ /F4 11.955 Tf 11.95 0 Td[(8) whereu(x)=KXi=1ui(x)pi(x),x2(3)]TJ /F4 11.955 Tf 11.95 0 Td[(9) isthereconstructedimage.Parameters,>0arexedandG~isaGaussianfunction. Using2)willgivethemodelthefollowingbenets:a)itensuresTVbaseddiffusion(q(x)=1)alongedgesandGaussiansmoothing(q(x)=2)inhomogenousregions.and,b)itemploysanisotropicdiffusion(1
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3.3ProposedModel Inthissection,wewillmakeadetailedpresentationoftheproposedwork.LetI:!RbeanimagedenedonarectangleR2.SupposetheimagecontainsKpatterns.Further,weassumethatforeachpixelx2:a)I(x)ands(x)arerandomvariables;b)Prob(s(x)=i)denotedbypi(x)istheprobabilitythatthepixelxbelongstothepatterniandc)theprobabilitydensityfunctionofI(x)giventhatxbelongstothei-thpatternisGaussianwithmeaneldui(x)andvariance2i,i.e. Prob(I(x)js(x)=i)=1=p 2iexp)]TJ /F4 11.955 Tf 10.49 8.08 Td[((I(x))]TJ /F3 11.955 Tf 11.95 0 Td[(ui(x))2 22i Forsimplicity,denoteProb(I(x)js(x)=i)byG(ui(x),i)foreachi=1,2,...,K.Thentheprobabilitydensityfunction(pdf)ofthepatternsmixtureimageIatanypixelxisgivenbytheformulaP(I(x)jU(x),P(x))=KXi=1G(ui(x),i)pi(x)(3)]TJ /F4 11.955 Tf 11.96 0 Td[(10) Thelikelihoodorthejointprobabilitydensityfunction(inthediscreteform)canbewrittenasYx2P(I(x)jU(x),P(x))=Yx2KXi=1G(ui(x),i)pi(x)(3)]TJ /F4 11.955 Tf 11.96 0 Td[(11) Theoptimalenergyforthemixturemodelisobtainedfromthenegativelog-likelihoodE[IjP,U]=)]TJ /F9 11.955 Tf 9.29 0 Td[(Zlog KXi=1G(ui(x),i)pi(x)!(3)]TJ /F4 11.955 Tf 11.96 0 Td[(12) forsome>0,providedthat(I(x)jP,U)isindependentof(I(y)jP,U)foranytwodistinctpixelsxandy.Theenergygivenintheformula(3-12)canbeapproximatedbyE[IjP,U]=KXi=1Zlog(p 2i)+(I(x))]TJ /F3 11.955 Tf 11.95 0 Td[(ui(x))2 22ipi(x)(3)]TJ /F4 11.955 Tf 11.96 0 Td[(13) withthesimplexconstraints(3-5)andforsomeconstant>0.Toobtaintherelation(3-13),weassumedthateachsoftownershippi(x)isclosertoahardonepi(x)' 21

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1i(x),fori=1,...,K.Thereforewecanwrite)]TJ /F3 11.955 Tf 9.3 0 Td[(log KXi=1G(ui(x),i)pi(x)!')]TJ /F3 11.955 Tf 21.92 0 Td[(log KXi=1G(ui(x),i)1i(x)!=)]TJ /F7 7.97 Tf 16.47 14.94 Td[(KXi=1(log(G(ui(x),i))1i(x)')]TJ /F7 7.97 Tf 29.09 14.95 Td[(KXi=1(log(G(ui(x),i))pi(x)=KXi=1(I(x))]TJ /F3 11.955 Tf 11.96 0 Td[(ui(x))2 22ipi(x)+1 2log(22i)pi(x) Toobtainthelengthtermofourenergymodel,weimposetwoconstraints[1,45]:a)eachprobabilitypi(x)hasatmosttwophases,i.e.pi(x)isclosetoeither0or1.b)thesoftboundariesareregular.Tocombinethesetwoconditions,weneedtousetheModica-Mortolatypeofenergy[33,46]with<<1controllingthetransitionbandwidthE[,pi]=Z9jrpi(x)j2+(pi(x)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pi(x)))2 ,i=1,...,K(3)]TJ /F4 11.955 Tf 11.96 0 Td[(14) Usingthe)]TJ /F1 11.955 Tf 6.78 0 Td[(-convergencetheory[1],itfollowsthatfor!0,E[,pi])166(!length(\,where)]TJ /F1 11.955 Tf 10.09 0 Td[(isthesetofedges,thatisn)-362(=[Ki=1i.WereferthereadertoModicaandMortola[33]foraproof(withsomeadequatemodication). Incorporatingallthreeenergyterms(tting,smoothing,length)weobtainedanewgeneralizedenergyfunctionaltobeminimized,withdifferentmeansandvariances,forthesoftMumford-ShahsegmentationwithKphases:E[P,UjI]=KXi=1Zjrui(x)j2+KXi=1Z(I(x))]TJ /F3 11.955 Tf 11.96 0 Td[(ui(x))2 22ipi(x)+1 2Zlog(22i)pi(x)+KXi=1Z9jrpi(x)j2+(pi(x)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pi(x))2 ,i=1,...,K(3)]TJ /F4 11.955 Tf 11.96 0 Td[(15) where=(1,...,K).TheenergytermE[U]=PKi=1Rjrui(x)j2whereisanadditiveconstant.Ourmodelalsousesafunctionalwithvariableexponentq(x)dened 22

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asin(3-8).Theenergyfunctionalbecomes:E[P,UjI]=KXi=1Zjrui(x)jq(x)+KXi=1Z(I(x))]TJ /F3 11.955 Tf 11.96 0 Td[(ui(x))2 22ipi(x)+1 2Zlog(22i)pi(x)+KXi=1Z9jrpi(x)j2+(pi(x)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pi(x)))2 ,i=1,...,K(3)]TJ /F4 11.955 Tf 11.96 0 Td[(16) Tosolve(3-16)wecomputetheEuler-Lagrangeequationsassociatedwiththisproblem. DeneV=(V1,...,VK)andv=(v1,...,vK).Then,withoutthesimplexconstraintonthemembershipP,foranygivenU,therstordervariationoftheenergyEwithrespecttoPisgivenby@E @fP=ZKXi=1Vidx+Z@KXi=1vidS(3)]TJ /F4 11.955 Tf 11.96 0 Td[(17) andVi=)]TJ /F4 11.955 Tf 9.3 0 Td[(18pi(x)+2 pi(x)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pi(x))(1)]TJ /F4 11.955 Tf 11.96 0 Td[(2pi(x))+(ui(x))]TJ /F3 11.955 Tf 11.95 0 Td[(I(x))2 22i+1 2log(22i),x2(3)]TJ /F4 11.955 Tf 11.96 0 Td[(18)vi=18@pi(x) @n,x2@(3)]TJ /F4 11.955 Tf 11.96 0 Td[(19) Therelation(3-17)canbewrittenintheform@E @fP=Vj+vj@(3)]TJ /F4 11.955 Tf 11.96 0 Td[(20) Infact,Pbelongstothe(K)]TJ /F4 11.955 Tf 11.95 0 Td[(1)-simplex.Considertheorthogonalprojection:TPRK!TPK)]TJ /F6 7.97 Tf 6.58 0 Td[(1(3)]TJ /F4 11.955 Tf 11.96 0 Td[(21) Foranyt2TPRK,(t)=t)]TJ /F4 11.955 Tf 13.15 8.08 Td[(1K K=t)]TJ /F9 11.955 Tf 12.62 0 Td[(1K(3)]TJ /F4 11.955 Tf 11.96 0 Td[(22) 23

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where=PKi=1ti Kand1K p K=(1,...,1) p Kisthenormaldirectionofthetangentplane.Therefore,theconstrainedgradientofEonthe(K)]TJ /F4 11.955 Tf 11.95 0 Td[(1)-simplexisgivenby@E @P=@E @fP=(V)]TJ /F9 11.955 Tf 12.62 0 Td[(1K)j+(v)]TJ /F9 11.955 Tf 12.62 0 Td[(1K)j@(3)]TJ /F4 11.955 Tf 11.96 0 Td[(23) andtheEuler-LagrangesystemofequationsonP,givenUisasfollows:Vi(x)=,x2(3)]TJ /F4 11.955 Tf 11.96 0 Td[(24)vi(x)=,x2@(3)]TJ /F4 11.955 Tf 11.96 0 Td[(25) whereViandviaregivenintherelations(3-18)and(3-19). TheEuler-Lagrangeequationforui(x)givenpi(x)is:div(q(x)jrui(x)jq(x))]TJ /F6 7.97 Tf 6.58 0 Td[(2rui(x)))]TJ /F9 11.955 Tf 11.96 0 Td[((ui(x))]TJ /F3 11.955 Tf 11.96 0 Td[(I(x)) 2ipi(x)=0,x2(3)]TJ /F4 11.955 Tf 11.96 0 Td[(26)q(x)jrui(x)jq(x))]TJ /F6 7.97 Tf 6.59 0 Td[(2@ui(x) @nj@=0,x2@(3)]TJ /F4 11.955 Tf 11.96 0 Td[(27) TheEuler-Lagrangeequationfor2iisgivenby:2i=Z(ui(x))]TJ /F3 11.955 Tf 11.95 0 Td[(I(x))2pi(x)=Zpi(x),(3)]TJ /F4 11.955 Tf 11.96 0 Td[(28) 3.4Algorithm Tondanoptimalsolution(U,P,)toproblem(3-16),weusedthealternatingminimization(AM)algorithm,whichisprogressive,i.eforeachstep(n+1),giventhepatternsUn=(uni;i=1,2,...,K)andtheownershipPn=(pni;i=1,2,...,K)ndPn+1=argminPE[PjUn,I](3)]TJ /F4 11.955 Tf 11.96 0 Td[(29) orequivalentlysolvethefollowingowequation:d(pi) dt=Lpi(I,ui,pi,i)(3)]TJ /F4 11.955 Tf 11.96 0 Td[(30) 24

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withLpi(I,ui,pi,i)=)]TJ /F4 11.955 Tf 9.3 0 Td[(18pi+2)]TJ /F6 7.97 Tf 6.59 0 Td[(1pi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pi)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(2pi))]TJ /F9 11.955 Tf 12.62 0 Td[(+(I)]TJ /F3 11.955 Tf 11.95 0 Td[(ui)2 22i+ 2log(22i))(3)]TJ /F4 11.955 Tf 11.96 0 Td[(31) and=1 KKXi=1)]TJ /F4 11.955 Tf 9.29 0 Td[(18pi+2 pi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pi)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(2pi)+(ui)]TJ /F3 11.955 Tf 11.95 0 Td[(I)2 22i+ 2log(22i)= KKXi=1(I)]TJ /F3 11.955 Tf 11.96 0 Td[(ui)2 22i+ 2KKXi=1log(22i)+2 KKXi=1(2p3i)]TJ /F4 11.955 Tf 11.96 0 Td[(3p2i)+2 K(3)]TJ /F4 11.955 Tf 11.96 0 Td[(32) since(KXi=1pi)=0(3)]TJ /F4 11.955 Tf 11.96 0 Td[(33) Following,basedonPnandUn,tondtheoptimalestimationUn+1=argminUE[UjPn,I](3)]TJ /F4 11.955 Tf 11.96 0 Td[(34) isequivalenttosolvetheassociatedowequation:d(ui) dt=Lui(I,ui,pi,i)(3)]TJ /F4 11.955 Tf 11.96 0 Td[(35) whereLui=div(q(x)jrui(x)jq(x))]TJ /F6 7.97 Tf 6.59 0 Td[(2rui(x)))]TJ /F9 11.955 Tf 11.95 0 Td[((ui(x))]TJ /F3 11.955 Tf 11.95 0 Td[(I(x)) 2ipi(x)(3)]TJ /F4 11.955 Tf 11.96 0 Td[(36) andq(x)isgiveninrelation(3-8). Fromtherelation(3-28),foreachstepn,giventhepatternsUn=(uni;i=1,2,...,K)andthemembershipPn=(pni;i=1,2,...,K),ndtheoptimaln=(ni;i=1,2,...,K). Summarizing,theupdating...(Un,Pn,n)!(Un+1,Pn+1,n+1)...isobtainedbysolvingthefollowingsystemofequations:pn+1i=pni+dtpLpi(I,uni,pni,ni) 25

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un+1i=uni+dtuLui(I,uni,pni,ni)n+1i=ni+dtsLi(I,uni,pni,ni)(3)]TJ /F4 11.955 Tf 11.96 0 Td[(37) whereLi=)]TJ /F9 11.955 Tf 9.3 0 Td[(Zpi(I)]TJ /F3 11.955 Tf 11.95 0 Td[(ui)2+2iZpi(3)]TJ /F4 11.955 Tf 11.96 0 Td[(38) anddtp,dtuanddtsarestepssizes,Lpi,Luiaredenedin(3-31)and(3-36). 3.5Experiments Toshowtheeffectivenessoftheproposedsegmentationmodel,fourexperimentsarereportedasfollows.Allthesimulationsaremadeonimageswhichcanbepartitionedintothreephases.Themodel(3-7)presentedin[45]isaspecialcaseoftheproposedmodel(3-16),whereq(x)=2andandthevariance2isaxedconstantforeachpartition.Formodel(3-16),thevariance2i,i=1,2,3couldtakedifferentvaluesanditisoptimizedduringtheimplementation. Figure3-1isacomparisonbetweenmodel(3-7)andourmodel(3-16)usinganMRIbrainimage.Figure3-1Aisthegivennoisyimage,contaminatedwithgaussiannoisewithmeanzeroandvariance2=0.005.Figure3-1B,Crepresentthesegmentationresultu(x)usingourmodel(3-16),respectivelymodel(3-7).InFigure3-1D,E,Farerepresentedthemeaneldsu1(x),u2(x),u3(x)ofthepatternsfortheproposedmodel(3-16)andonthethirdrow,inFigure3-1G,H,Iarethecorrespondingprobabilitiesp1(x),p2(x),p3(x).ThesoftMumford-Shahsegmentationresultu(x)showninFigure3-1B,Cisgivenbytheformula(3-9).Theresultsareobtainedafter100iterationsusingasparameters=3,=0.4,=0.01,=0.3,=0.1.Theoptimizedvariancesfortheproposedmodelobtainedafter100iterations,are:21=0.38265,22=0.56573,23=0.73654. Theproposedmodel(3-16)offersabettersegmentationresultthanmodel(3-7),whichweseethatitcontainsmorespuriousdots.Ofcourse,wecandecreasethespuriousdotsbychoosingbiggersmoothingcoefcient,butthentheedgesofthe 26

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partitionswillbeseriouslydamaged.Forourmodel,thevariance2ivariesfromonepatterntoanother,andthisfactpreventsthespuriousdotstoappear.Inthisway,themodelbecomesmorerobusttonoise. AsimilarcomparisonisshowninFigure3-2.Weexploretheinuenceofvarianceforasyntheticimagecorruptedwithgaussiannoisewithmeanzeroandvariance2=0.01.Thepurposeofthisexperimentistocomparetheresultsofourmodel(3-16),whenthevarianceisavaryingfunctionwhichneedstobeoptimizedwiththecaseofaconstantxedvariance.AsitcanbeseeninFigure3-2C,thesegmentationresultofthemodel(3-7)willgiveusanimagecontainingmorespuriousdots.Theoptimizedvariancesforourmodel,obtainedafter100iterationsare:21=0.38282,22=0.56057,23=0.73617. Figure3-3showsacomparisononq(x)forasyntheticimagecontaminatedwithgaussiannoisewithmeanzeroandvariance2=0.01.Wecomparetheresultsofourmodel(3-16),whenq(x)isavaryingfunctionontheinterval[1,2]withthecasewhenq(x)isaconstant,eitheroneortwo.Inourmodel,q(x)variesfrompixeltopixelandfromiterationtoiterationandprovidebetterresultsthanifq(x)isxedanddoesnotvaryduringtheimplementation.Toillustratethisfact,weconsideredtwocasesforq(x)constant:1)q(x)=1and2)q(x)=2.Incase1),Figure3-3D,thedenoisingismoreslowerandtakesmoreiterationsinordertoobtainasatisfactoryresult.Incase2),Figure3-3E,thedenoisingisfasterandleadstoalossoftheimagedetails.AsitcanbeseeninFigure3-3C,ourmodeloffersabettersegmentation,beingacombinationoftheTV-basedandisotropicsmoothing.FortheExperiments2and3,weusedthesamesetofparametersasfortheExperiment1.Theoptimizedvariancesforourmodel,obtainedafter100interationsare:21=0.38303,22=0.56075,23=0.73461. IntheExperiment4,wetesttheabilityofourmodeltoperformwellonanimagewithtexture.Wecompareourmodel(3-16)withmodel(3-7)usingasyntheticimagewithzebratexture,corruptedwithGaussiannoiseofmeanzeroanddifferentvariance 27

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oneachpartition.TheimageinExperiment4wascreatedbyoverlappingacircularsectionfromazebraimageonagraycircle,themeanintensityofthezebrasectionbeingequaltotheintensityofthegraycircle.Then,differentGaussiannoisewasaddedonallthreepartitions(black,gray,andzebra)withthesamemeanzeroandvariances0.0005,0.02and0.001.Fromgure3-4C,wecanseethattheproposedmodel(3-16)givesbetterresultsthanmodel(3-7)whichusesonlytheL2-normforthettingterm.Forourmodel,thevarianceisoptimizedduringthesegmentationprocessandgivesabettersegmentationresultthanifthevarianceisaxedconstantduringtheimplementation.Itcanbeseenthatsomespuriousdotsappear,whichcannotbesegmentedusingthemodel(3-7).Weincludedtwosegmentationresultsusingthemodel(3-7)with500,respectively1500iterations.AsitcanbeseeninFigure3-4D,Eforthemodel(3-7),ifweincreasethedegreeofsmoothing,thezebraimageisblurredbeforetheeliminationofthenoise.Tocomparethemodels,weusedthefollowingparameters:=6,=0.4,=0.01,=0.1,=0.05.Theoptimizedvariancesobtainedforourmodel,after500iterationsare:21=0.32075,22=0.45748,23=0.55782.WehavetestedtheperformanceofourmodelbycomputingthesignaltonoiseratioandtherelativeerrorforExperiments3and4.WedenotedbyIctheoriginalcleanimageandbyuthereconstructedimage.Withthisnotations,wedenedthesignaltonoiseratio(SNR)andtherelativeerror(ReErr)asfollows[21]: SNR=10log10jjIcjj22 jju)]TJ /F3 11.955 Tf 11.95 0 Td[(Icjj22,ReErr=jju)]TJ /F3 11.955 Tf 11.96 0 Td[(Icjj22 jjIcjj22(3)]TJ /F4 11.955 Tf 11.96 0 Td[(39) FromthecomputationalresultsandnumericalcomparisonresultsshowninTable3-1andTable3-2,weconcludethatourmodel(3-16)performsabetterdenoisingthanmodel(3-7). Table3-1. TheSNRandReErrforsyntheticimageinFigure3-3. q(x)variableq(x)=1q(x)=2 SNR24.1320.7615.46ReErr0.000010.000070.00080 28

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A B C D E F G H I Figure3-1. Thesegmentationresultsformodel(3-16)andmodel(3-7)foranMRIbrainimage.A)Givennoisybrainimage.B)Segmentationresultusingtheproposedmodel(3-16).C)Segmentationresultusingmodel(3-7).D),E),F)Themeaneldsu1(x),u2(x),u3(x)ofthepatternsformodel(3-16).G),H),I)Thecorrespondingprobabilitiesp1(x),p2(x),p3(x)formodel(3-16). Table3-2. TheSNRandReErrforsyntheticimageinFigure3-4. model(3-16)model(3-7),iter=500model(3-7),iter=1500 SNR17.698.104.04ReErr0.000280.023950.15558 3.6Conclusions Acriticalprobleminenergybasedmulti-phasesegmentationisthenon-convexityofthefunctionalenergy.Levelsetmethodshavebeensuccessfullyusedinmulti-phasesegmentation,buttheyfailtoworkforsoftsegmentationduetooverlappingandnoclear 29

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A B C Figure3-2. Comparisononvarianceforasyntheticimage.A)Syntheticnoisyimage.B)Segmentationresultwithvarying2i.C)Segmentationresultusing2ixedconstant. A B C D E Figure3-3. Comparisononq(x)resultsforasyntheticimage.A)Initialcleanimage.B)ImagecorruptedwithGaussiannoise.C)Thereconstructedimageresultusingq(x)variable.D)Thereconstructedimageresultusingq(x)=1.E)Segmentationresultusingq(x)=2. 30

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A B C D E F G H I J K Figure3-4. Segmentationresultsforazebraimage.A)Initialcleanimage.B)ImagecorruptedwithdifferentGaussiannoise.C)Segmentationresultusingproposedmodel(3-16).D),E)Segmentationresultusingmodel(3-7)at500,respectively1500iterations.F),G),H)Themeaneldsu1(x),u2(x),u3(x)ofthepatternsformodel(3-16).I),J),K)Thecorrespondingprobabilitiesp1(x),p2(x),p3(x)formodel(3-16). 31

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boundariesbetweendifferentphases.ThecombinationofthestochasticapproachesandthevariationalPDEmethodscanleadtomorepowerfulmodelsandefcientalgorithms. Ourwork[40]isageneralizationofShen'smodel(3-7)[45]usingMumford-Shahsoftsegmentation.Weassumethatineachpattern,thepixelintensityisarandomvariableindependentandidenticallydistributed(i.i.d.)fromaGaussiandistributionofmeanui(x)andvariance2i.Thevariancevariesfromonepatterntoanother,andthemeanisspatiallyvaryingdependingoneachpixeloftheimage.Inthisway,themodelbecomesmorerobusttonoise.Inaddition,theenergyfunctionalhasvariableexponentwhichensuresTVbaseddiffusionalongedgesandGaussiansmoothinginhomogenousregions,anditemploysanisotropicdiffusioninregionsinwhichthedifferencebetweennoiseandedgesisdifculttodistinguish.Theproposedframeworkcanbeappliedtorealandsyntheticimageswithgoodresults. 32

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CHAPTER4SEGMENTATIONANDDENOISINGOFIMAGESWITHMULTIPLICATIVENOISE 4.1Introduction Itiswellknownthatmultiplicativenoisesarefoundinmanyrealimages,suchasmedicalultrasoundimages,laserimagesandSARimages.Incomparisonwiththeadditivenoises,themultiplicativenoisesaremoredifculttoberemovedfromthecontaminatedimagebecauseoftheirmultiplicativenature.Thevariationalapproachtodenoisetheimagescontaminatedwithmultiplicative(speckle)noisewasproposedrstlybyRudin,LionsandOsher[43].Animportantmodel,whichcanbeappliedtothenon-texturedSARimagescorruptedbyGammanoise,isthevariationalmodeldevelopedbyAubertandAujol(AAmodel)[3].Recently,severalvariationalapproachesaredevotedtothemultiplicativenoiseremoval[19,23-25].Also,thesegmentationofimagescorruptedwithmultiplicativenoisehasnotbeenstudiedextensively.ManyvariationalmodelsweredevelopedtosegmentimagescontaminatedwithadditivenoisesuchastheMumford-Shahmodel[35],regioncompetitionmodel[53],geodesicactivecontour[7],andgeodesicactiveregion[36].However,ultrasoundimagesegmentationismoredifcultduetothespecklenoise.Inplus,ultrasoundimageshavepoorsignaltonoiseratioandhigherinhomogeneity. Thereareseveralmodelsdevelopedtosegmentimageswithmultiplicativenoisesuchasthepiecewiseconstantmodelandthepiecewisesmoothmodelintroducedandanalyzedinpaper[29].In[44],theauthorsdevelopedaregionbasedactivecontourmodelusingtheMLEapproach.ThemodelassumesthattheintensitiesofultrasoundimagesfollowtheRayleighdistribution.Thesolutionforthemodelsfrom[29]and[44]wasobtainedbyusingthelevelsetmethods.Anotherapproachforultrasoundimagesegmentationispresentedinpaper[22],wheretheauthorspresentsavariationalmodelbasedonFisher-Tippettdistribution. 33

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4.2ExistingModels Noiseremovalandimagesegmentationaretwokeystepsinimagevisionmodellingandanalysis.Multiplicativedenoisingproblemshavereceivedmoreattentionintherecentyears.Inamultiplicativenoisemodel,agivenimageIdenedonarectangleR2,isthemultiplicationofanoriginalcleanimageuandanoisen:I=un Withoutlossofgenerality,wecanassumethatuandnarepositiveinthenoisemodel.Itiswellknownthatmultiplicativenoisesarefoundinmanyrealworldimageprocessingapplications,suchasSARimages,laserimagesandmedicalultrasoundimages.Unlikeadditivenoises,themultiplicativeonesaremoredifculttoberemovedfromthecorruptedimagesbecauseoftheirmultiplicativenature.Theadditivenoiseremovalproblems,suchasthePDE-basedvariationalmethods,havebeenstudiedextensivelyoverthelastdecades.TheseincludetheRudin-Osher-Fatemi(ROF)model[42]andLysaker-Lundervold-Tai(LLT)model[32]. GivenanoisyimageI=u+ntheROFmodelcanbedescribedastheminimizationofthefunctionalZjruj+Z(u)]TJ /F3 11.955 Tf 11.96 0 Td[(I)2(4)]TJ /F4 11.955 Tf 11.95 0 Td[(1) ThersttermofthefunctionalistheTV-regularizationtermandthesecondisthettingtermwithasaweightedparameter.Thismodelpreserveswellthesharpedgesinimagedenoisinganditwasusedextensivelyspeciallyforimagescorruptedwithgaussiannoise.Incomparisontotheadditivenoise,themultiplicativenoiseremovalhasnotyetbeenstudiedcompletely.Asweknowsofar,thevariationalapproachwasproposedrstlybyRudin,LionsandOsher(RLOmodel)[43]astheminimizationoftheenergyfunctionalE(u)=Zjruj+1ZI u+2ZI u)]TJ /F4 11.955 Tf 11.96 0 Td[(12(4)]TJ /F4 11.955 Tf 11.95 0 Td[(2) 34

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whereI=unistheimagecontaminatedwithmultiplicativenoiseandthelasttwotermsarethedatattingtermswith1,2theweightedparameters.Recently,severalvariationalapproachesaredevotedtothemultiplicativenoiseremoval[19,23-25]. Animportantmodel,whichcanbeappliedtothenon-texturedSARimages,isthevariationalmodelproposedbyAubertandAujol(AAmodel)[3].TheauthorsproposedthefollowingrestorationmodelinfuZjruj+Zlogu+f u(4)]TJ /F4 11.955 Tf 11.95 0 Td[(3) whereubelongstoBV(),u>0andI=un,I>0inL1()isthegivenimage.isaregularizationparameter.The(AA)modelisspecicallydevotedtothedenoisingofimagescorruptedbyGammanoise,whichappearsmorefrequentlyinSARimages.Theauthorsprovedtheexistenceofaminimizerfortheproblem(4-3).Theauthorsinpapers[24]developedanewdenoisingmodel:minuZjruj+Z(u+Ie)]TJ /F7 7.97 Tf 6.58 0 Td[(u)(4)]TJ /F4 11.955 Tf 11.95 0 Td[(4) Thechoiceofthenewttingtermu+Ie)]TJ /F7 7.97 Tf 6.58 0 Td[(uisbasedontworeasons:oneisthattheexponentialtransformationpreservesimageedgeswell[19].Theotheristhatu+Ie)]TJ /F7 7.97 Tf 6.59 0 Td[(uisgloballyconvexforalluandI>0,whichensurestheuniquenessofthesolutionstothevariationalproblem(4-4). Anothervariationalmodelformultiplicativenoiseremoval,wasdevelopedinpaper[25].Thiswasmotivatedbytheformofthecorruptedimage(specicformedicalultrasoundimages)I=u+p un(4)]TJ /F4 11.955 Tf 11.95 0 Td[(5) wherenisazeromeanGaussianvariable.TheauthorsintroducedthettingtermE1(u)=R(I)]TJ /F7 7.97 Tf 6.58 0 Td[(u)2 uforremovingthespecklenoiseinultrasoundimages.The 35

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minimizationproblembecomesminuZjruj+Z(I)]TJ /F3 11.955 Tf 11.95 0 Td[(u)2 u(4)]TJ /F4 11.955 Tf 11.95 0 Td[(6) Themodelwasimplementedwithgoodresults.Thereareseveralmodels[22,29,44]onimagesegmentationfortheimagescontaminatedwithmultiplicativenoise.However,ultrasoundimagesaredifculttobesegmentedbecauseoflowsignal/noiseratiowhichreducesgreatlytheobservabledetailsinsuchimages.Consequently,theaccuracyandprecisionofthemeasurementsarecompromised.Oneofthemodelsdevelopedin[29]concernsthepiecewiseconstantsegmentation,wheretheauthorsproposedthefollowingmodel: Letu0:!R,u0=unbeagivenimagecontainingthemultiplicativenoisenandthedomainisboundedwithLipschitzboundaryanduispiecewiseconstant.DenotebyCincludedinthecontourwhichseparatesintotworegions1and2andlettheLipschitzfunctionbeonelevelsetfunctionwhichrepresentC.Thentheenergyinthetwo-phasecasepiecewiseconstantsegmentationisgivenby L(c1,c2,)=1Zu0 c1)]TJ /F4 11.955 Tf 11.95 0 Td[(12+2Zu0 c2)]TJ /F4 11.955 Tf 11.95 0 Td[(12+ZjrH()j(4)]TJ /F4 11.955 Tf 11.95 0 Td[(7) TheenergyminimizationproblemissolvedusingtheEuler-Lagrangeequationfortheunknownlevelsetfunction.Duetothefactthatultrasoundimageshavepoorsignaltonoiseratioandhigherinhomogeneity,thepiecewiseconstantMumford-Shahmodelpresentedin[29]isnotefcientforsegmentationofimageswithmultiplicativenoise.In[44],aregionbasedactivecontourmodelisdevelopedusingMLEwiththeassumptionthattheimagepixelsaremodeledasRayleighdistributedrandomvariablesandanitedifferenceapproximationoftheowwasderived.Bothmodelsmentionedaboveweretestedonreal(ultrasound)andsyntheticimages. 36

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4.3ProposedWork Inthissection,wemakeadetailedpresentationofourproposedvariationalmodel[41]forsimultaneousnoiseremovalandmultiphasesegmentationofultrasoundimages.First,letI:!Rbeanoisyimagedenedonanopen,bounded,smoothdomainR2.SupposethattheimageIcontainsKpatterns.Itiswellknownthatmedicalultrasoundimagescouldbestronglycorruptedbyaspecklenoise.Theseimagescanbemodeledascorruptedwithsignal-dependentnoiseoftheform(4-5)whereuistheoriginalcleanimageandnisazeromeanGaussiannoise. Theproposedmodelconsistsofminimizinganenergyfunctionalcontainingtwoparts:thedenoisingand,respectivelythesoftsegmentationpart.First,wedenotebyui(x)themeaneldoftheintensityofthepatterniandpi(x)theprobabilitythatthepixelx2belongstothepatterni,underthesimplexconstraints(3-5).Weassumethat,foreachpixelxwhichbelongstopatterni,thenoisyimagecanbemodeledbytheformulaI(x)=ui(x)+p ui(x)n(x),i=1,2,...,K.Then,foranyx2,n2(x)=KXi=1(I(x))]TJ /F3 11.955 Tf 11.95 0 Td[(ui(x))2pi(x) ui(x)(4)]TJ /F4 11.955 Tf 11.95 0 Td[(8) Theproposedmodelforsimultaneousdenoisingandsoftsegmentationoftheimagescorruptedwithspecklenoisereferstotheminimizationofthefollowingenergyfunctional:E[pi,uijI]=KXi=1Zjrui(x)jq(x)+KXi=1Z(I(x))]TJ /F3 11.955 Tf 11.95 0 Td[(ui(x))2pi(x) ui(x)+KXi=1Zjrpi(x)j2+Z KXi=1p pi(x)!2,i=1,...,K(4)]TJ /F4 11.955 Tf 11.95 0 Td[(9) withandweightedparametersandsubjecttothesimplexconstraints(3-5). 37

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Thereconstructedimageresultfortheproposedmodelatanypixelxfromisgivenbytheformulau(x)=KXi=1ui(x)pi(x)(4)]TJ /F4 11.955 Tf 11.96 0 Td[(10) Intheproposedwork,thesmoothingpartoftheenergyfunctionalcontainsavariableexponentdenedasinformula(3-8).Usingthisfunctionalwithvariableexponentwhichensuresanadaptivesmoothingwithfeaturepreserving.Forourworkmodel,wealsoseektominimizethetermPKi=1Rjrpi(x)j2+R(PKi=1p pi(x))2.Thatmeans,weimposetheconditionthatmembershipfunctionspi(x)aresmoothinsidethepatternanddiscontinuousacrossit.Inplus,foranyxfrom,underthesimplexconstraints(3-5),eachpatternmembershippi(x)isforcedtobeclosetotheverticesofthe(K-1)simplex,i.e.pi(x)isclosetoeither0or1. Tondanoptimalsolutionfortheminimizationoftheenergyfunctional(4-9),wecomputetheEuler-Lagrangeequationsassociatedwiththisproblem.DenoteU=(u1,...,uK)withthemembershipP=(p1,...,pK). Therstordervariationoftheenergyfunctionalgivenby(4-9)withrespecttothemembershipPiscomputedbyusingtheprojectionon(K-1)simplex.WecanwritetheequationsfortherstordervariationoftheenergyEunderP!P+P,withoutthesimplexconstraintsonPas@E=ZKXi=1Wipidx+Z@KXi=1wipidS(4)]TJ /F4 11.955 Tf 11.96 0 Td[(11) and Wi=)]TJ /F4 11.955 Tf 9.3 0 Td[(pi(x)+(I(x))]TJ /F3 11.955 Tf 11.96 0 Td[(ui(x))2 ui(x)+1 KPKi=1p pi p pi,x2(4)]TJ /F4 11.955 Tf 11.96 0 Td[(12) wi=@pi(x) @n,x2@(4)]TJ /F4 11.955 Tf 11.96 0 Td[(13) 38

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TakingW=(W1,...,WK)andw=(w1,...,wK),therelation(4-11)canbewritteninthefree-gradientform@E @fP=Wj+wj@(4)]TJ /F4 11.955 Tf 11.96 0 Td[(14) BecausePbelongstothe(K)]TJ /F4 11.955 Tf 11.95 0 Td[(1)-simplex,weconsidertheorthogonalprojection:TPRK!TPK)]TJ /F6 7.97 Tf 6.58 0 Td[(1 Foranyt2TPRK,(t)=t)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1K K=t)]TJ /F9 11.955 Tf 12.62 0 Td[(1K where=PKi=1ti Kand1K p K=(1,...,1) p Kisthenormaldirectionofthetangentplane. TheconstrainedgradientofEonthe(K)]TJ /F4 11.955 Tf 11.95 0 Td[(1)-simplexisgivenby@E @P=@E @fP=(W)]TJ /F9 11.955 Tf 12.62 0 Td[(1K)j+(w)]TJ /F9 11.955 Tf 12.62 0 Td[(1K)j@(4)]TJ /F4 11.955 Tf 11.96 0 Td[(15) Tosolvetheequation@E @P=0(4)]TJ /F4 11.955 Tf 11.96 0 Td[(16) isequivalenttosolvetheEuler-LagrangesystemofequationsonP,givenU:Wi(x)=,x2(4)]TJ /F4 11.955 Tf 11.96 0 Td[(17)wi(x)=,x2@(4)]TJ /F4 11.955 Tf 11.96 0 Td[(18) whereWiandwiaregivenintherelations(4-12)and(4-13). TheEuler-LagrangesystemofequationsonU,givenPis div(q(x)jrui(x)jq(x))]TJ /F6 7.97 Tf 6.59 0 Td[(2rui(x))+I(x)2 ui(x)2)]TJ /F4 11.955 Tf 11.95 0 Td[(1pi(x)=0,x2(4)]TJ /F4 11.955 Tf 11.96 0 Td[(19)q(x)jrui(x)jq(x))]TJ /F6 7.97 Tf 6.58 0 Td[(2@ui(x) @n=0,x2@(4)]TJ /F4 11.955 Tf 11.96 0 Td[(20) 39

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4.4Algorithm Toobtainanoptimalsolution(U,P)toproblem(4-9),weusedthe(AM)algorithm.Foreachstep(n+1),giventhepatternsUn=(uni;i=1,2,...,K)andthemembershipPn=(pni;i=1,2,...,K)ndPn+1=argminPE[PjUn,I](4)]TJ /F4 11.955 Tf 11.96 0 Td[(21) whereU=(u1,...,uK)andP=(p1,...,pK). Thisisequivalenttosolvetheowequation:d(pi) dt=Lpi(I,ui,pi,)(4)]TJ /F4 11.955 Tf 11.96 0 Td[(22) withLpi(I,ui,pi)=(I)]TJ /F3 11.955 Tf 11.95 0 Td[(ui)2 ui)]TJ /F4 11.955 Tf 11.95 0 Td[(pi+1 KPKi=1p pi pi)]TJ /F9 11.955 Tf 12.62 0 Td[((4)]TJ /F4 11.955 Tf 11.96 0 Td[(23) where=1 K KXi=1)]TJ /F4 11.955 Tf 9.3 0 Td[(pi+(I)]TJ /F3 11.955 Tf 11.96 0 Td[(ui)2 ui+1 KPKi=1p pi pi!=1 K KXi=1(I)]TJ /F3 11.955 Tf 11.95 0 Td[(ui)2 ui+1 KPKi=1p pi pi!(4)]TJ /F4 11.955 Tf 11.96 0 Td[(24) since(KXi=1pi)=0 Following,basedonPnandUn,theoptimalestimationUn+1=argminUE[UjPn,I](4)]TJ /F4 11.955 Tf 11.96 0 Td[(25) isobtainedbysolvingtheowequation:d(ui) dt=Lui(I,ui,pi,)(4)]TJ /F4 11.955 Tf 11.96 0 Td[(26) whereLui=div(qjruijq)]TJ /F6 7.97 Tf 6.59 0 Td[(2rui)+I2 u2i)]TJ /F4 11.955 Tf 11.96 0 Td[(1pi(4)]TJ /F4 11.955 Tf 11.96 0 Td[(27) 40

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andq(dependingonx)isgiveninrelation(3-8). Summarizing,theupdating...(Un,Pn)...!(Un+1,Pn+1)...isobtainedsolvingthefollowingsystemofequations:pn+1i=pni+dtpLpi(I,uni,pni)un+1i=uni+dtuLui(I,uni,pni)(4)]TJ /F4 11.955 Tf 11.96 0 Td[(28) wheredtpanddtuarestepssizes,Lpi,Luiaredenedabove. 4.5Experiments Toshowtheeffectivenessofourmodel,sixexperimentswereperformed.DenotebyNthenumberofiterationsforeachexperiment. Figure4-1isacomparisonbetweentheRLOmodel(4-2)andtheproposedmodel(4-9)usingasyntheticimagecontaminatedwithmultiplicativenoise.Figure4-1Aistheoriginalcleanimage,gure4-1BisthecontaminatedimagewithmultiplicativeGaussiannoiseofmeanzeroandvariance0.03.Figure4-1C,Drepresentthereconstructedimageresultu(x)usingourmodel,respectivelytheRLOmodel.Wehavetestedtheperformanceofourmodelbycomputingthesignaltonoiseratioandtherelativeerror(3-39)forbothmodels.WesummarizedtheresultsinTable4-1.FromthenumericalcomparisonresultsshowninTable4-1andfromthecomputationalresultsfromFigure4-1C,DwecanconcludethatourmodelperformsabetterandfasterdenoisingthanthetraditionalRLOmodelforthesamenumberofiterations.Theparametersusedforourmodelare=8,=0.8andthenumberofiterationsisN=50. Figure4-2showsacomparisonofareconstructedimageobtainedbyusingtheproposedmodelwithdifferentq(x).Wecomparetheresultsofourmodel(4-9),whenq(x)isavaryingfunctionontheinterval[1,2]andthecaseinwhichq(x)isaconstant,eitheroneortwo.Inourmodel,q(x)variesfrompixeltopixelandfromiterationtoiterationandprovidebetterresultsthanifq(x)isxedanddoesnotvaryinxandN.Toillustratethisfact,weconsideredtwocasesforq(x)constant: 41

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1)q(x)=1and2)q(x)=2.Incase1),Figure4-2C,thedenoisingismoreslowerandtakesmoreiterationsinordertoobtainasatisfactoryresult.Incase2),Figure4-2D,thedenoisingisfasterandleadstoalossoftheimagedetails.AsitcanbeseeninFigure4-2BandTable4-2,theproposedmodeloffersabetternoiseremoval,beingacombinationoftheTV-basedandisotropicsmoothing.Theparametersusedare=3,=0.3andthenumberofiterationsisN=50. IntheExperiment3,wecomparetheresultsbetweentheproposedmodelandRLOmodel(4-2),foranultrasoundimage.Theresultsareshownafter50iterationsandtheparametersusedforourmodelare=1,=0.4.Weincludedthemeaneldsforourreconstructedimagewiththecorrespondingprobabilitiesfortheproposedmodelandtheresultingminimizedenergyforeachmodel.Wecanseefromtheenergyplotforourmodelthatwereachedanoptimalsolutionafter50iterations.Theresultsshowthatourmodelperformsabetterandfasterdenoising,withfeaturepreservingthantheRLOmodel. Figure4-4representsacomparisonbetweenourmodel(4-9)andmodel(4-7).Figure4-4Aisthegivenimagecontaminatedwithmultiplicativenoisewithmeanzeroandvariance0.1,andwithcontrast17/60on[0,255]grayscale.Figure4-4Brepresentsthesegmentationresultu(x)usingourmodel,after50iterations,Figure4-4C,Drepresentthemeaneldsofthesegmentationresultforourmodel.Theparametersusedforthisexperimentforourmodelare=7,=0.7.Theresultsofmodel(4-7)areshowninFigure4-4E-Jafter50,respectively1200iterations.Aswecanseefromtable4-3andfromthereconstructedimagefromgure4-4B,ourmodelperformsabetterandfastersegmentationthanthemodel(4-7). IntheExperiment5,wecomparetheproposedmodelwithmodel(4-7)forthreetestimageswithdifferentlevelofnoiseanddifferentlevelofcontrast,representedinFigure4-5A-C.Figure4-5Aisaninitialimagewithcontrast17/33on[0,255]grayscaleandnoiseofvariance0.03,gure4-5Brepresentsaninitialimagewithcontrast17/60 42

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on[0,255]grayscaleandnoiseofvariance0.1andthelastinitialimagehascontrast17/138on[0,255]grayscaleandnoiseofvariance0.2.Theresultsareshownafter50iterations.Ingure4-5J-Lwehaveshownthecorrespondingenergiesfortheproposedmodel(4-9).Aswecanseefromthecomparisonresults,theproposedmodelperformswellthesegmentationforallsituationsgivenabove.Fortheproposedmodel,forthesethreedifferentcases,wehaveusedtheparameters=8,=0.8,=7,=0.7,andrespectively=5,=0.5.Forthecomparisonmodel(4-7),weusedthesameinitializationforalltestimages.FromtheExperiments4and5,wecanconcludethattheproposedmodelisfaster,morerobusttonoiseandcontrast. IntheExperiment6,wecompareourmodel(4-9)withmodel(4-7)foranultrasoundimage.Wehaveshownthenalreconstructedresultu(x)andthecorrespondingmeaneldsu1(x),u2(x)forbothmodels.Forourmodel,weusedtheparameters=1,=0.4.Wehaveincludedtheresultsforourmodelafter50iterationsandforthecomparisonmodel(4-7)after50,respectively1000iterations.Fromthisexperimentwecanconcludethatourmodelperformsabettersegmentationforthesamenumberofiterations.TheCPUtimeismeasuredinseconds. Table4-1. TheSNRandReErrforthesyntheticimageinFigure4-1. ProposedmodelRLOmodel SNR24.9621.48ReErr0.000010.00005CPUtime62.960000153.381415 Table4-2. TheSNRandReErrforsyntheticimageinFigure4-2. q(x)variableq(x)=1q(x)=2 SNR24.9621.2422.38ReErr0.000010.000050.00003 Table4-3. ThecomputingtimeforultrasoundthyroidimageinFigure4-3. ProposedmodelRLOmodel CPUtime39.36298684.0805575 43

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A BCD EFG HIJ Figure4-1. Noiseremovalresultsfortheproposedmodel(4-9)andtheRLOmodel.A)Syntheticimage.B)Syntheticimagewithnoise.C)Thedenoisedimageresultobtainedbyusingtheproposedmodel.D)ThedenoisedimageresultobtainedbyusingtheRLOmodel(4-2).E),F),G)Themeaneldsu1(x),u2(x),u3(x)fortheresultC).H),I),J)Thecorrespondingprobabilitiesp1(x),p2(x),p3(x)forourmodel. 44

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Table4-4. ThenumberofpixelsperpartitionforsyntheticimageinFigure4-4. InitialimageProposedmodelModel(4-7)Model(4-7)50iterations50iterations1200iterations U128847288471246728547U228753287534512429053 AB CD Figure4-2. Comparisononthevariableexponentq(x)forasyntheticimage.A)Syntheticimagecorruptedwithmultiplicativenoise.B)Thereconstructedimageusingq(x)variable.C)Thereconstructedimageusingq(x)=1xedinourmodel.D)Thereconstructedimageusingq(x)=2xedinourmodel. 4.6Conclusions Theproposedmodel[41]isanovelvariationalapproachforsimultaneoussegmentationanddenoisingofimagescontaminatedwithmultiplicativenoise.Byusingsoftsegmentationandanenergyfunctionalwithvariableexponent,themodelbecomesmorerobusttonoiseandperformsabettersegmentation.Toshowitsefciency,wecomparedourmodeltosometraditionalmodels,forbothreal(ultrasound)andsyntheticimages. 45

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A BC DE FG HI Figure4-3. UltrasoundthyroidimagedenoisingresultsfortheproposedmodelandtheRLOmodel.A)Ultrasoundthyroidwithnoise.B)Thedenoisedimageobtainedbyusingtheproposedmodel(4-9).C)ThedenoisedimageresultusingtheRLOmodel(4-2).D),E)Themeaneldsu1(x),u2(x)ofthepatternsforourmodel.F),G)Thecorrespondingprobabilitiesp1(x),p2(x)formodel(4-9).H),I)Theenergyversusiterationsfortheproposedmodel(4-9)andforRLOmodel(4-2). 46

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A BCD EFG HIJ Figure4-4. Segmentationcomparisonresultsforourmodel(4-9)andmodel(4-7).A)Syntheticimagewithnoise.B)Segmentationresultusingtheproposedmodel(4-9)after50iterations.C),D)Themeaneldsu1(x),u2(x)ofthesegmentationresultB).E)Segmentationresultusingmodel(4-7)after50iterations.F),G)Themeaneldsu1(x),u2(x)ofthepatternsformodel(4-7)after50iterations.H)Segmentationresultusingmodel(4-7)after1200iterations.I),J)Themeaneldsu1(x),u2(x)ofthepatternsformodel(4-7)after1200iterations. 47

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ABC DEF GHI JKL Figure4-5. Segmentationcomparisonresultsforourmodel(4-9)andmodel(4-7).A),B),C)Initialimageswithdifferentlevelofcontrastanddifferentlevelofnoise.D),E),F)Thecorrespondingsegmentationresultsusingtheproposedmodel(4-9).G),H),I)Thecorrespondingsegmentationresultsusingmodel(4-7).J),K),L)Theenergyversusiterationsfortheproposedmodel(4-9). 48

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A BCD EFG HIJ Figure4-6. Thyroidimagesegmentationresultsfortheproposedmodelandmodel(4-7).A)Ultrasoundthyroidwithnoise.B)Segmentationresultusingtheproposedmodel(4-9).C),D)Themeaneldsu1(x),u2(x)ofthepatternsforourmodel.E)Segmentationresultusingmodel(4-7)after50iterations.F),G)Thecorrespondingmeaneldsu1(x),u2(x)forE).H)Segmentationresultusingmodel(4-7)after1000iterations.I),J)Thecorrespondingmeaneldsu1(x),u2(x)forH). 49

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BIOGRAPHICALSKETCH IuliaMagdalenaPosircawasbornandraisedinBucharest,Romania.ShehaveaBachelorofSciencedegreeandaMasterofSciencedegreefromUniversityofBucharest.SheenteredthedoctoralprogramattheUniversityofFlorida,DepartmentofMathematicsin2004andstartedtheresearchworkwithDr.Chenin2006. 54