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Continuous Mixture Models for Feature Preserving Smoothing and Segmentation

Permanent Link: http://ufdc.ufl.edu/UFE0024915/00001

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Title: Continuous Mixture Models for Feature Preserving Smoothing and Segmentation
Physical Description: 1 online resource (110 p.)
Language: english
Creator: Subakan, Ozlem
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: bingham, color, continuous, directional, distributions, filters, fisher, gabor, gaussian, image, mixture, models, quaternions, segmentation, smoothing, watson, wishart
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Image smoothing and segmentation are fundamental tasks in computer vision. Although there are numerous algorithms that have been developed and applied to these tasks in various application domains, several challenges remain unconquered. In this dissertation, we consider the challenge of achieving smoothing and segmentation while preserving complicated and detailed features present in the image, be it a gray level or a color image. We present novel approaches that do not make use of any prior information about the objects in the image being processed, and yet produce promising results. The key idea here is to model the derived local orientation information via a continuous mixture of appropriate basis functions. We present several such models considering grayscale and color images separately. We propose two models for grayscale images; one involving a continuous mixture over the covariance matrices of Gaussian basis functions, and another involving a continuous mixture over the mean direction vectors of Watson basis functions. For color image processing, we introduce a novel Quaternionic Gabor Filter (QGF) which can combine the color channels and the orientations in the image plane. The local orientation information in the color images can be extracted using the QGFs. We show that these filters are optimally localized both in the spatial and frequency domains and provide a good approximation to quaternionic quadrature filters. In a second logical step, we propose continuous mixtures of appropriate hypercomplex exponential basis functions to model this local orientation information. We derive closed form solutions for the proposed models. These continuous mixture models are then used to construct spatially varying kernels which are convolved with the color image or the signed distance function of an evolving contour (placed in the color image) to achieve feature preserving smoothing or segmentation, respectively. We compare and quantitatively validate the proposed models with numerous experimental results on real images including the images drawn from Berkeley Segmentation Data Set. These comparisons revealed that the techniques we developed are superior than the state-of-the-art algorithms described in the literature. Finally, the dissertation is concluded with a list of potential directions for future research.
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 Ozlem Subakan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Vemuri, Baba C.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024915:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024915/00001

Material Information

Title: Continuous Mixture Models for Feature Preserving Smoothing and Segmentation
Physical Description: 1 online resource (110 p.)
Language: english
Creator: Subakan, Ozlem
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: bingham, color, continuous, directional, distributions, filters, fisher, gabor, gaussian, image, mixture, models, quaternions, segmentation, smoothing, watson, wishart
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Image smoothing and segmentation are fundamental tasks in computer vision. Although there are numerous algorithms that have been developed and applied to these tasks in various application domains, several challenges remain unconquered. In this dissertation, we consider the challenge of achieving smoothing and segmentation while preserving complicated and detailed features present in the image, be it a gray level or a color image. We present novel approaches that do not make use of any prior information about the objects in the image being processed, and yet produce promising results. The key idea here is to model the derived local orientation information via a continuous mixture of appropriate basis functions. We present several such models considering grayscale and color images separately. We propose two models for grayscale images; one involving a continuous mixture over the covariance matrices of Gaussian basis functions, and another involving a continuous mixture over the mean direction vectors of Watson basis functions. For color image processing, we introduce a novel Quaternionic Gabor Filter (QGF) which can combine the color channels and the orientations in the image plane. The local orientation information in the color images can be extracted using the QGFs. We show that these filters are optimally localized both in the spatial and frequency domains and provide a good approximation to quaternionic quadrature filters. In a second logical step, we propose continuous mixtures of appropriate hypercomplex exponential basis functions to model this local orientation information. We derive closed form solutions for the proposed models. These continuous mixture models are then used to construct spatially varying kernels which are convolved with the color image or the signed distance function of an evolving contour (placed in the color image) to achieve feature preserving smoothing or segmentation, respectively. We compare and quantitatively validate the proposed models with numerous experimental results on real images including the images drawn from Berkeley Segmentation Data Set. These comparisons revealed that the techniques we developed are superior than the state-of-the-art algorithms described in the literature. Finally, the dissertation is concluded with a list of potential directions for future research.
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 Ozlem Subakan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Vemuri, Baba C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024915:00001


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Therearemanypeoplewhohavehelpedmealongthewaywithmydissertation,andIwishtotaketheopportunitytothankthemthroughthispage.First,Iwouldliketoexpressmygratitudetomysupervisor,ProfessorBabaVemuri,notonlyforhispatientguidanceandusefulcritiquesbutalsofortheplentyoffreedomhegavemethroughouttheprogressofthisdissertation.IowehimalotforourlongmeetingsonearlyFridaymornings,aswellasforallofmyunexpectedvisits.Iamthankfultohimformanycriticalandfruitfuldiscussions,bywhichIgottheinspirationandmotivationtopursuethiswork.IalsoappreciatehisencouragementsandnancialsupportduringmyPhD.Next,IwanttoextendmythankstoProfessorEduardoVallejos.Heintroducedmetotheplantbiology,andgavemetheopportunitytotaketheMRIscansforplantrootsandworkwithrootMRIs.Hehasalwaysbeenextremelypatientandhelpful.IamalsogratefultohimforthenancialsupportsheprovidedduringmyPhD.Ithasbeengreatpleasureworkingwithhim.Furthermore,myspecialthanksgotomycommitteemembersProfessorsArunavaBanerjee,JereyHoandAnandRangarajan.Ihavelearnedalotthroughthevaluablecoursestheyoered,aswellastheextensiveandhelpfuldiscussionstheybroughtduringourweeklyseminars.Ialsowanttothankthemforpointingmetopossibleimprovementsonthisdissertation.ThisresearchwassupportedbyNIHEB007082andbySMCRSPthroughagrantfromtheUSAgencyofInternationalDevelopment.IalsoreceivedatravelawardfromtheUFOceofResearchandGraduateProgramsandtheUFDivisionofSponsoredResearch.IwouldalsoliketothankmyformerandcurrentlabmatesBingJian,FeiWang,AngelosBarmpoutis,TingChen,MeizhuLiu,SanthoshKodipaka,O'neilSmith,AdrianPeter,KarthikGurumoorthyandmanyothers. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1OrganizationoftheDissertation ........................ 13 1.2Notation ..................................... 14 2CONTINUOUSMIXTUREMODELSFORGRAYSCALEIMAGES ...... 17 2.1PreviousWork ................................. 17 2.2LocalOrientationInformation ......................... 21 2.3ModelingDerivedLocalOrientationusingaMixtureofWisharts ..... 22 2.4ModelingDerivedLocalOrientationusingaMixtureofBinghams ..... 27 2.5ConvolutionKernelsforSmoothingandSegmentation ............ 31 3EXPERIMENTALRESULTSANDVALIDATIONONGRAYSCALEIMAGES 40 3.1ExperimentsonGrayscaleImageSmoothing ................. 40 3.2ExperimentsonGrayscaleImageSegmentation ............... 42 4CONTINUOUSMIXTUREMODELSFORCOLORIMAGES .......... 60 4.1PreviousWork ................................. 60 4.2Quaternions ................................... 64 4.3QuaternionicGaborFilters ........................... 65 4.4AContinuousMixtureontheOrientationSpace ............... 70 4.5AContinuousMixtureontheUnitQuaternionSpace ............ 72 5EXPERIMENTALRESULTSANDVALIDATIONONCOLORIMAGES ... 80 5.1ExperimentsonColorImageSmoothing ................... 80 5.2ExperimentsonColorImageSegmentation .................. 82 6CONCLUSIONS ................................... 100 REFERENCES ....................................... 102 BIOGRAPHICALSKETCH ................................ 110 6

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Table page 1-1Mathematicalnotationsandsymbolsusedinthedissertation. .......... 15 3-1PSNRvaluesfordenoisedimagesindB ....................... 58 3-2F1-measure(Dice'scoecient)valuesforthesegmentationresults ........ 58 5-1ThePSNRsofthedenoisedcolorimagesfordierentalgorithms ......... 99 5-2F1-measure(Dice'scoecient)valuesforthecolorimagesegmentationresults 99 7

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Figure page 1-1Asegmentationexample ............................... 16 1-2Animagesegmentationexamplewithmultipleregionsofinterest ........ 16 1-3Anexampleofsmoothinganoisyimage ...................... 16 2-1CurveevolutionexperimentswithaGaussiankernelandwiththeBMWkernel 35 2-2Derivinglocalorientationinasyntheticimagewithandwithoutnoise ...... 36 2-3Weightvectorsonasyntheticimagewithandwithoutnoise ........... 37 2-4Meansandstandarddeviationsoferrorsinorientationestimationonasyntheticimage ......................................... 38 2-5Convolutionkernelsonasyntheticimage ...................... 39 3-1Smoothingresultsontheboatimage ........................ 46 3-2Methodnoiseexperimentondenoisingoftheboatimage ............. 47 3-3SmoothingresultsontheBarbaraimage ...................... 48 3-4MethodnoiseexperimentondenoisingoftheBarbaraimage ........... 49 3-5Segmentationresultsonazebraimage ....................... 49 3-6Segmentationresultsonanimage .......................... 50 3-7Asegmentationexperimentonaleopardimage .................. 51 3-8Segmentationexperimentonacaseimage ..................... 51 3-9SegmentationresultsoftheBMWkernelontheBerkeleySegmentationDataSet 52 3-10Segmentationexperimentonatestimage ...................... 53 3-11Segmentationresultsontheplaneimage ...................... 54 3-12Segmentationexperimentonthebirdimage .................... 55 3-13Segmentationexperimentontheelephantsimage ................. 56 3-14Segmentationexperimentontheowerimage ................... 57 3-15SensitivityanalysisusingtheF-measurescoresfortheparametersofdierentalgorithms ....................................... 59 4-1AQuaternionicGaborFilter. ............................ 76 8

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....................... 76 4-3Quaternion-convolutionoftheimageofBarbarawithaQuaternionGaborFilterbank .......................................... 77 4-4ApplicationofaQuaternionicGaborFilterbankacrossequalluminance .... 78 4-5AQuaternionGaborFilterexperimentonasyntheticcolorimagewithequal(R+G+B)=3values ................................ 78 4-6Convolutionkernelsonarealcolorimage ...................... 79 5-1Denoisingabutteryimage ............................. 85 5-2Methodnoiseoutputsonanimageofabuttery .................. 86 5-3Denoisingtwoparrots ................................ 87 5-4Methodnoiseexperimentonanimageoftwoparrots ............... 88 5-5Restorationofthenoisymandrillimage ...................... 89 5-6Methodnoiseoutputsofthedenoisedmandrillimage ............... 90 5-7DenoisingexperimentonanimagefromtheBerkeleyDataSet .......... 91 5-8Methodnoiseexperimentonanimageofthehorses ................ 92 5-9Resultsofinpaintingashnetinanimage ..................... 93 5-10Aninpaintingexperimentontheimageofaparrotinacage ........... 93 5-11Asegmentationexperimentonatigerimage .................... 94 5-12Segmentationresultsontheparadeimage ..................... 95 5-13Asegmentationexperimentonthestarshimage ................. 96 5-14Segmentationoutputsofdierentalgorithmsonthebualoimage ........ 97 5-15Asegmentationexperimentontheastronautsimage ............... 97 5-16F-measureplotofthesensitivityanalysis ...................... 98 9

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Imagesmoothingandsegmentationarefundamentaltasksincomputervision.Althoughtherearenumerousalgorithmsthathavebeendevelopedandappliedtothesetasksinvariousapplicationdomains,severalchallengesremainunconquered.Inthisdissertation,weconsiderthechallengeofachievingsmoothingandsegmentationwhilepreservingcomplicatedanddetailedfeaturespresentintheimage,beitagrayleveloracolorimage.Wepresentnovelapproachesthatdonotmakeuseofanypriorinformationabouttheobjectsintheimagebeingprocessed,andyetproducepromisingresults. Thekeyideahereistomodelthederivedlocalorientationinformationviaacontinuousmixtureofappropriatebasisfunctions.Wepresentseveralsuchmodelsconsideringgrayscaleandcolorimagesseparately.Weproposetwomodelsforgrayscaleimages;oneinvolvingacontinuousmixtureoverthecovariancematricesofGaussianbasisfunctions,andanotherinvolvingacontinuousmixtureoverthemeandirectionvectorsofWatsonbasisfunctions.Forcolorimageprocessing,weintroduceanovelQuaternionicGaborFilter(QGF)whichcancombinethecolorchannelsandtheorientationsintheimageplane.ThelocalorientationinformationinthecolorimagescanbeextractedusingtheQGFs.Weshowthattheseltersareoptimallylocalizedbothinthespatialandfrequencydomainsandprovideagoodapproximationtoquaternionicquadraturelters.Inasecondlogicalstep,weproposecontinuousmixturesofappropriatehypercomplexexponentialbasisfunctionstomodelthislocalorientationinformation.Wederiveclosed 10

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Finally,thedissertationisconcludedwithalistofpotentialdirectionsforfutureresearch. 11

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Beginninginthelate70's,computervisionstartedtogainamorefocusedattentionaftercomputerscouldmanagethestorageandcomputationoflargedatasets,suchasthosefromimages.Togetherwiththeadvancesinthedevelopmentandmanufacturingoftheelectroniccomponentswhichresultedintheemergenceofmachineswithincreasingcomputingpower,morememory,andsuperiorgraphicalscreens,onlyfewyearswereneededtoseethetremendousamountofgrowthincomputervisionresearch.Withinthiseld,analyzingtheimagecontenthasbeenverypopular,sinceitbecamevitalinmanyapplications,suchasimageanalysisforpersonalandnationalsecurity,medicalimaging,communicationandentertainment,computer-aideddiagnosisetc.Thepurposeofimageanalysisistogivethemachinestheperformanceofthehumanvisualsystemtointerpret,understandandusetheimagesfordecisionsandfurtherprocessing.Actually,thisisaverydicultproblem,andafterseveraldecadesofresearch,creatingcomputerintelligenceintheimageanalysisisstillanopensubject. Themotivationsthatdriveusinthisworkcomprisetwomajordicultiesincomputervision,therstofwhichinvolvesremovingthenoiseandenhancingthedatatowardsabetterinterpretationoftheimage.Theseconddicultyliesindeterminingwhichpartsofanimageprovidetherelevantandnecessaryinformationforfurtherprocessing.Amongothervisiontasksaddressingthisproblem,imagesegmentationisprobablyoneofthemostsignicant,sinceitisaprecursorinnumerousstagesofcomputervision.Looselyspeaking,theaiminimagesegmentationistopartitiontheimageintohomogeneousregionswithrespecttoagivenmeasure.Foranexample,seethesegmentationresultsofasquirrelimageinFigure 1-1 Inimagesegmentation,measuresforhomogeneityarenotwell-dened.Giventhenumerousvarietiesofnaturalimages,itiscommontohavemultipleregionsofinterestswithdierentcontextualinformation(seeFigure 1-2 ).Similarly,inimagedenoising,the 12

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1-3 Awiderangeofmethodshavebeenproposedforimagedenoisingandimagesegmentation.Weprovidetherelevantpreviousworkintheintroductionofthefollowingchapters.Withthegoalofpreservingfeatures/detailsinanimage,weproposemodelsforanalyzinglocalorientationinformationatalatticepoint,andthenincorporatethisinformationintoimagesmoothingandsegmentationkernels.WefocusontheGaborlterstoextractthelocalorientationinformation,astheywereshowntosuccessfullymodelthebehaviorofthereceptiveeldsinthemammalianprimaryvisualcortex[ 1 ].Themaincontributionsofthisdissertationcanbelistedasfollows: 2 ,weintroducecontinuousmixturemodelsforquantifyingthederivedorientationinformationviaGaborltersingrayscaleimages.Wepresenttwomodelsforthispurpose;onemodelisoverthecovariancematricesofGaussianbasisfunctions,andtheotherisacontinuousmixtureoverthemeandirectionvectorsofWatsonbasisfunctions.Usingtheanalyticformsforthesemodels,wedevelopconvolutionkernelsforsmoothingandforsegmentation.Tothe 13

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Chapter 3 addressesthesmoothingandsegmentationtasksongrayscaleimages,usingthemodelsproposedinChapter 2 .Weshowtheexperimentalresultsalongwiththequantitativeevaluationdepictingthemeritsoftheproposedformulations. InChapter 4 ,weconsidercolorimages,andproposetwocontinuousmixturemodelsforcolorimagesmoothingandcolorimagesegmentation.Inordertoprocesscolorimagesinaholisticway,wechoosetorepresentthemusinghypercomplexnumbers,specicallyquaternions.Thisrepresentationfurtherrequiresanappropriatetechniquetoextractthelocalorientation.(Ingrayscaleimages,Gaborlterswereusedtoextracttheorientationinformation.)ThispursuitleadsustointroduceanovelQuaternionGaborFilterforusewithcolorimages.Weanalyzethepropertiesoftheproposedlter.Thederivedorientationinformationisthenrepresentedbycontinuousmixturemodels;wepresenttwosuchmodels,oneisontheunitquaternionspaceandtheotherisontheorientationspace.Derivingtheclosedformsolutionsfortheseintegrals,weformulatetheconvolutionkernelsforcolorimageprocessing. Chapter 5 providestheapplicationsoftheproposedmodelsonthecolorimagedenoisingandsegmentation.Promisingresultsareobtainedonsyntheticimagesaswellasonrealdatasets.Weextendourevaluationwithseveralquantitativevalidationexperiments. ThisdissertationisconcludedwithanoutlookinChapter 6 .Itspurposeistosummarizethecontributionsandshowpossibleresearchdirectionsforfurtherimprovementsbasedontheproposedwork. 1-1 .Wewilluseboldfacelowercaseletterstodenotethevectors.Matriceswillbedenotedbyboldfaceuppercaseletters.Theithcomponentofavectorwiswrittenaswi,whereas 14

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Table1-1. Mathematicalnotationsandsymbolsusedinthedissertation. NotationExplanation E[]ExpectationoperationjjDeterminantofamatrixtr()Traceofamatrixvec()Thecolumn-by-columnvectorizationofamatrixInnnidentitymatrix()TTransposeofavectorormatrixkkL2normoftheargument()ComplexorhypercomplexconjugateThevectorcrossproductRThesetofallrealnumbersS11-dimensionalunitsphereinR2Sn1(n1)-dimensionalunitsphereinRnPnThemanifoldofnnsymmetricpositivedenitematrices.HThespaceofHamiltonianquaternionsSO(3)Special-orthogonalgroupof33orthogonalmatricesS()andV()Scalarandvectorpartsofaquaternion,respectivelyI0ThemodiedBesselfunctionoftherstkindandzerothorder0F1()ThehypergeometricfunctionofthematrixargumentRe[]andIm[]Realandimaginarypartsoftheargument 15

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B C Asegmentationexample. A )Originalimage[ 2 ]. B )Segmentationoutputobtainedusingthegeodesicactiveregionstechniquein[ 3 ]. C )Segmentationoutputobtainedusingthecueintegrationmethodin[ 4 ]. B C Animagesegmentationexamplewithmultipleregionsofinterest[ 5 ]. A )Originalimage. B )Segmentationperformedbyahumansubject. C )Boundarymapofthesegmentationin B B C Anexampleofsmoothinganoisyimage. A )Originalimage. B )NoisyimagewithaGaussiannoiseofstandarddeviation35. C )Denoisingoutputobtainedusingcoherenceenhancingdiusionalgorithmin[ 6 ]. 16

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Imagedenoising/smoothinghasatwofoldpurposeinearlyvisualinformationprocessing:(1)removalofnoisewhichsimplyhampersthemanualandcomputationalanalysis,(2)enhancementoflocaldiscontinuities,terminationsandbifurcations.Bothofthesepurposesshouldbeachievedwithoutsacricingtheusefuldetailsintheimages.Manyoftheavailablealgorithmslocallysmooththeimagealongoneorseveraldirectionschosentofavorsmoothingalongedgesbutnotacrossit.However,theyfailtopreservecomplexlocalstructuressuchasjunctions. Thereisawidespectrumofimagesmoothingalgorithms,someofwhichdatebacktothe1970'sandarebasedonlinearsystemtheory[ 7 ].Inthepastfteenyears,therehasbeenaurryofactivityintheappliedmathcommunityonimagesmoothingtechniquesmotivatedbytheworkofPeronaandMalik[ 8 ].Toovercomethelimitationsofthelinearmethodswhichleadtoisotropicdiusion,PeronaandMalikproposedapartialdierentialequationbasedmethoddescribingananisotropicdiusionlter.Theanisotropywasachievedviaascalar-valuedfunctiondenedontheimagegradienteldi.e.,ascalardiusivitycoecient.Thiscoecientincludesaxedgradientthresholdwhichallowsfordiscriminationoftheedgecontoursandthehomogeneousareas.Perona-Malikformulationgaveimpetustoseveralanisotropicdiusionlters,mostofthemaddressingmoregeneral 17

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9 10 ]fordetailsonsomeofthesetechniques.Itshouldbenotedthatnoneofthesemethodsaddressedtheissueofpreservingfeaturesthatrepresentedcomplexlocalgeometriessuchasjunctions,corners.In[ 11 ],ajunctionpreservinglteringtechniquewasintroducedusingamorphologicalapproach.Thistechnique,however,requiresthejunctionstobedetectedpriortosmoothing.Morerecently,TschumperleintroducedanimageregularizationPDEwhichtakesthecurvatureconstraintsintoaccount,andappliedittomulti-valuedimages[ 12 ].Hepresentsimpressiveresultsforcurvaturepreservation.Whatisuncleariswhetherthistechniquecanpreservejunctionswherethecurvatureisnotdened.Moreover,suchjunctionsarenotrepresentablebyrank-2tensors. Inthesamevein,itisdesirableforimagesegmentationtechniquestopreservethecomplexlocalgeometrywhiledetectingtheregionboundaries.Tomesofliteraturehavebeenwrittenonimagesegmentation,andthisareaofresearchhasalonghistoryspanningthepastthreedecades.VariationalformulationofthisproblemwaspopularizedbyKassetal.intheirseminalworkontheso-calledsnakesa.k.a.activecontourmodels[ 13 ].MumfordandShahproposedaregion-basedvariationalformulationofthisproblemearlierin[ 14 ],andthisparadigmwaslaterpopularizedbyTsaietal.inanactivecontourframework[ 15 ].Thesnakesmodelthatconstitutesaclosedcurveexpressedasanarbitrarilyparameterizedcurvewasprimarilydesignedasaninteractivesegmentationmodelandprovedtobequiteusefulandgeneralinthiscontext.AnalternativemodelcalledthegeometricactivecontourinalevelsetframeworkwasthenproposedinthepioneeringworksofMalladietal.[ 16 { 18 ],andCasellesetal.[ 19 ].Thismodelinvolvedaclosedcurverepresentedinanimplicitformthatallowedforeaseinmodelingshapeswitharbitraryunknowntopologies.Followingthesemodels,avariationalformulationofthegeometricactivecontourmodel,calledthegeodesicsnakes,wasindependentlyintroduced 18

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20 ]andKichenassamyetal.[ 21 ].Overthepastdecadeandahalf,therehavebeenseveralapproachestosegmentation,someofwhichareimprovementsoverthegeodesicactivecontoursaswellasthetraditionalsnakes,andsomeofwhicharegraph-basedglobaloptimizationapproaches.Formoreonvariationalformulationsoftheimagesegmentationproblemthatledtoimprovementsoftheoriginalproposalsoftheactivecontourmodelandalsoforgraph-basedtechniques,wereferthereaderto[ 22 ].Despitetheplethoraoftechniquesthathavealreadybeenproposedinthecomputervisioncommunity,segmentationpreservingcomplexlocaldetailsremainselusive. Inthischapter,wepresenttwocontinuousmixturemodelsbothofwhichareadaptive,convolutionbasedapproachesforsmoothingandsegmentation.Firstofthesemodelshasbeenpresentedinthecontextofthediusion-weightedMRsignalattenuationbyJianandVemuriin[ 23 { 25 ],andlaterusedinthecontextofimagesmoothingandsegmentationbySubakanetal.in[ 26 ]and[ 27 ].Thesenewandinnovativeapproachesaordthepreservationofthecomplicatedlocalgeometriesoftheboundariesofobjectsinrealsceneswithoutusinganypriorinformation.Bothmodelsconsistoftwomainstages.Intherststage,thelocalorientationinformationisextracted.Severaltechniquescanbeemployedforthispurpose.Oneofthemostpopular,andtheoneadoptedforthiswork,isbasedontheapplicationofGaborlters.Gaborltersarewell-knownquadraturelterswhichhavebeenwidelyusedinimageprocessingapplicationsincludingregistration[ 28 ],texturesegmentation[ 29 30 ]andedgedetection. Theorientationinformationateachlatticepointisthenrepresentedbyacontinuousmixturemodel.Continuousmixtureispreferredhereoverthenitemixturemodelbecauseoneneednotspecifythenumberofcomponentsinthemixtureexplicitly.Inthiscontext,wepresenttwocontinuousmixturemodels.TherstmodelrepresentsthelocalgeometryusingacontinuousmixtureoforientedGaussians.ThecontinuousmixturerepresentationiscastastheLaplacetransformofthemixingdensityoverthespaceofcovariance(positivedenite)matrices.Thismixingdensityisassumedtobein 19

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31 32 ],whoseLaplacetransformevaluatestoaclosedformexpressioncalledtheRigauttypefunction[ 23 ]:ascalar-valuedfunctionoftheparametersoftheWishartdistribution. Thesecondmodelisacontinuousmixturemodelontheorientationspace.ConsideringthefactthattheorientationinformationderivedbyGaborltersisantipodallysymmetricanditsdomainistheunitcircle,weuseacontinuousmixtureoverthemeandirectionvectorsoftheantipodallysymmetricWatsonfunctions.ThemixingdensityisassumedtobeamixtureofBinghamdensities.WederivetheclosedformsolutionforthiscontinuousmixturemodelwhichevaluatestoamixtureofmodiedBesselfunctionsoftherstkindandzerothorder. Theweightsinbothmixturemodelsarethencomputedusingasparsedeconvolutiontechnique.Inthesecondstage,weconstructtheconvolutionkernelsforsmoothing/segmentationusingtheseweightswithinthecontinuousmixturekernelofthecorrespondingmodel.Forsmoothing,wecompareourmethodswiththeedgeenhancinganisotropicdiusionmethodin[ 33 ],thecurvaturepreservingimageregularizationalgorithmin[ 12 ],andthenon-localmeans(NL-means)algorithmin[ 34 ].Forquantitativevalidation,wepresentPSNR(PeakSignal-to-NoiseRatio)valuesofthedenoisedimagesandalsotheresultsobtainedbythe\methodnoise"experimentproposedin[ 34 ]. Forsegmentation,werstpresentqualitativecomparisonswithtworecentmethods.OneofthesemethodsisasuccessfulgraphtheoreticapproachpresentedrecentlybySchoenemannandCremers[ 35 ].Theyintroducedanenergyminimizationframeworkwhichemployscurvatureconstraintsinagraph-theoreticformulationinvolvingminimumratiocyclesonproductgraphs.TheothermethodisaprominenttexturesegmentationapproachbyRoussonetal.,whopresentedavariationalformulationinalevel-setframeworkincorporatingasetoffeaturesobtainedfromthestructuretensor[ 36 ].Theirmethodis,however,restrictedtothe2-classsegmentationproblem.Incontrast,ourmethodisnotrestrictedtothe2-classsegmentationproblem. 20

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37 ]andthemeanshiftsegmentationapproachin[ 38 ].Bothofthesemethodsarestate-of-the-artmethods.Multiscalesegmentationalgorithmuseslocalgroupingcuesacrossthemultiplescalesoftheimagetocapturecoarseandneleveldetails.Thetechniquein[ 38 ]isawell-knownmodedetectionandclusteringapproachbasedonthemeanshiftprocedureinthejointspatial-rangedomain.Foranobjectivecomparison,wepresentthebestF-measurescoresachievedbythesemethodsandourmodels. Thekeystrengthandthenoveltyofourmethodslieinthepresentationofcontin-uousmixturemodelsforconvolution-based,spatiallyvarying,adaptiveapproachestosmoothingandsegmentationwhichpreservethecomplicatedgeometriesoftheobjectsinrealsceneswithoutusinganypriorinformation.Furthermore,tothebestofourknowledge,thisisthersttimethataconvolution-basedapproachisbeingemployedforfeaturepreservingsegmentation. Theremainderofthischapterisorganizedasfollows:WegiveabriefoverviewoftheGaborltersinSection 2.2 .ContinuousmixturemodelsareintroducedinSections 2.3 and 2.4 .WepresenttheconvolutionkernelsforgrayscaleimagesmoothingandimagesegmentationinSection 2.5 39 ],whichhasbeenwidelyusedforfeatureextractioninreportedliterature.DuetotheirGaussianenvelopes,theyhaveamainadvantageofachievingtheminimumspace-frequencyproductspeciedintheuncertaintyprincipleinspatialandfrequencydomainssimultaneously[ 40 41 ].Therefore,theyareoptimalintermsofspace-frequencylocalization.Additionally,theyexhibittheexibilityofbeingtunabletoanyfrequencyororientation,andtheycanformarelativelygoodapproximationofawaveletframe.Suchtuningisparticularlyappropriatewheneveroneis 21

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ThecomplexorientedGaborlterwithanon-sphericalGaussianenvelopefunctionisalinearshift-invariantlterwithanimpulseresponsegivenby:h(x;$;;g)=1 2p 2(RgTx)T1RgTxexp(i$RgTx); wherexisthespatialcoordinatevector,$isthecenterfrequencyofthelter,isadiagonalcovariancematrixwhichdeterminesthefrequencybandwidthalongtheaxesinCartesiancoordinatesandRgisarotationmatrixwhoserstcolumnisaunitvectorg.Notethattheresultinglterhasanellipsoid-likeGaussianenvelopedeterminedby,anditsorientationisgivenbyg. ThelocalorientationinformationisderivedasthemagnitudeofGaborlterresponsesfromanimageandisgivenby:G(x;$;;g)=jjh(x;$;;g)I(x)jj 2.4 ).InSection 2.3 ,wepresentamixtureofWishartsmodel,calledRigautkernel,andinSection 2.4 ,weproposeaBinghamMixtureofWatsonsmodel,dubbedtheBMWkernel,formodelingthederivedorientationinformation. 22

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wheredF=f(K)dKdenotestheunderlyingprobabilitymeasurewithrespecttosomecarriermeasuredKonPn,G(x;g)isthemagnitudeoftheGaborlterresponsewithanorientationgatthelatticepointx,G0isthemaximallterresponseatthislatticepoint.Hence,Equation 2{3 impliesacontinuousmixtureofGaussianfunctionswithf(K)beingamixingdensity.For2Dimages,n=2.Equation 2{3 canberewrittenasfollows: ThisintegralisexactlytheLaplacetransform(matrixvariatecase)oftheprobabilitymeasureFonPn,whereLfdenotestheLaplacetransformofafunctionfwhichtakesitsargumentassymmetricpositivedenitematricesfromPn.ForthedenitionoftheLaplacetransformonPn,wefollowthenotationsin[ 42 ]. 42 ]Iff(K)isafunctionofapositivedenitennmatrixK,theLaplacetransformoff(K)atacomplexsymmetricmatrixB(Notethat,inourcase,B=ggT.)isdened[ 43 ]as, 23

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44 ]Amatrix-valuedrandomvariableYwithashapeparameter,asymmetricpositivedenitematrix,andascaleparameterpinn1 2;1;issaidtohaveWishartdistributionp;,ifithasthefollowingprobabilitydensityfunction: 44 ].Itistypicallyusedforstudyingthedistributionofthecovariancematrixformedfromasamplefromamultivariatenormaldistribution[ 42 ].IfanrsmatrixZisN(0;Ir)(i.e.normallydistributedwithE[Z]=0,andIristhecovariancematrixofthevectorz=vec(ZT)wherevec(A)denotesthematrixAincolumn-wisevectorizedform.Irisrridentitymatrix.IristheKroneckerproductofIrand,whichgivesamatrixwithoccurringrtimesonthe 24

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42 ]. Moreover,substitutingthegeneralprobabilitymeasureinEquation 2{3 withtheWishartmeasurep;allowsustosolvetheintegralinaclosedform.Jianetal.[ 23 ]observedthatifthemixingdensityfischosenastheWishartdistribution,thentheLaplacetransforminEquation 2{4 existsinaclosedformyielding whereInisnnidentitymatrixandisthescaleparameteroftheWishartdistribution.Wishartdistributionisaunimodaldistribution,thereforeonecannotexpectittoresolvetheorientationalheterogeneitythatisencounteredinthepresenceofcomplexlocalgeometry.Therefore,weuseadiscretemixtureofWishartdistributionsforthemixingdensityinEquation 2{4 ,i.e.weset dF=NXi=1widpi;i:(2{8) WeassumethatalltheWishartcomponentsinthemixturehavethesameshapeparameter,i.e.fpigNi=1havethesamevalue.InChapter 3 ,wepresentthetuningcurveswithdierentvaluesforthisparameteranddiscussthemodel'ssensitivitytotheparametersettings.Inordertoestimatethenumericalscaleoftheeigenvaluesofi,werstuseasingleGaussianmodelG(x;g)=G0=exp[gTg]andthensolveforusinglinearregression.Thetraceoftheresultingisusedasagoodestimationforthetraceofiinthecontinuousmixturemodel.Wextheratiobetweenthelargerandthesmallereigenvalues(e.g.6)sothattheeigenvaluesoficanbedeterminedonapixelbypixelbasis.Furthermore,thisrotationalsymmetryleadstoatessellationwhereNunitvectorsevenlydistributedontheunitspherearechosenastheprincipaleigenvectorsofi,andhencecircumventstheproblemofdiscretizingPn. 25

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ForMmeasurementswithgj;j=1;2;:::;M,themathematicalmodel canbeformulatedasthesolutiontoalinearsystemAw=y,wherey=(G(x;g)=G0)containsthenormalizedmeasurements,AisanMNmatrixwith andw=(wi)istheunknownweightvector.Thiscanbecastasadeconvolutionproblemformulatedinageneralformas whererepresentscertainnoisemodel.Weassumethatthemeasurementerrorsarei.i.d.andnormallydistributed.SincethemaximizationofthelikelihoodfunctionunderaGaussiannoisemodelforalinearmodelisequivalenttominimizingasum-of-squareserrorfunction,aleastsquaresminimizationtechniquecanbeused.Somenumericalissuesremaintobeaddressed.Forexample,thematrixAcanbeill-conditioned,whicheectsthestabilityofthesystem.Besides,thelinearsysteminEquation 2{11 canbeunder-determinediftherearelessmeasurements(inourcase,Gaborlterresponses)thanthenumberofcomponentsinthemixtureofWisharts,i.e.ifM
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2{11 ,orimposeadditionalconstraints.SincetheweightsfwigNi=1representthe(area)fractionscorrespondingtotheheterogeneityoftheorientationwithinalatticepoint,oneexpectsthemtobenonnegative.Moreover,itisreasonabletoassumethatwhasasparsesupportsinceitisunlikelytohavealargenumberofspikesatanylatticepointinanimage.Undertheseconstraints,weuseasparsedeconvolutionmethod:non-negativeleastsquares(NNLS)minimizationwhichachievesanaccurateandsparsesolutionfor minkAwyk2subjecttow0: JianandVemurihaveinvestigatedseveraldeconvolutionschemesandhaveshownthatthisdeconvolutionmethod,namelyNNLS,outperformsmanyothermethodsinachievingaccuracy,stabilityandsparsity[ 24 25 ].Afterobtainingtheweightvectorw,thederivedlocalorientationinformationcanbeexpressedasinEquation 2{9 NotethatthelocalorientationinformationisobtainedbyMGaborlterswithorientationsontheunitcircle.Inthenextsection,weproposeacontinuousmixturemodelonSn1.Wederivetheclosedformsolutionforn=2,whichevaluatestoamixtureofmodiedBesselfunctionsoftherstkindandzerothorder. 45 { 47 ].Aunitrandomvectorvissaidtohavethe(n1)-dimensionalvonMises-FisherdistributionifithasthefollowingprobabilitydensityfunctionwithrespecttotheuniformdistributiononSn1: 27

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32 ]For0andjjjj=1, 2I0()whereI0isthemodiedBesselfunctionoftherstkindandzerothorder. 48 { 50 ]distributionisoneofthesimplestmodelswithantipodalsymmetry,anditsdensitywithrespecttotheuniformdistributiononSn1isgivenasfollows: 32 ]For0andjjjj=1, 28

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Equation 2{15 isacontinuousmixtureofWatsonfunctionswithf(v)beingamixingdensity.Solvingforthemixingdensitythatbestexplainsthegiven(derived)orientationdataG(x;g)leadstoaninverseproblem.Followingaparametricstatisticaltreatment,weimposeaBinghamdistributiononthemeandirectionvectorv.Binghamdistribution[ 51 ]isageneralizationofWatsondistribution;whileWatsondistributionsarerotationallysymmetricabouttheirmodaldirections,Binghamdistributionsarenotnecessarilyrotationallysymmetricaboutanyaxis.Thesehavetheprobabilitydensities dB(v)=1F1(1=2;n=2;)1exp[vTv];(2{16) whereisasymmetricnnmatrix,1F1isaconuenthypergeometricfunctionofmatrixargumentasdenedbyHerz[ 43 ],andBdenotesaBinghamdistributionwithparametermatrix.Infact,Binghamdistributionisthesphericalanalogueofthen-variatenormaldistribution;essentially,itcanbeobtainedbythe\intersection"ofazero-meannormaldensitywiththeunitsphereinRn: dB(v)=N(v;0;) 2vT1v]; where=1 21. WeobtainthefollowingclosedformsolutionforEquation 2{15 (inthecaseofn=2)usingaspectraldecompositionoftheparametermatrixinconjunctionwitha 29

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2cos21 2cos2]esin2[1 2cos21 2sin2]e=2d;(2{18) whereg=(cos;sin)TistheorientationoftheGaborlter,(cos;sin)Tistheprincipalaxisoftheconcentrationmatrix,andistheratiooftheeigenvaluesof. WeobservedthattheintegrandinEquation 2{18 isthekernelofsome2-wrappedvonMisesdistributionobtainedbydoublingtheanglesforn=2.ThisallowsustoexpressEquation 2{18 as 2I0(jjujj) 2);(2{19) whereI0()denotesthemodiedBesselfunctionoftherstkindandzerothorder,and 2cos21 2cos21 2sin21 2sin2375:(2{20) Furthersimplicationyields wherea=1 2.ItcanbeseenthatBinghamdistributionwithparametermatrixforn=2hasasinglemodeinthedirection(cos;sin)Twiththeconcentrationratioof.Inordertomodelorientationalheterogeneity,wechooseadiscretemixtureforthemixingdensityasinSection 2.3 ;substitutetheprobabilitymeasureinEquation 2{15 with dF=NXi=1widBi(v):(2{22) Theclosedformsolutionforthismodelcanbegivenas: 30

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whereai=i1 2.TheconcentrationvaluesfigNi=1andtheprincipalorientationsfigNi=1oftheparametermatricesfigNi=1aretreatedasthebasis,andfwigNi=1formtheunknownweightvectortobeestimated.WechooseNunitvectorsevenlydistributedontheunitsphereastheprincipaleigenvectorsoffigNi=1.Asbefore,notethatNdoesnotcorrespondtothenumberofpeaksorWatsonfunctionsmodelingthelocalstructure.Weassumethatalltheihavethesamevalue.Weexperimentedwithdierentvaluesofiandpresentedthetuningcurvesforthevariationsofthisparameterforsensitivityanalysis(seeChapter 3 ).Asiincreases,thedistributionbecomesmoreconcentratedabouttheprincipalorientationsi.TheunknownweightvectorwcanbeestimatedbysolvingalinearsystemAw=y,wherey=(G(x;g)=G0)containsthenormalizedmeasurements,AisanMNmatrixwith AsdiscussedinSection 2.3 ,weuseanonnegativeleastsquaresminimizationmethodtosolveforw. Inthenextsection,wedeveloptheconvolutionkernelsforthepurposesofimagesmoothingandimagesegmentation.Foreachmodel,RigautandBMW,wepresenttheconvolutionfunctionwhichleadstoaspatiallyvaryingkernel.NotethatthepropertyofbeingspatiallyvaryingisduetothedependencyofiintheRigautkernelandweightvectorwinbothmodelsonspatiallocation. 31

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2.3 ,i.e.dF=PNi=1widpi;i.UsingLaplacetransformoftheWishartdistribution,theRigautkernelisgivenas Whenp!1,thismodelreducestoadiscretemixtureoforientedGaussianswithweightvectorw.AlsonotethatsincetheweightvectorwandthefigNi=1changewithregardtothelocalorientationinformationateachlatticepoint,thisformulationleadstospatiallyvaryingconvolutionkernels. ForsegmentationusingtheBMWmodel,wesetthekernelatalatticepointtothefollowing: 2.4 ,i.e.dF=PNi=1widBi(v).FollowingthederivationwhichledtoEquation 2{23 ,theBMWkernelisgivenas wheretheparametermatrixiforn=2hasasinglemodeatthedirection(cosi;sini)Twiththeconcentrationratioofi.ai=i1 2,andistheanglethatthecoordinatevectormakeswiththex-axis. ThekeyideaoflevelsetmethodsistorepresentanevolvingcurveCbythezerolevelsetofaLipschitzcontinuousfunction:!R.So,C=f(x;y)2:(x;y)=0g.WechoosetobenegativeinsideCandpositiveoutside.CisevolvedusingthedescribedRigautkernelconvolutionortheBMWkernelconvolution;i.e.isconvolvedlocallywiththecorrespondingkernelinanarrowbandingalgorithm.Thelevel-setupdateequationis 32

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Theupdatestopswhennofurtherchangesinthezerolevelsetareobserved.Thefeature/junctionpreservingpropertyisachievedduetothenatureofthecharacteristicresponseoftheconvolutionkernel.WhentheconvolutionkernelR(x)forapointx(e.g.bifurcationpoint)intheimagehashighvalues,convolutionwithatthispointresultsinhighvalues;consequently,thecurvepropagatesincludingthispoint.Ontheotherhand,ifGaborlterresponsesarerelativelylowforsomepointintheimage,thenthisleadstolowvaluesintheR(x);thus,evolvingcurvedoesnotincludethispoint.Toelucidatethis,weshowtheconvolutionofthesigneddistancefunctionofanevolvingcontour(placedinFigure 2-1A )withaGaussiankernel.Initialcontourplacedinthetestimageisaclosed-curvesimilarinshapetotheobjectofinterest(seeFigure 2-1A ).AsshowninFigure 2-1C ,ajaggedclosedcontourbecomessmootherafteriterativelyconvolvingwithaGaussiankernel(=2).Ifwecontinuetheevolutionwiththiskernel(seeFigure 2-1D ),thecurveshrinkstoacircle,thentoapoint,andthennallyvanishes.SincetheGaussiankerneldoesnothaveanyinformationregardingtheimagestructure,thecurvedoesnotclingtothefeaturesofinterest.However,whentheGaussiankernelisreplacedwithourBMWkernel(orRigautkernel),thecurvecanclingtotheregionswithorientedfeatures(seeFigure 2-1G ).Initialcontourforthissyntheticimageisdeliberatelychosentobeacircularclosed-curve(seeFigure 2-1E )toshowthatthekernelR(x)canstillpulltheevolvingcontourtowardthefeaturesintheimage,asshowninFigure 2-1H Forsmoothingusingtheaforementionedkernels,theupdateequationchangesto whereI()istheimagefunctiontobesmoothed. NowwecanillustratetheproposedkernelsonthesyntheticimageofanXjunctionshowninFigure 2-2A .Weconvolvetheimagewith81Gaborlterswithorientations 33

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2-2B ).Thepeaksintheplotareconsistentwiththegroundtruthorientationsobservedatthebifurcationpoint.ThesameexperimentiscarriedoutonanoisyversionofFigure 2-2A ,whichisshowninFigure 2-2C .Thepeaksinthisnoisyversionareslightlydeviatedfromthegroundtruthorientations(seeFigure 2-2D ). ThederivedorientationdatashowninFigure 2-2 aremodeledusingtheproposedcontinuousmixtures,namelytheRigautkernelandtheBMWkernel.ThediscretizationNinthiscaseissetto321.TheweightswintheEquation 2{9 andEquation 2{23 areshowninFigure 2-3 .BothRigautandBMWmodelsgenerateonlytwopositiveweights,whichareaccuratelyfoundatornearthegroundtruthorientations. Toprovideamorerealisticcomparisonandillustration,wecreatednoisyprolesfortheXjunctioninFigure 2-2A .Thenoisevariancesareinf0:01;0:02;0:03;:::0:10g.Foreachnoiselevel,thenoisyprolehas100samples.Figure 2-4 showsthemeansandstandarddeviationsoftheerrorsinthedetectedorientationscomparedtothegroundtruthforthissimulation.Eveninthepresenceofstrongnoise,bothmodelsresultinsmallerrorsinthedetectedorientation;however,RigautmodelperformsslightlybetterthanBMW.NotethattheBMWmodelgivessmallererroranglesthantheRigautmodeldoes,inthepresenceofrelativelylownoise. Figure 2-5 illustratestheconvolutionkernelsR(x)inEquation 2{26 andEquation 2{28 thatcanbeusedforsmoothingandsegmentationafterdeterminingtheweightsshowninFigure 2-3 .Figure 2-5 canbeinterpretedasfollows:TherearetwocomponentsinboththeRigautkernelandtheBMWkernelfortheXjunction.Oneofthesetwopositivecomponentshasanorientationof45asillustratedinFigures 2-5A and 2-5B ,andtheotherhasanorientationof135asseeninFigures 2-5C and 2-5D .Noticethatthegreencolordenoteshighervaluesinthekernelcomponentcomparedtotheblue(seethecolorbarsassociatedwitheachgure).Thisgreencolorisobservedtocorrectlyindicate 34

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2-5 ;however,inthecomponentsoftheBMWkernel,aredcolorisobservedalongthegroundtruthorientations,whichmeansthattheBMWkernelhasstrongervaluescomparedtotheRigautkernel. B C D E F G H CurveevolutionexperimentswithaGaussiankernelandwiththeBMWkernel. A D )CurveevolutionwithaGaussiankernelof=2. E H )CurveevolutionwiththeBMWkernel. 35

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B C D Derivinglocalorientationinformationinasyntheticimagewithandwithoutnoise. A )Syntheticimagewithmultipleorientations. B )Derivedlocalorientationinformationatthebifurcationpointin A C )Imagewithzero-meanGaussianwhitenoiseofvariance0.07. D )Derivedlocalorientationinformationatthebifurcationpointin C 36

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B C D Weightvectorsonasyntheticimagewithandwithoutnoise. A )& C )wvectorintheRigautkernelforthederivedlocalorientationinformationinFigures 2-2B and 2-2D ,respectively. B )& D )wvectorintheBMWkernelforthederivedlocalorientationinformationinFigure 2-2B andFigure 2-2D ,respectively.Thex-axisshowstheindicesofw,whilethey-axisshowsthecorrespondingnumericalvaluesofw. 37

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MeansandstandarddeviationsoferrorsinorientationestimationonthesyntheticimageinFigure 2-2A 38

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B C D Convolutionkernelsonasyntheticimage. A C )2DviewoftheRigautkernelcomponentsatthebifurcationpointinFigure 2-2A B D )2DviewoftheBMWkernelcomponentsatthebifurcationpointinFigure 2-2A .Numericalvaluesintheconvolutionkernelsareembeddedinthegures. 39

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Inthischapter,weanalyzetheperformanceofthesmoothingandsegmentationframeworksproposedinthepreviouschapterandcompareourresultswithseveralstate-of-the-arttechniquesongrayscaleimageprocessing. 52 ]thatcontaincomplexlocalgeometriessuchasjunctions,cornersetc.Wecompareourmethodswiththeedgeenhancinganisotropicdiusionmethodin[ 33 ],thecurvaturepreservingimageregularizationalgorithmin[ 12 ],andalsothenon-localmeans(NL-means)lteringalgorithmin[ 34 ].Forobjectivevalidationofourmethods,wepresentthebestPSNRvaluesachievedbyeachmethodonthetestimageswithdierentnoiselevels.ParametersofeachmethodwerechosensoastoreachitsbestPSNRvalue.PSNRisdenedas: PSNR=10log102552 whereistheimagedomain,I0isthenoise-freeidealimage,and^Iisitsestimateobtainedfromthedenoisingmethod.Table 3-1 showsthePSNRvaluesachievedbytheedgeenhancinganisotropicdiusion(EED)[ 33 ],thecurvaturepreservingregularization(CPR)[ 12 ],theNL-meansltering[ 34 ],theRigautandtheBMWkernels,respectively. Inallofourexperiments,weusedthesamenumberofmeasurementsfortheRigautandtheBMWmodels;i.e.thesizeoftheGaborlterbank,M,is41forallexperiments.Thetessellationcontains321vectorsontheunitcircle.Hence,thesizeofmatrixAis41321,andtheunknownofthisunder-determinedsystem,whichistheweightvectorw,isa321-dimensionalvector.Notethatthissizedoesnotcorrespondtotheexpectednumberofdierentorientationsatalocation.TheconcentrationparameteristhesameforallcomponentsinthemixtureofBinghamdistributionsoftheBMWmodel.Similarly, 40

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3-15 ).Inthefollowingexperiments,thevaluesthatyieldthebestPSNRvalueswereselected. Therstexperimentisontheimageofaboatwithseveralsharpjunctions.TheNL-meansalgorithmachieveslowerPSNRvaluethaneitherRigautorBMWkernels.ThedenoisingwiththeNL-meansalgorithmseemstocausewashouteectonthisimage.Forabettervisualassessment,weshowazoomed-inregionofinterest(seeFigure 3-1 )afterrunningthecompetingalgorithmsonthewholeboatimage. Wetestedtheaforementionedsmoothingmethodsusingthe\methodnoise"criterion[ 34 ]whichisthedierencebetweenthenoisyimageandthedenoisedone.Thismeasurehasbeenpreviouslyadoptedbythecommunityforexperimentalvalidationpurposesunderdierentnames.Ifamethodissuccessful,themethodnoiseshouldlooklikerandomnoiseandcontainaslittleimageinformationaspossible.Asvisuallyevident,theNL-means,Rigaut,andBMWmodelsgeneratemethodnoiseimageswhichlooklikerandomnoiseandhavealmostnostructureinformation(seeFigures 3-2D 3-2E ,and 3-2F ).However,RigautandBMWmodelsgivehigherPSNRvaluescomparedtothecompetingmethodsasshowninTable 3-1 .Figure 3-3 showstherestorationabilityofourmethodswhenappliedtotheimageofBarbarawithaGaussiannoiseofrelativelyhighvariance(withaPSNRof12.7).Thenoisyimageissatisfactorilysmoothedbyourfeaturepreservingsmoothing,whereastheedgeenhancingdiusionmethodleadstosignicantblurring,andhencelosestheoriginallocalgeometryoftheimage.Curvaturepreservingregularizationalgorithmgivesabetterresultalthoughsomeregionswereunevenlysmoothed(seethetexturesonthetrousers).TheNL-meanslteringresultsinthelossofmanydetailsonBarbara'sfaceandotherplaces.Thedicultyinthisimageisthatthescenehassignicantheterogeneityintexture.Theorientationalcomplexityofthelocaltextureisaccuratelycapturedby 41

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ThemethodnoiseexperimentfortheBarbaraimageisillustratedinFigure 3-4 .TheimagestructurecanbeeasilyseeninFigure 3-4D obtainedusingtheNL-meansmethod,alsothePSNRofthisdenoisedimageislowcomparedtoourmethods.However,weobservethatthemethodnoiseofthesmoothingachievedbytheBMWmodel(seeFigure 3-4F )containsinsignicantimageinformationeventhoughthenoiselevelofthetestimageissignicantlyhigh. 5 ]withassociatedgroundtruthsegmentationsandaquantitativeevaluationofoursegmentationmethods. Thersttestimage(seeFigure 3-5 )containstwozebrasinanaturalscene.Thelow-contrastregionswhereasuccessfulsegmentationisquitedicultwhenusingstandardactivecontourorgraphcutsbasedmethodswereclearlysegmentedbybothofourtechniques;e.g.seethefeetofthetwozebras.Notethepresenceofjunctionsinthezoomed-inview. Inthenextexample,wecompareourmethodswithaveryrecentandsuccessfulgraphtheoreticapproachbySchoenemannandCremers,andthereforewechoose 42

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3-6 depictstheresultsfrom5dierentsegmentationmethods:(i)thepiecewiseconstantversionoftheMumford-Shahsegmentationscheme[ 53 ],(ii)theMumford-Shahscheme[ 15 ],(iii)elasticratiotechniquebySchoenemannetal.[ 35 ],(iv)ourRigautkerneland(v)ourBMWkernelrespectively.Oncarefulexamination,Figures 3-6D and 3-6E areslightlybetterinsegmentation(seethedetailsaroundthetail)than 3-6C .TechniquesinFigures 3-6A and 3-6B failinthisdicultscene. InFigure 3-7 ,weshowaleopardimagewhichhasbeenexperimentedwithbymanyothersegmentationapproachesinthepast.Thisisatexturedimage,andthereforewewouldliketocompareourmethodswitharecenttexturesegmentationmethodproposedbyRoussonetal.[ 36 ].Recoveryoftheleopard'stail,whichwasmissedoutintheothercompetingapproaches,provesourmethodstobefullycompetitivetotherecentapproaches[ 36 54 ].AlsonotethattheBMWmethodachievesabetterresultincapturingtheposteriorlegoftheleopard. InFigure 3-8 ,weshowthesegmentationresultsonanotherimagetakenfromSchoenemannandCremers[ 35 ].Notetheaccuratesegmentationofthesling-onstrapattachedtothecaseinFigures 3-8B and 3-8C ascomparedtothatobtainedbythecompetingmethod[ 35 ]showninFigure 3-8A InFigure 3-9 ,wepresentmoreexperimentalresultsobtainedbyourmodel.WechoseseveraldicultscenesfromtheBerkeleySegmentationDataSet.Notethatthesurferwasaccuratelysegmentedoutofthebumpyoceanbackground.Similarly,theleopardandthetreesweresegmentedwhilepreservingthefeaturesproperly.Weobtainedresultssignicantlyclosetothesegmentationsperformedbyhumansubjects. Figures 3-10 through 3-14 containmoresegmentationresultsonseveralimagesfromtheBerkeleySegmentationDataSet[ 55 ].Forcomparisonpurposes,weprovidethesegmentationresultsperformedbyhumansubjects,themultiscalenormalizedcutimagesegmentationalgorithmin[ 37 56 ],andthemeanshiftalgorithmin[ 38 ],alongwithour 43

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3-2 showsthehighestF1-measure(orDice'sCoecient)valuesobtainedbythemultiscalenormalizedcutalgorithm,themeanshiftsegmentation,theRigautkernel,andtheBMWkernelmethodsonthisimageset. Towardsaquantitativevalidationalongwithasensitivityanalysis,Figure 3-15 presentstheF-measurescoresoftheabovementionedsegmentationmethodson20imagesdrawnfromBerkeleySegmentationDataSet,includingtheimagesetsinFigures 3-9 through 3-14 44

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ExperimentationshowedthattheF-measurescoresuctuatesignicantlywithrespecttothenumberofregionsparameterinthemultiscalenormalizedcutsalgorithm,andalsowithrespecttotheresolutionparametersofthemeanshiftalgorithm.However,thescoreschangeonlyslightlywithrespecttothechangesinpoftheRigautkernel,depictingtheinsensitivityofthesegmentationresultstothisparameter.AsimilarargumentisvalidforchangesintheconcentrationparameteroftheBMWkernel.Ontheotherhand,weobservedchangesinF-measurescoresduetothethresholdparameterinthederivedorientationinformation.Gaborlterresponsesbelowthethresholdaresetto0.Thishelpstodeterminethelevelofdetaildesiredinthesegmentation.ThethresholdparametercanbeseenasascaleparameterforGaborlters,thereforeitisreasonabletoobservechangesrelatedtothisparameter.Alowthresholdpercentageresultsinover-segmentation,leadingtohighrecall,lowprecisionvalues,andconsequentlylowF-measurescores.Nevertheless,thisparameterdoesnotcausetheF-measurescorestouctuate.Therefore,ahighthresholdvaluecanbeexpectedtoyieldsegmentationsofthemoredominantobjectsinascene,comparedtoalowervalue.Inourexperiments,weobservedthatforagiventhresholdvalue,theF-measurescoreschangedonlyslightlywithrespecttothechangesinporchangesin.Thisshowsthatourmethodshavelesssensitivitytotheparametersettingsofourmodels.AlsonoticethattheF-measurescoresobtainedbyourmethodsareconsiderablyhigherandstablerthanthoseachievedbythecompetingmethods. 45

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B C D E F Smoothingresultsontheboatimage[ 52 ]. A )Zoomed-inregioninanoisyversionofboatimage(Gaussiannoise,=0,=0:14), B )outputoftheedgeenhancinganisotropicdiusion, C )outputofthecurvaturepreservingregularizationmethod, D )outputoftheNL-meanslter, E )outputoftheRigautmodel, F )outputoftheBMWmodel. 46

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B C D E F Methodnoiseexperimentondenoisingoftheboatimage[ 52 ]. A )Boatimage(512x512pixels)withaGaussiannoiseof=0,=0:14, B )themethodnoiseoftheedgeenhancinganisotropicdiusion, C )themethodnoiseofthecurvaturepreservingregularizationmethod, D )themethodnoiseoftheNL-meanslter, E )themethodnoiseoftheRigautmodel, F )themethodnoiseoftheBMWmodel. 47

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B C D E F DenoisingresultsontheBarbaraimage[ 52 ]. A )BarbaraimagewithGaussiannoiseofzeromeanand0.07variance.Denoisedimagesobtainedfrom: B )theedgeenhancinganisotropicdiusion, C )thecurvaturepreservingregularizationmethod, D )thenon-localmeansltering, E )theRigautmodel, F )theBMWmodel. 48

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B C D E F MethodnoiseexperimentondenoisingoftheBarbaraimage[ 52 ]. A )Barbaraimage(ofsize256x256pixels).Methodnoiseof: B )theedgeenhancinganisotropicdiusion, C )thecurvaturepreservingregularizationmethod, D )theNL-meansltering, E )ourRigautkernelmethod,and F )ourBMWkernelmethod. B Segmentationresultofatexturedimageusing A )ourRigautkernel-basedconvolutionlter, B )ourBMWkernel-basedconvolutionlter. 49

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C D E Segmentationresultsobtainedfrom A )piecewiseconstantMumford-Shah, B )piecewisesmoothMumford-Shah, C )elasticratiobySchoenemannandCremers[ 35 ], D )theRigautkernel, E )theBMWkernel.(Figures A B & C werereproducedfrom[ 35 ],(Figure7inpage6),withcopyrightpermissionofc2007IEEE.) 50

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B C Segmentationoftheleopardimage.Resultsobtainedfrom A )Roussonetal.[ 36 ], B )ourRigautkernel-basedconvolutionlter, C )ourBMWkernel-basedconvolutionlter.(Inputimage:courtesyofJunmoKim) B C Segmentationexperimentonacaseimage[ 35 ].Resultsobtainedby A )thetechniquein[ 35 ], B )ourRigautkernelmethod, C )ourBMWkernelmethod. 51

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SegmentationresultsoftheBMWkernelontheBerkeleySegmentationDataSet[ 5 ].(rst&thirdrows)Segmentationsperformedbyhumansubjects(fromthegroundtruthdatainBerkeleySegmentationDataSet[ 5 ]),(second&fourthrows)SegmentationsobtainedbyourBMWkernel. 52

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B C D E F SegmentationexperimentonatestimagefromtheBerkeleySegmentationDataSet[ 5 ]. A )Originalimage.Segmentationsperformedby: B )ahuman(takenfromtheBerkeleySegmentationDatasetandBenchmark[ 5 ]), C )themultiscalenormalizedcutimagesegmentationalgorithmin[ 37 ], D )themeanshiftalgorithmin[ 38 ], E )ourRigautkernel, F )ourBMWkernel. 53

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B C D E F Segmentationresultsontheplaneimage[ 5 ]. A )Originalimage.Segmentationsperformedby: B )humans(takenfromtheBerkeleySegmentationDatasetandBenchmark[ 5 ]), C )themultiscalenormalizedcutimagesegmentationalgorithmin[ 37 ], D )themeanshiftalgorithmin[ 38 ], E )ourRigautkernel, F )ourBMWkernel.(Theimagesareofsize481x321pixels.) 54

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B C D E F Segmentationexperimentonthebirdimage[ 5 ]. A )Originalimage.Outputsobtainedfrom B )ahumansegmentation(takenfromtheBerkeleySegmentationDatasetandBenchmark[ 5 ]), C )themultiscalenormalizedcutimagesegmentationalgorithmin[ 37 ], D )themeanshiftalgorithmin[ 38 ], E )ourRigautkernel, F )ourBMWkernel.(Theimagesareofsize481x321pixels.) 55

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B C D E F Segmentationexperimentontheelephantsimage[ 5 ]. A )Originalimage.Resultsobtainedfrom: B )humansegmentations(takenfromtheBerkeleySegmentationDatasetandBenchmark[ 5 ]), C )themultiscalenormalizedcutimagesegmentationalgorithmin[ 37 ], D )themeanshiftalgorithmin[ 38 ], E )ourRigautkernel, F )ourBMWkernel.(Theimagesareofsize481x321pixels.) 56

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B C D E F Segmentationexperimentontheowerimage[ 5 ]. A )Originalimage.Segmentationsperformedby: B )humans(takenfromtheBerkeleySegmentationDatasetandBenchmark[ 5 ]), C )themultiscalenormalizedcutimagesegmentationalgorithmin[ 37 ], D )themeanshiftalgorithmin[ 38 ], E )ourRigautkernel, F )ourBMWkernel.(Theimagesareofsize481x321pixels.) 57

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PSNRvaluesfordenoisedimagesindB ImageEEDCPRNLMeansRigautBMWNoisyImage Boat22.527.026.727.827.917.2Barbara19.220.820.722.122.712.7Clown18.425.422.326.827.315.7Cameraman24.227.825.928.328.019.5House28.731.831.732.131.920.3 Table3-2. ImageMS-NcutsMeanShiftRigautBMW Birds0.450.670.720.69Plane0.510.610.810.88Hawk0.470.540.810.84Elephants0.520.680.830.85Flowers0.480.620.860.82 58

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59

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57 ].Toretrievethelocalgeometryofvectorvaluedimages,Weickertproposedtoextendhiscoherenceenhancingdiusionusingacommondiusiontensorforallimagechannels[ 6 ].Later,Kimmeletal.introducedadiusionPDEcalledBeltramiow[ 58 ]whichinvolvestheminimizationoftheglobalareaofthesurfacerepresentingthevectorvaluedimagewithrespecttothesurfacemetric.In[ 59 ],Tangetal.extendedtheirdirectiondiusionframeworktosmoothingonlythechromaticitychannelofcolorimages,andcombineditwiththescalaranisotropicdiusionappliedtothebrightnesschannelofthecolorimage.SeveralextensionsoftheMumford-Shahfunctionalhavebeenproposedin[ 60 61 ]forvariationalrestorationandedgedetectionofcolorimages.Theseextensionsarebasedonageometricmodelofimagesasmanifolds. 60

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12 ].Formoreonmultichannelimagerecovery,wereferthereaderto[ 62 63 ]. Avastamountofresearchhasbeenperformedonimagesegmentationduringthepastthreedecades;variational[ 14 22 ],statistical[ 64 { 66 ],combinatorial[ 67 68 ],curveevolutionbased[ 13 18 { 21 69 70 ]techniquesareonlysomeexamples.Ontheotherhand,colorimagesegmentationisarelativelynascentareaincomputervision.Theliteratureoncolorimagesegmentationisnotasextensiveasthatongray-valuedimagesegmentation.Somepublishedmethodsdirectlyapplytheexistinggraylevelsegmentationmethodstoeachchannelofacolorimageandthencombinetheoutputsinsomewaytopresentanalsegmentationresult.Inthecolorsnakesmodel[ 71 ],SapiroextendsthegeodesicactivecontourmodeltothecolorimagesbasedontheideaofevolvingthecontourwithacouplingtermbasedontheeigenvaluesoftheRiemannianmetricoftheunderlyingmanifold.Chanetal.extendtheChan-Vesealgorithmforscalarvaluedimagestothevector-valuedcase[ 72 ].Intheirwork,inadditiontotheMumford-Shahfunctionaloverthelengthofthecontour,theminimizationinvolvesthesumofthettingerrorovereachcolorcomponent.Assumingnocorrelationbetweenfeaturechannels,Broxetal.proposeanenergyminimizationframeworkwheretheenergyfunctionalisthesumoftheconditionalprobabilitiesofthecomputedfeaturesofanimage:colorchannels,opticalowcomponentsandtexturechannels[ 73 ].In[ 74 ],colorimagesarehandledasthreeseparatemonochromeimages.In[ 75 ],anextensionoftheVoronoitessellationtopseudo-metricspacesisappliedtocolorimages,whereEuclideandistanceinLabcolorspaceisusedtocomputethecolordierences.Bertellietal.presentavariationalframeworkbasedonpairwisepixelsimilarities;theyuseL2distancesintheLabcolorspacewithoutanycouplingbetweenthechannels[ 76 ].In[ 38 ],amodedetectionandclusteringapproachemployingthemeanshiftprocedureispresentedinthejointspatial-rangedomainwithaEuclideanmetric. 61

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77 78 ].Thekeyinnovationofourworkhereisauniedapproachtocolorimagerestorationandsegmentationusing(i)anovelquaternionGaborlter(QGF)toextractthelocalorientation,and(ii)continuousmixturesontheunitspheretomodelthederivedorientation[ 79 ]. Amajorturningpointintheeldofmathematics,specically,inalgebra,wasthebirthofnoncommutativealgebraviaHamilton'sdiscoveryofquaternionsin1843.Thisdiscoverywastheprecursortonewkindsofalgebraicstructuresandhashadanimpactinvariousareasofmathematicsandphysics,includinggrouptheory,topology,quantummechanicsetc.Morerecently,quaternionshavebeenemployedinbioinformatics,computergraphics[ 80 ],navigationsystems[ 81 ]andcodingtheory[ 82 ].Incomputergraphics,quaternionrepresentationoforientationsfacilitatedcomputationallyecientandmathematicallyrobustapplications(suchasavoidingthegimballockinEuleranglerepresentation).Inimageprocessing,quaternionshavebeenusedtorepresentcolorimages[ 83 84 ].AnimagesegmentationmethodthatemploysquaternionicextensionofPCAwiththequaternionrepresentationofcolorhasbeenpresentedin[ 85 ].Huietal.usedstandardGaborltersoncolorimagesrepresentedusingreducedbiquaternionstoperformimagesegmentation[ 86 ].Quaternionicrepresentationofcolor,togetherwiththeextensionoftheFouriertransformtohypercomplexnumbers,hasledtoapplicationsincolorsensitiveltering[ 87 ],edgedetection[ 77 88 ]andcrosscorrelationofcolorimages[ 78 ].ThehypercomplexFouriertransformwasrstdenedbyDelsuc[ 89 ]innuclearmagneticresonance.Later,dierentdenitionsforthequaternionicFouriertransform(QFT)havebeenintroducedin[ 90 ]and[ 91 ]independently.Basedontheirdenitionof 62

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92 ].TheyextendedtheGaborlterbyusingtwoquaternionbasisiandjtoreplacethesinglecomplexnumberiinthedenitionofthecomplexGaborlter.However,theydidnotapplytheirltertocolorimagessincetheirdenitionofQFTassociatestheimaginaryunitsiandjtothelocalorientationsintheimageplane,whichhasnorelationshiptothecolorchannelsinacolorimage.In[ 93 ],analternativedenitionforQFTwasproposed,whichutilizessimpleformulaefortheFouriertransformofcomplex-valuedsignalsthatcanbecomputedeciently.WefollowthisalternativeQFTtointroduceanoveldenitionfortheQuaternionicGaborFilterswhichcanbeemployedtoextractfeaturesfromcolorimageswithoutconictinginterpretationsbeingassignedtothehypercomplexunits[ 79 ].WefurthertestQGFsfortheoptimalitywithrespecttothetwo-dimensionaluncertaintyprinciple.AnothercontributionofthischapteristheformulationofcontinuousmixturemodelswhichincorporatetheQGF-derivedlocalorientationintothesmoothingandsegmentationkernels. Continuousmixturemodelshavebeenpresentedinvariouscontexts[ 23 { 27 ].Inthischapter,weproposecontinuousmixturestomodelthelocalorientationinformationextractedviatheproposedQGFs.Weintroducetwosuchmodelsandderiveclosedformsolutionsforthecontinuousmixtureintegrals,whicharelateremployedindevelopingconvolutionkernelsforfeature/detailpreservingrestorationandsegmentationofcolorimages[ 79 94 ].Theproposedspatially-varyingkernelsdonotuseanypriorinformation,andyetyieldhighqualityresults. Therestofthischapterisorganizedasfollows:InSection 4.2 ,webrieydescribethequaternionalgebraandquaternionFouriertransform,andtheninSection 4.3 wepresentanoveldenitionforQGFs.InSection 4.4 ,weintroduceacontinuousmixturemodelforquantifyingthederivedorientationinformationtoperformcolorimagesmoothing. 63

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4.5 ,weproposeanothercontinuousmixturemodelonthederivedorientationinformationforuseinsegmentation. Higherdimensionalcomplexnumbersarecalledhypercomplexanddenedas whereikisorthonormaltoilfork6=linanN+1dimensionalspace.TheHamiltonianquaternionsareunitaryR-algebra;thebasicalgebraicformforaquaternionq2His: whereq0;q1;q2;q32R,theeldofrealnumbers,andi;j;karethreeimaginarynumbers.Hcanberegardedasa4-dimensionalvectorspaceoverRwiththenaturaldenitionofadditionandscalarmultiplication.Thesetf1;i;j;kgisanaturalbasisforthisvectorspace.Hismadeintoaringbytheusualdistributivelawtogetherwiththefollowingmultiplicationrules: IfwedenotethescalarandvectorpartsofaquaternionqbySqandVq,respectively,theproductoftwoquaternionsqandpcanbewrittenas wheretheandindicatethevectordotandcrossproducts,respectively.Theconjugateofaquaternion,denotedby,simplynegatesthevectorpart,q=q0q1iq2jq3k.Thenormofaquaternionqiskqk=p 64

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Usingtheinnerproduct,theanglebetweentwoquaternionscanbedenedas: cos=Sqp and isthelengthoftheshortestgeodesicbetweentwounitquaternionsqandp.Itcanalsobecalledtheangleofrotationmetricforquaternions.Anyquaternioncanbewritteninpolarform whereisaunitpurequaternion. Quaternionrepresentationofcolorimagepixelshasbeenproposedindependentlyin[ 83 84 ].Theyencodethecolorvalueofeachpixelinapurequaternion.Forexample,apixelvalueatlocation(n;m)inanRGBimagecanbegivenasf(n;m)=R(n;m)i+G(n;m)j+B(n;m)kwhereR;GandBdenotethered,greenandbluecomponentsofeachpixel,respectively.This3-componentvectorrepresentationyieldsasystemwhichhaswell-denedandwell-behavedmathematicaloperationstoapplyoncolorimagesholistically. 89 ].Later,Ell[ 90 ]andBulowandSommer[ 91 ]independentlyintroducedthequaternionFouriertransform,respectivelyas 65

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In[ 93 ],anotherdenitionforQFTwasproposedwiththemotivationofgeneralizingthestandardcomplexoperationalformulaeforconvolutionincolorimages: whereisaunitpurequaternion.Intherestofthisdissertation,wewillfollowthisabovedenition.ForcolorimagesinRGBspace,ischosenas1 FirstweprovetheModulationTheoremforthecontinuousQFT. Proof. Inthefollowing,weintroduceanovelQuaternionicGaborFilter. 66

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4-1 depictsaquaternionicGaborlterwithorientationforillustrationpurposes. LetusconsidertheQFTofanisotropicGaussianin2D.QFTofananisotropicGaussiancanbeevaluatedsimilarly. 22e2(ux+vy)dx=NZRZRe(x+2u2)=22dxey2 22e2222u2e2vydy=N2ZRey2 22e222u2e2vydy Aftersomealgebraicmanipulations,weobtainthatQFTfg(x;y)g=e222(u2+v2),i.e.anun-normalizedGaussianin(u;v)-space,withbeingaconstant.Hence,usingtheQFTofaGaussiantogetherwiththeModulationTheoremforQFT,wecanconcludethatquaternionicGaborltersshownaboveareshiftedGaussiansinthehypercomplexfrequencydomain,i.e.if 2x2y2 2y2e2(u0x+v0y);(4{14) thentheQFToffis: Figure 4-1 depictsaquaternionicGaborlterwithorientation=4,forillustrationpurposes.ForanapplicationofQGF,considertheFigure 4-2 .IfweapplyaQGFwithanorientationof=4toanimage,thenweobtainhighresponseswhereverthereare=4orientedfeatures.Figure 4-2B illustratesthemagnituderesponseofsuchahorizontally 67

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InanalogytoGaborlters,weconsiderthequaternionicanalyticsignalwhichhasbeendenedin[ 92 ]toworkwithQGFs.Forpositivefrequenciesu0andv0,themainamountoftheGaborlter'senergyinEquation 4{15 isintheupperrightquadrant.Hence,QGFsprovideapproximationtoquaternionicanalyticsignal.InordertoshowthatQGFsareoptimallylocalizedinbothquaternionicspatialandfrequencydomainssimultaneously,wewillsimplyextendthedenitionoftheuncertaintiesforquaternion-valuedfunctionswhichhasalsobeendonein[ 95 ].Thespatialandfrequencyuncertaintiesxanduofaquaternion-valuedsignalf(x;y)canbegivenas: (x)2=RR2f(x;y)f(x;y)x2dxdy TheuncertaintiesoftheQGFgiveninEquation 4{14 canbeevaluatedusingtheabovedenitionsandtheiranalogsforyandvtobe x=x 2p 2p Thus,QGFsareshowntoachievetheminimumproductofuncertaintiesdenedin[ 40 ] xyuv=1=162:(4{18) InFigure 4-3 ,weshowthescalarandvectorpartsofthesumresponsesobtainedfromapplicationof13orientedQGFstotheimageofBarbara.Weconvolvethequaternionrepresentationofthecolorimagewitheachquaternion-valuedlter,andthenillustratethesumovereachunit.Notethattheconvolutioninvolvesquaternionmultiplication.ColortransitionsinthecoupledchannelsGB,RBandRGshowthemselvesinthecomponentsofthevectorpartoftheQGFresponses. 68

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4-4 ,wedemonstratethattheproposedQuaternionicGaborFilterscanextractthelocalorientationinformationfromaconstantluminanceimageaswell.Figure 4-4A showsasyntheticcolorimagewhereallpixelshavethesameluminancevalue,butthechromaticityinsidetheobjectdiersfromthechromaticityoutside.Theluminancechannelshowsthatallpixelshavethesamevalue(seeFigure 4-4B ).Weapplied10QGFstothequaternionrepresentationofthiscolorimage.Thesumofthemagnituderesponsesof10QGFsisshowninFigure 4-4D .Althoughablack-and-whiteversion(Figure 4-4C )oftheinputimageisauniformgraywithoutanychangesinorientation,theproposedQGFssuccessfullyderivetheorientationinformationinthecolorversion,showingthattheyarewellsuitedforanalyzingcolorimagesandtheresultisnotagrayscaleimageprocessing. WehavechosentheunitpurequaterniondirectioninQGFas1 4-5A showsacolorimagewhere(R+G+B)=3isthesameforallpixels.AsshowninFigure 4-5C ,theproposedframeworkcanaccuratelyextracttheorientationinformation. NotethatboththeimageinFigure 4-4 andtheimageinFigure 4-5 cannotbesegmentedordenoisedusingagray-levelimageprocessingtechniquebecausetheobjectsinthecolorimagesdonotappearintheirgray-valuedversions.However,ourcolorimageprocessingframeworkcandetectobjectsinsuchimages,yieldingaccuratesegmentationandsmoothinginthelatersteps,andthisframeworkisnotsensitivetoequalluminance. 69

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wheredF=f()ddenotestheunderlyingprobabilitymeasurewithrespecttotheuniformdistributiondonS1.GvH;v=i;j;kdenotethei;jandkcomponentsofthevectorpartinthequaternion-valuedresponse,respectively.Weonlymodelthecomponentsofthevectorpart.Scalarpartofthelterresponsecanberegardedasasmoothedsecondderivativeoftheinitialimage,andcanbeofuseforedgedetection. Inordertoavoidanill-posedinverseproblemwhichrequiresrecoveringadistributiondenedonthecirclegiventhemeasurementsGiH(x;;),weimposeamixtureofvonMisesdistributionsonasaprior.ThevonMisesdistributionshaveasignicantroleinstatisticalinferenceonthecircle,analogoustothatofthenormaldistributionsontheline.Forstatisticalpurposes,anyvonMisesdistributioncanbeapproximatedbyanormaldistributionwrappedaroundthecircumferenceofthecircleofunitradius.isdistributed 70

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1 2I0()ecos()d;(4{20) whereandarethemeandirectionandtheconcentrationparameter,respectively.I0isthemodiedBesselfunctionoftherstkindandzerothorder[ 32 ]. Thisdistributionisunimodalandsymmetricabout=.determinesthedegreeoftheclusteringaroundthemode;i.e.arelativelylargervalueofmeanshigherclusteringaroundthemode.Inordertohandleorientationalheterogeneityweneedamultimodaldistribution.Therefore,wechoosethepriortobeadiscretemixtureofvonMisesdistributions: dF=NXn=1wn1 2I0(n)encos(n)d:(4{21) PluggingthismeasureintoEquation 4{19 ,weobtainourmodelgivenasfollows: 2I0(n)encos(n)ecos()d:(4{22) However,notethatthisisstillacontinuousmixturemodel.Nherecorrespondstotheresolutionofthediscretizationofthecircle;itdoesnotcorrespondtothenumberofmodes(peaks)characterizingthelocalgeometryorthenumberofdominantlocalorientations.WeobservedthatthekernelofthevonMisesdistributioncanbeutilizedtoderiveaclosedformsolutionforthecontinuousmixtureintegral,leadingto: WecanformulatethecomputationofthisanalyticformasthesolutiontoalinearsystemAw=y,wherey=fGvH(x;;m)gMm=1containsthemeasurementsobtainedviaanapplicationofMQGFstotheimage,AisanMNmatrixwith 71

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minkAwyk2subjecttow0: Asparsesolutioniswhatisexpectedateachimagelatticepointsincelocalimagegeometrydoesnothavealargenumberofedgesmeetingatajunction.Oncewisestimatedforthegivendataateachlatticepoint,wecanconstructtheconvolutionkernelforcolorimagesmoothing.TheupdateequationforimagechannelIv;v=R;G;Bisgivenasfollows:Ivt+1(x)=Ivt(x)Qv(x); whereQv(x)istheconvolutionkernelontheright-handsideofEquation 4{23 forthecorrespondingGvH(),wvistheweightvectorobtainedfromEquation 4{25 usingthecorrespondingGvH()measurements,andtheorientationistheanglethatthecoordinatevectorxmakeswiththex-axis.Thisformulationyieldsaspatiallyvaryingconvolutionkernelbecausethewvectordependsonlocation;itisestimatedateachlatticepointxinanimage.Moreover,theweightswandhencetheconvolutionkernelisdierentforeachcolorchannelIi.Notethatthisframework(namedasQGvM{QuaternionicGaborlterswithvonMisesdensity)handlesthecouplingbetweenthecolorchannelsthroughtheapplicationofquaternionicGaborlterstothequaternionrepresentationofthecolorimage. 72

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Thespaceofunitquaternions isthe3-sphereinH,itformsagroupundermultiplicationandpreservesthehermitianinnerproduct.Anappropriatechoiceforthekernelfunctionsisthereforeexp(cos(d(q;p))),whered(q;p)=2cos1(S(qp))isthelengthoftheshortestgeodesicbetweenquaternionsqandp,asgivenin( 4{7 ).Thustheproposedmodelisgivenby, wheredF:=f(q)dqdenotestheunderlyingprobabilitymeasurewithrespecttotheuniformdistributiondqonS3.GmaxHisthemaximalmagnituderesponseamongallresponsesatanimagelocation.Inordertoavoidanill-posedinverseproblemwhichrequiresrecoveringadistributiondenedonthemanifoldofunitquaternionsgiventhemeasurementsGH(x;;),weimposeamixtureofBinghamdistributionsonqasaprior.Manifoldoftheunitquaternionsdouble-coversSO(3).Double-coveragecanbeinterpretedasantipodal-symmetry;thus,Binghamdistributionisanaturalchoiceforquaternionpriors.Forstatisticalpurposes,Binghamdistributionischaracterizedasthehypersphericalanalogueofthen-variatenormaldistribution;essentiallyitcanbeobtainedbythe\intersection"ofazero-meannormaldensitywiththeunitsphereinRn.Letqbea4-dimensionalrandomunsignedunitdirection.qisdistributedasBL;AifithastheBinghamdensity[ 96 ]givenby, 73

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43 ]. Here,wemakeausefulobservationwhichhelpsderivingananalyticsolutionfortheproposedcontinuousmodel:UsingtherelationshipbetweenS3andSO(3),Prentice[ 96 ]hasshownthatqhasaBinghamdensityifandonlyifthecorrespondingrotationmatrix,Q,inSO(3)hasamatrixFisherdistribution.Arandom33rotationmatrixQissaidtohaveamatrixFisherdistributionFFifithasthefollowingpdf: 2dF;(4{31) wherePistherotationmatrixcorrespondingtotheunitquaternionwiththeanglebeingtheorientationoftheQGFandtheaxisbeing=1 dF=NXi=1wi0F1(3=2;FiFiT=4)1etr(FiTQ)dQ(4{32) isadiscretemixtureofmatrixFisherdensitiesovertherotationmatrixQwithrespecttotheuniformdistributiononSO(3).WechoosetochangethepriortothismixtureofmatrixFisherdensitiesbecausethematrixFisherdensityisunimodalandwillnotbeabletohandleorientationalheterogeneity.Onceagain,themodelinEquation 4{31 isstillacontinuousmixturemodel.NherecorrespondstotheresolutionoftheSO(3)discretizationandnotthenumberofdominantlocalorientations.WeobservedthatthekernelofthematrixFisherdistributioncanbeutilizedtoderiveaclosedformsolutionfor 74

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2;1 4FiP Inordertocomputethisanalyticform,wecanwriteitasthesolutiontoalinearsystemAw=y(asinSection 4.4 ),wherey=fkGH(x;;j)kgMj=1=GmaxHcontainsthenormalizedmeasurementsobtainedviaanapplicationofMQGFstothecolorimage,AisanMNmatrixwith 2;1 4hFiPj andw=(wi)istheunknownweightvector.Theweightsinthemixturecanbesolvedusingasparsedeconvolutiontechnique,anon-negativeleastsquares(NNLS)minimizationwhichyieldsanaccurateandsparsesolution.Oncewisestimatedforthegivendataateachlatticepoint,wecanconstructtheconvolutionkernelforcolorimagesegmentation.WerepresentanevolvingcurveC(inacurveevolutionframework)bythezerolevelsetofaLipschitzcontinuousfunction:!R.So,C=f(x;y)2:(x;y)=0g.WechoosetobenegativeinsideCandpositiveoutside.Cisevolvedusingthefollowingupdateequation: whereK(x)istheconvolutionkernelobtainedfromEquation 4{33 bysettingthematrixPtotherotationmatrixcorrespondingtotheanglethatthecoordinatevectorxmakeswiththex-axis.Notethatthisformulation(namedasQGmF{QuaternionicGaborswithmatrixFisherdensity)yieldsaspatiallyvaryingconvolutionkernelsincethewvectorisestimatedateachlatticepointinanimage.Figure 4-6 illustratesthe33convolutionkernelsfordierentlocationsinarealimage. 75

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B C AQuaternionicGaborlter(sizeoftheltermaskis128128)withanorientationof=4: A )thescalarpartofthelter, B )onecomponentfromthevectorpartofthelter, C )2Dviewofthescalarpart. B C D E F ApplicationoftheQGFinFig. 4-1 to A )acolorimage(fromtheBerkeleyDataSet[ 55 ]): B )themagnituderesponse, C F )thescalar,i,j,andkpartsofthelterresponse,respectively. 76

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B C D E F G H I ImageofBarbaraisquaternion-convolvedwithQGFsofdierentorientations: A )colorimage[ 52 ], B )sumofthemagnituderesponses, C F )scalar,i,j,kpartsofthesumoftheQGFresponses,respectively, G I )GB,RB,RGimages,respectively. 77

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B C D ApplicationofaQuaternionicGaborFilterbankacrossequalluminance: A )asyntheticcolorimagewheretheobjectandthebackgroundareofequalluminance, B )luminancechannel, C )agrayscaleversionof A D )thesumofthemagnituderesponsesof10QGFsappliedtothecolorimagein A B C AQuaternionGaborFilterexperimentonasyntheticcolorimagewithequal(R+G+B)=3values. A )Asyntheticcolorimagewhere(R+G+B)=3isthesameeverywhere. B )(R+G+B)=3values. C )ThesumofthemagnituderesponsesoftheQGFsappliedtothecolorimagein A 78

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Convolutionkernelsonarealcolorimage[ 97 ].Numericalvaluesoftheconvolutionkernelsforthreelocationswithdierentorientationalheterogeneityareshownintheguretogetherwithavisualillustration. 79

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52 ].Wecompareourdenoisingresultswiththreeprominenttechniques:Weickert'scoherenceenhancingdiusion(CED)forcolorimages[ 6 ],theBeltramiowproposedby[ 58 ],andthecurvaturepreservingregularization(CPR)proposedby[ 12 ].Inthedenoisingexperiments,foreachalgorithmtheoutputsthathavethehighestPSNRvaluesareshown.ParametersofeachmethodwerechosensoastoreachitsbestPSNRvalue.WecomputethePSNRontheRGBchannelsofthecolorimage.WealsoreportthePSNRvaluesontheluminancechanneloftheYCbCrrepresentationoftheRGBimage,sincethehumaneyeismoresensitivetolumainformationinacolorimage.PSNRforRGBdomainisdenedas:PSNR=10log102552 (5{1)MSE=1 3jjXx2Xv=R;G;B(Iv0(x)^Iv(x))2 Inallofourexperiments,weusethesamenumberofmeasurementsforourmodel;i.e.thesizeoftheQuaternionGaborFilterbank,M,was21forallexperiments.N,theresolutionofthediscretizationoftheunitcircleforthemixingdensity,wassetto81.Hence,thesizeofmatrixAis2181,andtheunknownofthisunder-determinedsystem,whichistheweightvectorw,isan81-dimensionalvector.Notethatthissizedoesnotcorrespondtotheexpectednumberofdierentorientationsatapixel.Theconcentration 80

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5-1 .OurunsupervisedandadaptivemethodproducesthehighestPSNRsinallcases. InFigure 5-1 ,weillustratethepotentialofourapproachwithabutteryimagecorruptedbyadditivewhite-Gaussiannoise,havingahighstandarddeviation(Figure 5-1B ,=35).Ourmethodpreservesimportantgeometricfeatures/detailsandtheoriginalcolorcontrastswithoutproducingundesirableartifacts(seeFigure 5-1F ).However,bothinFigure 5-1C andinFigure 5-1D ,wecannoticethecolorartifactsinatregions,whichlooklikearticialtextureeects.Thecoherenceenhancingdiusioncreatesbereectsonthebackground.Thecurvaturepreservingregularizationperformsbetter,howeveritcreatesacolorbleedingaroundtheedgesofthewings(seezoomed-inviewinFigure 5-1E ).Bothqualitativelyandquantitatively,ourtechniqueoutperformsthecompetingmethods. AnothercomparisonispresentedinFigure 5-3 withmulti-coloredparrots.ThenoisyimagehasaPSNRvalueof17.62inRGBdomain.Inthisexperiment,competingmethodsgeneratedblurredimages.AlthoughtheBeltramiowgivesaslightlylowerPSNRthanthecoherenceenhancingdiusion,itsmoothestheatregionsbetterandproducesavisuallymorepleasingimage(Figure 5-3D ).WecannoticesomecolordiusingeectinFigure 5-3E .Ouralgorithm,however,isabletoremovethenoise,preservethecolorandtheorientationdetailswithoutanycolorblendingproblems(seethepatcharoundtheeyeintheclose-upviewinFigure 5-3F ),aswellasachievethehighestPSNRvalue. WeshowthemethodnoiseoutputsoftheQGvMtechniqueandthecompetingalgorithmsonthebutteryandtheparrotsimagesinFigure 5-2 and 5-4 .QGvMproduces 81

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5-2B andalsothepatcharoundtheeyeisvisibleinFigures 5-4B 5-4D ). Wetestedourquaternion-basedframeworkonthemandrillimageandobtainedbetterresultsbothvisuallyandquantitatively,asshowninFigure 5-5 .Notethatthecompetingmethodsblurredthetextureontheface,especiallyonthehair.Inthistestcase,thoughthemethodnoiseoutputs(seeFigure 5-6 )ofthesealgorithmslooklikerandomnoise,itisstillpossibletoseethesmalldetailsfromtheimagegeometry. Ininpainting,wecompareourresultswiththedirectapplicationofthecurvaturepreservingPDEasproposedbyTschumperlein[ 12 ].Toll-inthemissing/desiredimageregions,weapplytheiterativeconvolutionofourspatially-varyingkernelontheregionstoinpaint,withoutusinganytexturesynthesisorreconstructiontechniqueasapost-processingstep.WeillustratehowourtechniquecanbeusedtoreconstructregionswhichwerelostorremovedfromdigitalphotographsinFigure 5-9 andFigure 5-10 alongwiththecomparisons.Inbothexperiments,ourmethodgeneratesabetterresult.NotethattheshnetisstillnoticeableinFigure 5-9B ,similarlythecageinFigure 5-10B .Inaddition,parrot'stoeisover-diusedbythecurvaturepreservingregularization,whereasourresultlooksvisuallymoreappealing. 38 ].Wecomparewiththisalgorithmsinceitpresentsatoolforfeaturespaceanalysis.Inthefollowingexperiments,foreachalgorithmthesegmentationsthatyieldthehighestF-measurevaluesareshown. InFigure 5-11 ,weshowsegmentationresultsobtainedbyvariousrecentmethods.Ourmodelyieldsasegmentationsignicantlyclosetothehumangroundtruth.Thezoomed-inviewsbothinFigure 5-11D andFigure 5-11E showthedetailedsegmentation 82

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5-11I hasalmostallofthetailsegmented,manyjunctions,edgesarenotpreservedinthesegmentation. IntheQGmF,wecanadjustthelevelofdetails/features,whichrevealthemselvesintheoutputoftheQGFappliedtothecolorimages.Todothis,weintroduceathresholdparameteronthemagnitudeofthelterresponses.Arelativelylowthresholdresultsinasegmentationcapturingthelowcontrastdetailsinsmallscales.Figure 5-12C illustratessuchanexamplewherethethresholdissetto0:005.ThemeanshiftalgorithmachievesasuccessfulresultasshowninFigure 5-12D .However,uniformregionsarenotconsistentlypreserved,e.g.theskyismis-segmented;theboundariesdividetheregionswhichareactuallycomposedofconnectedcomponents,ascanbeseenbetweentheclouds.Moreover,thecrowdonparadeismis-segmentedwiththeground.Figure 5-12E showsabettersegmentationusingourQGmFmethod(notethatthemanridingthehorseandthecrowdareclearlysegmented,alsonotetheaccuratelocalizationoftheboundarybetweenthebarricadeandthepavement).Figure 5-12F showsthepixelscorrectlylabeledbytheQGmFasbelongingtothesegmentationboundary. AnothervisualcomparisonisprovidedinFigure 5-13 .Sincethemodedetectioncalculationsinthemeanshiftalgorithmaredeterminedbyglobalbandwidthparameters,thealgorithmtendstomisssmall-scaledetailsinsomeplacesorover-segmenttheuniformregions(seethesmallareasonthestarshwhicharemis-segmentedasbeingapartoftheouterregioninFigure 5-13B ).Ontheotherhand,theQGmFmaintainscoherencewithintexturedregionswhilepreservingthesmallscaledetailsaroundtheboundariesasshowninFigure 5-13D .Onceagain,alowthresholdvalueresultsinover-segmentation(seeFigure 5-13C ). InFigure 5-14B ,notetheregionswhichhavealmostequalluminancebutdierentchromaticity.BothFigure 5-14C andFigure 5-14D areover-segmented;however,Figure 5-14E showsahighqualityresultwhichisveryclosetothehumansegmentation(see 83

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5-14A ).InFigure 5-15B ,themeanshiftsegmentationalgorithmmis-segmentstheheadsoftheastronauts,andtheboundariesoftheastronautontheleftaremissed.Asvisuallyevident,theQGmFperformsbetterthanthecompetingmethod. Inordertohaveaquantitativeevaluationofourapproach,wepresentthehighestF1-measure(orDice'sCoecient)scoresofourmethodandthecompetingmethodfortheaboveimages,asshowninTable 5-2 .Furthermore,inFigure 5-16 wepresentasensitivityanalysisusingtheF1-measureson100testimages(includingtheimagesabove)drawnfromtheBerkeleySegmentationDataSet[ 5 ].F1-measure,commonlyknownastheF-measure,istheevenlyweightedharmonicmeanofprecisionandrecallscores.Precisionandrecallarepreferredasmeasuresofsegmentationqualitybecausetheyaresensitivetounderandover-segmentation.ThehumansegmentationsfromtheBerkeleySegmentationDataSetwereusedasthegroundtruthintheevaluation.Sincetherearemultiplehumansegmentationsperimage,wecomputetheF-measurescoresagainsteachofthesesegmentationsandthentaketheaverage.Theboundariesbetweentwosegmentationsarematchedbyexamininganeighborhoodwithinaradiusof=2.IntheQGmF,wetestedtheeectofthethresholdparameter(forvaluesin[0:005;0:05])ontheQGFresponses.Forthemeanshiftsegmentationalgorithm,wetestedtheeectofthekernelbandwidthparameters:hs,spacebandwidth;andhr,rangebandwidth.Theydeterminetheresolutionofthemodeselectionandtheclustering.Wetestedfor3dierenthsvaluesin[7;10;20].Ineachcurveforthemeanshiftalgorithm,x-axisshowsthevariationsofthehrvaluesin[4;20]arrangedinascendingorderfromlefttoright.ExperimentationshowedthattheF-measurescoreschangesignicantlywithrespecttothebandwidthparametersinthemeanshiftsegmentationalgorithm,makingitdiculttochoosetherangeoftheparameterswhichcanprovidegoodresults.IntheQGmF,weobservedthatalowthresholdvalueforQGFresultsinover-segmentationwhichischaracterizedinthecurvesbylowF-measure,whereasanylevelofdetailforsegmentationcanbeachievedby 84

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B C D E F Denoisingabutteryimage. A )Originalimage[ 97 ]. B )NoisyimagewithaGaussiannoiseofstandarddeviation35.Denoisedimagesusing C )thecoherenceenhancingdiusion, D )theBeltramiow, E )thecurvaturepreservingregularization, F )ourmethod. 85

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B C D E MethodnoiseoutputsondenoisingofabutteryimageinFigure 5-1 .Methodnoiseoutputsof A )thenoisyimageandtheoriginalnoise-freeimage, B )thecoherenceenhancingdiusion, C )theBeltramiow, D )thecurvaturepreservingregularization,and E )ourmethod. 86

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B C D E F Denoisingtwoparrots. A )Originalimage[ 98 ]. B )NoisyimagewithaGaussiannoiseofstandarddeviation35.Denoisedimagesobtainedfrom C )thecoherenceenhancingdiusion, D )theBeltramiow, E )thecurvaturepreservingregularization,and F )ourmethod. 87

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B C D E MethodnoiseexperimentondenoisingoftwoparrotsinFigure 5-3 .Methodnoiseoutputsfor A )thenoisyimageandtheoriginalnoise-freeimage, B )thecoherenceenhancingdiusion, C )theBeltramiow, D )thecurvaturepreservingregularization,and E )ourmethod. 88

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B C D E F Restorationofthenoisymandrillimage. A )Originalimage[ 52 ]. B )NoisyimagewithaGaussiannoiseofstandarddeviation35.Denoisedimagesobtainedfrom C )thecoherenceenhancingdiusion, D )theBeltramiow, E )thecurvaturepreservingregularization,and F )ourmethod. 89

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B C D E MethodnoiseoutputsofthedenoisedmandrillimageinFigure 5-5 .Resultsfor A )thenoisyimageandtheoriginalnoise-freeimage, B )thecoherenceenhancingdiusion, C )theBeltramiow, D )thecurvaturepreservingregularization,and E )ourmethod. 90

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B C D E F Restoringthenoisyhorsesimage. A )Originalimagefrom[ 5 ]. B )NoisyimagewithaGaussiannoiseofstandarddeviation35.Denoisedimagesobtainedfrom C )thecoherenceenhancingdiusion, D )theBeltramiow, E )thecurvaturepreservingregularization,and F )ourmethod. 91

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B C D E MethodnoiseexperimentondenoisingofthehorsesinFigure 5-7 .Methodnoiseoutputsfor A )thenoisyimageandtheoriginalnoise-freeimage, B )thecoherenceenhancingdiusion, C )theBeltramiow, D )thecurvaturepreservingregularization,and E )ourmethod. 92

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B C Inpaintingashnetin A )acolorimage,using B )thecurvaturepreservingregularization,and C )ourmethod. B C Inpaintingacagein A )acolorimage(courtesyofD.Tschumperle), B )withthecurvaturepreservingregularization,and C )withourmethod. 93

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B C D E F G H I Asegmentationexperimentonatigerimage[ 5 ]. A )Originalimage. B )FinaldistancefunctionobtainedbyourQGmFmethod. C )RegionsextractedbytheQGmFmethod.Resultsobtainedfrom D )theQGmFmethodwithalowthresholdvalueof0.01, E )theQGmFmethodwithathresholdvalueof0.02, F )ahumansegmentation(fromthegroundtruthintheBerkeleySegmentationDataSet[ 55 ]), G )themeanshiftalgorithm, H )thecueintegrationmethodin[ 73 ]usingonlyGaborfeatures, I )thecueintegrationmethodin[ 73 ]usingthetexturefeaturesobtainedbystructuretensorandthecolorchannels. 94

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B C D E F Segmentationoftheparadeimage[ 5 ]. B )Segmentationperformedbyahumansubject(fromthegroundtruthintheBerkeleySegmentationDataSet[ 5 ]). C )SegmentationresultoftheQGmFmethodwithalowthresholdvalueof0.005. D )Segmentationresultofthemeanshiftalgorithm. E )SegmentationresultoftheQGmFmethodwithathresholdvalueof0.02. F )Truepositives(TP)mapof E withrespectto B 95

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B C D Segmentationoutputsofdierentalgorithmsonthestarshimage[ 5 ]. A )Humansegmentation(fromthegroundtruthintheBerkeleySegmentationDataSet). B )Outputofthemeanshiftalgorithm. C )OutputoftheQGmFmethodwithathresholdof0.005. D )OutputoftheQGmFmethodwithathresholdof0.025. 96

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B C D E Segmentationoutputonanimage[ 5 ]withcloseluminancevalues. A )Humansegmentation(fromthegroundtruthdata[ 5 ]). B )Luminancechannelofthecolorimage. C )OutputoftheQGmFmethodforthecolorimage(QGFthreshold=0.005). D )Outputofthemeanshiftsegmentation. E )OutputoftheQGmFmethod(QGFthreshold=0.025). B C Segmentationoftheastronautsimage[ 5 ]. A )Segmentationoutputgivenbyahumansubject(fromthegroundtruthinBerkeleySegmentationDataSet[ 5 ]). B )Outputofthemeanshiftsegmentation. C )OutputoftheQGmFmethod. 97

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F-measureplotshowingthesensitivityanalysisfortheevaluationofthemeanshiftsegmentationalgorithmandtheQGmFconvolution-basedkernelmethod.FortheQGmF,x-axisshowsthevariationsofthethresholdparameterforQGFresponses,arrangedinorderfromlefttoright,whiley-axisshowsthecorrespondingF-measurevalue.ThethresholdforQGFvarieswithin[0:005;0:05].Forthemeanshiftsegmentationalgorithm,thecorrespondingvaluesforthespacebandwidthparameter(hs)areshownintheplot,pointsalongeachcurvecorrespondtothevariationsoftherangebandwidthparameter(hr)in[4;20]. 98

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ThePSNRsofthedenoisedcolorimagesfordierentalgorithms ImagePSNRMethodCEDBeltramiCPROursNoisyImage ButteryLuminance26.4527.3725.1428.1822.32RGB24.4824.8423.1126.3317.71ParrotsLuminance29.0128.9528.9130.0322.30RGB26.9526.8526.7527.7017.62MandrillLuminance25.1825.3325.4327.2021.63RGB22.5322.5222.2723.5119.28ClownLuminance27.7928.7430.1231.522.92RGB25.4526.2827.2527.6818.37BarbaraLuminance27.8129.3530.8331.422.33RGB24.5025.2025.1025.3017.59HorsesLuminance27.728.0428.1729.2121.83RGB26.0426.3226.2927.7320.20PeppersLuminance31.030.5730.6532.4722.04RGB27.9527.0827.2028.2820.06 Table5-2. ImageQGmFMeanShift Astronauts0.740.56Starsh0.810.52Parade0.760.65Bualo0.860.67Tiger0.830.64 99

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Inthisdissertation,weaddressedtheproblemoffeature/detailpreservingsmoothingandsegmentation,andpresentedseveralcontinuousmixturemodelsforcapturingthelocalgeometryofanimageinthepresenceofnoiseviaspatiallyvaryingconvolutionlters.Thestrategytoaccomplishthisobjectivewastocapturethecomplicatedlocalgeometrycontainedatalatticepoint,followedbytheincorporationofthisinformationintospatiallyvaryingconvolutionlters.Additionally,thesamelterswheniterativelyappliedtoadistanceeldrepresentationofanactivecontouryieldedfeaturepreservingsegmentationsoftheinputimages. Ingrayscaleimagesmoothingandsegmentation,thelocalorientationinformationwasobtainedviatheapplicationofGaborlterstothedata.OneoftheproposedlterswasderivedusingthelocalorientationinformationexpressedasacontinuousmixtureofzeromeanGaussianfunctionsandassumingthemixingdensitytobeamixtureofWisharts,leadingtotheRigautKernel.ThesecondlterwasderivedexpressingthesamedataasacontinuousmixtureofWatsonfunctionswiththemixingdensityassumedtobeamixtureofBinghamsleadingtotheBMWkernel. Wepresentedresultsthatdepictedaccuratesmoothing/segmentationofscenescontainingavarietyofcomplexlocalgeometries.Tothebestofourknowledge,thisisthersttimethataconvolutionbasedapproachwasusedforfeaturepreservingsegmentation.Smoothingresultsfromanapplicationofourspatiallyvaryinglterstoimagedatawerecomparedwithstate-of-the-artdenoisingmethodsandobservedtodepictsuperiorperformance.Inthecontextofimagesegmentation,weappliedourfeature/detailpreservinglterstoimagesofvaryingcomplexityincludingimagesfromtheUCBerkeleydatasetanddepictedsuperiorperformanceovercompetingstate-of-the-artmethods. Wealsopresentedquaternion-basedframeworksforfeature/detailpreservingrestorationandsegmentationofcolorimages.Werstintroducedanovelquaternionic 100

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TheproposedmethodshandlethecouplingbetweenthecolorchannelsthroughtheapplicationofQGFstothequaternionrepresentationofcolorimages.Thisprocessalsointegratesthecolorinformationandthetextureinformation.Thisuniedinformationguidesthepropagationoftheactivecontourduringsegmentation.Similarly,inrestorationprocesses,imagechannelsdonotevolveindependentlywithdierentsmoothinggeometriesbecausetheorientationspaceandthecolorcomponentsarelinkedthroughtheQGFs.Ofcourse,thereisstillspaceforimprovementinthemethodsdescribedinthisdissertation.Forinstance,weenvisionthattheupdateequationofthecolorimagesmoothingprocesscanbemodiedtoperformaquaternion-convolutionofcolorimagewithaquaternion-valuedkernel.Anotherpossibleresearchdirectioninvolvesexploringthewaystomodelthefullquaternion-valuedQGFresponseusingquaternion-valuedbasisfunctionstogetherwiththedistributionsontheunitquaternionspace.Thismaybeaccompaniedwithextensionsofthesparsedeconvolutiontechniquestohypercomplexsystems.Futureresearchwillencompassthesearchforsuchformulationstodiscovernewvaluabletoolsforcolorimageprocessing. 101

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OzlemNurcanSubakanreceivedherB.S.andM.S.degreesincomputerengineeringfromtheBilkentUniversity,Ankara,TurkeyinMay2003andSeptember2005,respectively.ShereceivedherPh.D.degreeincomputerengineeringfromtheUniversityofFlorida,Gainesville,FLinAugust2009.Shewasawardedseveralprestigiousscholarshipsduringhereducation.Herresearchinterestsincludecomputervision,imageprocessing,medicalimageanalysisandmachinelearning. 110