Citation
Continuous Mixture Models for Feature Preserving Smoothing and Segmentation

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

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering
Computer and Information Science and Engineering
Committee Chair:
Vemuri, Baba C.
Committee Members:
Rangarajan, Anand
Ho, Jeffrey
Banerjee, Arunava
Vallejos, Carlos E.
Graduation Date:
8/8/2009

Subjects

Subjects / Keywords:
Computer vision ( jstor )
Curvature ( jstor )
Data smoothing ( jstor )
Gabor filters ( jstor )
Image enhancement ( jstor )
Image filters ( jstor )
Image processing ( jstor )
Images ( jstor )
Matrices ( jstor )
Quaternions ( jstor )
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
bingham, color, continuous, directional, distributions, filters, fisher, gabor, gaussian, image, mixture, models, quaternions, segmentation, smoothing, watson, wishart
Genre:
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Computer Engineering thesis, Ph.D.

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. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2009.
Local:
Adviser: Vemuri, Baba C.
Statement of Responsibility:
by Ozlem Subakan.

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University of Florida
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University of Florida
Rights Management:
Copyright Subakan, Ozlem. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
489208731 ( OCLC )
Classification:
LD1780 2009 ( lcc )

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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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[1] J.P.JonesandL.A.Palmer,\Anevaluationofthetwo-dimensionalgaborltermodelofsimplereceptiveeldsincatstriatecortex,"JournalofNeurophysiology,vol.58,no.6,pp.1233{1258,December1987. [2] M.RoussonandR.Deriche,GeometricLevelSetMethodsinImaging,VisionandGraphics.Springer,2003,ch.DynamicSegmentationofVectorValuedImages. [3] N.Paragios,\Geodesicactiveregionsandlevelsetmethods:Contributionsandapplicationsinarticialvision,"Ph.D.dissertation,I.N.R.I.A./UniversityofNice-SophiaAntipolis,2000,retrievedinMay23,2009,from http://www.inria.fr/RRRT/TU-0636.html [4] M.Rousson,\Cueintegrationsandfrontevolutionsinimagesegmentation,"Ph.D.dissertation,UniversityofNice-SophiaAntipolis,2004,retrievedinMay2009,from http://www.box.net/public/static/z541eovrio.pdf [5] P.Arbelaez,C.Fowlkes,andD.Martin,\TheBerkeleySegmentationDatasetandBenchmark,"RetrievedinMay16,2009,from http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/segbench/ ,2007. [6] J.Weickert,\Coherence-enhancingdiusionofcolourimages,"in7thNationalSymposiumonPatternRecognitionandImageAnalysis,1997,pp.239{244. [7] A.RosenfeldandA.C.Kak,DigitalPictureProcessing.Orlando,FL,USA:AcademicPress,Inc.,1982. [8] P.PeronaandJ.Malik,\Scale-spaceandedgedetectionusinganisotropicdiusion,"IEEETransactionsonPatternAnalysisMachineIntelligence,vol.12,no.7,pp.629{639,1990. [9] G.AubertandP.Kornprobst,MathematicalProblemsinImageProcessing:PartialDierentialEquationsandtheCalculusofVariations.Springer-Verlag,2002. [10] G.Sapiro,GeometricPartialDierentialEquationsandImageAnalysis.CambridgeUniversityPress,2001. [11] V.Caselles,B.Coll,andJ.Morel,\Junctiondetectionandltering:Amorphologicalapproach,"inIEEEInternationalConferenceonImageProcessing,vol.1,1996,pp.I:493{496. [12] D.Tschumperle,\Fastanisotropicsmoothingofmulti-valuedimagesusingcurvature-preservingpde's,"InternationalJournalofComputerVision,vol.68,no.1,pp.65{82,2006. [13] M.Kass,A.Witkin,andD.Terzopoulos,\Snakes:Activecontourmodels,"Interna-tionalJournalofComputerVision,vol.1,no.4,pp.321{331,1988. 102

PAGE 103

D.MumfordandJ.Shah,\Boundarydetectionbyminimizingfunctionals,i,"inProceedingsofIEEEConferenceonComputerVisionandPatternRecognition,1985,pp.22{26. [15] A.Tsai,A.J.Y.Jr.,andA.S.Willsky,\Curveevolutionimplementationofthemumford-shahfunctionalforimagesegmentation,denoising,interpolation,andmagnication,"IEEETransactionsonImageProcessing,vol.10,no.8,pp.1169{1186,2001. [16] R.Malladi,J.A.Sethian,andB.C.Vemuri,\Atopologyindependentshapemodelingscheme,"inSPIEProceedingsonGeometricMethodsinComputerVi-sionII,vol.2031.SPIE,July1993,pp.246{256. [17] R.Malladi,J.Sethian,andB.C.Vemuri,\Evolutionaryfrontsfortopology-independentshapemodelingandrecovery,"in1stEuropeanConferenceonComputerVision,1994,pp.3{13. [18] R.Malladi,J.A.Sethian,andB.C.Vemuri,\Shapemodelingwithfrontpropagation:Alevelsetapproach,"IEEETransactionsonPatternAnalysisandMachineIntelligence,vol.17,no.2,pp.158{175,1995. [19] V.Caselles,F.Catte,T.Coll,andF.Dibos,\Ageometricmodelforactivecontoursinimageprocessing,"NumerischeMathematik,vol.66,pp.1{31,1993. [20] V.Caselles,R.Kimmel,andG.Sapiro,\Geodesicactivecontours,"InternationalJournalofComputerVision,vol.22,pp.61{97,1997. [21] S.Kichenassamy,A.Kumar,P.Olver,A.Tannenbaum,andA.Yezzi,\Gradientowsandgeometricactivecontourmodels,"inIEEEInternationalConferenceonComputerVision,1995,pp.810{815. [22] N.Paragios,Y.Chen,andO.Faugeras,HandbookofMathematicalModelsinCom-puterVision.Secaucus,NJ,USA:Springer-VerlagNewYork,Inc.,2005. [23] B.Jian,B.C.Vemuri,E.Ozarslan,P.Carney,andT.Mareci,\Anoveltensordistributionmodelforthediusion-weightedmrsignal,"Neuroimage,vol.37,no.1,pp.164{176,2007. [24] B.JianandB.Vemuri,\Multi-berreconstructionfromdiusionmriusingmixtureofwishartsandsparsedeconvolution,"InformationProcessinginMedicalImaging,vol.20,2007. [25] B.JianandB.C.Vemuri,\Auniedcomputationalframeworkfordeconvolutiontoreconstructmultiplebersfromdiusionweightedmri,"IEEETransactionsonMedicalImaging,vol.26,no.11,pp.1464{1471,2007. [26] O.N.Subakan,B.Jian,B.C.Vemuri,andC.E.Vallejos,\Featurepreservingimagesmoothingusingacontinuousmixtureoftensors,"inIEEEInternationalConferenceonComputerVision,RiodeJaneiro,Brazil,Oct.2007. 103

PAGE 104

O.N.SubakanandB.C.Vemuri,\Imagesegmentationviaconvolutionofalevel-setfunctionwithaRigautkernel,"inIEEEConferenceonComputerVisionandPatternRecognition,Anchorage,Alaska,June2008. [28] J.Liu,B.C.Vemuri,andF.Bova,\Ecientmulti-modalimageregistrationusinglocal-frequencymaps,"MachineVisionandApplications,vol.13,no.3,pp.149{163,2002. [29] D.F.Dunn,W.E.Higgins,andJ.Wakeley,\Texturesegmentationusing2-dgaborelementaryfunctions,"IEEETransactionsonPatternAnalysisandMachineIntelligence,vol.16,no.2,pp.130{149,1994. [30] T.P.Weldon,W.E.Higgins,andD.F.Dunn,\Ecientgaborlterdesignfortexturesegmentation,"PatternRecognition,vol.29,no.12,pp.2005{2015,1996. [31] J.Jost,RiemannianGeometryandGeometricAnalysis.Berlin:Springer-Verlag,2002. [32] K.V.MardiaandP.E.Jupp,DirectionalStatistics,2nded.JohnWileyandSonsLtd.,2000. [33] J.Weickert,\Coherence-enhancingdiusionltering,"InternationalJournalofComputerVision,vol.31,no.2-3,pp.111{127,1999. [34] A.Buades,B.Coll,andJ.-M.Morel,\Anon-localalgorithmforimagedenoising,"IEEEComputerSocietyConferenceonComputerVisionandPatternRecognition,CVPR,vol.2,pp.60{65,June2005. [35] T.SchoenemannandD.Cremers,\Introducingcurvatureintogloballyoptimalimagesegmentation:Minimumratiocyclesonproductgraphs,"inIEEEInternationalConferenceonComputerVision,RiodeJaneiro,Brazil,Oct.2007. [36] M.Rousson,T.Brox,andR.Deriche,\Activeunsupervisedtexturesegmentationonadiusionbasedfeaturespace,"inIEEEConferenceonComputerVisionandPatternRecognition,2003,pp.699{706. [37] T.Cour,F.Benezit,andJ.Shi,\Spectralsegmentationwithmultiscalegraphdecomposition,"IEEEConferenceonComputerVisionandPatternRecognition,vol.2,pp.1124{1131,2005. [38] D.ComaniciuandP.Meer,\Meanshift:Arobustapproachtowardfeaturespaceanalysis,"IEEETransactionsonPatternAnalysisandMachineIntelligence,vol.24,no.5,pp.603{619,May2002. [39] D.Gabor,\TheoryofCommunication,"JournalofInstituteofElectricalEngineers,vol.93,pp.429{457,1946. 104

PAGE 105

J.G.Daugman,\Uncertaintyrelationforresolutioninspace,spatialfrequency,andorientationoptimizedbytwo-dimensionalvisualcorticallters,"JournaloftheOpticalSocietyofAmericaA,vol.2,no.7,pp.1160{1169,1985. [41] G.H.GranlundandH.Knutsson,SignalProcessingforComputerVision.KluwerAcademicPublisher,1995. [42] R.J.Muirhead,AspectsofMultivariateStatisticalTheory.JohnWiley&Sons,1982. [43] C.S.Herz,\Besselfunctionsofmatrixargument,"TheAnnalsofMathematics,vol.61,no.3,pp.474{523,1955. [44] G.LetacandH.Massam,\Quadraticandinverseregressionsforwishartdistributions,"TheAnnalsofStatistics,vol.26,no.2,pp.573{595,1998. [45] P.Langevin,\Magnetismeettheoriedeselectrons,"AnnalesdesChimieetdesPhysique,vol.5,pp.70{127,1905. [46] R.vonMises,\Uberdieganzzahligkeitderatomgewichteundverwandtefragen,"PhysikalischeZeitschrift,vol.19,pp.490{500,1918. [47] R.Fisher,\Dispersiononasphere,"RoyalSocietyofLondon.SeriesA,MathematicalandPhysicalSciences,vol.217,no.1130,pp.295{305,1953. [48] E.Dimroth,\UntersuchungenzummechanismusvonblastesisundsyntexisinphyllitenundhornfelsendessudwestlichenchtelgebirgesI.Diestatistischeauswertungeinfachergurteldiagramme,"MineralogyandPetrology,vol.8,no.2,pp.248{274,1962. [49] A.E.Scheidegger,\Onthestatisticsoftheorientationofbeddingplanes,grainaxes,andsimilarsedimentologicaldata,"U.S.GeologicalSurveyProfessionalPaper,vol.525C,pp.164{167,1965. [50] G.S.Watson,\Equatorialdistributionsonasphere,"Biometrika,vol.52,pp.193{201,1965. [51] C.Bingham,\Anantipodallysymmetricdistributiononthesphere,"TheAnnalsofStatistics,vol.2,no.6,pp.1201{1225,1974. [52] R.Jain,R.Kasturi,andB.G.Schunck,Machinevision.NewYork,NY,USA:McGraw-Hill,Inc.,1995. [53] T.F.ChanandL.A.Vese,\Activecontourswithoutedges,"IEEETransactionsonImageProcessing,vol.10,no.2,pp.266{277,2001. [54] J.Kim,J.W.F.III,A.J.Y.Jr.,M.Cetin,andA.S.Willsky,\Nonparametricmethodsforimagesegmentationusinginformationtheoryandcurveevolution,"inIEEEInternationalConferenceonImageProcessing,2002,pp.797{800. 105

PAGE 106

D.Martin,C.Fowlkes,D.Tal,andJ.Malik,\Adatabaseofhumansegmentednaturalimagesanditsapplicationtoevaluatingsegmentationalgorithmsandmeasuringecologicalstatistics,"inIEEEInternationalConferenceonComputerVision,vol.2,July2001,pp.416{423. [56] T.Cour,F.Benezit,andJ.Shi,\Multiscalenormalizedcutssegmentationcode,"RetrievedinSeptember4,2008,from http://www.seas.upenn.edu/timothee/software/ncut multiscale/ncut multiscale.html ,2008. [57] P.BlomgrenandT.F.Chan,\ColorTV:Totalvariationmethodsforrestorationofvector-valuedimages,"IEEETransactionsonImageProcessing,vol.7,no.3,pp.304{309,1998. [58] R.Kimmel,R.Malladi,andN.A.Sochen,\Imagesasembeddedmapsandminimalsurfaces:Movies,color,texture,andvolumetricmedicalimages,"InternationalJournalofComputerVision,vol.39,no.2,pp.111{129,2000. [59] B.Tang,G.Sapiro,andV.Caselles,\Colorimageenhancementviachromaticitydiusion,"IEEETransactionsonImageProcessing,vol.10,no.5,pp.701{707,2001. [60] A.Brook,R.Kimmel,andN.A.Sochen,\Variationalrestorationandedgedetectionforcolorimages,"JournalofMathematicalImagingandVision,vol.18,no.3,pp.247{268,2003. [61] L.Bar,A.Brook,N.Sochen,andN.Kiryati,\Deblurringofcolorimagescorruptedbyimpulsivenoise,"ImageProcessing,IEEETransactionson,vol.16,no.4,pp.1101{1111,April2007. [62] N.P.Galatsanos,M.N.Wernic,A.K.Katsaggelos,andR.Molina,\Multichannelimagerecovery,"inHandbookofImageandVideoProcessing,A.Bovik,Ed.ElsevierAcademicPress,2005. [63] R.Lukac,B.Smolka,K.Martin,K.Plataniotis,andA.Venetsanopoulos,\Vectorlteringforcolorimaging,"IEEESignalProcessingMagazine,vol.22,no.1,pp.74{86,Jan.2005. [64] S.GemanandD.Geman,\Stochasticrelaxation,gibbsdistributionsandthebayesianrestorationofimages,"IEEETransactionsonPatternAnalysisandMachineIntelli-gence,vol.6,no.6,pp.721{741,November1984. [65] S.C.ZhuandA.Yuille,\Regioncompetition:unifyingsnakes,regiongrowing,andbayes/mdlformultibandimagesegmentation,"IEEETransactionsonPatternAnalysisandMachineIntelligence,vol.18,no.9,pp.884{900,Sep1996. [66] D.Cremers,M.Rousson,andR.Deriche,\Areviewofstatisticalapproachestolevelsetsegmentation:integratingcolor,texture,motionandshape,"InternationalJournalofComputerVision,vol.72,no.2,pp.195{215,April2007. 106

PAGE 107

Y.Boykov,O.Veksler,andR.Zabih,\Fastapproximateenergyminimizationviagraphcuts,"IEEETransactionsonPatternAnalysisandMachineIntelligence,vol.23,p.2001,1999. [68] T.SchoenemannandD.Cremers,\Acombinatorialsolutionformodel-basedimagesegmentationandreal-timetracking,"IEEETransactionsonPatternAnalysisandMachineIntelligence,vol.99,no.1,2009. [69] A.Tsai,J.Yezzi,A.,I.Wells,W.,C.Tempany,D.Tucker,A.Fan,W.Grimson,andA.Willsky,\Model-basedcurveevolutiontechniqueforimagesegmentation,"inIEEEConferenceonComputerVisionandPatternRecognition,vol.1,2001,pp.463{468. [70] M.RoussonandN.Paragios,\Priorknowledge,levelsetrepresentations&visualgrouping,"InternationalJournalofComputerVision,vol.76,no.3,pp.231{243,2008. [71] G.Sapiro,\Colorsnakes,"ComputerVisionandImageUnderstanding,vol.68,no.2,pp.247{253,1997. [72] T.F.Chan,B.Y.S,andL.A.Vese,\Activecontourswithoutedgesforvector-valuedimages,"JournalofVisualCommunicationandImageRepresentation,vol.11,pp.130{141,2000. [73] T.Brox,M.Rousson,R.Deriche,andJ.Weickert,\Unsupervisedsegmentationincorporatingcolour,texture,andmotion,"inComputerAnalysisofImagesandPatterns,2003,pp.353{360. [74] P.F.FelzenszwalbandD.P.Huttenlocher,\Ecientgraph-basedimagesegmentation,"InternationalJournalofComputerVision,vol.59,no.2,pp.167{181,2004. [75] P.A.ArbelaezandL.D.Cohen,\Ametricapproachtovector-valuedimagesegmentation,"InternationalJournalofComputerVision,vol.69,no.1,pp.119{126,2006. [76] L.Bertelli,B.Sumengen,B.Manjunath,andF.Gibou,\Avariationalframeworkformultiregionpairwise-similarity-basedimagesegmentation,"IEEETransactionsonPatternAnalysisandMachineIntelligence,vol.30,no.8,pp.1400{1414,August2008. [77] T.EllandS.Sangwine,\Hypercomplexfouriertransformsofcolorimages,"IEEETransactionsonImageProcessing,vol.16,no.1,pp.22{35,January2007. [78] C.Moxey,S.Sangwine,andT.Ell,\Hypercomplexcorrelationtechniquesforvectorimages,"IEEETransactionsonSignalProcessing,vol.51,no.7,pp.1941{1953,July2003. [79] O.N.SubakanandB.C.Vemuri,\Quaternion-basedcolorimagesmoothingusingaspatiallyvaryingkernel,"in7thInternationalConferenceonEnergyMinimization

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[80] K.Shoemake,\Animatingrotationwithquaternioncurves,"inSIGGRAPH'85:Proceedingsofthe12thAnnualConferenceonComputerGraphicsandInteractiveTechniques.NewYork,NY,USA:ACM,1985,pp.245{254. [81] J.B.Kuipers,QuaternionsandRotationSequences:APrimerwithApplicationstoOrbits,AerospaceandVirtualReality.PrincetonUniversityPress,August2002. [82] B.A.Sethuraman,B.S.Rajan,S.Member,andV.Shashidhar,\Full-diversity,highratespace-timeblockcodesfromdivisionalgebras,"IEEETransactionsonInformationTheory,vol.49,pp.2596{2616,2003. [83] S.-C.PeiandC.-M.Cheng,\Anovelblocktruncationcodingofcolorimagesbyusingquaternion-moment-preservingprinciple,"IEEEInternationalSymposiumonCircuitsandSystems,ISCAS,vol.2,pp.684{687,May1996. [84] S.Sangwine,\Fouriertransformsofcolourimagesusingquaternionorhypercomplexnumbers,"ElectronicLetters,vol.32,no.21,pp.1979{80,1996. [85] L.ShiandB.Funt,\Quaternioncolortexturesegmentation,"ComputerVisionandImageUnderstanding,vol.107,no.1-2,pp.88{96,2007. [86] W.Hui,W.Xiao-Hui,Z.Yue,andY.Jie,\Colortexturesegmentationusingquaternion-gaborlters,"inIEEEInternationalConferenceonImageProcessing,Oct.2006,pp.745{748. [87] S.SangwineandT.Ell,\Colourimageltersbasedonhypercomplexconvolution,"IEEProceedingsonVision,ImageandSignalProcessing,vol.147,no.2,pp.89{93,Apr2000. [88] S.Sangwine,\Colourimageedgedetectorbasedonquaternionconvolution,"Elec-tronicLetters,vol.34,no.10,pp.969{71,1998. [89] M.A.Delsuc,\Spectralrepresentationof2DNMRspectrabyhypercomplexnumbers,"JournalofMagneticResonance,vol.77,no.1,pp.119{124,1988. [90] T.Ell,\Quaternion-fouriertransformsforanalysisoftwo-dimensionallineartime-invariantpartialdierentialsystems,"Proceedingsofthe32ndIEEEConfer-enceonDecisionandControl,vol.2,pp.1830{1841,Dec1993. [91] T.BulowandG.Sommer,\Daskonzepteinerzweidimensionalenphaseunterverwendungeineralgebraischerweitertensignalreprasentation,"inMustererkennung1997,19.DAGM-Symposium.London,UK:Springer-Verlag,1997,pp.351{358. [92] T.BulowandG.Sommer,\Multi-dimensionalsignalprocessingusinganalgebraicallyextendedsignalrepresentation,"inAFPAC'97:ProceedingsoftheInternational

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[93] S.SangwineandT.A.Ell,\Thediscretefouriertransformofacolourimage,"inImageProcessingII:MathematicalMethods,AlgorithmsandApplications,J.M.BlackledgeandM.J.Turner,Eds.,2000,pp.430{441. [94] O.N.SubakanandB.C.Vemuri,\Colorimagesegmentationinaquaternionframework,"in7thInternationalConferenceonEnergyMinimizationMethodsinComputerVisionandPatternRecognition,EMMCVPR2009,Bonn,Germany,August2009. [95] T.Bulow,\Hypercomplexspectralsignalrepresentationsforimageprocessingandanalysis,"Ph.D.dissertation,UniversityofKiel,1999,advisor-GeraldSommer. [96] M.J.Prentice,\Orientationstatisticswithoutparametricassumptions,"JournaloftheRoyalStatisticalSociety.SeriesB(Methodological),vol.48,no.2,pp.214{222,1986. [97] V.Racko,\Digitalimageprocessinganddatacompression,"RetrievedinMay12,2009,from http://www.ktl.elf.stuba.sk/projects/dim/images/ ,2009. [98] K.Wojciechowski,B.Smolka,H.Palus,R.Kozera,W.Skarbek,andL.Noakes,Inter-nationalConferenceonComputerVisionandGraphics,Warsaw,Poland,September2004,(intheProceedingsofComputationalImagingandVision).Secaucus,NJ,USA:Springer-VerlagNewYork,Inc.,2006. 109

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