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Variational Models for Simultaneous Image Segmentation and Noise Removal

Permanent Link: http://ufdc.ufl.edu/UFE0044953/00001

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Title: Variational Models for Simultaneous Image Segmentation and Noise Removal
Physical Description: 1 online resource (54 p.)
Language: english
Creator: Posirca, Iulia M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: image -- noise -- segmentation -- variational
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Iulia M Posirca.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Chen, Yunmei.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044953:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044953/00001

Material Information

Title: Variational Models for Simultaneous Image Segmentation and Noise Removal
Physical Description: 1 online resource (54 p.)
Language: english
Creator: Posirca, Iulia M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: image -- noise -- segmentation -- variational
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Iulia M Posirca.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Chen, Yunmei.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044953:00001


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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