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Competitive Mixture of Local Linear Experts for Magnetic Resonance Imaging

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Firstofall,IwouldliketothankmyPh.D.advisor,Dr.JoseC.Principe.Heledmeintothisfabulousadaptiveworldwhich,Ithink,willaectmywholelife.Hisbroadknowledge,hisdeepinsightandhisdevotionhaveencouragedmethroughoutmyPh.D.career.Withouthisguidanceandadvice,thisdissertationwouldnothavebeenpossible.IwouldliketothankDr.JereyR.Fitzsimmons,Dr.YijunLiuandDr.JohnG.HarrisfortheirtimeandpatienceservingasmyPh.D.committeemembers.Theiradvicesandcommentsimprovedthedissertationtoabetterquality.IfeelverygratefulforDr.JereyR.FitzsimmonsandDr.YijunLiufortheirconsecutivesupportinphased-arrayMRIareaandfunctionalMRIarearespectivelyinmyPh.D.career.IwouldalsoliketothankDaveM.Petersonforthedatacollection,supervisiononmyhardwareexperienceandhelpfuldiscussionallthetime.IwouldalsoliketothankDr.DenizErdogmusforbringinghisbrillianceanddriveforresearchintoourwork.IwouldalsoliketothankDr.ErikG.Larssonforbringingmeintoscienticresearch.IwouldalsoliketothankDr.MargaretM.Bradleyforprovidingmeaninterestingprojecttoworkwithandsupportingme.IwouldalsoliketothankDr.GuojunHeforhiscollaborationandvaluablecomments.Throughoutmyresearchandcoursework,IhavebeenhavingalotofinteractionwithCNELcolleagues.IwouldespeciallyexpressmythankstoDr.Sung-PhilKimforhisinsightfulcommentsandcollaboration.IalsohavebenetedalotfromourlonghoursofdiscussionfrombigpicturestothespeciedtopicswithMustafaCan iv

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v

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page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. ix LISTOFFIGURES ................................ x ABSTRACT .................................... xiv CHAPTER 1INTRODUCTION .............................. 1 1.1LiteratureReviewofMagneticResonanceImaging .......... 1 1.1.1HistoryofMRI ......................... 1 1.1.2fMRI ............................... 1 1.1.3ImageReconstructioninPhased-ArrayMRI ......... 2 1.2MagneticResonanceImagingBasics .................. 3 1.2.1InteractionofaProtonSpinwithaMagneticField ..... 3 1.2.2MagnetizationDetectionandRelaxationTimes ........ 4 1.2.3MagneticResonanceImaging .................. 6 1.3Maincontributionandintroductiontoappendix ........... 7 2STATISTICALIMAGERECONSTRUCTIONMETHODS ........ 11 2.1OptimalReconstructionwithKnownCoilSensitivities ....... 11 2.2Sum-of-squares(SoS) .......................... 11 2.2.1SNRAnalysisofSoS ...................... 12 2.2.2Conclusion ............................ 14 2.3ReconstructionMethodsUsingPriorInformationonCoilSensitivities 15 2.3.1SingularValueDecomposition(SVD) ............. 17 2.3.2BayesianMaximum-Likelihood(ML)Reconstruction ..... 18 2.3.3LeastSquares(LS)withSmoothnessPenalty ......... 21 2.4ResultsandDiscussion ......................... 24 3SUPERVISEDLEARNINGINADAPTIVEIMAGERECONSTRUC-TIONMETHODS,PARTA:MIXTUREOFLOCALLINEAREXPERTS 33 3.1LocalPatternsinCoilProle ..................... 33 3.2CompetitiveLearning .......................... 34 3.3MultipleLocalModels ......................... 34 vi

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............................. 36 3.5TheNonlinearMixtureofLocalLinearExpertsforPhased-ArrayMRIReconstruction .......................... 39 3.6Results .................................. 40 4SUPERVISEDLEARNINGINADAPTIVEIMAGERECONSTRUC-TIONMETHODS,PARTB:INFORMATIONTHEORETICLEARN-ING(ITL)OFMIXTUREOFLOCALLINEAREXPERTS ....... 56 4.1BriefReviewofInformationTheoreticLearning(ITL) ........ 56 4.2ITLBridgedtoMRIReconstruction ................. 57 4.3ITLandRecursiveITLTraining .................... 58 4.4Results .................................. 60 5UNSUPERVISEDLEARNINGINfMRITEMPORALACTIVATIONPAT-TERNCLASSIFICATION .......................... 63 5.1BriefReviewoffMRI .......................... 63 5.2UnsupervisedCompetitiveLearninginfMRI ............. 65 5.2.1TemporalClusteringAnalysis(TCA) ............. 65 5.2.2NonnegativeMatrixFactorization(NMF) ........... 66 5.2.3AutoassociativeNetworkforSubspaceProjection ...... 68 5.2.4OptimallyIntegratedAdaptiveLearning(OIAL) ....... 69 5.2.5CompetitiveSubspaceProjection(CSP) ........... 70 5.2.5.1hardcompetition ................... 71 5.2.5.2softcompetition .................... 72 5.2.6AlgorithmAnalysis ....................... 75 5.2.7fMRIApplicationwithCompetitiveSubspaceProjection .. 76 5.3Results .................................. 78 5.4Discussion ................................ 85 6CONCLUSIONSANDFUTUREWORK .................. 88 6.1Conclusions ............................... 88 6.2FutureWork ............................... 89 APPENDIX AMRIBIRDCAGECOIL ........................... 94 BMEASURINGTHESIGNAL-TO-NOISERATIOINMAGNETICRES-ONANCEIMAGING:ACAVEAT ...................... 100 B.1Introduction ............................... 100 B.2TheSignal-to-NoiseRatio(SNR) ................... 101 B.3MeasuringtheSignal-to-NoiseRatio ................. 104 B.4Illustration ............................... 106 vii

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.......................... 109 CQUALITYMEASUREFORRECONSTRUCTIONMETHODSINPHASED-ARRAYMRIMAGES ............................ 111 C.1ImageQualityMeasureReview .................... 111 C.2Methods ................................. 112 C.2.1TraditionalSNRmeasures ................... 112 C.2.2LocalnonparametricSNRmeasure .............. 113 DMRIIMAGERECONSTRUCTIONVIAHOMOMORPHICSIGNALPROCESS-ING ...................................... 115 D.1DataModel ............................... 115 D.2Homomorphicsignalprocessing .................... 115 D.3NumericalResults ............................ 117 D.4ConcludingRemarks .......................... 121 EHOMOSENSE:AFILTERDESIGNCRITERIONONVARIABLEDEN-SITYSENSERECONSTRUCTION .................... 124 E.1Introduction ............................... 124 E.2Method ................................. 124 E.3ResultsandDiscussion ......................... 126 E.4Conlusion ................................ 127 FHYBRID1DSENSE,AGENERALIZEDSENSERECONSTRUCTION 129 F.1Introduction ............................... 129 F.2Method ................................. 129 F.3ResultsandDiscussion ......................... 130 F.4Conclusion ................................ 131 GTRAJECTORYOPTIMIZATIONINK-TGRAPPA ........... 133 G.1Introduction ............................... 133 G.2Method ................................. 133 G.3ResultsandDiscussion ......................... 135 G.4Conclusions ............................... 137 REFERENCES ................................... 138 BIOGRAPHICALSKETCH ............................ 147 viii

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Table page A{18legbirdcagecoilparameters ........................ 95 D{1Normalizedentropyof(a)SoS,(b)homomorphicsignalprocessing,and(c)contrast-enhancedhomomorphicsignalprocessing. ........... 121 G{1k-tpatterncomparisonink-tGRAPPAinreductionfactor4cardiacimages. 135 ix

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Figure page 1{1Theprincipleofmagneticmoment,(a)Protonspin,(b)Angularprocessionofaprotonspininanexternalmagneticeld. ............... 4 1{2Blockdiagramofmagnetizationdetectionbyareceivercoil. ....... 5 2{1Thefourelementphased-arraycoil. ..................... 16 2{2Performanceofthefouralgorithms,SVD(circle),ML(square),LS(star),SoS(triangle),shownintermsofimagereconstructionSER(dB)versusmeasurementSNR(dB).Clearly,MLandLSperformalmostidenticallyoutperformingSVDandSoS,whichalsoperformidentically. ....... 25 2{3Thevivioimageobtainedfroma)Coil1b)Coil2c)Coil3d)Coil4.Thecoilsensitivityestimatesforf)Coil1g)Coil2h)Coil3i)Coil4,andj)thereconstructedimageobtainedusingtheSoSreconstructionmethod. 27 2{4Theratioofthemaximumsingularvaluetotheaverageofthesmallerthreesingularvaluesofthemeasurementmatricesfor5x5non-overlappingregionsa)summarizedinahistogramandb)depictedasaspatialdistrib-utionovertheimagewithgrayscalevaluesassignedinlog10scale,brightervaluesrepresentinghigherratios. ...................... 28 2{5Thereconstructedimagesusinga)SVDb)MLc)LSd)SoSapproaches. 29 2{6TheestimatedlocalSNRlevelsofthereconstructedimagesusinga)SVDb)MLc)LSd)SoSapproaches,wherethetopleftregionisthenoisereference.Noticethatin(a)-(d)theSNRlevelsareoverlaidontherecon-structedimageofthecorrespondingmethod.Topreventthenumbersfromsqueezing,theseimagesarestretchedhorizontally.Thetopleftcornerofeachimageisusedasthenoisepowerreference. .............. 32 3{1Blockdiagramofthelinearmultiplemodelmixtureandlearningscheme. 37 3{2Blockdiagramofthenonlinearmultiplemodelmixtureandlearningscheme. 40 3{3Transversecrossectionsofahumanneckasmeasuredbythefourcoilsfromonetrainingsample. .......................... 46 3{4Coronalcrossectionsofahumanneckasmeasuredbythefourcoilsusedasthetestingsample. ............................. 47 x

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..................................... 48 3{6Adaptivelearningperformance,(a)Learningcurveofwinnermodelsforthemodelnumber4,8,16,(b)Learningcurveofthelinearmixtureofcompetitivelinearmodelssystemforthemodelnumber4,8,16. ...... 49 3{7Learningcurveofthenonlinearmixtureofcompetitivelinearmodelssys-temforthemodelnumber4. ......................... 50 3{8Thereconstructionimage,(a)Fromonetransversetrainingsamplebynonlinearmixtureoflocallinearexperts,(b)TheSNRperformanceofthereconstruction. ................................ 50 3{9Pixelclassicationforthemodelnumber2,4,8,16. ............ 51 3{10ReconstructedimagesandtheirSNRperformancesfromthemixtureofcompetitivelinearmodelssystemwiththemodelnumber16andthecoilnumber4,36. ................................. 52 3{11Reconstructedtestimagesforacoronalcrossectionfromahumanneck,(a)SoSwithoutwhitening(b)SoSwithwhitening(c)Linearmixtureofmodels,(d)Nonlinearmixtureofmodels. .................. 53 3{12SNRperformancesofthereconstructedtestimagesforacoronalcrossec-tionfromahumanneck,(a)SoSwithoutwhitening(b)SoSwithwhitening(c)Linearmixtureofmodels,(d)Nonlinearmixtureofmodels. ..... 54 3{13Imagequalitymeasure,(a)-(b)Thetworeconstructionsbynonlinearmix-turesofmodelsusingtwonearidential4coilsamples,(c)Thenoisepowerfromthesubtrationofthetworeconstructionimagesin(a). ....... 55 4{1Blockdiagromofthenonlinearmultiplemodelmixtureandlearningscheme. 58 4{2Histogramofoutputerrorfromthewell-trainedMLPnetworkbyMSE. 59 4{3Adaptivelearningperformance,(a)Theinformationpotentiallearningcurve,(b)Thekernelvarianceanealingcurve. ............... 60 4{4Thereconstructionimagesofthecoronalimageby(a)ITLtrainingand(b)MSEtraining. ............................... 61 4{5TheSNRperformanceofthereconstructionimagesofthecoronalimageby(a)ITLtrainingand(b)MSEtraining. ................. 62 5{1Blockdiagramofautoassociativenetwork. ................. 69 5{2Theblockdiagramofcompetitivesubspaceprojectionmethodology. ... 71 xi

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............. 77 5{4Thelearningcurveinthesecondphaseoftrainingfromcompetitivesub-spaceprojectionforM=1;2;3(Themeansquareerror(MSE)isnormal-izedbytheinputsignalpower). ....................... 79 5{5TheprojectionaxesforthenumberoftheprojectionaxesM=2andmodelnumberK=3afterthesecondphasetrainingofcompetitivesub-spaceprojectioniscompleted. ........................ 80 5{6TheclustercentroidsformodelnumberK=4andprojectionaxesM=2. 82 5{7TheclustercentroidsformodelnumberK=3andprojectionaxesM=2. 83 5{8Thefourbasisimages(1-2upperrowand3-4lowerrowfromlefttoright)aredeterminedbyNMFusingrealfMRIdata. ............... 84 5{9TheencodingtimeseriescorrespondstofourbasisimagesbyNMFusingrealfMRIdata. ................................ 85 5{10Thetemporalmaximaplotfortemporalclusteringanalysis(TCA)method. 86 5{11Functionalregionlocalizationby(a)temporalclusteringanalysis(b)non-negativematrixfactorizationand(c)competitivesubspaceprojection. .. 87 A{1Transmitonlybirdcagecoilowchart. ................... 96 A{2Receivercoilowchart,withC1,C2aretheparallelcombinationofa20pFcapacitoranda1-15pFadjustablecapacitor;C3,C8aretheparallelcombinationofa4.7pFcapacitor,a91pFcapacitor,anda39pFcapacitor;C4istheparallelcombinationofa3.9pFcapacitoranda1-15pFadjustablecapacitor;C5,C7aretheparallelcombinationofa91pFcapacitoranda39pFcapacitor;C6istheparallelcombinationofa18pFcapacitoranda1-15pFadjustablecapacitor ......................... 97 A{3Schematicrepresentationofasingletransmit/receiveswitchingcircuitforprotectionofthereceivingpreamplier. ................... 98 A{4Blockdiagramofthequadraturetransmitcoil,andreceive-onlyphasedarraysetup. .................................. 99 B{1Syntheticdataexample.(a)Originalnoisystepfunctionsignalxn,(b)transformed(squared)signalyn,and(c)thetrueandthemeasuredSNRlevels. ..................................... 107 xii

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...................................... 109 B{3ReconstructionimagesandtheirSNRperformance.(a)logrithmofSoS,(b)SNRoflogrithmofSoS. ......................... 110 B{4ReconstructionimagesandtheirSNRperformance.(a)medianlteredSoS,(b)SNRofmedianlteredSoS. .................... 110 D{1Canonicformforhomomorphicsignalprocessor. .............. 115 D{2Photographofthephasedarraycoil,transmitcoil,andcabling. ..... 117 D{3Vivosagittalimagesofcatspinalcordfromcoil1-4andthespectralestimateofSoS. ................................ 118 D{4(Upperrow)Spatialdistributionofthecoilsensitivitiesforfourcoilsig-nals.(Lowerrow)Spectraldistributionofthecoilsensitivitiesforfourcoilsignals. ..................................... 119 D{5Thereconstructionimagecontrastversusthehigh-passltercutofre-quencyandthestopbandmagnitude. .................... 120 D{6High-passltertoeliminatecoilsensitivities. ................ 121 D{7Reconstructedimages.(a)Sum-of-squares(sos),(b)homomorphicsignalprocessing,(c)contrast-enhancedhomomorphicsignalprocessing,and(d)reconstructionfromthelteredcoilsensitivities. .............. 122 D{8Thepdfdistributionofthereconstructedimages. ............. 123 E{1SoSofaxialphantomdata. ......................... 126 E{2High-passandlow-passlterwithorder4andcutofrequencyat64. .. 127 E{3CentralPElinefromReconstructionsofhomoSENSEofMSE=0.23%,SENSEofMSE=2.19%comparedwithSoS. ................ 128 F{1Reconstructionofvariabledensityimagingwith64ACSlinesandR=4;(a)SENSE,MSE1.96%;(b)Hybrid1dSENSE,MSE1.71%;(c)SoS ... 131 G{1k-ttrajectoryink-tGRAPPA. ....................... 136 G{2k-tpatterncomparisonink-tGRAPPAinR=5cardiacimages. .... 137 xiii

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Magneticresonanceimaging(MRI)isanimportantcontemporaryresearcheldpropelledbyexpectedclinicalgains.MRIincludesmanyinterestingspecialties.Re-centlythedataacquisitiontimeinscanningpatientsbecameacriticalissue.ThecollectiontimeofMRIimagescanbereducedatacostofdevicecomplexitybyusingmultiplephased-arraycoils,whichbringtheproblemofadequatelycombiningmulti-plecoilimages.Inthisdissertation,theproblemofcombiningimagesobtainedfrommultipleMRIcoilsisinvestigatedfromastatisticalsignalprocessingpoint-of-viewwiththegoalofimprovingsignal-to-noise-ratio(SNR)inthereconstructedimages.Anewadaptivelearningstrategyusingcompetitivelearningaswellaslocallinearex-pertsisdevelopedbytreatingtheproblemasfunctionapproximation.Theproposedmethodhastheabilitytotrainonasetofimagesandgeneralizeitsperformancetopreviouslyunseenimages. TovalidatetheeectivenessoftheadaptivemethodinMRIimaging,thecom-petitivemixtureofexpertswasalsotestedintheextractionofinformationfromfunctionalMRI(fMRI)images.Theproblemistolocalizethefunctionalpattern xiv

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xv

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1 ]andPurcelletal.[ 2 ]in1946,whichledtotheirNobelPrizein1952.TherelaxationtimesoftissuesandtumorswerefoundtobedierentbyDama-dianin1971[ 3 ].ThisdiscoveryopenedapromisingapplicationareaforMRI.In1973Lauterbur[ 4 ]proposedmagneticresonanceimagingusingthebackprojectionmethod,forwhichhesharedtheNobelprizein2003.Ernstetal.[ 5 ]introducedtheFourierTransformofthek-spacesamplinginto2Dimaging,resultinginthemodernMRItechniqueandasharedNobelprizein1991. 6 ].Thepurposeofthistechniqueistounderstandhowfunc-tionalregionsinsidethebrainrespondtoexternalstimuli.Theinformationbetween 1

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thefunctionalregionsofthebrainandthecognitiveoperationshasbeeninvesti-gated[ 7 ].Thetemporalpartitionedactivitydemonstratesfunctionalindependencewithrespecttothelocalizedspatialityinsidethebrain[ 8 ].Thechallengesremaininlocalizingbrainfunctionwhenthereisnoprioriknowledgeavailableaboutatimewindowinwhichastimulusmayelicitresponse.Thusthereisnotimingforthebrain'sresponsetoalign.Thespatialactiveregionscanstillbelocatedaccordingtothetemporalresponseactivatedbyasinglestimulus[ 9 ].ThereforefMRIprovidesamethodtounderstandthemappingbetweenbrainstructuresandtheirfunctions. 10 ].Theyshowedthatthismethodlosesonly10%ofthemaximumpossiblesignal-to-noise-ratio(SNR)withnoprioriinfor-mationofthecoils'positionsorRFeldmaps.Thisresultsetsthefoundationofphased-arrayimagereconstructionanddemonstratesitsprevalenceintheindustry.BasedonSoS,asubstantialbodyofresearchhasfocusedonsophisticatedtechniquesforphaseencodingtogetherwiththeuseofgradientcoils.Thisworkincludesthesen-sitivityencodingforfastMRI(SENSE)technique[ 11 ]andsimultaneousacquisitionofspatialharmonics(SMASH)imaging[ 12 ].Bothmethodsreducethescanningtimebyundersamplingalongthegradient-echodirectionfromk-spaceinparalleldatacol-lection.Debbinsetal.[ 13 ]suggestedaddingtheimagescoherentlyaftertheirrelativephaseswereproperlyadjustedbyanothercalibrationscan.Thismethodincreasedtheimagingratebyreducingdemands,suchasbandwidthandmemory,whileitkept

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muchoftheSNRperformancecomparedtoSoS.Walshetal.usedadaptivelterstoimproveSNRintheimage[ 14 ].KellmanandMcVeighproposedamethodthatcanusethedegreesoffreedominherenttothephasedarrayforghostartifactcancellationbyaconstrainedSNRoptimization[ 15 ].Thismethodalsoneedsaprioriinforma-tiononreferenceimageswithoutdistortiontoestimatecoilsensitivities.Bydderetal.proposedareconstructionmethodthatestimatedthecoilsensitivitiesfromthesmoothenedcoilimagestoreducenoiseeects[ 16 ].ABayesianmethodusingitera-tivemaximumlikelihoodwithaprioriinformationincoilsensitivitieswaspresentedrecentlybyYanetal.[ 17 ].Recentlyimagereconstructionmethodsincorporatinglo-calcoilsensitivityfeatureshavebeenproposedsuchasparallelimagingwithlocalizedsensitivities(PILS)[ 18 ],localreconstruction[ 19 ],etc. 1.2.1InteractionofaProtonSpinwithaMagneticField 20 21 ].Protonspinexpressesprocessingapositivecharge.Thisangularprocessioncreatesaneectivecurrentloop,whichgeneratesitsowneld,calledamagneticmoment(Fig. 1{1(a) ).TheinteractionofthemagneticmomentwithanexternalmagneticeldBtendstoaligntoB.ThisalignmentisanangularprocessionconsideringBistheaxis,determinedbytheBlochequation ThegeometricalrepresentationinFig. 1{1(b) demonstratesthattheprotonspinrotatesleft-handedaroundBwiththemagnitudeofxed.FromEqn.( 1{1 ),the

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(b) Theprincipleofmagneticmoment,(a)Protonspin,(b)Angularproces-sionofaprotonspininanexternalmagneticeld. Larmorprocessionformulaisderived whereisthegyromagneticratiooftheprotonand!isnamedtheLarmorfrequency.ItisshownthattherotationfrequencyoftheprotonmagneticmomentisdeterminedbothbytheexternaleldBandprotonnature.Basedonthebiologicalabundanceofhydrogen(63%),thisprotonistakenasthemeasurednucleiwithgyromagneticratioequalto42:58MHz=T.

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Figure1{2. Blockdiagramofmagnetizationdetectionbyareceivercoil. withthethermalenergykT,wherekisBoltzmann'sconstantandTabsolutetem-perature.Thus,netmagnetizationM0cannotprovidedetectablesignals.Therefore,another 1{2 ).Theuxchangeduetothemagnetizationprocessioncanbedetectedbytheelectromotiveforce(emf)inducedinthevicinityreceivercoilgivenbyreciprocityprinciple dt(MBrf)d3r(1{3) whereBrfisthemagneticeldfromthereceivercoil. However,spinningmagnetizationisaectedbytherelaxationtimesincludinglongituderelaxationtimeT1andtransversalrelaxationtimeT2.Thelongitudere-laxationtimeT1determinesthespeedofalignmentbacktodirectionofthestaticeldB,whichisduetotheinteractionfrommagnetizationMandexternaleldB;whilethetransversalrelaxationtimeT2measuresthedephasingeectofthespin-spindecaycausedbytheinteractionamongspins.

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where(z)istheonedimensionalspindensityandk(t)= Insteadoftheabove1Dimaging,twodimensionalspatialextensioniseasytoaccomplishbyaddingonemoreencodingdirection.Supposethatphased-arraycoilsconsistingofnccoilsareusedfortheparalleldatacollectionandlet(xk;yk);k=1;;ncbethecoordinateofthekthcoil.Letx;y;zbeorthogonalunitvectorsthatspantheCartesiancoordinatesystemunderconsideration,andsupposethatasuitablegradientmagneticeldisappliedtoenableselectiveexcitationofathinsliceparalleltothe(x;y)plane,sayz=z0.Atagivencoordinatex;y;z0andtimet,letGx(t)andGy(t)bethestrengthoftheexternalmagneticeldanddene

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Then,foragivenkthreceivercoil,thereceivedtimedomainsignalcanbewrittenas whereisaconstant,and(x;y)isproportionaltothe"transversemagnetiza-tion"(whichisessentiallythequantityofinterestintheimaging),Ck(x;y)isthesensitivityandek(t)isnoisebothfromthekthcoil. Equation( 1{7 )showsthatthereceivedsignalx(t)isequaltothe2DFouriertransformofthemultiplicationofthetruepixelvalue(x;y)andthecoilsensitivityCk(x;y)sampledatkx(t)andky(t). AftertheinverseFourierTransformisappliedtothereceivedk-spacesignalxk,theresultingspatialsignalsk(i;j)fromcoilkatcoordinate(i;j)istheobservedby wherenk(i;j)iscomplex-valued(Gaussian),widesensestationary(WSS),zero-mean,spatiallywhitenoise,whichispossiblycorrelatedacrosscoilswithcovariancematrixQ(spatiotemporallyconstantduetotheWSSassumption[ 10 22 23 ]).Notethatthenoisecorrelation,ifproperlycompensatedfor,doesnotposealimitationtotheachievableimagequality[ 24 ].Inthissignalmodel,thespecicvaluesofthecoilsensitivitiesare,ingeneral,notknown.However,someaprioriknowledgeintheformofstatisticaldistributionsorstructuralconstraints(suchasspatialsmoothness)maybeavailable.

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ThedissertationdemonstratesthatSoSisanoptimallinearcombinationbaseonthesignal-to-noise-ratio(SNR)analysiswhiletheoptimalityishardtosatisfyinpractice.Thisdisadvantagedrivesustoresearchnovelreconstructionmethods.Byincorporatingthelocalsmoothnesspropertyofcoilsensitivities,threestatisticalim-agereconstructionsareproposed,namedassingularvaluedecompositionmethod,Bayesianmaximum-Likelihoodreconstruction,andleastsquareswithsmoothnesspenalty.Thesemethodsgainsa1-2dBSNRimprovementcomparedwithSoS.Al-thoughthestatisticalmethodsgiveanalyticalsolutions,theydon'thavethecapabil-itytomanipulatehistoricaldata.Therefore,chapter3switchestoadaptivelearningmethodstoextractfeaturesfromhistoricalscanningimages.Oncetheadaptivenet-workiswelltrained,itcanbegeneralizedtootherunknownscanningimages.Com-petitivelearningcombinedwithlocallinearexpertsisproposedinthisdissertationtoimplementthedivide-and-conquerstrategyinthisfunctionapproximationcase.Suchcompetitivelearningtopologyincorporatesintelligenceintotheadaptivenet-worktodecouplethesubtaskswhichhasweakcorrelationsinbetween.Bytrainingaconsiderableamountofsamples,theSNRimprovementinthetestimageissigni-cant.Chapter4furtherthisideatoinformationtheoreticallearning,wheretheerrorcriterionchangesfrommeansquareerrortoRenyi'sentropy.Thisisanextensionfromsecondorderstatisticstohigherorderstatistics.Thiscompetitivelearningideaisextendedfromsupervisedlearningtounsupervisedlearningbyproposingcompet-itivesubspaceprojectioninchapter5.ItisappliedinfunctionalMRIareaandhelpstolocatetheactivatedspatialandtemporalpatternsinsidebrain.Asasummary,conclusionsconsistingofdiscussionandsomeproposedfutureworkaredescribedinchapter6. Besidesthemainbodyofthedissertation,somecomplimentaryworkisworthtomention.Asweknow,theMRIscanningwithphasedarraycoilsspecifydierentcoilcongurationtodierentpartsofpatientsorphantoms.Thusthecoildesign

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needscarefulconsiderationinacertainscanningcase.AppendixAbrieydescribesthefourelementbirdcagecoilusedinthedatacollection. Medicalimagequalityisalwaysatoughbutinterestingtopicwhichmeasurestheamountoftrueobjectinformationextracted.Thedicultyisduetolackofknowledgeofthetruevisualsystem,thenoiseandtheblurringeect.NormallySignal-to-noiseratio(SNR)measurestheimagenoisewhilecontrast-to-noiseratio(CNR)measurestheblurringeect.Incaseofpixelbasedimagereconstruction,blurringeectisignorableandSNRisnormallyusedastheimagequalitymeasureinfullsampleddatacase.However,nonlineartransformationchangesthesecondorderstatistics.ThusSNRmeasurementmaygiveafakeimagequalityevaluation.AppendixBdescribestheproblemindetail.Inordertoconquerthisproblem,appendixCgivesaimagequalityspeculationusingnonparametricpdfestimation. Exceptfortheproposedstatisticalimagereconstructionmethodsallinimagespace,appendixDgivesanotherperspectiveinmodelingthisprobleminspectraldomain.Homomorphicsignalprocessinghelpsbridgesthelteringprocessbetweenspectraldomainandtheimagedomain.Thenalquantitativeentropyinimagequalityisalsoaninterestingmeasurement. ThefollowingthreeAppendixchaptersdescribesmyinternworkinInvivoCor-poration.Inpartialparallelacquisition(PPA),itattractsmuchinteresttosamplethek-spaceusingvariabledensity.Naquistsamplingisusuallyatlowfrequencyandun-dersamplingisusuallyathighfrequency.ThusNaquistsamplingconservestheimageenergyandleadstohighSNRinnalreconstruction;theundersamplingreducesthescanningtimebytheaccelerationfactor.However,thecombinationfromtherecon-structionsofthetwopartsseparatelyisachallenge.Theringeectisobviousinnalreconstructionifthetwopartsarenaturallyseparately;ontheotherhand,lteringthetwopartsmayincorporatebiasintothenalreconstructionaswell.AppendixEgivesanoptimallterdesignstrategytominimizethebiaseectwithsmoothing

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lter.AppendixFextendsthek-spacesamplingtoanarbitrarytrajectory.Thusthepartialparallelimagereconstructionisgeneralizedbyaninverseprobleminhybridspace.Dynamicimagingwithundersamplingisahotspot.Peopleareinterestedindierentreconstructionmethods,suchask-tBLAST,k-tSENSEandk-tGRAPPA,etc.Littleworkisdoneonhowthek-tsamplingtrajectoryaectsthereconstructionperformance.AppendixGgivesanoptimalsearchcriterioninndingtheoptimalk-ttrajectoryrelatedtok-tGRAPPAmethod.

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^(i;j)=cH(i;j)Q1s(i;j) whereHdenotestheconjugate-transpose(Hermitian)operation,c(i;j)isthevectorofcoilsensitivitiesands(i;j)isthevectorofmeasurementsforpixel(i;j).TheSNR-optimalityofthisreconstructionmethodamongalllinearcombinerscanbeproved,forexample,byapplyingtheCauchy-Schwartzinequality[ 25 ].TheSNRforthisreconstructionmethodcanbedeterminedtobejj2jjcjj2=2,where2isthenoisepower(ofbothrealandimaginaryparts). 10 ],isextensivelyimplementedintheindustryduetoitshighimagereconstructionqualityandsimplemathcalculation.Thismethodestimatesthecoilsensitivityckatthekthcoilas^ck 11

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BasedonthecoilsensitivityestimatedinEqn.( 2{10 ),TheSoSreconstruction^canbeinterpretedasanoptimallinearcombination ^=PNk=1^cksk wheres=[s1;;snc]Tcontainsallsignalelementsinnccoils.Inmostpracti-calcases,thenoiseacrosscoilsiscorrelated,assumingspatialwidesensestationar-ity(WSS).ThecoilvectorsneedstobeprewhitenedbythenoisecovariancematrixQbeforeusingthebasicSoSreconstruction.Thus,awhitenedSoSiswrittenas ^=q Maximum-ratiocombining(optimalcombining):Ifthecoilsensitivitiesckareknown,theoptimalestimateofcanbeshowntobe ~=PNk=1cksk where()standsforthecomplexconjugate.Aneatandself-containedderivationofthisresultcanbefoundin,forexample[ 10 26 ],althoughitalsofollowsdirectlybyusingsomestandardresultsonminimumvarianceestimationtheory[ 27 ].Wecaneasilyestablishthat~isunbiased,i.e.,E[~]=,whereE[]standsforstatisticalexpectation.ThentheSNRin~isequalto[ 10 26 27 ]

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^=vuut (ThisSoSestimatecanbeinterpretedasanoptimallinearcombinationaccordingtoEq.( 2{5 )butwithckreplacedbysk=q PNk=1jskj2[ 16 ].)Clearly,ifthenoiselevelgoestozerotheSoSestimateconvergesto^!q PNk=1jckj2whichisingeneralnotequalto.Therefore,SoSreconstructiontypicallyyieldsseverelybiasedimages,eveninthenoise-freecase.Unlessckisconstantforallcoils(whichiscertainlynotthecaseinpractice),thisbiasdependsonthecoilnumberkandhenceitcannotbecorrectedforifckisunknown.Also,ckaretypicallynotconstantoveranentireimage,andthereforethebiaswillbelocation-dependent,whichmayimplyseriousartifactsintheimage. WenextanalyzethestatisticalpropertiesoftheSoSmethod.ForahighinputSNR,theexpressionfor^inEq.( B{15 )canbewritten: ^=vuut PNk=1jckj2(2{8) where
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images.TheSNRof^isobtainedas: PNk=1jckj22 whichisequaltothesameastheSNRforoptimalcombiningwithknowncoilsen-sitivities(seeEq.( 2{6 )).Therefore,fromapureSNRpointofview,SoSisoptimalforhighinputSNR. 28 ].ThishighSNRperformanceensuresthenalreconstructionimagequality,whichisthemostimportantvirtueofSoS.Second,itgivesanunbiasedestimateinthenoise-freecase.Aswecansee,ifthenoiselevelgoestozerotheSoSestimateconvergesto^!q PNk=1jckj2.WithckestimatedinEqn.( 2{10 ),PNk=1jckj2isone.ThusthisSoSestimate^approachesthetruepixelvalue,whichalsoexplainsthereasonwhychoosingsuchcoilsensitivityestimatorinEqn.( 2{10 ).Besides,SoSdoesn'tneedanypriorinformation.Ononehand,withnoneedforprescanorotherinformationaboutthemagneticeld,datacollectionissim-plied.Ontheotherhand,nostatisticalassumptionconcerningthecoilsensitivitiesispredeterminedandthusreducesthemodelingerror. However,thewidely-usedsum-of-squaresmethodhasitsowndisadvantages.ThoughithastheasymptoticalSNRoptimalityproperty,theconditionforthisop-timality,whichisthehighmeasurementSNRcondition,isnotalwayssatisedinpractice[ 28 ],especiallyinphased-arrays,wherethecoilsmeasureonlyaportionoftheimage.Thiscreatestheproblemofconsideringpurenoisepixelsequallyweightedtopixelswithactualsignal.AnotherpotentialdisadvantagefortheSoSmethodandotherSoSbasedmethods(e.g.,SENSE&SMASH)lieswithinthestatistical

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assumptionofspatialwidesensestationarity(WSS)ofthenoise.Since,ingeneral,thenoisecovariancematrixQisnotknownapriori,aregionconsistingofpurenoisepixelsmustbeusedtoestimateitempirically.Thisoftenrequiresamanualselec-tionofthenoisypixelsoranotherreferencescancontainingonlynoise,undertheadditionalassumptionsthatthenoisestatisticsarestationarywithineachimagingtrialandareindependentfromtheobjectbeingimaged.Ifthenoiseexhibitslocalpropertiesinthespatialdomain(e.g.,thenoisestatisticsdiersfromthesignalregiontothebackgroundregion),thenoisecovarianceestimatedfromtheglobalspaceoracertainlocalspacedistortsorignoressomeeectiveinformationandthushurtsthereconstruction. D{3(a) D{3(d) takenusingthe4-coilphasedarrayshowninFig. 2{1 (4.7T,TR=1000ms,TE=15ms,FOV=105cm,matrix=256128,slicethickness=2mm,sweepwidth=26khz,1average).RegardingtheSoSasalinearcombinationmethodology,theequivalentcoilsensitivityestimatesproducedbythisalgorithmarefoundinEqn.( 2{10 ).TheseestimatedcoilsensitivityprolesgeneratedbytheSoSarealso

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Figure2{1. Thefourelementphased-arraycoil. showninFig. D{4(a) D{4(d) ,aswellasthereconstructedimageestimate(Fig. D{7(a) ).NoticeinFig. D{4(a) D{4(d) thatthefourspatialcoilsensitivityprolesexhibitasmoothbehaviorasafunctionofthespatialcoordinates. Asimilarstructuralbehaviorofthecoilsensitivityproleshasalsobeenob-servedinimagesofvariousotherobjects,includingphantomsandhumantissues.Thisobservationisthemainmotivationbehindthetwoassumptionsstatedabove.Thethreereconstructionmethodsthatareproposedbelowtakeadvantageofthisstructuralqualityofthecoilsensitivitiesoverspacetogenerateoptimalresultsinastatisticalarraysignalprocessingframeworkundertheassumptionsstated.

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whereisthevectorofpixelvaluesintheregion,S=[s1;;snc]isthemeasurementmatrixofsizeNnc,andNisthenoisematrix(ofthesamesizeasS)consistingofindependentsamplesacrosspixels,butpossiblycorrelatedacrosscoils. Intheidealnoise-freecase,Shasrankone,andtheleftandrightsingularvectorsofSareandc,respectively.However,thepresenceofnoiseincreasestherankofS;hencetheleftsingularvectorandtherightsingularvectorcorrespondingtothemaximumsingularvaluewillyieldtheleastsquaresestimatesofandc[ 25 ].Specically,if isthesingularvaluedecomposition(SVD)ofS,thenu1andv1minimizejjSu11vT1jj2(inEqn.( 2{11 ),UandVareorthonormalsingularvectormatricesandisadiagonalmatrixthatcontainsthesingularvaluesindescendingorder).Theestimateoftheimageinregionistherefore^=1u1andthecorrespondingcoilsensitivityvectorestimateforthisregionis^c=v1giventheunitenergyconstraintonc.Theproceduremustberepeatedforallregionsinthewholeimage.Usingeigenvalueperturbationtheory,theasymptoticSNRofthismethodcanbefoundtobeidenticaltothatofoptimallinearcombining.Thesecondassumptionusedinthisapproach(besidesA1)isthat

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D{3 .Fig. 2{4(a) showsthehistogramoftheratiobetweenthelargestsingularvalueofthelocalmeasurementmatrixtothemeanoftheotherthreesingularvalues(therearefoursingularvaluessincetherearefourcoils).Sincethereareveryfewsmallsingularvalueratios,weconcludethatinmostlocalregionstherank-onemeasurementmatrixassumptionaccuratelyholds.Infact,thenoise-onlyregionsdominantlycontributetothesmallsingular-value-ratios.Toillustratethisfact,inFig. 2{4(b) wealsopresentthesingular-value-ratioasafunctionofspatialcoordinateforthecat-spineimage,using55squarelocalregions. ^=argmaxp(Sj)=Zp(S;cj)dc=Zp(Sjc;)p(c)dc(2{12)

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Herep(Sj)istheconditionalpdfofthemeasurementmatrixgiventheimage,p(S;cj)isthejointpdfofthemeasurementmatrixandthecoilsensitivityvec-torconditionedontheimage,p(Sjc;)istheconditionalpdfofthemeasurementmatrixgiventhecoilsensitivityvectorandtheimage,andnally,p(c)isthepdfofthecoilsensitivityvector. AGaussiannoisedistributioncanoftenbejustiedbyinvokingthecentrallimittheorem[ 29 ].Inaddition,MLformulationswithGaussiandisturbancetermstendtogiverisetomathematicallyconvenientexpressions,ofteninaleast-squaresform,whichareoftenintuitivelyappealing.(Forinstance,itisnothardtoshowthatthemax-SNRreconstructionofEqn. 2{1 isequivalenttoMLifthenoiseisGaussian.)IfwefurtherassumethatthedensityofcisalsoGaussianwithmeanandcovariance,theconditionalpdfoftheobserveddatabecomes 2{12 )bymultiplyingitwithp(Sj)toresultinareconstructionthatisoptimalinthemaximumaposteriori(MAP)sense.2

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TheincorporationofaprioriknowledgeaboutmodelparametersviaBayesianstatisticshastheadvantagethattheuncertaintyinthevaluecanbecontrolledbyadjustingthecovariancematrix.Forexample,asituationwithlittleinitialknowl-edgeaboutthevalueofccanberepresentedbyamatrixwithlargeeigenvalues.Ontheotherhand,setting=0resultsinaleast-squareoptimalestimationofcorrespondingtoc=. Theaboveintegralresultisaproductofanexponentialfunctionmultipliedwithandeterminant,wherebothhaveaQT 2{14 )withrespecttoboththeparametersandc.Forthispurpose,weuseacyclicalgorithm:1:begin initialize 0;T;i=02:computec0:c0argmincF(c;0)3:do ii+14:compute:i+1argminF(;ci)4:ci+1argmincF(c;i)5:until F(i+1;ci+1)F(i;ci)
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wherestandsfortheKroneckerproductand ~Bi=(QT 2c(2{18) InanMRIapplication,wemayobtainandceitherviaanalyticalmodelingoftheelectromagneticeldsassociatedwiththecoils,orviacalibrationscansofaphantomwithknowncontrasts;bymodulatingtheparameter,wecandirectlyinuencetheaccuracyofthepriorknowledgeofc.Sucheortstocomputethecoilsensitivitypatternsmustusethenite-dierencetime-domain(FDTD)method,whichisacomputationalmethodtosolveMaxwellsequations.FDTDdividestheproblemspaceintorectangularcells,calledYee-cells,andusesdiscretetime-steps[ 30 31 ].ThisapproachhasbeensuccessfullyemployedtocomputethesensitivitypatternsoftransmitandreceivecoilsforMRI[ 32 ].Thenoisecovariance,ontheotherhand,canbeestimatedfromthecoilimagesusingportionsoftheframethatdonothaveanysignal. Sinceaclosed-formexpressionforthesolutionofthisreconstructionalgorithmisnotavailable,itisdiculttoobtainanasymptoticSNRexpression.Nevertheless,sincethesolutionisthexed-pointoftheiterations,perturbationmethodscouldbeusedtoobtainanSNRexpression,possiblyaftertediouscalculations. 1{8 )andassumptionA2,asimpleandintuitiveapproachistosolveapenalizedleast-squares(LS)problemtoreconstructtheimagefromthecoilmeasurements.RecallthatLSmethodscoincidewithMLiftheerrorisGaussian.Anaturalsmoothnesspenaltyfunctionisonethatattemptstominimizetherstandsecondorderspatialderivativesofthecoilsensitivities.

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However,suchanapproachalonedoesnotsolvetheproblem,becausetheoptimalsolutionofapenalizedLScriteriontendstoyieldimageswithlargeintensity.Thisissobecausedecreasingtheamplitudeofthecoilsensitivityproledecreasesitsderiv-ativesaswell,causingthereconstructedimagetobescaledupbythesameamount.Therefore,itappearsnecessarytoalsoimposeapenaltyonthetotalenergyoftheimage.Theresultingpenalizedleastsquarescriterion,whichhastobeminimizedtoobtaintheoptimalreconstructedimage,isgiveninEqn.( 2{19 ) wherenowdenotesthevectorofpixelvaluesforthewholeimage;hencenoparti-tioningisrequiredhere.NotethatthepenaltyterminthisLSformulationcanbe

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interpretedasaBayesianprior. 2{19 )withrespecttotheoptimizationvariablesW=[T;cT1;:::;cTnc]Tis @@J @c1...@J @cnc377777775=(1123)G0+1G1+2G2+3G3G0=266666664@J0 wheredenoteselement-wisevectorproduct.In( 2{20 ),AiandBiarenon-symmetricsparseToeplitzmatricesthatarisefromthematrixformulationoftherstandsecondorderdierences.Inparticular,A1andA2are(M-1)xMand(N-1)xNmatriceswith1sonthemaindiagonaland-1sintherstupperdiagonal,andB1andB2are(M-2)xMand(N-2)xNmatriceswith1sonthemaindiagonal,-2sintherstupperdiagonal,and1sinthesecondupperdiagonal.Allotherentriesofthesematricesarezeros.SimilartothecaseoftheBayesianreconstructionalgorithm,obtaininganasymptoticSNRexpressionforthisalgorithmshouldbepossiblealthoughitisalgebraicallycomplicated. 32 ].

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Ingeneral,leastsquarescriteriacanbeshowntobeequivalenttothemaximumlikelihoodprincipleiftheprobabilitydistributionsunderconsiderationareGaussian,orperhapsothersymmetricunimodalfunctionswherethepeakofthedistributioncorrespondstoitsmeanvalueaswell[ 27 ].Besidesthethreestatisticalreconstruc-tions,aimagereconstructionmethodbasedonspectraldecompositionisworthtomentioninAppendixD. Theimageintensityestimatevectorsofallfouralgorithmsarenormalizedtounitysuchthatinthecomparisonwiththegroundtruth(whichisavailableinthissetup)usingsignal-to-errorratio(SER)withoutconsideringscalingerrors.TheSER

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Figure2{2. Performanceofthefouralgorithms,SVD(circle),ML(square),LS(star),SoS(triangle),shownintermsofimagereconstructionSER(dB)versusmeasurementSNR(dB).Clearly,MLandLSperformalmostidenticallyoutperformingSVDandSoS,whichalsoperformidentically. isdenedasSER(dB)=10log10(jjjj2=jj^jj2),where^isthenormalizedesti-mateobtainedusingthecorrespondingalgorithm.TheresultsofthisMonte-CarloexperimentonthedescribedsyntheticdataarepresentedinFig. 2{2 intermsofav-eragereconstructionSERversusmeasurementSNRforallalgorithms.Theseexperi-mentsshowthatallfouralgorithmsasymptotically(astheSNRapproachesinnity)achieveequivalentreconstructionSERlevels.ForlowSNR,however,althoughtheSVDandSoSyieldthesamelevelofSERperformance,theMLandLSalgorithmsprovideaslight(about0.6dB)gaininSER.

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Asasecondcasestudy,allfouralgorithmsareappliedtothemultiplecoilimagespresentedinFig. D{3(a) D{3(d) ,whicharecollectedbythecoilarrayshowninFig. 2{1 withthepreviouslyspeciedmeasurementparameters(InAppendixA,adetailedphasedarraycoilisintroduced).Forthetwoiterativemethods(MLandLS),theSoSestimateofthecoilsensitivityprolesandimageintensitiesareutilizedasinitialconditions.Inaddition,forbothSVDandMLalgorithms,55non-overlappingregionsinwhichthecoilsensitivityisassumedtobeconstantareused,andthescaleambiguityforthesolutionofeachregionisresolvedbynormalizingthepowerofthereconstructedsignalforthatregiontothatoftheSoSreconstruction.TheMLalgorithmusesanoisecovarianceestimateQobtainedfromapurelynoiseregionofthecoilimages,andinanad-hocmanner,thecovarianceofthecoilsensitivitydistributionisassumedtobe=I.Alsoquiteheuristically,intheLSalgorithm,allthreeweightparametersaresettoi=0:1. 2{6 ,arectangularregionatthetopleftcorner,whichconsistsofpurenoise,isselectedasthereferencenoisepowerregion.TheSNRintheotherrectangularregions,asshowninFig. 2{6 ,arecalculatedbydividingthesignalpowerintheselectedregionbythenoisepowerestimatedfromthereferenceregion.Thevaluesarethenconvertedtodecibelsusingthe10log10()formula.

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(b) (c) (d) (e) (f) (g) (h) (i) Thevivioimageobtainedfroma)Coil1b)Coil2c)Coil3d)Coil4.Thecoilsensitivityestimatesforf)Coil1g)Coil2h)Coil3i)Coil4,andj)thereconstructedimageobtainedusingtheSoSreconstructionmethod. astheestimatedlocalSNRlevelsofthesereconstructedimagesarepresentedinFig. 2{5 & 2{6 .BycomparingtheSNRestimatesinFig. 2{6(a) 2{6(d) ,weobservethattheSVDandSoSmethods,ingeneral,produceimageswithequalSNRlevels(althoughSVDisobservedtobemoresensitivetonoiseandmeasurementartifactsasdiscussedbelow),whereastheMLapproachimprovestheSNRbyupto2dBandtheLSapproachimprovestheSNRbyupto3dBovertheperformanceofSoS.However,thecorrelationbetweenSNRandimagequalitywillbeexplainedinAppendixBandC. Atrstlook,aclearartifactintheSVDreconstructedimageshowninFig. 2{5(a) isvisible.Althoughthisartifactisnotasvisibleintheotherthreereconstructed

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(b) Theratioofthemaximumsingularvaluetotheaverageofthesmallerthreesingularvaluesofthemeasurementmatricesfor5x5non-overlappingregionsa)summarizedinahistogramandb)depictedasaspatialdistributionovertheimagewithgrayscalevaluesassignedinlog10scale,brightervaluesrepresentinghigherratios. images(Fig. 2{5(b) D{7(a) )duetothesmallsizeofthegures,uponcloserexami-nation,weseethatthishorizontalartifactalsoexistsintheseimages.Thereasonforthisartifactisidentiedasahorizontalmeasurementartifactthatexistsinallfourcoilmeasurementsatthatlocation(moststronglyseenintherstcoil).Thisartifact,alongwithmeasurementnoise,isampliedintheSVDreconstructionmethodtothehighlyvisiblelevelinFig. 2{6(a) .ThereasonforthisamplicationofnoiseandoutlierscanbeunderstoodbyinvestigatingFig. 2{4(b) .Theratiosofthemaximumsingularvaluestominimumonesarenotaslargeinthetophalfofthecoilmeasure-mentimageasthesameratiosinthebottomhalfoftheimage.Consequently,A3isnotasstronglysatisedinthetophalfasthebottomhalf.ThiscausestheSVDalgorithmtopasstheexistingmeasurementnoisetothereconstructedimagewithsomeamplication.Theartifactinthemeasurementsisalsoampliedintheprocess.

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(b) (c) (d) Thereconstructedimagesusinga)SVDb)MLc)LSd)SoSapproaches. Phased-arrayMRIresearchhasexperiencedanincreasedinterestinthelastdecadeduetothepotentialgainsinbothimagingqualityandacquisitionspeed.Althoughmanyalgorithmshavebeenproposedforphased-arrayMRimagerecon-struction,inadditiontotheperhapsmostcommonlyusedsum-of-squaresalgorithm,theseapproachesarenotbasedonastatisticaloroptimalsignalprocessingframe-work. Inthischapter,theproblemofcombiningimagesobtainedfrommultipleMRIcoilsisstudiedfromastatisticalsignalprocessingpoint-of-viewwiththegoalofimprovingSNRinthereconstructedimages.Inordertopursuethisapproach,cer-tainmodelassumptionsmustbemade.Idevelopedasetofassumptionsthatwereobservedtoholdondatacollectedfromrealmeasurements,andthreealternativeal-gorithms,stemmingfromwell-establishedstatisticalsignalprocessingtechniques,andfoundedontheseassumptionswereproposed.Thenewproposedmethods,namelysingularvaluedecomposition,maximum-likelihood,andleast-squareswithsmooth-nesspenalty,wereevaluatedonsyntheticandrealdatacollectedfromafour-coilphasedarrayusinga4:7Tscannerforsmallanimals.Aquantitativeanalysisofthereconstructedimagesobtainedusingmeasurementsofacatspinalcordrevealedthat

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itispossibletoimprovethequalityofthenalimages(intermsoflocalSNR)byupto2dBusingthemaximum-likelihoodapproachandupto3dBusingtheleast-squaresapproach. TheSNRisaconvenientandwidelyusedqualityassessmentinstrumentforMRimages.Theuseofthesingularvaluedecompositionandleast-squaresmethodsstatis-ticallymakesensewhenthissecondorderquantityisutilizedforqualityassessment.Ontheotherhand,otherquantitativemeasuressuchassignal-to-contrastratiomightbemorerepresentativeofimagequalityasperceivedbyahumanobserver.Inthatcase,alternativeoptimizationcriteriaforoptimalreconstructionofthecoilmeasure-mentsmustbederived.Thesealternativecriteriamustbeconsistentwiththedesiredqualitymeasure,aswellasbeingsucientlysimple. Therearestillunsolvedissues,however.Forexample,iftheoriginalmeasure-mentsalreadyhavehighSNR,thenthereconstructedimageusingSoSperformsclosetomaximumratiocombining;thereforeafewdBofgaininreconstructionSNRmaynotbevisibletothehumaneye.Withthemaximum-likelihoodapproach,Iusedthestandardcircular-Gaussiannoisemodel;yetIendedupwitharelativelycomplicatedexpressionthatneedstobemaximized.Moreaccuratestatisticalsignalmodelsmightimprovetheperformanceoftheapproach;nevertheless,computationalcomplexityisalwaysaconcernforMRI. Therefore,thedisadvantagesofSoSreconstructionandotherstatisticalimagereconstructionmethodsdrivemetoresearchfurtherthistopic.Allthesemethods,withoutexception,reliedonbuildingalgorithmsbasedonstatisticalorstructuralassumptionsaboutthesignalmodel.Theseapproacheswereeitherheuristicorsta-tisticalinnature.AnadaptivesignalprocessingframeworkhasnotyetbeenstudiedforphasedarrayMRI.Inthenextchapter,Iproposetotackletheimagereconstruc-tionprobleminmultiple-coilMRIscenariosbyacompetitivemixtureofexperts.Theexpectedgainsfromthisapproachincludethefollowing:thereisnoneedtopropose

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ordiscoversignalmodelsthatdescribethemeasurementswell(amustinstatisticalsignalprocessingapproaches)andthelocalstructureoftheinputspaceisnaturallyextractedfromthedata.Thusthekeydicultytoestimatethecoilsensitivitiesisavoided.Moreover,adaptivesystemsaremoreexibleandrobusttoinconsisten-ciesandnonstationaritiesinthedataastheycanbeupdatedon-linewhileinuse.Withameaningfuladaptationparadigmadaptivesystemsareabletoapproximateoptimalstatisticalsignalprocessingapproaches(tothelimitssetbythetopology)whilerequiringlessdesigneort.However,theadaptiveframeworkrequiresadesiredresponseforadaptationoperation,aswillbediscussedbelow.

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(b) (c) (d) TheestimatedlocalSNRlevelsofthereconstructedimagesusinga)SVDb)MLc)LSd)SoSapproaches,wherethetopleftregionisthenoisereference.Noticethatin(a)-(d)theSNRlevelsareoverlaidonthereconstructedimageofthecorrespondingmethod.Topreventthenumbersfromsqueezing,theseimagesarestretchedhorizontally.Thetopleftcornerofeachimageisusedasthenoisepowerreference.

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33 ].Thisfeatureisdescribedbycoilsensitivityproles,andexplainstheB1eldmapgeneratedfromthecoilgeometry.Duetothespatialcongurationofphased-arraycoils,thesensitivitiesofthecoilsarerestrictedtoaniteregionofspace.ThislocalcoilsensitivityfeatureisusedinrecentMRIimagereconstructionsuchasparallelimagingwithlocalizedsensitivities(PILS)[ 18 ],localreconstruction[ 19 ],etc. Besidesthesensitivitymaplocality,itisofinterestwhetherthethermalnoisegeneratedinthereceivercoilspossesseslocalproperty.ThethermalnoiseVnoiseinthecoilsisexcitedbytheimagedlossybodyinthecoilvicinity,wherethermsvoltageofthenoiseisgivenbyNyquist'sformula[ 34 35 ] wherekistheBoltzmann'sconstant,TBthetemperatureofthebody,ftheband-widthofthepreamplierattachedtothecoilandRLtheequivalentlossresistanceofthecoil.Foragivendesignedcoilsystem,thethermalnoiseVnoiseshouldsolelydependonRL.ThelossresistanceRLisaectedbymanyfactors,i.e.,thegeome-triesofthecoilandbody,theirpositionsrelativetoeachother,theconductivityandcomplexpermittivityofthedielectricandthecoilcoupling.TheloadaectsRLbytransferringtheuncoupledcoilsintocoupledandnallyinuencethegeneratednoise.Thiseectproducesthelocalnoisepropertydistinctinthedesiredimageregionand 33

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backgroundimageregion.However,duetocomplexlocalstructureinsidetheimageregion,thenoisepropertyishardtoestimatethere.Basedontheselocalpatternsincoilproles,anadaptivesignalprocessingstrategyisproposedtoextractlocalfeaturesandincorporatethemintoimagereconstruction. 36 ],vectorquantization[ 37 ],andtimeseriesprediction[ 38 ],etc.Theyemploycompeti-tionamongtheprocessingelements(PEs)bylateralconnectionoracertaintrainingrule.Thesimplecompetitivelearningwiththewinner-take-all(WTA)activationruleleadstoaPEunderutilization[ 39 ].Twootherschemes,namedasfrequency-sensitivecompetitivelearning(FSCL)andtheself-organizingmap(SOM),areaddressedtosolvethisproblem.FSCLincorporatestheconsciencetermintothetrainingtodriveallthePEsinsidethenetworkintocompetition[ 40 ].TheSOMmethodproposedbyKohonenusesasoftcompetitionschemetoadaptnotonlytheactivationPEbutalsoitsneighborhood[ 41 ].Competitivelearninghasalsobeenusedinimageprocessingasimagecompression[ 42 ],imagesegmentation[ 43 ]andcolorimagequantization[ 44 ].

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mixturesoflocallinearmodelsinclassicationgeneralizebetterperformancethanradialbasisfunction(RBF)andMLPwithacomparablenumberofparameters[ 45 ]. Functionapproximationisaninterestingareatouselocalmodelsinrepresentinganonlinearinput-outputmapping.Theideaoflocallinearmodelingwasappliedinpredictingchaotictimeseries,wherethenonlineardynamicsislearnedbythelocalapproximation[ 46 47 ],aswellasinthenonlinearautoregressivemodelparameterestimationgivenaMarkovstructure[ 48 ].Jacobsetal.proposedamixtureofexpertsnetworkfollowedbyagatednetwork,usedinmultispeakervowelrecognition[ 49 ].Thisdivide-and-conquerstrategyrstprovidesanewviewofeithermodularversionofmultilayersupervisednetworkoranassociativeversionofcompetitivelearning.Theauthorscontrolthecouplingamongthemultiplemodelsbyusinganegativelogprobabilitycostfunction.However,thegatingweightsareinput-basedandnotadaptedtooptimality.Fancourtintroducedacooperativefashiontothemixtureofexpertsnetwork,whereboththemodelparametersandgatingweightsaretrainedbytheExpectation-Maximization(EM)algorithm[ 50 ].Thismethoddeterminestheproportionofthedatumwhichbelongstoasinglelinearmodelaccordingtoitsposteriorprobabilityandcombinesthelinearwienersolutionineachmodel.ThesoftcompetitionactuallyprovidesanewdatasetstrictlyfollowingtheassumedGaussiandistributionforeachmodel,itmayincorporatethemodelingerrorintothenalestimation. Anotherexampleinmodularnetworksisappliedtothelocaltrainingoftheradialbasisfunction(RBF).Itshowsthatthek-nearestneighbormethod(KNN)orRBFnetworkcanbegeneralizedtoalocallearningmodelbasedondierentkernelselection[ 51 ].Alocalmodelnetworkswhichincorporateslocallearningtoaradialbasisfunction(RBF)isproposedalsointhedivide-and-conquerstrategy[ 52 ].AGrowingMulti-ExpertsmethodisproposedasanovelmodularnetworkwhichaddsalocallinearmodeltotheRBFatthegatingnetworkstage[ 53 ].Inspiteofthe

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networktopology,itdeploysaredundantexpertsremovalalgorithmtoremovetheredundantmodelsinordertoexploittheoptimalnetworkarchitecture.Amongthesemethods,theparameterchoicesofthekernel(usuallyGaussiankernel)needscarefulconsideration.BasedonthesmoothtailsoftheGaussiankernel,thedecompositionoftheinputspaceissomewhatoverlapped,whichmeansthatthelocalpropertiesarenotfullysatised.Besides,thekernelparameterisadaptedbythepureinputmapping,whichdoesn'ttaketheinput-outputmappingintoaccount.NowIproposetoimplementthecompetitivemixtureoflocallinearexpertsmethodintoimagereconstructioninphased-arrayMRI.Itachievessimplelocalpropertiesforeachmodelwhilethetrainingisbasedontheinput-outputmappingwhilethegatednetworkhastheuniverseapproximationability. 50 ].InthecaseofMRI,itispossibletoobtainasequenceoftrainingimagesfromaphantomstatisticallyrepresentativeoftheactualobjectstobeimaged,orequivalently,atrainingsetofimagesfromasubjectinthebeginningofthesession.Thesupervisedadaptivetrainingprocesslearnsthesesamplepropertiesandstorestheminsystemweights.Oncetheimagereconstructionsystemistrained(calibrated)withthissetofimages,itcanthenbeutilizedforreconstructingimagesfromscansofothersubjects(e.g.,tissues).Thedesiredhighqualityoutputisformedbyutilizingastandardimagereconstructionalgorithm(suchasSoS)onmultiplescansofthetrainingimagesetandaveragingthereconstructedimagestogenerateahighSNRdesiredoutputimagewithdi;jasthedesiredoutputforpixel(i;j).Analternativetomultipleimagesistoincreasethescantimetoimprovethequalityofasinglescanimage.Intraining,thenetwork

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Figure3{1. Blockdiagramofthelinearmultiplemodelmixtureandlearningscheme. isexpectedtomaptheinputssi;jobtainedfromsinglescanimagestothecleandesiredoutputdi;j.Notethatthetrainingisnotnecessarilyrelatedtotheimagetobedetectedlater(i.e.,itcancomefromaphantomplacedontheMRI)becausethegoalistodeterminethespatialproleofthecoils,whichislargelyunrelatedtotheobjectbeingimaged. AschematicdiagramoftheproposedimagereconstructiontopologyisdepictedinFig. D{1 .Thistopologyconsistsofmultiplelinearmodelsoperatingonthecoilmeasurementvectorsthatspecializeindierentregionsofthemeasurementvectorspace.Forpixel(i;j),modelmproducesanoutputxmthatisthelinearcombinationoftheinputvectorsi;j:xmi;j=wmTsi;j,wherewmarethemodelweightsform=1;;M,Mbeingthenumberoflinearmodels.Theinputvectorsi;jmayconsistofonlythecoilmeasurementsforpixel(i;j)inatraininginputimageoritmayincludethecoilmeasurementsforpixel(i;j)anditsneighbors(inwhichcasetheneighborhoodradiusmustbespecied).Theneighborhoodistypicallyasmallqqsquareregioncenteredatpixel(i;j).

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TheselinearmodelsaretrainedcompetitivelyusingtheLMSalgorithm[ 54 ](inawinner-take-allfashion),wherethecriterionofthecompetitionistheoutputmeansquareerror(MSE)evaluatedoveraneighborhoodofpixels(whichisanrrregioncenteredatthe(i;j)pixel[ 50 ].Forthenthepoch wherem=argminmPr2k=1(d(k)xm(k))2isthewinningmodelindexchosenfrom[1;;M],r2isthenumberofpixelsinthelocalregion,1isthestepsizeandonlythemodelwiththesmallestMSEisupdated.Ther2localregionhastheeectofnoisesuppressionincasethecurrentpixelisverynoisy,andpreventsthewrongmodelselection.Thisprocedureisrepeatedformultipleepochsuntiltheweightvectorsofallmodelsconverge.Asaresult,thecompetitivelearningphasemapsthegrayscalecoilamplitudeimagestomultiplelocalexpertsbasedonthespatialclusteringofcoilvectorsandtheirprojectiontothedesiredresponse. Themultiplemodeloutputs,whichcapturethespatiallocalfeaturesofthecoilimages,arethencombinedtoproduceanestimateoftheimageintensityatthe(i;j)pixelusing ^i;j=gTi;jxi;j(3{3) wherexi;j=WTsi;jisthevectorofoutputsfromthemultiplemodels(W=[w1;;wM]),andthemixingweightsarealsolinearcombinationsoftheinput,i.e.,gi;j=Vsi;j.OncethemultiplelinearmodelsaretrainedwithcompetitiveLMS,themixingmatrixparametersVcanbetrainedwithLMSusingtheoutputsofthecompetitivemodelsastheinputandthesamedesiredoutput,asillustratedin

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Fig. D{1 where2isthestepsize.Alternatively,Vcanbedeterminedusingtheanalyticallinearleastsquaresolution. Since,itisassumedthatboththemodeloutputsandthemixtureweightsarelinearfunctionsoftheinputs,inthetestphase,theoutputoftheproposedmixtureoflinearexpertscanbewrittenas ^i;j=sTi;jVTWTsi;j=sTi;jGsi;j(3{5) WenotethatEqn.( 3{5 )implicitlyusedtoreconstructatestingimagesissimilartothewhitenedSoSreconstructiongivenin( 2{4 )

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Figure3{2. Blockdiagramofthenonlinearmultiplemodelmixtureandlearningscheme. asinglehiddenlayerandalinearoutputlayer.DuetotheuniversalapproximationcapabilitiesofMLPs,itisexpectedthatthisnewtopologywillimprovethenalSNRbybetteremphasizingtheoutputsofthelinearmodelsthatarerelevantanddeemphasizingtheoutputsofthosemodelsthatarenotrelevantforthecurrentpixel.Thisnonlinearmixturemodel(withMinputs,Lhiddenprocessingelements,andonelinearoutput)andtheadaptationstrategyareshowninFig. 3{2 TheoutputoftheMLPisgivenby^i;j; wheref()isthesigmoidshapednonlinearfunctionofthehiddenlayer.TheMLPweightsV1,v2,b1,b2aretrainedwitherrorbackpropagationaccordingtotheMSEcriterion[ 55 ].TheinputstotheMLParetheoutputsofthelinearmodelsandthedesiredoutputisthesamedi;jthatisusedtotrainthelinearmodelscompetitively.

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andcoronal(9measurements)fastcollectionhumanneckimagesina4-coilMRIsys-tem(fastspinecho(FSE),TE=15ms,TR=150ms,ETL=2,FOV=40cm,slicethickness=5mm,matrix=160128,NEX=1).Sampleimagesfromthefourcoilsforbothcross-sectionsareshowninFig. 3{3 & 3{4 ,whichisnoisyduetotheshortscanningtimeforeachsample.Alltransversecross-sectionmeasurementsareusedfortrainingandoneofthecoronalcross-sectionmeasurements(theonethatisshowninFig. 3{4 )isusedfortestingtheresultingnetwork.ThedesiredreconstructedimageisestimatedbyaveragingtheSoSreconstructionforeachtrainingcoilim-agesample(Fig. 3{5(a) ).ItshighSNRperformancedemonstratesacleanandlownoisedesiredresponse(Fig. 3{5(b) ).Bothtrainingandtestingdatasetsconsistofmagnitudeimagesnormalizedto[1;1]beforeprocessing. Thetrainingprocedure,madeintwostages,needssomespecialdiscussion.Intherststage,theweightsinthelocallinearexpertsaretrainedcompetitivelybyLMS.ThenumberofcompetitivemodelsisselectedtobeM=4(aswillbeexplainedbelow).Theinputvectorsi;jcorrespondstoonlyonepixel(i;j).Thetrainingofthelocalmodelsstopsafter20epochswiththestepsize1=0:01demonstratedinFig. 3{6(a) .Aftertheweightsintherststagearewelltrained,themultipleexpertoutputsaretakenastheinputtothesecondcombinationstageanddi;jusedintherststageisagaintakenasthedesiredresponse.Thelinearmixturenishestrainingin5epochswith2=0:01byLMSalgorithmshowninFig. 3{6(b) .Alternatively,thenonlinearmixturenetworkisa3-layerMLPnetworkwithonelinearoutputPE,5hiddenPEsandMinputPEscorrespondingtoMmultipleexperts.Thetrainingstopsin30epochswiththestepsize2=0:005bythebackpropagationalgorithmshowninFig. 3{7 .Thereconstructedimagecalculatedfromonetrainingsampleofthetransversecross-sectionsbythetrainednonlinearmixtureoflocallinearexpertsshowsapeakSNRof33dB(Fig. 3{8 ),whichisstill12dBlowerthanthepeakSNR

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ofthedesiredresponseimage(Fig. 3{5 ).ThisSNRgapmeansthatthereisstillroomforimprovementinfutureresearch. Thecoilmeasurementsofthetestimage(coronalcross-section)arecombinedusingSoS(withoutandwithwhitening)aswellastheproposedmixturemodelnet-work.Sinceareference(agroundtruth)isnotavailableinMRI,typicallytheimagequalityismeasuredbytheempiricalSNRmeasure,whichinfactdoesnotconformtothetraditionaldenitionofSNRinsignalprocessing.TheprocedureforcomputingtheSNRisasfollows:1.Findareferenceregioninthereconstructedimagewherethereisnosignal(i.e.,apurenoiseregion).2.Computethevarianceofthenoiseinthisreferenceregion.3.Forallotherregions,computethesignalpower(whichincludesboththeactualsignalandtheremainingnoiseinthatregion).4.CalculateSNRinaregionastheratioofthepowerofthesignalinthatregiontothevarianceofthenoiseinthereferenceregion.ConvertSNRtodB.Inordertooptimallyconguretheproposedmethod,parameteranalysisisaddressednow.First,Idemonstratethespecializationofthelocallinearmodelsoftherststagedur-ingtraining.Theimportantquestionconcerningthenumberoflocallinearmodelsisalsoaddressed.Fig. 3{9 showsthespatialdistributionofthepixelsforasampleimageusingtrainedmultiplelocalmodelsforthecaseswhereM=2;4;8;16.Onecanobservethatthe2-modelsystembasicallysegmentstheimageintonoiseandsignalregions.AsMisincreased,theadditionalmodelshelpsegmentthesignalandnoiseregionstosmallerpartitionsdependingontheirlocalstatistics.Asexpected,asthenumberofmodelsisincreased,theMSEofthewinningmodelsintrainingconvergestoprogressivelylowervalueasshowninFig. 3{6(a) .However,theoverallMSEofthenaloutput(afterthecombinationofthemultiplemodeloutputs)doesnotdecreasesignicantlywhenthenumberoflocallinearmodelsisincreasedabove

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3{6(b) ).Duetoadditionalcomputationalloadandgeneraliza-tionconsiderations,oneshouldinpracticeselectthesmallestnumberofmodelsthatyieldsatisfactoryperformance.Inmeasuringtheperformance,themodelingMSEandtheSNRofthereconstructedimagecanbemonitoredsimultaneouslytomakeadecision.InourMRIsystem,forexample,M=4isalogicalchoice,andwillbeusedintherestoftheexperiments. Next,Istudytheeectofincreasingthespatiallterorderq(theneighborhood)foreachcoilimageinreconstructingthecenterpixelvalue.Ingeneral,theinputvectorsi;jforreconstructingpixel(i;j)canconsistofallpixelsinaneighborhoodofpixel(i;j)fromallcoils.Forexample,ifa33regioncenteredaroundpixel(i;j)isselected,thenthecompetinglinearcombinersbecome9-tap2-dimensionalspatialFIRltersoneachcoil,yieldingatotalofp=433=36inputvalues,assumingnc=4coils.Thisextensionoftheinputvectortoincludeneighboringpixelsinthereconstructionofthecenterpixelallowsdesigningcompetingminimum-MSEspatialltersforeachcoil.IncreasingthesizeofthesespatiallterswillintroduceadditionalsmoothingcapabilitiesthathelpincreaseSNR.Asanillustration,theperformanceofthelinearmixtureoflocallinearexpertsapproachisdemonstratedonthecoronalcross-sectionreconstructionusingnoisymeasurementsfrom4coils.ComparedtotheSNRperformanceofthecompetitivelinearexpertsusingonlythecenterpixel(4-dimensionalinputvector)showninFig. 3{10(a) & 3{10(c) ,theSNRobtainedbyusinga36-dimensionalinput(i.e.,a33neighborhoodfortheFIRlters)showninFig. 3{10(b) & 3{10(d) isupto4dBhigherinthesignalregions.Thisnoisesuppressionisachievedatthecostofsomeblurringofthesharpdetailsinthereconstructedimage,duetothelow-passlteringeectoftheincreased-lengthspatiallters;sosmallermasksormultipleinputmultipleoutputcombinationmethodareadvisable. NowIanalyzetheadvantageofnonlinearcombinationofexperts.Sinceboththelinearandnonlinearmixturesoflocallinearexpertsusethesamedrivingmodels,the

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MSEfromthecompetitivewinnermodelsintrainingcanberegardedasalowerboundfortheMSEoftheoverallsystemforbothparadigms,whichisaround295forM=4caseshowninFig. 3{6(a) .However,theselectionofthewinnerrequirestheknowledgeofthedesiredresponse,whichisnotavailableduringtesting.Therefore,Iresorttoalinearornonlinearcombinationadaptedinthetrainingset.ThecombinationphasecombinesnotonlythewinnerexpertcorrespondingtothecurrentpixelbutalsotheotherM1experts,wherethewinnerchangeswithintheMmodelswhentheadaptationgoesfromonepixeltoanother.Thus,theMSEoftheoverallnetworkshouldbeworsethanthatofthewinners.ThenalMSEcomputedinnonlinearmixturesdemonstratesacloservaluetothelowerbound(MSEofaround350)thantheMSEinthelinearmixture(around530)inFig. 3{7 & 3{6(b) ,whichrespondstohigherSNRinreconstruction. Finally,thenonlinearmixtureoflocallinearexpertsiscomparedtoSoS,whitenedSoS,andlinearmixtureoflocallinearexperts.Thecomparisonisbasedonthere-constructionofthesamecoronalcross-sectionimagefrom4coilmeasurements.ForwhitenedSoS,thewhiteningcovariancematrixisestimatedfromthecoronalcross-sectionmeasurements.ThereconstructedimagesandtheestimatedSNRlevelsarepresentedinFig. 3{11 & 3{12 .FocusingonlyontheSNRlevelsinthesignalregions,weobservethatthenonlinearmixtureoflocallinearexpertsapproachimprovedtheperformanceupto4dB,5dB,and15dBoverthatofthelinearmixtureoflocallinearexperts,SoSwithwhitening,andSoSmethods,respectively.Thelightwhiteregionattheupper-leftcornerisusedasthenoisereferenceincomputingtheSNRlevels. Asweknow,thedenitionofSNRasameasureforimagequalityisnotap-propriate.Furthermore,onemightarguethatthenoiseisonlysuppressedinthebackgroundregion(i.e.notaectingthesignalregion).Inspiteoftheabsenceofanytheoreticaljustication,thiswaspracticallydemonstratedwrong.Twonearlyidenticalsamplesofthehumanspinalcordimagecollectedbyfourcoilsarecollected.

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ThetworeconstructedimagesandthenoisegivenbythesubtractionareshowninFig. 3{13 .Itcanbeseenthatthenoiseisevenlydistributedwithnocorrelationneitherwiththebackgroundnorwiththesignalregion;onthecontrary,theyarespatialevenlydistributeddespiteoftheresiduestructure.ThustheSNRmeasureaccessesboththenoisyandsignalregions.

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(b)Coil2 (c)Coil3 (d)Coil4 Transversecrossectionsofahumanneckasmeasuredbythefourcoilsfromonetrainingsample.

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(b)Coil2 (c)Coil3 (d)Coil4 Coronalcrossectionsofahumanneckasmeasuredbythefourcoilsusedasthetestingsample.

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(b)SNRofthedesirereconstructedimage Desiredreconstructedimage,(a)estimatedbyaveragingtheSoSrecon-structionforeachcoilimagesample,(b)SNRperformanceoftheesti-mateddesire.

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Figure3{6. Adaptivelearningperformance,(a)Learningcurveofwinnermodelsforthemodelnumber4,8,16,(b)Learningcurveofthelinearmixtureofcompetitivelinearmodelssystemforthemodelnumber4,8,16.

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Figure3{7. Learningcurveofthenonlinearmixtureofcompetitivelinearmodelssystemforthemodelnumber4. (b) Thereconstructionimage,(a)Fromonetransversetrainingsamplebynonlinearmixtureoflocallinearexperts,(b)TheSNRperformanceofthereconstruction.

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(b)M=4 (c)M=8 (d)M=16 Pixelclassicationforthemodelnumber2,4,8,16.

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(b)M=4,p=36 (c)M=4,p=4 (d)M=4,p=36 ReconstructedimagesandtheirSNRperformancesfromthemixtureofcompetitivelinearmodelssystemwiththemodelnumber16andthecoilnumber4,36.

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(b) (c) (d) Reconstructedtestimagesforacoronalcrossectionfromahumanneck,(a)SoSwithoutwhitening(b)SoSwithwhitening(c)Linearmixtureofmodels,(d)Nonlinearmixtureofmodels.

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(b) (c) (d) SNRperformancesofthereconstructedtestimagesforacoronalcrossec-tionfromahumanneck,(a)SoSwithoutwhitening(b)SoSwithwhiten-ing(c)Linearmixtureofmodels,(d)Nonlinearmixtureofmodels.

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(b) (c) Imagequalitymeasure,(a)-(b)Thetworeconstructionsbynonlinearmixturesofmodelsusingtwonearidential4coilsamples,(c)Thenoisepowerfromthesubtrationofthetworeconstructionimagesin(a).

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Asweknow,entropyisusedtomeasuretheuncertaintyofarandomvariableduetoaitspdf.Shannonrstlydenedtheaverageinformationofarandomvariable,namedShannonentropy,whichisformulatedin 56

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whereE[]andf()aretheexpectionandpdfofarandomvariablex,respectively.WiththeTaylorseriestheoryatthepointx=0,Shannonentropycanbeexpandedas ThusEqn.( 4{2 )showsthattheShannonentropyisaexpectationoftheweightedsumofallordermomentswhiletheweightsdependsonthehigherorderderivativesofpdf.Thusentropycontainsthecombinedinformationofallthemoments.Ifhigherordermomentsmakesenseinsomecases,theentropymeasureasaadaptationcriterionismoresuitablethanMSE.InsteadofShannonentropy,RenyientropyismorewidelyusedduetoitsmathematicalattractivityandgeneralitywhereShannonentropyisonlyoneofitsspecialcase.TheresearchofincorporatingtheinformationtheoreticquantitiesintotheadaptivetrainingguidedbyDr.PrincipeinourCNELlabhasbeenfornearlyadecade.Theup-to-datecontributiontoincorporateminimizingRenyientropycriterioninsupervisedlearningstrategyisshownin[ 56 ]. 4{1 .ThetopologyisthesameasinFig. D{1 exceptthattheoptimizationstrategyinthesecondstageisalternatedbyinformationtheoreticlearninginsteadofminimizingMSE. Therstquestiontoaskiswhetherornotthistrainingmethodcompassinghigherordermomentsisneeded.Thepdfoftheerrordistributionofthewell-trainedMLPnetworkbyminimizingMSEgivesthedetailofvalidity.IfthepdfoftheerrordemonstratesGaussianityorcanbesimplydescribedbytherstandsecondorderstatistics,thereisnoreasonweshouldincorporatethisITLtrainingideaintothisnetwork.However,Fig. 4{2 showsasuper-Gaussiandistributionwithkurtosisof

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Figure4{1. Blockdiagromofthenonlinearmultiplemodelmixtureandlearningscheme. 27:4.Thepdfshowsslimtailanddominantmainlobewithtwopeaksinsidewithonedominantandanotherhaving17%peakvalueoftherstone.Sincethepdfisnotunimodel,itcan'tbeexactlydescribedbythelowerordermomentsandtheapplicationofITLstrategytothisproblemissuitable. 1logZ+1f(x)dx(4{3) SpecialinterestisfocusedonRenyi'squadraticentropy(=2)hereforsimplicity.ParzenwindowisusedtoestimatethepdfwithGaussiankernels,andsimpliestheRenyi'squadraticentropyas ^H2(x)=log[1 wheretheinformationpotentialV(x)isdenedasV(x)=1

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Figure4{2. Histogramofoutputerrorfromthewell-trainedMLPnetworkbyMSE. weightwgivenby w=@V @w=1 However,sincetheprobabilisticdensityfunctionestimatedateachsamplepointneedstoutilizethewholedataset,thecomputationalloadofITLtrainingisofo(N2).InourcaseofMRItrainingimagecaseofN=45160128=921600samples,o(N2)isnotapplicable.Thusarecursiveentropyestimator,whichgreatlyreducesthecomputationalload,isdenedas ^fk+1(x)=(1)^fk(x)+G(xxk+1;2)(4{6) Thentheinformationpotentialanditsderivativeareprovidedtodrivetheadaptation

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(b) Adaptivelearningperformance,(a)Theinformationpotentiallearningcurve,(b)Thekernelvarianceanealingcurve. ThusthecomputationalloadoftherecursiveITLtraininggreatlyreducetoo(NL),where IntheMRIreconstructionproblem,supposeforgettinglengthL=20,thecom-putationalloadiso(NL)=o(92160020)=o(18432000).Consideringthatthealgorithmneedsalargenumberofepochs(forexample,150epochs),thematlabcodestilltakesalongtime.Fortunately,matlabprovidesinferfacestoexternalroutineswritteninotherlanguages,calledMATLABApplicationProgramInterface.Speci-cally,theCengineroutineallowscallingthecomponentsofaCMEX-le.ThustherecursiveITLcodewaswritteninCandbasedonanself-denedinterfacefunction,matlabcancallthiscorecodeinCtogreatlysavecomputationaltime.ThoughtheITLcodeconsistslotsofmatrixcomputation,thisCMEXcodeusesonlyapproxi-mately1=20executiontime,whichmakesthecodefeasible.

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(b) Thereconstructionimagesofthecoronalimageby(a)ITLtrainingand(b)MSEtraining. measurements)fastcollectionhumanneckimagescollectedina4-coilMRIsystemusedinthepreviouschapter.Allthetraining,testinganddesiredsamplesareexactlythesameasbeingusedbefore. Fig. 4{3(a) showsthelearningprocessforthenormalizedinformationpotentialVoftheoutputerrorvs.numberofepochs.ThenormalizedinformationpotentialVremainsbetween(0;1).Itshowshowmucherrorremainsintheoutputduringtraining,wherethehigherinformationpotential,thelowererror.V=1denotesazerooutputerrorideallywhichmeansthattheoutputperfectmatchesthedesire.Inourcase,thenalV=0:58showsastillquitelargeerrorafterconvergence.Atthesametime,thekernelsizeannealingfrom0:1!0:005isshowninFig. 4{3(b) .SincetheParzenwindowpdfestimatorcanbeconsideredasaconvolutionbetweenthetruepdfandthekernelfunction,thekernelsizeannealingcanavoidthelocalminimatrapintheprocessingoftrainingandnallyachieveasolutionclosetoglobaloptima.

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(b) TheSNRperformanceofthereconstructionimagesofthecoronalimageby(a)ITLtrainingand(b)MSEtraining. ThenalreconstructiontestingimagegivenfromtheITLlearningshowsapeak2dBhigherSNRthanthatoftheMLPtrainingusingminimizingMSEcriterion,showninFig. 4{4 & 4{5 .ThenIcanconcludethattheinformationofthehigherordermomentsinthenonlineartrainingneedstobeconsidered.ActuallytheITLtrainingtakesthispointintoconsideration.

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57 ].However,itwasonlyinthelastdecadethattherapiddevelopmentoffunctionalmagneticresonanceimaging(fMRI)techniquesalloweddynamicmappingofthebrainprocesseswithnespatialresolution.Theinformationbetweenfunctionalbrainregionsandcognitiveprocesseshasbeinginvestigated[ 7 ],andthetemporalsegmentedactivitydemon-stratesfunctionalindependencewithrespecttothelocalizedbrainanatomy[ 8 ].SofarthemainmethodologyinfMRIistosegmenttheactivationregionintermsofthetemporalresponsegivenbytheexternalperiodicstimulus.Suchstimulusalternatesbetweentaskandcontrolconditionsgivingasupervisedbaselineforthetemporalresponse.Plentyofmethodshavebeenproposedtoaddressthisproblem,whichcanberoughlycategorizedintomodel-basedandmodel-independent.Correlationanaly-sis(CA),asamodel-basedmethod[ 58 59 ],combinesthesubspacemodelingofthehemodynamicresponseandtheuseofthespatialinformationtoanalyzefMRIseries.However,themodel-basedmethodsarenoteectiveinneuronalpatternanalysiswhenthetemporalinformationisnotavailable.Thusvariousmodel-independentmethodswereproposed,includingprincipalcomponentanalysis(PCA)[ 60 61 ],independentcomponentanalysis(ICA)[ 62 63 ]andclusteringmethods[ 64 65 66 ]toquantifythefMRIresponses. 63

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Howeverthechallengeremainsinlocalizingbrainfunctionwhenthereisnoaprioriknowledgeavailableaboutthetimewindowinwhichastimulusmayelicitare-sponse[ 67 68 ].Insuchcasesthere'snotimingforthebrainresponse,soconventionalsegmentationwithstimulusisimpossible.Inaddition,thefMRIsignalissubjecttohighlevelofnoise,especiallyfornon-repeatablephysiologicaleventsorrelativelylongevents(comparedtocognitiveprocesses)inthebrain,suchasthosefollowingeatinganddrinking.Atemporalclusteringanalysis(TCA)methodwasproposedtorevealthebrainresponsefollowingeating[ 9 ].Thisisaspace-timemethodologythattriestobridgethegapbetweenspatiallocalizationsandtemporalresponses.However,TCAcanstillbeimprovedanditsperformanceishamperedbyseveralassumptionsthatarenotnecessarilysatisedbycognitivesignalsmeasuredbyfMRI.Newmethodsarerequiredfordealingwiththesechallenges. Subspaceprojectionmethodasusedintheimagedeconvolutionseemstobesuit-ableforthistask.Theyhavetheadvantageofdatacompressionandnoisecancelationandarewidelyimplementedinimageprocessing,suchasimagecompression[ 69 ],hy-perspectralimageclassication[ 70 ],etc.Theoptimallinearsubspaceprojectionintermsofpreservingenergyisthewellknownprincipalcomponentanalysis(PCA)[ 71 ].However,inmanycasestheglobalPCAisnotoptimalinparticularwhenthedatadis-tributionsarefarfromGaussian.Competitivelearningisknownforitspowerfullocalfeatureextractionasdemonstratedinthelaterchaptersofthisdissertation.ItcanalsobeappliedinunsupervisedmodeasinvectorquantizationcombinedwithPCA[ 72 73 ].Haykinetal.proposedtheOIAL(optimallyintegratedadaptivelearning)method,whichgivessmallerMSEandhighercompressionratio[ 74 ].However,OIALdoesn'ttakethebiasamongmodelsintoconsideration,whichleadstosub-optimalresults.Fancourtetal.combinedthemixtureofexpertsandPCAintoacooperativenetworktosegmenttimeseriesandimages[ 75 ].Bothmethodsaresensitivetotheinitialconditionespeciallywheninputisinahighdimensionalspace.TheSOM

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methodproposedbyKohonenaddressedasoftcompetitionschemetoadaptnotonlytheactivationPEbutalsoitsneighborhood[ 41 ],whichisagoodwaytosolvetheinitialconditionproblem.Weincorporatethisideaintosubspaceprojections,namedcompetitivesubspaceprojection(CSP)method,torepresentdataoptimallynotonlyintermsoflocalprojectionaxesbutalsolocalclustercentroids.ThismethodologyintroducesforthersttimethecompetitivelearningintofMRIimageprocessing.Theadvantageofthismethodliesinthefactthatitdoesn'tneedanypriorinformationoftimecoursesegmentation,sinceitisself-organizing.Theunsupervisedvectorspacerepresentationoptimallyclustersvectoroftimeseries,whichgivesoptimalspatialtask-orientedsegmentation.Thissegmentationisuncorrelatedwithimagecontentandhasagoodnoiserejectionperformance. 5.2.1TemporalClusteringAnalysis(TCA) 9 ].TCAeectivelyextractsthestatisticalpropertiesfroma3-dimensionaldataspace(the2-dimensionalspatialimageplusthetimedimension)andformsaprob-abilisticsequenceovertimewhereeachelementNmax(t)ofthesequencerepresentsthenumberofpixelswhichreachmaximumvaluethroughoutthetimeseries.GiventhefMRIimageofsizeMNatdiscretetimet,wheret=1;;L,andthepixelvaluei;j(t)atinstanttwithi=1;;Mandj=1;:::;N,thetemporalmaximaresponseNmax(t)canbewrittenas wheref(i;j(t))=1ifi;j(t)i;j(t);8t;t6=t;0otherwise.Thismethodim-plicitlyassignsprobabilityP(i;j;^t)=1topixel(i;j)atthepeaktimeof^t,whileitassignsprobabilityP(i;j;t)=0forallothertimeinstants.Next,f(i;j(t))

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ateachpixelateachtimeinstantissummedtoobtainthetemporalmaximare-sponseNmax(t)=PMi=1PNj=1P(i;j;t).Thisquantityisameasureofgroupingac-tivation(possiblyduetoacommoncause)sinceitassumesthatfunctionalresponsehappensnotinaseparatevoxelbutinagroupofvoxels.Suchgroupofvoxelscanbedistinguishedbythetemporalmaximaresponseduetotheirsimilartemporalpeaks. Thismethodhasbeensuccessfullyappliedtomappingthebrainactivitiesfol-lowingglucoseingestion.Itprovidesadeterministicanalyticalsolutionwithstraight-forwardcomputations.However,ithaslimitation.Firstly,itissuitableforevent-relatedfMRIfunctionlocalizationwheretheresponseisdemonstratedonlyinshorttimepeaks.Itisnotsuitableforothertask-relatedfMRIproblems.Secondly,itisaectedbyimpulsivenoiseandoutliersmodelingfalsetemporalmaxima,thusyieldwrongestimatesfortheresponsetimeandregion. 76 ].AKLnonnegativedatamatrixS,whereeachcolumnisasamplevector,canbeapproximatedbyNMFas whereEistheerrorandWandHhavedimensionsKRandRL,respectively.WconsistsofasetofRbasisvectors,whileeachrowofHcontainstheencodingcoecientsforeachbasis.ThenumberofbasesisselectedtosatisfyR(K+L)
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columnofSisavectorized2DspatialfMRIimageofdimensionK=MN,andthenumberofcolumnsrepresentsthenumberofimagesamplesalongthediscretetimeaxis.GiventhefactorizationinEqn.( 5{2 ),eachbasisfunctionwr,whichistherthcolumnofWwherer=1;;R,isconsideredtobeavectorized2DlocalfeatureimageofdimensionMN;thecorrespondingvectorhr,whichistherthrowofH,codestheintensityandthetimingoftheactivationforthecorrespondingbasisimagewrinthereconstructionoftheNMFapproximation.Iftheencodingvectorhrdemonstratessparsity,i.e.,ifitpeaksoccasionally,thesepeaksmightbecorrelatedwiththeresponsetime(tothestimulus).Inaddition,thecorrespondingbasisimageswillalsohighlightthespatialdetailsoftheresponseofthebraintotheparticularstimuli.Thus,thedecompositionofSintoWandHjointlyprovidestheanswertowhenandwherefunctionalregionsact. ThedecompositionofSintoWandHcanbedeterminedbyoptimizinganerrorfunctionbetweentheoriginaldatamatrixandthedecomposition.TwopossiblecostfunctionsusedintheliteraturearetheFrobeniusnormoftheerrormatrixjjSWHjj2FandtheKullback-LeiblerdivergenceDKL(SjjWH).Thenonnegativityconstraintcanbesatisedbyusingmultiplicativeupdaterulesdiscussedin[ 77 ]tominimizethesecostfunctions.Inthisdissertation,wewillemploytheFrobeniusnormmeasure,forwhichthemultiplicativeupdaterulesthatconvergearegivenbelow, whereAa;bdenotestheelementofmatrixAatathrowandbthcolumn.Ithasbeenprovenin[ 77 ]thattheFrobeniusnormcostfunctionisnonincreasingunderthisupdaterule.

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wherethebasisvectorwiisorthonormaltoeachother.Theoptimallinearsub-spaceprojectionintermsofthesecondordermomentisprincipalcomponentanaly-sis(PCA).Itpreservesmaximumvarianceoftheprojectedrandomvariable(namedprincipalcomponents)withtheconstraintoforthogonalaxes.PCAsolutioncanbeachievedbysingularvaluedecomposition(SVD).TheorthonormalweightmatrixWcanbealsoestimatedbyanunsupervisedHebbianlearningstrategyknownasgeneralizedHebbianalgorithm(GHA)andadaptiveprincipalcomponentsextrac-tion(APEX)[ 71 ]. Fromanotherperspective,PCAcanbeconsideredasminimizingthereconstruc-tionmeansquareerror(MSE)byaconstrainedlinearprojection.ThusHebbianlearningisequivalenttoanautoassociativenetwork,asshowninFig. 5{1 [ 78 ].Theoutputofthehiddenlayeristheprojectedrandomvariableyandthedesiredre-sponseisnothingbuttheoriginaldataitselfx.MinimizingMSEbetweenxandthereconstructionWWTxallowsLMS(leastmeansquare)adaptationforweightma-trixW.Thusanunsupervisedmodelisequivalentlysovledbyasupervisedlearningscheme,whichiscomputationallyattractive.

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Figure5{1. Blockdiagramofautoassociativenetwork. 78 ].Theroutineofthismethodislistedasthefol-lowing:1.InitializeKtransformmatricesW1;;WK.2.Foreachtraininginputvectorx,

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wherePi=WTiWi,andb)updatetransformmatrixWiaccordingto whereisalearningparameter,andZ(x;Wi)isalearningrulethatconvergestotheMprincipalcomponentsoffxjx2Cig.3.Repeatforeachtrainingvectoruntilthetransformationconverges. 5{2 .Itconsistsofmulti-ple(K)autoassociativenetworkscorrespondingtoKpatternstobeclassied.Whenainputvectorxentersthesystem,theKexpertscompeteintermsofMSEbetweeninputxandthereconstruction~x.ThewinningexpertischosenbasedonaspecicminimumMSEcriterion.Thewinningexpertanditsneighborhoodareadaptedus-ingLMSwiththereconstruction~xasthedesiredresponsegiveneachx.AftertheadaptationforthewholeCSPnetworkconverges,theinputdataisclassiedtoKpatternscorrespondingtoKautoassociativenetworks.

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Figure5{2. Theblockdiagramofcompetitivesubspaceprojectionmethodology.

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itemsJ1(W;b)andJ2(W)aregivenby 2jjx~xjj2=1 2(jjxjj22xTWWTx+xTWWTWWTx+2xTWWTb2xTb+bTb)J2(W)=MXi=1jjwijj2MXj=1;j6=i[wTiwj](5{7) wherexistheinputvectorcontainingneighborhoodpixels,~x=W~y+b=WWTx+bisthereconstruction,~y=WTxistheprojectionvectorwhichhaslessdimensionthanthatofxandwiistheithcolumnvectorofmatrixW.Basedonthema-trixlemmaof@aTa=@w=2J(a;w)awhereJi;j(a;w)=@aj=@wi,theadaptationcriterionatthenthiterationiswrittenas wi(n)=[~yi(n)(~x(n)x(n))+x(n)(wi(n)T~x(n)~yi(n))+(2wi(n)MXj=1;j6=iP(wi;wj)wj(n))]b(n)=[x(n)~x(n)](5{8) whereP(wi;wj)isMMmatrixwithPr;s(wi;wj)=0ifr6=s;Pr;s(wi;wj)=sign(wi(r)wj(r))ifr=s.sign()isasignfunctionandr;s=0;;M1. Wenoticethatthelengthoftheprojectionvector~yislessthanthevectorlengthofxduetoitssubspacedimensioncompression.Thisdimensionreductionresultsintheerrorbetweentheinputxandthereconstruction~x,whereitistheerrorwhichdrivestheadaptation.Afullspaceprojectionin~ycauseserrorzeroandmakesnosenseforreconstruction.

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1.Choosesmallrandomvariables.2.Useglobaleigenvectorsplussmallrandomperturbationsaddedtoeachclass.3.ArbitrarilydividethedataintoKclasses,estimatethelargestLeigenvectors.4.ArbitrarilydividethedataintoKclasses,estimatethesmallestLeigenvectors.However,inhighdimensionalspace,noneoftheaboveestimationfortheinitialconditionassuresconvergenceforallpatterns. Inthecompetitivestrategy,theperformanceofhardcompetitiondependsonhowwelldatatstheinputspace.Ifdataonlycoverspartofspacewithacertainstructure,initialWandbforsomemodelsmaystayfarfromthedatastructure.Thusthesemodelweightswillnotbeabletowinadaptationandleadtonullmodels.Inanextremecasewhereaspecicmodelalwayswinsadaptation,nocompetitionstrategyisapplied.Sincethenumberofsamplesneededtotinputspaceexponentiallyincreaseswithdimension,highdimensionaldataislikelytohaveaninitialconditionproblem. Softcompetitionisanalternativetosolvetheinitialconditionproblem.Insoftcompetition,notonlythewinningmodelbutalsoitsneighboringmodelsareadapted.HerewearenotinterestedinpreservingthetopologymappinglikeSOMsincesubspaceprojectionisabletopreservethecomplexstructureofdatawhiletheinterestliesattheadaptationrobustnesssoftcompetitionprovides.Theadaptationmethodologyconsistsoftwoindependentphasesinsoftcompetition.Robustnessisachievedintherstphase,whichdealswiththetopologicalorderingoftheweightsanddriveallmodelweightsspatiallyclosetodata.Thesecondphaseisaconvergencephase.Itnallytunesthemodelweightstothelocalstructureofinputwithamuchsmallerstepsizecomparedwiththatintherstphase. EachmodeladaptationismodiedbyaGaussianweightingfunction wi(n)=i(n)[~yi(n)(~x(n)x(n))+x(n)(wi(n)T~x(n)~yi(n))]b(n)=i(n)[x(n)~x(n)](5{9)

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wherei(n)istheweightingfunctionoftheithmodel,whichhasageneralformof i(n)=exp(di(n)2 wheredi(n)isadistancemeasureshowinghowclosetheithmodeltsthelocalclusterand(n)2isthekernelwidth.Inordertoderivetheproperdi(n)and(n)2incompetitivesubspaceprojection,afewcriterionsneedstobesatised:1.Ineachadaptation,thewinningmodelwithleastMSEshouldgetthelargestadaptationstepwhiletheothermodeladaptationsdependonhowwelltheyttheinput.2.Intherstphaseoftraining,theweightingfunctionshouldbecontrolledinagivendynamicrangesuchthatallmodelsarerobustlyadaptedindependentofdatastructure.3.Inthesecondphaseoftraining,theweightingfunctionshouldnallyapproximatelyshrinktoadeltafunctioncenteredatthewinningmodeltoachievethenalwinner-take-allfashion.Basedonthelistedcriteria,di(n)and(n)2aresuchthattheweightingfunctionisgivenby i(n)=exp((ei(n)ei(n))2 1 wherethereconstructionerrorinmodeli,thewinningmodeliandnearestneighbortothewinningmodeliareei(n)=~xi(n)x(n),ei(n)=~xi(n)x(n)andei(n)=~xi(n)x(n),respectively;f()isanonlineartruncationfunctiontocontroltheextremelargeei(n)forstableconvergence;lrepresentsascalarproportionaltotheepochindex.Ifi=i,i(n)=1givesthelargeststepsize;otherwiseintherstepochwherek=1,i(n)isalwaysintherangeof[0.3679,1],whereassuresthetuningoftheneighboringintherstphaseoftraining.Inthesecondphaseoftraining,alargenumberofepochsareneededforne-tuningtheinput.Thuslwill

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nallyapproachalargeintegerandexponentiallyshrinktheweightingfunctiontoapproximateadeltafunction. 73 ]whichtreatsndingspatiallocationsandshapesofclustersastwoindependentprocesses,CSPcouplesthetwoaspectsofclusterrepresentationandadaptsthemsimultaneouslytooptimallyrepresentdataspace.Secondly,thissubspaceprojec-tionmethodperformnoisesuppression.Finally,thismethodtrainsthecompetitivesystembysupervisedtraininginsteadofunsupervisedtraining.Thisisdonebyes-timatingthedesiredresponsewiththeautoassociator.Thereforethecomputationalloadisgreatlyreduced. Thereexistthreeissueswhichneedsfurtherdiscussion.Therststatesthatex-plicitorthogonalityisconstrainedtotheprojectionweightW.Withoutanorthogo-nalityconstraint,thedatacanstillbeprojectedintoasubspacewhiletheprojectioneciencyisnotguaranteed.TakeWforoneexpertforexample,anynewweightmatrix~W=WRrotatedbyRwhereR=R1satisesthesameprojectionerrorforthatexpert.Thusthisisequivalentfordatainsidethiscluster.However,somedataoutsidethisexpertmayfollow~WotherthanWtocausemisclassication. Thesecondissueistheweightingfunctioni.Thisweightingfunctionisderivedbythethreebasicsoftcompetitioncriterionslogically.AlthoughitisspeciedtoCSP,

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iitselfcanbegeneralizedtootherunsupervisedlearningproblems.ItisacomprisebetweenthesoftmaxactivationfunctioninfuzzyclusteringandtheusuallyusedGaussiankernelinself-organizingmap(SOM).Ononehand,theproposedweightingfunctionsharestheadvantageoffuzzyclusteringwhereallthesoftcompetitionisdeterminedbythewholeclustercenterstatistics.ThisinformationismoreaccuratethantheGaussiankernelinSOM.Ontheotherhand,itincorporatesthetruncationnonlinearfunctionintoiwhichmaintainstheexibilityofhardcompetitionasSOMdoesaftershrinking. Anotherissueisthescaleambiguityforinputs.Asisknown,subspaceprojectionisindependentofthenormwhileanyEuclideandistanceclusteringmethods,e.g.LBGandK-means,takenormintoaccount.ThisproposedcompetitivesubspaceprojectionmethodisamixtureofprojectionmethodandEuclideandistancemethodsandthushasscaleambiguityforinputs.Aclusterisdenedbyitslocalstructure,whichmeanshowthelocaldataisgrouped.Itscenter,projectionaxesandprojectedvariancearethelinearrepresentationofclusteritselfwhichlinearlydeterminethestructureandshapeofcluster.Thus,ascalarmultiplicationcanbeconsideredasacentershiftandexpandedorshrinkingprojectedvariancewiththesameaxes.Ifthecenterisshifted(originalcenterisnonzero),thisscalarmultiplicationshouldgenerateanewclusterwhileitisarguingifthecenterisunchanged(originalcenteriszero).Thusthecompetitivesubspaceprojectionwhichtakesthenormjointlywiththesubspaceprojectionisreasonable.

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(b) (c) (d) Threedimensionalsyntheticdata,(a)projectedtoitsrstandseconddimension,wherethethirddimensionisinsignicantinclassication(b)clusteringdatain(a)byk-means,(c)clusteringdatain(a)byoptimallyintegratedadaptivelearning(OIAL),(d)clusteringdatain(a)bycom-petitivesubspaceprojection(CSP).Theintersectedlinesin(c)and(d)representthetwoprojectionaxesforeachcluster. thechangeofitspixelintensitythroughtime.Somepixelsinthebackgroundnoiseandpartinsidethebrainmaynotrespondtothestimulusandthustheirintensitiesuctuatesinasmalldynamicrangeduetoscanningnoise.Otherpixelsmayre-spondtoitwithdelayanddemonstrateacertaintimecoursestructureintime.Theclassicationusingtheproposedcompetitivesubspaceprojectioncansegmentthedierentpatternsbasedonthistimestructuredierenceamongpixels.Thepatternswhichhavespecictimestructuresgiveinformationofspatiallocationandtemporalresponsepeakofagivenfunctionaleect.

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ClusteringPerformanceComparison:Thesyntheticdatausedtodemon-stratetheclusteringperformanceisgeneratedwithcomplexstructure.Thedatawhichisthreedimensionalisprojectedtoitsrstandseconddimensionsforclassi-cationsincethethirddimensionisinsignicantinsegmentation(Fig. 5{3(a) ).Thedatastructureconsistsoffournaturalclusterswheretheirshapesaretwoapproxi-materectangles,onecircleandoneellipse.Thetworectangleclustersapproximatethecircleclusterfromdierentdirectionswhiletheellipseclusterstayscompara-tivelyfarfromtheotherthree.Threeclusteringmethodsarecomparedbasedonthesyntheticdata,whicharekmeans,optimallyintegratedadaptivelearning(OIAL)andcompetitivesubspaceprojection(CSP).ItisshowninFig. 5{3(b) thatkmeansgroupsdatawellexceptthatitmisclassiessomesamplesinsideitsneighboringendsideofthetworectangleclusters.Thereasonisthatk-meanscannotreectcomplexclusteringboundaryinclassication.Fig. 5{3(c) revealsthatthesegmentationwhichOIALdoesconformstoseparatingtheinputspaceintomultipleconeswiththever-texattheoriginduetothefactthatthein-betweenclustercentroiddistancesarenotconsidered.WecanseethatCSPgivesareasonableclusteringttingitsnaturalstructureintermsofsubspaceprojection(Fig. 5{3(d) ).Herethepreservedsubspacedimensionistwowhiletheprojectionaxestotherstandseconddimensionsofthesyntheticinputaredemonstrated.

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Figure5{4. ThelearningcurveinthesecondphaseoftrainingfromcompetitivesubspaceprojectionforM=1;2;3(Themeansquareerror(MSE)isnormalizedbytheinputsignalpower). Thefunctionalimagesforma3Dmatrixofsize6464750.Theneachpixelvectorispreprocessedwithmean-removalfromitspixelintensitywithnormalizedstandarddeviation(mean-removedalongtimeforeachpixelindependently).Themean-removalalongtimeseriesforeachpixeleliminatesthein-betweencorrelationduetoimagestatistics,whichleadstoanindependenttimeresponsesforpixelinten-sities.Thenormalizationprocessreducestheeectofthescalesofthevectornormandfurtheravoidsdivergenceintrainingphase. Thepreprocessedpixelvectorxi;jatspatiallocation[i;j]istakenastheinputofcompetitivesubspacenetworkwithunsupervisedlearningwith=0:01andthetrun-cationfunctionf(a;b)=5bifa>5bandaotherwise.Thesoftcompetitionadaptsthewinningmodelanditsneighborhoodmodelswithdierentstepsize.There'stwophasesintraining.Therstphaseusesalargestepsize(1=0:02)andsmallnumberofepochs(epochnumber=10)totrainthemodelweightswithMSEcriterion.In

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Figure5{5. TheprojectionaxesforthenumberoftheprojectionaxesM=2andmodelnumberK=3afterthesecondphasetrainingofcompetitivesubspaceprojectioniscompleted. therstfewepochs,theweightingfunctioni(n)insoftcompetitionassureseverymodeltobeadaptedtoavoidnullmodels.Thustheclusterweightsaretopologicallyorderedaroundtherangeofinputvectorlocalization.Thissoftcompetitionstagegivesanapproximateclusteringestimatewhichissoftlyspeciedbythedataitself.Afterthetrainingintherststageconverged,asmoothcompetitionstrategyisusedwithasmallstepsize(2=0:005)andamuchlargerepochnumber300areusedtoslowlytraintheweightstopreciselycapturetheinputstructure.Thedesiredpatterninspatialandtemporaldomainisachievedafternalconvergence. Therstproblemintrainingistodeterminehowtochoosetheminimumnum-berofhiddenlayerMinautoassociatorwithoutsacricingperformance.Thisisequivalenttondhowmanyminimumprojectionaxesareneededtoextractthetime

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structure.Asweknow,thefMRIimageseriesxi;jformavectorspaceofhighdi-mensionT.Clusteringxi;jisdividingthespaceintomultiplepatterns,whereeachpatternresemblesastimulatedtemporalresponseplusnoiseoronlynoiseininactiveregions(thepreprocessingexcludesimagecontentinterference).Thisisdeterminedbytheadditivenoiselevelofdata.Ifthenoiselevelisprettyhigh,noisemaybedom-inantinmainaxesandtheeectivetimestructuremayhavetobeextractedfromthesecond,orevenmoreinsignicantaxis.Fig. 5{4 demonstrateshowthelearningcurveintheprecisesecondphaseoftrainingisaectedbythenumberofprojectionaxisM.Itshowsthatthenalmean-squareerror(MSE)reduces5:7%andonly1:5%whenMincreasesfrom1to2andfrom2to3respectively.Wecanconcludethattheclusteredtimeinformationcanbewellrepresentedbyusingthersttwoaxes.ThisisalsoillustratedinFig. 5{5 .Theusefultimestructureisexhibitedinthesecondaxeswhiletherstaxisischosentorepresentnoise.ThusonlytwoprojectionaxisM=2isenoughforcompetitioninthiscase.Besides,incasesofthetimestructureisdramaticallydominatedbynoise,thetwoprojectionaxesarestillpreferredforalgorithmstability. AnotherimportantquestionishowtodeterminethenumberofmodelsKneededforcompetition.Onepatternisneededforinactivepixelsandatleastonemorepatternforthetaskactivatedpixels.However,wedon'tknowhowmanypatternsarestimulatedinadvance.WehavetopredetermineamodelnumberK,sayK=4,tocheckifanymodelsinsideareactuallysubdividedfromnaturalonecluster.TheclustercentroidsofK=4modelsareshowninFig. 5{6 .Itisdemonstratedthattheclustercentroidsfrommodel1and2overlapsomewhat,whichmeansthatthemodel1and2shouldcomefromonenaturalclusterandcouldbecombined.Thusonlythreemodelsarenecessaryforthistask. ThepurposeofthetaskusedinourfMRIstudyistoidentifytheneuralcorrelatesunderlyingeye-blinkingandapplyournewanalysisapproachestodissociateneuronal

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Figure5{6. TheclustercentroidsformodelnumberK=4andprojectionaxesM=2. responsefromtheresponseinducedbymotion(i.e.theeye-blinkingperse)attheeyesandbythephysiologicalbackground(i.e,theperiodiccardiovasculareects).Fromamethodologicalpointofview,thepurposeofthetaskdesignandtheanalysisprocedure(i.e.,Iwasblindtothetimingoftheprotocol,inotherwords,Iamnotawareoftheexacttimewhenthesubjectwasperformingtheeye-blinkingtask)istoidentifyatimewindowfortheresponsewhenthereisnoaprioriknowledgeaboutthetimingoftaskon-set. First,accordingtoourresults,therearetwopeaksfound,whichareconsistentwiththeactuallyrecordedprotocol(blindtome).SoallthsesmethodsseemtobeabletodetecttimewindowsbutourmethodsaremoresensitivethanTCA.Finally,thelocalizationdetectionderivedfromthetemporalclusteringanalysis(TCA),non-negativematrixfactorization(NMF),competitivesubspaceprojection(CSP)meth-odsarecompared.Thenonnegativematrixfactorizationmethodwithfourbases

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Figure5{7. TheclustercentroidsformodelnumberK=3andprojectionaxesM=2. givesthetimeseriesclassicationandtheircorrespondingspatialclassicationinFig. 5{8 & 5{9 whilethecompetitivesubspaceprojectionmethodwithK=3andM=2givesthetimeseriesclassicationinFig. 5{7 andtheircorrespondingspatialclassicationinFig. 5{11 .Itshowsthatbothmethodslocalizestwoactivatedre-gions.Oneregionisaroundeyeswherethetaskofblinkingattwoseparateinstantscorrespondstothetwopeaksinitsclustercentroidatsample136and366.ThisisclosetotheresultgivenbytheTCAmethod(sampleindex126and367)showninFig 5{10 .However,TCAcan'tlocalizetheblinkingtaskfocusingontheeyeregionasNMFandCSPdoes,showninFig. 5{11 .TheseoutlinersinTCAareduetothenoiseinference.Itdemonstratestheproposedmethodhasmorerobusttonoiserejec-tionperformance.AnotherregionlocatedbyNMFandCSPisatthecentercorticalregionwithperiodicoscillationwhileTCAcompletelyignoresit.

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Figure5{8. Thefourbasisimages(1-2upperrowand3-4lowerrowfromlefttoright)aredeterminedbyNMFusingrealfMRIdata. OurresultsfurthersuggestthatthelocalizationbasedonTCAmaynotbesen-sitiveenough,evenfordetectingthestrongmotion-relatedchangesinfMRIsignals.However,wefailedtodetectanycorticalactivationusingeithermethodbutinsteadournewmethodsdetectedboththeresponseinducedbyblinkingperseattheeyesandresponseinducedbyphysiologicalbackgroundnoise,i.e.,theperiodiccardiovas-culareectsinthecentralspinaluid(CSF).ButourmethodsarestillbetterinthelocalizationofchangesinfMRIsignalthanTCA.Thereasonwhywedidnotdetectanyblinking-relatedcorticalactivationmaybelyinginthefactthatourmethodsarenotsensitiveenoughtodetectsmallBOLDresponseovershadowedbythelargernon-BOLDeectsinducedbybothmotionandperiodicphysiologicalnoises.Afur-therstepistoreneourmethodbasedonalteringprocedure,soastoremovenoisecomponentsrstdenedandthentodetecttheBOLDresponse.

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Figure5{9. TheencodingtimeseriescorrespondstofourbasisimagesbyNMFusingrealfMRIdata. FunctionalMRIanalysisprovidesavaluabletoolforunderstandingbrainactiv-ityinresponsetoexternalstimuli.Inthisdissertation,werstlyincorporatesoftcompetitionasatoolforextractingtemporalandspatialactivationsinsequences

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Figure5{10. Thetemporalmaximaplotfortemporalclusteringanalysis(TCA)method. offMRIimagesthataretakenfromsubjectswhoareexposedtotask.IthasbeenshownthattheconclusionsdrawnfromCSP,NMFandapreviouslyproposedmethod(TCA)areconsistentinndingpeakresponseswhilebothCSPandNMFaremorerobusttonoiseinterferencethanTCAandCSPcangeneralizetoanyresponse. There'sstillroomtoimproveCSPinclassifyingfMRIimages.Theimplicitorthogonalityconstraint,theimagesupportreduction,andsomealternativeinfor-mationtheoreticaloptimizationcriteriawillbeconsideredinfutureresearch.AlsoitsapplicationwillbeextendedtotraditionalfMRIproblemwithknowntemporalreferenceinourfuturework.

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(b) (c) Functionalregionlocalizationby(a)temporalclusteringanalysis(b)nonnegativematrixfactorizationand(c)competitivesubspaceprojec-tion.

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3{11 Intheproposal,reconstructionresultsareshownforsinglesnap-shotMRIim-ages,therefore,theSNRlevelsandtheimagequalityispoorandinsucientforpracticalpurposes.However,notethatinpracticemanyreconstructedsnap-shotsliketheseareaveragedinordertogethigherSNRlevels.Therefore,thedemonstrated 88

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improvementsintheSNRofsinglesnap-shotsdirectlyinuencetwoimportantfac-tors:1.thenalreconstructedimagequalityimprovesassumingthesamenumberofaveragedsnap-shots.2.TheimageacquisitiontimeisreducedassumingaxedSNRsothatfewersnap-shotsarenecessary.However,thismeasuredimprovementinSNRmaynotextendtolessnoisyimages.Thereareseveralissuesthatneedtobefurtherdiscussed.Thedesiredresponseiscurrentlyestimatedbytheaverageofthereconstructionfromeachsample.Itmaybepossibletoreplacethelengthyprocedurebyasinglenearoptimalimagewithalongscanningtime.Therelativelyheavycomputationalloadoftrainingneedstobeconsidereddespiteperformanceimprovements. Finally,Iwouldliketoaddressthenecessityofthedesiredresponsethatcanbeconsideredasashortcomingofthistechnique.Ithasbeendemonstratedthatasystemtrainedinonesetofimagesperformswellonanother,butitistooearlytoquantifythegeneralizationofthetechniquetoanyimagecollectedinthesameMRImachine.EectivelyIbelievethattheimprovedmethodisextractingthespatialanisotropyofthecoils,butitisdoingsobyusingimagesthatdescribethespatialEMeld.Thereforefurtherworkisnecessarilytoquantifytheseeorts. GeneralizationTheapplicabilityforthemethodispredictedinitsgeneraliza-tionability.Theimplicitpowerofadaptivelearningrestsonitssystemidenticationcapabilitytoextractthesysteminformationandstoreitinthenetworkweights.Thisinformationofthesystemisestimatedfromthesampleswithoutanystatisticalassumption.Thenthisnetworkcanbealiatedtootheruntrainedsampleswhichimplicitlyhavethesamedatadistribution.Thuswegainthefreedomofpredictingunknowndata,whichhasthesamemetricinherentinnature,withtrainingonagivenamountofsamples.Thecompetitivemixtureoflocallinearexpertscapturestheconstantspatialanisotropiccoilsensitivitiesandnoisepropertiesfromtraining

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imagesdirectly.Howwellthiscompetitivelocaltrainingmethodlearnstheinnercoilstructureneedstobeexperimentallyvalidated.Experimentsareproposedwiththreephantomsrepresentingthreedierentimagecontentsscannedasthreetrainingsetsinagivenscanningsystem.Eachtrainingtrainsacompletelyseparatecompetitivemixturenetwork.Thedierentthreenetworkweightsshouldbeapproximatelycon-stantbecausetheyreectthefeaturesofthephased-arraycoils.Ifthisresultholds,thegeneralizationoftheproposedalgorithmisvalidated. 3{6 ,theerrordecreasesveryfastintherstfewiterationandthenuctuates.Thismeansthatthelocallinearexpertsarenotbeingproperlyadaptedprobablyduetoerrorsintheselectionofthewinners.Softcompetitionmaybeanalternative.Futureworkshouldinvestigatehowtocombinemultipleinputmultipleoutputmodelswithoutloosingthesharpnessofthereconstruction,buthighercomputationalcomplexitycanbeexpected.

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considerincorporatingthehigherorderpolynomialintothemultiplemodelsforbet-terfunctionapproximationsinlocalregions.However,thetrainingforsuchmultiplenonlinearmodelsisdicultduetomanylocalminima.

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reconstructionwithoutcouplingwiththeunrelatedmodeloutputs.However,thecostpaidislosingtheabilitytotrackthelocalmetricbetweeninputanddesire. Besides,suchphantomcanbeusedasaqualitymeasureofthereconstructionmethods.AproposedmethodisgivenbasedonthestandardNEMASNRcalculationasthefollowing.Twonearidenticaltestingphantomimagesarereconstructedandsubtractedtowardeachothertoeliminateanyimagestructure.Thenthepurenoisewithp

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signalpowerbythelocalnoisepowerfromthissubtractiongivesthelocalSNR,whichtrulyrepresentsthelocalnoiselevel. Thesecondreasonwhythetrainingisnotwidespreadinindustryisbasedonthecurrentlongtrainingtimeandunguaranteedalgorithmrobustnessduetolocalmin-ima.Whattheindustryprefersisadirectanalyticalsolutionlikethesum-of-squaresmethod.Thecurrentdevelopmenttowardschipdesignforthefastconvergenceoftrainingthelargeamountofimagepixelsisoptimistic.However,thelocalminimaproblemfornonlinearsystemisstillopen.Wehopetondsomeasymptoticoptimalandconstrainedoptimalconditionforthecompetitivelocallinearmodelinthenearfuture. ThelastreasonfocusesonimprovingtheimageSNRwhileincreasingscanningtimebymultiplecoils.Asweknow,thescanningspeedcanbefurtherimprovedbyparallelimaging,whichundersamplesink-space.Thisparallelimagingideacanbefurtherintroducedintotheadaptivetrainingmethodology,wheretheadvantagesofadaptivetrainingispreservedwhilefurtherimprovetheimagingspeed.

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Atransmitonlybirdcagecoiltocreateanhomogeneousexcitationprolewascoupledwitha4channelphasedarrayreceivercoiltocollecttheimagedata.TheguidanceforthisconstructionwasprovidedbyDavidM.PetersonintheRFcoillaboratoryatAdvancedMagneticResonanceImagingandSpectroscopy(ARMIS)locatedintheUniversityofFlorida'McKnightBrainInstitute.InordertointegratetransmitandreceivercoilstotheSiemensAllegraMRIsystem,acustomsetofTRswitchesandsoftwarecoilcongurationsleswereimplemented. A8polequadraturebirdcagecoilwasselectedtoprovideahomogeneoustransmiteld.CoilsareevaluatedinthetransceivemodeandthenconvertedtoTxonlybytheadditionofdecouplingtrapsthatareturnedonintheRxmodel.Thetransmitonlybirdcagecoilwith8legsismeasuredusingBirdcageBuilderv1.0areshowninTable. A{1 Eachcoilwaslaidoutfrompiecesofcopperstrip(3M,Minneapolis,MN)ontheappropriateformer.ThedimensionsforeachcoilwereoptimizedinordertomaximizetheSNR,whilepreservingthehomogeneityofthecoilstoanacceptablelevel. Thecapacitors(AmericanTechnicalCeramics[ATC],1000V)werethenequallydistributedandtheresonantmodewasveriedusingaHewlettPackard(HP)8752networkanalyzerandtheneareldprobeset.Integermultiplesofhalf-wavecablesweremadewithcabletraps,dependinguponthedesiredlength. Acabletrapisanarrowbandequivalentofwrappingthecablearoundaferritecore.However,sinceferritesarenotconducivemagnetically,thesetrapsarecon-structedfromdiscretecomponents.Thecableiswrappedintoaninductiveloop,andacapacitorthatisresonantwiththeshieldinductancewasplacedfromoneendof 94

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TableA{1. 8legbirdcagecoilparameters Resonantfrequency 123.20MHz Coilradius 10cm RFshieldradius 12cm Leglength 31.50cm Legwidth 1.30cm Endringlength 8.84cm Endringwidth 1.30cm Calculatedcap 5.02pF Calculatedlegselfinductance 275.99nH Calculatedend-ringselfinductance 54.95nH Calculatedlegeectiveinductance 102.54nH CalculatedEnd-ringseg.eectiveInductance 67.27nH thelooptotheother.Thismadeahighimpedanceblockforunbalancedcurrentsontheshield.Thetwocableswerethenattachedtothedrivepoints,whichweredirectlycoupledtothecoil,90degreeapart,allowingforquadraturemodeoperation.Eachcoilwasloadedwithasalineandcoppersulfatephantomthataccuratelyrepresentedtheanatomythateachcoilwasdesignedtoaccommodate.Thecapacitorineachdrivelegwassplitintotwocapacitors,onewithrelativelyhighreactanceandtheotherofrelativelylowreactance.Thelowerreactancepointwasusedforimpedancematching.Thematchwasobtainedbychangingthecapacitanceofthecoilatthedrivebreakwithoutsignicantlyshiftingfrequency.Thegroundedsideofthecablewasattachedbetweenthedrivebreaksandanyimaginarypartoftheimpedancewasthencanceledwithsomeseriescapacitanceorinductance,ifnecessary,toobtaina50-ohminputrealimpedance.Intheseparticularcases,theimaginaryimpedance-compensatingelementswerenotrequired.Thisprocedureproducedaquasi-balancedmatchcongurationwithminimalcomponents. Thematchingisolationisveryloaddependent.TheowchartisdemonstratedinFig. A{1 .Thereceivercoilconsistsoffourdecoupledloopwithsizeof6.1cmx7.0cmwithosetof2:1cm.TheowchartisdemonstratedinFig. A{2

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FigureA{1. Transmitonlybirdcagecoilowchart. A{3 willbeusedtopresentthegeneralmethodforcreatingaT/Rswitchforthereceivephasedarray. ComponentsL1,C1,C2arethephaseshiftersforadjustingphasebetweenthecoilandthepreamplierforreceivemode.Theisolationbetweenthetransmitterandthereceivecoilreliessolelyonthetrapcircuitsonthereceivecoilduringtransmitandthetrapcircuitsonthetransmitcoilduringreceive.L2isadjustedwithC3whenD1isactiveinordertoprovideahighimpedanceblock.C3shouldbea-j50Ohmstoproducea50-Ohmtransmissionlineequivalentand90-degreephaseshiftwhencombinedwithL2,L3andtheinductanceofD1.L4isachokeforDCandshouldhaveatleast1000Ohmsofreactance,ifasuitableinductorcannotbefounda

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FigureA{2. Receivercoilowchart,withC1,C2aretheparallelcombinationofa20pFcapacitoranda1-15pFadjustablecapacitor;C3,C8arethepar-allelcombinationofa4.7pFcapacitor,a91pFcapacitor,anda39pFcapacitor;C4istheparallelcombinationofa3.9pFcapacitoranda1-15pFadjustablecapacitor;C5,C7aretheparallelcombinationofa91pFcapacitoranda39pFcapacitor;C6istheparallelcombinationofa18pFcapacitoranda1-15pFadjustablecapacitor parallelLCtrapresonantatthefrequencyofinterestwillproducethesameresults.Thiscircuitwasemployedfourtimesinordertogetfourchannelsforthesystem. TheT/Rschematicwasconvertedtoadouble-sidedcircuitboardusingciccard(Holophase,DavieFl)withthecomponentsononesideandagroundplaneontheother.TheboardwasmanufacturedbyAdvancedCircuits(Boulder,CO).Com-ponentswerethenplacedontheboardwithtuningdoneontheHP8752networkanalyzer(Hewlett-Packard,SantaRosa,CA).OncetheT/Rswitchwascomplete,coilstoevaluatereceiveonlyphasedarrayhadtobeconstructed. Uponconstructionofthehardware,apowersplitterisusedtoprovidetheRFpowertotheexcitationcoil.Atransmissionmethodwasusedandcompletedbyconnectingatwo-way,90-degreesplitter,thusallowingfortransmissioninquadrature.ThismethodisshowninFig. A{4

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FigureA{3. Schematicrepresentationofasingletransmit/receiveswitchingcircuitforprotectionofthereceivingpreamplier. Priortotesting,thecoilshadtobeconguredinsoftware.Avechannelcoilcongurationwasprogrammed(onetotransmit,fourtoreceive).

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FigureA{4. Blockdiagramofthequadraturetransmitcoil,andreceive-onlyphasedarraysetup.

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21 ].Itisalsoconsideredtobeanextremelyimportantinstrumentforthestudyofotherpartsofthenervoussystem(suchasthespinalcord),aswellasvariousjoints,thethorax,thepelvisandtheabdomen.Becauseoftherecentinterestinsignalprocessingforimprovingtheimagequality,whichisoftenquantiedbytheestimatedsignal-to-noiseratio(SNR),itisimperativetounderstandhowthismeasurecanbeaectedbynonlinearsignalprocessingoperations. 10 23 22 ]),animprovementinSNRcanalwaysbetranslatedintoanincreaseinacquisitionspeedandthereforebeusedtoreducetheimagingcostandmotionartifacts.Therefore,improvingtheSNRinMRimageshasbecomeextremelycriticalforreducingmotionartifactsandinapplicationswhereimagingspeedisamajorconcern.Suchapplicationsinclude 100

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imagingofdynamicprocesses,suchastheheart[ 79 ].Also,sinceanimprovementinSNRcansignicantlycutimagingtimes,itcanincreasethecost-eectivenessofMRIequipmentinahospitalenvironmentaswellasdecreasebreath-holdingdurationsandotherdiscomfortsforpatients. Evaluationofthequalityofareal-worldimageisoftenasubjectivetask,andperhapsduetotheabsenceofmoresophisticatedindicators,theSNRappearstobeoneofthemostpopularlyusedmeasuresofthequalityofanMRimage.IngeneraltheSNRdoesnotmeasurebiaserrors(whichareoftensignicant),andfurthermorethereisnotalwaysaclearcorrelationbetweentheSNRandtheimagequalityasvisuallyperceivedbyahumanobserver,whichismorerelatedtothecontrastinabroadsense(see,e.g.,[ 80 ,Ch.7]foradiscussionofvisualimagequality).InthiscommunicationwedemonstratethattheSNRcanbemanipulatedbynonlinearoperationsonthedata,andthatitissometimesalsodiculttomeasureobjectively.WethereforebelievethatcautionshouldbeexercisedwhentheSNRisthesolequalitymeasureofareconstructedimage,oroftheimprovementoeredbyasignalprocessingalgorithm,whichpossiblyemploysanonlinearoperationatsomestageofprocessing. Inthisstudy,weassumethatareal-valuedMRimageisalreadyobtainedfromtherawk-spacedataandthatnecessarycorrectionstoreducephasedistortionsmayhavebeenappliedasdiscussedin[ 81 82 83 ].However,sincetheresultsonthedistortionofSNRundernonlinearoperationsaretrueingeneral,similareectsareexpectedtooccurifnonlineartechniquesareemployedwhenreconstructingimagesfromtherawk-spacedata.

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modeloftheform wheresisasignalofinterest,andeisnoise.Weassumethatbothsandearerandomvariables.Also,throughoutthispaperweassumeforsimplicitythatallsignalsandnoisearereal-valued, SNRx=Efs2g whereEfgstandsforstatisticalexpectation. Letusconsiderthefollowingnonlineartransformationofx: wherek!isthefactorialofkandf(k)(x)isthekthderivativeoff(x),assumingthatallderivativesoff(x)arewell-dened.TheSNRinyisequalto: SNRy=Eff2(s)g wherebyconvention0f(s)=f(1)(s)and0f2(s)=(0f(s))2,andwheretheapproxima-tionisvalidwhenSNRx1.WeconcludethatSNRy>SNRxexactlywhen 84 81 85 ]formorediscussiononthestatisticsofthenoiseinMRimages.

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andthereforenonlineartransformationscanimprovetheSNRinasignal,providedthatthefunctionf(x)andthestatisticaldistributionofsaresuchthat( B{5 )holds. Ingeneral,theconditionsonf(x),underwhich( B{5 )holds,dependonthedistributionofs.However,wecaneasilystudyafewspecialcases.If,forexample,f(x)=x2,then0f(x)=2xandhence SNRy WeconcludethatSNRy>SNRxifandonlyif Forzero-meanrandomsignalss,( B{7 )holdsexactlywhen where(s)istheKurtosisofs.(ForaGaussiandistribution,(s)=0;distributionsforwhich(s)>0arecalledsuper-Gaussian,anddistributionsforwhich(s)<0arecalledsub-Gaussian.)Thismeansthatiftheprobabilitydistributionoftheimageishighlysuper-Gaussian(mostnaturalimagesareinthisclass),i.e.,denseraroundthemeanandheavieratthetails(e.g.,Laplacian)thenthesquare-operation(suchastheoneusedincreatingmagnitudeimages)coulddeceivinglydemonstrateanimprovementinSNR. Aninterestingquestioniswhetheritispossibletondafunctionf(x)suchthatSNRy>SNRxregardlessofthedistributionofs.Withoutlossofgenerality,consideraunit-powersignals(i.e.,Efs2g=1).Thenfrom( B{5 )SNRy>SNRxiff2(s)>0f2(s)foralls.Thiscanbeachievedifwechoosef(s)=aswhereasatises1=e(lnjaj)2a2s=0f2(s).Hence,anonlinearityintheformofanexponentialfunctionwithexponentaintherange1=e
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TheSNRinfxngisSNRx=Efs2ng=Efe2ng. Foragivenimage,theSNRisusuallyestimatedbyusingamoment-basedesti-matoroftheform whereNsandNnarethenumbersofpixelsinthesignalandthenoiseregion,re-spectively.ForareasonablyhighSNRxandforalargenumberofmeasuredpixels,

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wehavethat whereweusedtheassumptionthatthenoiseenhaszeromeanandisindependentofsn(thisequationwasdiscussedinmoredetailbyHenkelman[ 82 ]).Notefrom( B{11 )thatthemeasuredSNRisalwayslargerthanthetrueSNR.However,whenSNRxishigh,estimatingitvia( B{10 )ingeneralgivesreliableresults. WenextdiscusshowthemeasuredSNRcanchangewhenthesignalfxngistransformedviaaquadraticnonlinearfunction. FromtheanalysisinSection B.2 weknowthatforaconstantsignal,SNRySNRx=4andhencetheSNRinfyngislessthanthatinfxng.(Thisisnaturalsincethesign,orthephaseforcomplexdata,islostwhenthetransformation( B{12 )isapplied.)Nevertheless,theSNRinfyng,asmeasuredvia( B{10 )canbemuchlargerthantheSNRmeasuredfromtheoriginalimagefxng.Tounderstandwhythisisso,consider

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themeasuredSNRinfyng,assumingthatSNRx1: 1 1 Thisexpressionessentiallybehavesas(dSNRx)2.Therefore,weexpectthemeasuredSNRinfyngtobemuchlargerthanitactuallyis;i.e.,thesquaringin( B{12 )makesthesignalappeartoanobserverasifitweremuchlessnoisy. B{1(a) for2=1.InFigure B{1(b) weshowthesignalafterthenonlineartransformation( B{12 ).Finally,inFigure B{1(c) weshowthetrueSNRfortheoriginalsignal,themeasuredSNRfortheoriginalsignal(asdenedvia( B{10 )),thetrueSNRforthetransformedsignal,andthemeasuredSNRinthetransformedsignal(asdenedvia( B{13 )),forsomedierentvaluesof1=2.ThetrueSNRinthetransformedsignalynisapproximately6dBlowerthantheSNRintheoriginalsignalxn,when1=2ishigh.(WecanseethatthemeasuredSNRofxnconvergestothetrueSNRinthiscase;cf.( B{11 ).)This6dBdierenceinSNRbe-tweeny2nandx2ncorrespondstothetheoreticalvalueof1/4describedinSection B.2

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(b) (c) Syntheticdataexample.(a)Originalnoisystepfunctionsignalxn,(b)transformed(squared)signalyn,and(c)thetrueandthemeasuredSNRlevels. Ontheotherhand,themeasuredSNRinfyngappearsmuchlargerthanthetrueSNR,whichcorroboratesthendingsofSection B.3 86 ].Thedatacollectedfromaphasedarrayoffourcoilsiscombinedusingthesum-of-squares(SoS)techniqueto

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yieldareconstructedimage.Letykbetheobservedpixelvaluefromcoilk: whereisthe(real-valued)objectdensity(viz.theMRcontrast),ckisthe(complex-valued)sensitivityassociatedwithcoilkfortheimagevoxelunderconsideration,andnkiszero-meancomplex-valuednoise.TheSoSreconstructionforthisvoxelisobtainedvia WeconsidertwodierentnonlinearoperationsontheSoSreconstruction:naturallogarithmandmedianltering(MF).Theformernonlinearoperationsimplygener-atesanewimagebymodifyingthepixel-by-pixelvaluesbyapplyingthelogfunction.Thelatteroneisastandardnonlinearimageprocessingtechniquethatisrobusttooutliers,whichisoftenusedtoimproveSNR.Inmedianltering,eachpixelvalueissimplyreplacedbythemedianofthevaluesofitsneighboringpixels(hereweusea55regioncenteredatthepixelofinterest). InFigure B{2 B{3 & B{4 wepresenttheimagesprovidedbySoS,log-SoS,andMF-SoSaswellastheircorrespondinglocalSNRestimatesusingthereferencenoiseregionshownintheupper-rightcorner.ObservethatalthoughtheMF-SoSimageexhibitsanimprovedSNRcomparedtotheoriginalSoS,theimageactuallylooksworse.Ontheotherhand,theSNRofthelog-SoSimagedecreased,yetthedynamicrangeandthesignalcontrastinlow-powerregionsareimproved.

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(b) ReconstructionimagesandtheirSNRperformance.(a)SoS,(b)SNRofSoS. 87 ]forideasalongtheselines).

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(b) ReconstructionimagesandtheirSNRperformance.(a)logrithmofSoS,(b)SNRoflogrithmofSoS. (b) ReconstructionimagesandtheirSNRperformance.(a)medianlteredSoS,(b)SNRofmedianlteredSoS.

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88 ].Hotellingtracecriterion(HTO)isusedfortheoptimalclassicationoftheimagingsystem[ 89 ].ThechannelizedHotellingOb-server(CHO)providesagoodsignaldetectionperformanceapproximatingthehumanobserver[ 90 ].Theobservermodelbasedqualityaccessroughlyestimateshowthetwoormoreobjectclassesareseparatedbylikelihoodratiotest.However,itisnotsuitableasaquantitativemeasuretorankmedicalimagequalityevenwithsimilarcontent. TheSNRisconsideredasaprevalentmeasuretoevaluatetheperformanceofamagneticresonance(MR)images.Thoughitdoesn'tprovideinformationfortheimageresolutionandtheimageblur,itgiveshowmuchnoisecorruptsthesignalinsidetheregionofinterest(ROI).Thismeasureisgenerallyusedtoestimateintrinsicphysicsunderlyingthemagneticresonanceimaging(MRI)scanningsystemincludingtheradiofrequency(RF)coildesignandthesystemparameterselection[ 91 92 ].ThepurposeistondaguidelineforaqualitymeasureinMRIbySNR[ 93 ]. 111

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Inthelastdecades,multiplephased-arraycoilsareusedforfastimaging.TheincreasedequipmentcomplexityincreasestheSNRwhileequivalentlyreducethescan-ningtimewhichhasthebenetsofreducingthemotionartifactsoftheimage.Thesum-of-squares(SoS)method,proposedbyRoemeretal.[ 10 ],setsthefoundationofphased-arrayimagereconstructionanditisprevalentintheindustry.BasedonSoS,asubstantialbodyofreconstructionmethodshavebeenproposedtoreconstructcoilimages.However,noeortshasbeenreportedonthecomparisonamonghowtoeectivelyevaluatethereconstructionperformancebyaproperSNRmeasure.Thedicultieslieintwoaspects.Ononehand,thereconstructionisimplicitlynonlin-ear.AnynonlineartransformcanunlimitedlyincreasetheSNRestimatedbyasingleacquisitionimagesamplewithoutaectingthetruesignalperformanceinROI[ 94 ].ThusafairevaluationhowthenonlineartransformaectstheimagequalityinsideROIneedstobestudied.Ontheotherhand,thoughthenoisepropertiesarewellstudiedin[ 95 ],thenoisestatisticsafterthenonlineartransformcanbearbitrarilyanything,wherenoparametricestimationcanbeusedhere.Inthispaper,westudythereconstructedimagequalityproblemandincorporateanon-parametricnoisesta-tisticsmeasureintotheSNR. C.2.1TraditionalSNRmeasures 96 ].Forasingleimageacquisition,theSNRSNRsingleiscomputedas whereMsistheaveragesquarerootofsignalpowerinROI,SDbistheensemblestandarddeviationofthenoiseselectedinthebackgroundregion.Thebackground0.655factorisduetotheskewednoisedistributionbasedonthemagnitudeim-agefromtheFouriertransform[ 97 ].ThisfashionofSNRcalculationisconsistent,

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wherenoscannerstabilityneedstobeconsidered,suchthatnoimageregistrationisrequired.However,ithasthedisadvantagethatartefactsfromghostimagesandnon-uniformitiescanbeprojectedintothebackgroundareas.Besides,itishardtovalidatethenoisepowerinthebackgroundareaisequivalenttothatinthesignalregionbasedonthisSNRmeasurement. AnotherwaytocalculateSNRisaccordingtothedualacquisition whereMs1istheaveragesquarerootofsignalpowerinROIontherstimage,SD12istheensemblestandarddeviationintheROIonthesubtractionimage.Ithastheoppositeadvantageanddisadvantagecomparedwiththesingleacquisition. wherei;jisthekthsubregionand^Mand^SDdenotethenonparametricestimatesofthesquarerootofsignalpowerandnoisepower.Sincethesignalandnoiseonthereconstructedimagealreadypassthroughanonlinearsystem,theirprobabilityden-sityfunction(pdf)cannotbeuniformdistributed.Thustheirrstorderandsecond

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orderstatisticscannotbeestimatedasanensembleaverage.However,nonparamet-ricParzenwindowprovidesawaytoestimatepdffromdatasamples.Therefore,thenoisepower^SD12;i;jcanbeestimatedusingParzenwindowas ^SD12;k=s Z(xkxk)2f(xk)dxk=vuut wherexkisthesubtractiondatasampleandNkistotalnumberofsamplesinthekthsubregion,f(xk)isthepdfofxk,G()istheGaussiankernelusedtosmooththepdfestimationwithkernelwidth2k.Thenthesquarerootofsignalpower^Ms1;kisestimatedas ^Mx1;k=s Zx12kf(x1k)dx1k2^SD212;k=vuut wherex1kisthedatasampleinthekthsubregionfromtherstimage.

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whereisthe(real-valued)objectdensity(viz.theMRcontrast),ckisthe(ingeneralcomplex-valued)sensitivityassociatedwithcoilkfortheimagevoxelunderconsider-ation,andekiszero-meannoisewithvariance2k.Weassumeinthisappendixthatthenoiseiswhite;atthepriceofsomeadditionalnotationallourresultscaneasilybeextendedtonoisewithageneralcovariancestructure. 98 ].Asignalmodeledasaproductoftwocomponentscanbesplitbyusinghomomorphicsignalprocessing.TheMRIsignaljskjisrepresentedbytheprod-uctoftwopositivecomponents,thetruepixelandthesensitivityjckj(0
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Thelogarithmfunctionrstlytransformsthemultiplicationofandckintoanaddition. logjskj=log+logjckj(D{2) Thelinearsystemseparatesandjckjbyseparatingtheassumeddierentspectralofeachcomponent.Themosteectiveinformationinthetruepixelimageisatthesharpboundarybetweenbonesandmusclesorbetweenbonesandtissuesbecauseofdierentwaterpercentagesinside.Thustheeectiveismostlyahigh-frequencysignal.Themagnitudeofthecoilsensitivitiesjckj,relatedtothecoilsignals,isrelativelyslow-varyinginsignalareaandmostlyalow-frequencysignal.Thoughtheymayhavesomeoverlapinthelowfrequencydomain,onecouldpartlylteroutthecoilsensitivitiesbypassingthetwo-dimensionalFouriertransformofthelogarithmofthecoilimagethroughahigh-passlter.ThenaninverseFouriertransformrecoversthetruepixelsignalfromthefrequencydomaintotheoriginalspatialdomain.Thethirdstepisanexponentialfunctionthateliminatestheeectofthelogarithm.Theoutput^skfromthehomomorphicsignalprocessorforeachcoilisconsideredasamultiplesampleofpixelimage.Thus,thereconstructionissimpliedtoaverage^sk Someimageprocessingmethodsareimplementedtoimprovethereconstructedimagequality.TheGaussianshapedfrequencydomainlter,whichhasthesameshapeinthespatialandfrequencydomains,isusedtoremovenoiseinthenoisearea.Anonlineargammafunctionisusedtoweighttowardthehigherpixels.Thoughthenonlineartransformintroducesbias,itincreasestheimagecontrast. Thecriterionofthelterselectioninhomomorphicsignalprocessorisakeyproblem.SNRinthehomomorphicsignalprocessingmethodisnotasuitablecrite-rion;onthecontrary,thelowerSNRisthecostofthemethodtogainhigherimage

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FigureD{2. Photographofthephasedarraycoil,transmitcoil,andcabling. contrastbecausepartoftheenergyislteredoutinthesignalareawhilethenoiseisnotaectedmuchduetoitsapproximatelyuniformspectralinthefrequencydo-main.Besides,theMMSEcriterion(minPkjsk^ckj2)doesn'tgivetheoptimalsolutionbecauseofthecomputationalcancellationduetothewayofsplittingandck.Weproposetheeectivemaximumimagecontrastinthereconstructedimageasacriteriontochoosethehigh-passlter.Thiscriteriondenotestheeectiveinfor-mationofinterestcomparedwithSNRandgivesusgoodresultswhichalsohasthedisadvantageofmanuallyspecifyingtheimagecontrastareaofinterest. D{2 (TR=1000ms,TE=15ms,FOV=105cm,matrix=256128,slicethick-ness=2mm,sweepwidth=26khz,1average)[ 86 ].Figs. D{3(a) D{3(b) D{3(c) ,and D{3(d) showthecollectedfourcoilimages,wherecoils1,2focusontheupperpartoftheimageandcoils3,4emphasizethelowerpartoftheimageduetodierentcoillocations.

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(b)2 (c)3 (d)4 (e)SoS Vivosagittalimagesofcatspinalcordfromcoil1-4andthespectralestimateofSoS. ThespectraldistributionoftheSoSestimateofthetruepixelimageisshowninFig. D{3(e) (alltheguresinthefrequencydomainareshownin[0 2{10 ,andtheirspectraldistributionsareshowninFig. D{4 .Wecanseethatthecoilsensitivitiesareslow-varyingintheeectivesignalareaandshowtheirlow-passproperty.

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(b)Coil2 (c)Coil3 (d)Coil4 (e)Coil1 (f)Coil2 (g)Coil3 (h)Coil4 (Upperrow)Spatialdistributionofthecoilsensitivitiesforfourcoilsignals.(Lowerrow)Spectraldistributionofthecoilsensitivitiesforfourcoilsignals. Thusahigh-passlterisdesignedtolterthecoilsensitivities.Thecutofre-quencyandthestopbandmagnitudeofthelterarechosenbasedontheeectivemaximumimagecontrastcriterion(thelterorderadjustmentisnotconsideredforsimplicity).Fig. D{5 showsthattheimagecontrastsurfacehasaglobalmaximumandthemagnitudeatthepeakisovertwotimeshigherthanthatinSoS.BasedonthelterwithpeakcontrastinFig. D{6 ,thetruepixelimageisreconstructedbythelteroutputsforeachcoil.Theproposedmethoddemonstratesvisuallybetter

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FigureD{5. Thereconstructionimagecontrastversusthehigh-passltercutofre-quencyandthestopbandmagnitude. reconstructionresultsthanSoSmethodinFig. D{7 .Fig. D{7(d) showstherecon-structedlteredcoilsensitivities,indicatingthattheeectiveinformationofthehighcontrastimageisnotlteredoutbytheproposedmethod.Thisisbecausethoughtheenergyisdominantinlow-passband,theeectiveinformationoftheimageismainlyinhigh-passband.TheprobabilitydensityfunctiondistributionsofthesereconstructedpixelsareshowninFig. D{8 .Itshowsthatthecontrast-enhancedho-momorphicsignalprocessingmethodwhichhastheattestpixeldistributioninthemiddleofintensityscale(between50and100)givesthebestimagecontrast(similartohistogramequalization).Thismethodalsoshowsagainof10%innormalized

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(b)Coil2 (c)Coil3 (d)Coil4 High-passltertoeliminatecoilsensitivities. TableD{1. Normalizedentropyof(a)SoS,(b)homomorphicsignalprocessing,and(c)contrast-enhancedhomomorphicsignalprocessing. Method (a) (b) (c) Entropy 0.8064 0.8714 0.8924 entropycomparedtotheSoSmethodcomputedby(Table. D{1 ), logNscaleXf(^)log(f(^))(D{4) where^isthereconstructedpixel,f()isthepixeldistribution,NscaleisthepixelintensityupperboundandEisthenormalizedentropy.

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(b)b (c)c (d)d Reconstructedimages.(a)Sum-of-squares(sos),(b)homomorphicsig-nalprocessing,(c)contrast-enhancedhomomorphicsignalprocessing,and(d)reconstructionfromthelteredcoilsensitivities. range.However,theimagequalityisnotaectedinthedesiredsignalareawithhighcontrast.

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FigureD{8. Thepdfdistributionofthereconstructedimages.

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99 ],GeneralizedSMASH[ 100 ],andGeneralizedSENSE[ 101 ]considerthisproblemgloballybysolvinghugematrixequations,whichistimeconsuming.Parallelprocessingtodierentaccelerationfactorregions,suchasMadore'smethod[ 102 ],reducesthereconstructiontimebutcanhavearingingartifactatedges.King[ 103 ]addressedasmoothingltertoseparatevariabledensityhigh-passandlow-passdata,thussuppressingtheringartifact.Therecombinationofthetwoimagescouldstillleadtoanintensitybiasbetweenthehigh-passandlow-passcomponents.ThischapterdiscussesalterdesignstrategynamedashomoSENSEinvdSENSEtoreducebiaswhilestilllteringouttheringeect. 124

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R.Finallythecombinationoflow-passandhigh-passreconstructionsgivesthenalimageateachpixellocation. TwokeyissuesexistinvdSENSE.First,thehigh-passlterdesignshouldprovideenoughinformationforthesensitivitymapusedforreconstructioninthehigh-passpart.Itcanbeeasilyseenthattheidealhigh-passlterwiththecutofrequencyattheborderoftheACSlinesfailssincenoNyquistsampledcenterk-spaceiscontainedinsidethehigh-passpart.ThesensitivitymappreliminarilyestimatedfromrawACSlinescan'treectthetruecoilmappingeectinthehigh-passbandandthusthehigh-passSENSEreconstructiondoesn'thaveaclearunwrappingeect.Therefore,thelteredhigh-passpartmustincludeatleastpartofthescaledcenterfullsampledk-space.Second,thetwoimagesLPandHPshouldbecombinedinawaywhichdoesnotover-weighteitherloworhighfrequencyinformation. TheproposedHomoSENSE,whichiscorrelatedtohomomorphicimageprocess-ing,providesanenergybalancecriteriontoinstructboththelterdesignandthenalcombination.Thenalreconstructionshouldhaveenergyequaltoanestimateoftheenergyoffullysampleddata.Theassumptionismadethatthedistributionofenergyink-spaceisfairlysmooth,sothattheequallyspacedundersampledhigh-passspectrumhas1/Renergyofthefullspectrum.WithscalingfactorsintheIFFTtakenintoaccount,energybalanceincoilkcanbewritteninEqn.( E{1 ), whentheenergyconservingcombination=p E{2 ). (FHP(i;j)2+FLP(i;j)2)=1(E{2)

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FigureE{1. SoSofaxialphantomdata. Basedonthiscriterion,thehigh-passltercanbedesignedas followedbylow-passlter whereFHPstandardisastandardhigh-passlterink-spacewithpassbandmagnitudeone,stopbandmagnitudezero,cutofrequencyclosetoACSboundaryandarbitrarylterorder. E{1 .Thek-spacesamplesaredecimatedinouteraccelerationfactoroffourbesidethecentral64ACSlines.Thehigh-passandlow-passltersbasedonButterworthlteroforder4andcutofrequencyatboth107and151

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FigureE{2. High-passandlow-passlterwithorder4andcutofrequencyat64. areshowninFig. E{2 .ThecentralPElinefromthereconstructionsusingSENSEandhomoSENSEisdemonstratedinFig. E{3 .ItdemonstratesthathomoSENSEgiveslessMSEcomparedtoSENSEwithlesslowfrequencybiasinthereconstruction.Thereconstructiontimeofbothmethodsisaboutthesame.

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FigureE{3. CentralPElinefromReconstructionsofhomoSENSEofMSE=0.23%,SENSEofMSE=2.19%comparedwithSoS.

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101 ]providesaglobaltwodimensionalsolutiontothisproblembyreplacingthematrixinverseofsizeN2N2,whichhasthecomputationalloadofO(N6),withaconjugategradi-ent(CG)thusthecomputationalloadreducingtoO(LN3)giventheimagesizeandepochnumberL.SPACERIP[ 99 ]andGeneralizedSMASH[ 100 ]furtherdecomposethe2Dk-spaceinto1Dk-spaceusingSMASHmodelingbysolvingthegeneralizedk-spacetrajectoryprobleminhybrid(k;r)space.ThisimprovementreducesthematrixsizetoNNwiththecomputationalloadO(N3)evenwithoutanyitera-tivealgorithm.However,bothmethodsincorporatemodelingerrorwhenexpressingthecoilsensitivitiesaslinearcombinationoforthogonalsets.Theproposedmethod,namedHybrid1dSENSE,istosolveSENSEinhybrid(k;r)spacebysolvingthein-verseconvolutionequations,whichavoidstruncationerrorsasin[ 99 100 ]andhighcomputationalloadin[ 101 ]. 129

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Givenanccoilarraywithspatialcoilsensitivityci(x;y)andtrueMRimage(x;y)wherei=1;;nc,theacquireddatainhybrid(k;r)spaceisgivenby wherexandkyisthefrequencyencoding(FE)directioninimagespaceandthephaseencoding(PE)directionink-spacerespectively,~ci(x;ky)and~(x;ky)aretheFouriertransformofcoilsensitivityci(x;y)and(x;y)atFEpositionxalongPEline.Suchexpressionisreformulatedinmatrixformincludingmultiplecoilprolesas wheres(x;ky)=[s1(x;ky);;snc(x;ky)]istheundersampledhybridspacevec-torofsizeM1nc1withM1eectivesamplesinPEdirectionofeachcoilimage,Pu(x;ky)=[Pu1(x;ky);;Punc(x;ky)]inwhicheachelementisanon-symmetricToeplitzmatrixwith~ci(x;ky)intherstcolumnandits(N1)thordercircularleftshiftedversionintherstrow.ThusaleastsquaresolutiongivesthehybridspacereconstructioninEqn.( F{3 ), whichgivesthenalreconstructedimagefollowedbya1DFFT. F.3 .ItdemonstratesthatHybrid1dSENSE

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(b) (c) Reconstructionofvariabledensityimagingwith64ACSlinesandR=4;(a)SENSE,MSE1.96%;(b)Hybrid1dSENSE,MSE1.71%;(c)SoS givesslightlybetterMSEcomparedtoSENSEwithlesslowfrequencybiasinthereconstruction.Howeverthereconstructionisaround20timeslongercomparedtotraditionalSENSEalgorithminthiscase.

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formulatedintounderdeterminedleastsquaresolutionineachcoil.Andfurthermul-tiplecoilsbalancethisunderdeterminedissuebyincreasingtimesnumberofequationswhilemaintainingthesamenumberofunknowns.Theprocessingistimeconsumingduetohugematrixinversion;however,parallelprocessingcandramaticallyreducetheproblembyincreasingtheprocessorcomplexitysinceeachspatiallocationinFEdirectionisdecoupledalready.Still,there'sroomforimprovement.Thespatiallocalredundancyoftheimagerequiresadditionalconsiderationeitherby1DdecompositionwithMarkovRandomField(MRF)or2Ddecompositioninfurtherresearch.

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104 ],TSENSE[ 105 ],k-tBLASTandk-tSENSE[ 106 ]andk-tGRAPPA[ 107 ].Thefurtherchoiceofsamplingpatternink-tspaceaectsSNR,unwrappingartifact,andtemporalresolution.Onedesirestooptimizethesamplingpatterngivenaprioriknowledgeoftheimagedynamics,suchasbreathingmotionorcontrastinjection.TsaoJ.etal.givetheoptimalsamplingpatternsfork-tBLASTandk-tSENSEbyqualitativelyanalyzingthepointspreadfunction(PSF)inx-fspace[ 108 ].Thispaperfocusesontheoptimizationstrategyappliedtok-tGRAPPAandconjecturesaquantitativecriterionfortrajectoryoptimization.Parametersinthecriterioncanbeadjustedtoaccommodatespatialandtemporalscalesofchange. G.3 .WecanseeeachperiodicpatternisasquareblockoflengthequaltotheaccelerationfactorRandtheacquireddataareonthemaindiagonalwherealltheodiagonalpointsaremissingdatareadytobeestimatedbythenearest 133

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neighborsinbothkandtdirections.Inthek-ttrajectoriesconsideredhere,thetorderofkcolumnsamplingintheblockcanbeanypermutationof1;2;;R.Thetrajectoryoptimizationproblemisroughlycomparedbetweenk-tSENSEandk-tGRAPPA.Ink-tSENSE,theauthorsfocusedonpointspreadfunctionsuchthatpositionscontaininglargesignalsinx-fspaceoverlapwithpositionscontainingsmallsignals,inotherwords,thedesiredtrajectorypatternshouldhavethetemporalltereect.ThustheperspectiveofPSFconstrainingthesamplingpatternink-tspaceexhibitstheexplicitperiodicity.Thek-tGRAPPAmethoduseslocalinterpolationanddoesn'tneedtheglobalx-fPSFanalysis.Thusfromthelocalinterpolationpointofview,thesamplingpatternhasahigherdegreeoffreedom. Acriterionforevaluatingsamplingpatternspositsthatthetemporalorspatialcorrelationisinverselyproportionaltothedistancebetweendata[ 108 ].Thusthemethodisbasedonoptimizingtheoveralldistancesfrommissingpointstotheknownpointsduetok-tGRAPPAinterpolationstrategyinFig. G.3 .SincethesamplingpatternofsizeRRisrepeatedalongtimeandspace,onlyonepatternistakenintoconsideration.Twocriteriaareproposedtojudgethepatternselection1.Foreachmissingdatapointinsidethepattern,theaveragedistancemeasuretoalltheneighboringknowndataissmall.2.Foroverallmissingdatapointsinsidethepattern,thedistancemeasuredistributiontendstobeuniform.Criterion1meansthatthesamplingpatternputsamissingdatapointasclosetoitsneighboringknowndataaspossible.Applicationofcriterion2avoidstheextremecasewheresomemissingdatapointsareclosetoitsneighborsandareverywellestimatedwhileotherpointsarefarfromneighborsandarepoorlyestimated.Thereforethereconstructionisconstrainedinbalancebycriterion2.TheaverageofL2normdistanceforallR2Rmissingpointsisaconstantduetotheperiodicityofpatternandthusisanunsuitablemeasure.ThereforeacriterionusinginverseofL2norm

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TableG{1. k-tpatterncomparisonink-tGRAPPAinreductionfactor4cardiacimages. [k,t]pattern [1,2,3,4] [1,2,4,3] [1,3,2,4] [1,3,4,2] [1,4,2,3] [1,4,3,2] RMSE 9.59% 10.49% 10.44% 10.45% 10.5% 9.56% Crietrion 0.9242 0.9186 0.9186 0.9186 0.9186 0.9242 distanceisproposedtoevaluatethesetofpossiblek-ttrajectoriesinEqn.( G{1 ), maxpatternvuut wherexandtaretheweightingfactorsinPEdirectionandtemporaldirectioncorrespondingtothespatialandtemporalcorrelationsseparately,istheorderofnonlineartransfertoadjustthedistancedistribution,dn()denestheEuclideandistancefrommissingdatantoitsnearestneighbors,kxandktrepresentscoordinatesinPEdirectionandtemporaldirectionrespectively. G{1 demonstratesthecaseR=4,wheretheoptimaltrajectoriesaccordingtotheoptimizationcriterion(x=t=1;=2forR=4;5)producethelowestRMSE.Patterns[1;2;3;4]and[1;4;3;2]havethehighestcriterionvalue.IntheR=5casethetrajectoryoptimizationcriterionpredictsoptimalpatternsof[13524]and[14253],correspondingwiththelowestRMSEvaluesof10:06%and10:07%showninFig. G.3

PAGE 151

FigureG{1. k-ttrajectoryink-tGRAPPA.

PAGE 152

FigureG{2. k-tpatterncomparisonink-tGRAPPAinR=5cardiacimages.

PAGE 153

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PAGE 162

RuiYanwasborninChongqing,China,onJanuary1st,1978.HereceivedhisB.E.degreeintheDepartmentofWirelessCommunicationsfromBeijingUniversityofPostsandTelecommunications,in1999.HecontinuedhisgraduatestudyintheTrainingCenterfromBeijingUniversityofPostsandTelecommunicationsbetween1999-2000andintheDepartmentofElectricalandComputerEngineeringfromtheOldDominionUniversitybetween2000{2001.In2001,hejoinedtheDepartmentofElectricalandComputerEngineeringattheUniversityofFloridatopursueaPh.D.inmachinelearningandmedicalimagingwithaM.S.degreeobtainedin2003.UndertheguidanceofDr.JoseC.PrincipeintheComputationalNeuroEngineeringLaboratory,hisresearchismainlyfocusedonadaptivesignalprocessingappliedtomedicalimaging.HeisamemberoftheIEEESignalProcessingSocietyandalsoastudentmemberoftheIEEE. 147


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COMPETITIVE MIXTURE OF LOCAL LINEAR EXPERTS FOR MAGNETIC
RESONANCE IMAGING
















By

RUI YAN
















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2006


































Copyright 2006

by

Rui Yan




































This work is dedicated to those who devote their '. /. f, enthusiasm and ,, '. .:1, to

scientific research.















ACKNOWLEDGMENTS

First of all, I would like to thank my Ph.D. advisor, Dr. Jos6 C. Principe. He

led me into this fabulous adaptive world which, I think, will affect my whole life. His

broad knowledge, his deep insight and his devotion have encouraged me throughout

my Ph.D. career. Without his guidance and advice, this dissertation would not have

been possible.

I would like to thank Dr. Jeffrey R. Fitzsimmons, Dr. Yijun Liu and Dr. John G.

Harris for their time and patience serving as my Ph.D. committee members. Their

advices and comments improved the dissertation to a better quality. I feel very

grateful for Dr. Jeffrey R. Fitzsimmons and Dr. Yijun Liu for their consecutive

support in phased-array MRI area and functional MRI area respectively in my Ph.D.

career.

I would also like to thank Dave M. Peterson for the data collection, supervision

on my hardware experience and helpful discussion all the time. I would also like to

thank Dr. Deniz Erdogmus for bringing his brilliance and drive for research into our

work. I would also like to thank Dr. Erik G. Larsson for bringing me into scientific

research. I would also like to thank Dr. Margaret M. Bradley for providing me an

interesting project to work with and supporting me. I would also like to thank Dr.

Guojun He for his collaboration and valuable comments.

Throughout my research and coursework, I have been having a lot of interaction

with CNEL colleagues. I would especially express my thanks to Dr. Sung-Phil Kim

for his insightful comments and collaboration. I also have benefited a lot from our

long hours of discussion from big pictures to the specified topics with Mustafa Can









Ozturk. The sleepless nights with projects going on with Mustafa Can Ozturk, Anant

Hegde and Jianwu Xu are also unforgettable.

Final thanks go to my parents, who had faith in me and ahv-- supported me.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ...................... ......... ix

LIST OF FIGURES ................................ x

ABSTRACT ................... .............. xiv

CHAPTER

1 INTRODUCTION .................... ....... 1

1.1 Literature Review of Magnetic Resonance Imaging .......... 1
1.1.1 History of MRI .......... ....... ....... 1
1.1.2 I\! RI ............................. 1
1.1.3 Image Reconstruction in Phased-Array MRI ......... 2
1.2 Magnetic Resonance Imaging Basics ..... . . . 3
1.2.1 Interaction of a Proton Spin with a Magnetic Field . 3
1.2.2 Magnetization Detection and Relaxation Times . . 4
1.2.3 Magnetic Resonance Imaging . . . 6
1.3 Main contribution and introduction to appendix . . .. 7

2 STATISTICAL IMAGE RECONSTRUCTION METHODS . ... 11

2.1 Optimal Reconstruction with Known Coil Sensitivities ...... ..11
2.2 Sum-of-squares (SoS) .................. ..... .. .. 11
2.2.1 SNR Analysis of SoS .................. ..... 12
2.2.2 Conclusion ................... . .. 14
2.3 Reconstruction Methods Using Prior Information on Coil Sensitivities 15
2.3.1 Singular Value Decomposition (SVD) . . ..... 17
2.3.2 B i, i ,i Maximum-Likelihood (MilI) Reconstruction ..... .18
2.3.3 Least Squares (LS) with Smoothness Penalty . . 21
2.4 Results and Discussion .................. ..... .. 24

3 SUPERVISED LEARNING IN ADAPTIVE IMAGE RECONSTRUC-
TION METHODS, PART A: MIXTURE OF LOCAL LINEAR EXPERTS 33

3.1 Local Patterns in Coil Profile ................ .. .. 33
3.2 Competitive Learning. .................. ... .. 34
3.3 Multiple Local Models .................. ... .. .. 34


. .... 34
3.3 Multiple Local Models .. . . . . .... 34









3.4 The Linear Mixture of Local Linear Experts for Phased-Array MRI
Reconstruction ........ . . ........... 36
3.5 The Nonlinear Mixture of Local Linear Experts for Phased-Array
MRI Reconstruction .................. ....... .. 39
3.6 Results ................... ... ....... 40

4 SUPERVISED LEARNING IN ADAPTIVE IMAGE RECONSTRUC-
TION METHODS, PART B: INFORMATION THEORETIC LEARN-
ING (ITL) OF MIXTURE OF LOCAL LINEAR EXPERTS ...... ..56

4.1 Brief Review of Information Theoretic Learning (ITL) . ... 56
4.2 ITL Bridged to MRI Reconstruction ................ .. 57
4.3 ITL and Recursive ITL Training ................ ... 58
4.4 Results ................... ... ....... 60

5 UNSUPERVISED LEARNING IN -i\ RIl TEMPORAL ACTIVATION PAT-
TERN CLASSIFICATION .................. .. 63

5.1 Brief Review of fMRI .......... . . .... 63
5.2 Unsupervised Competitive Learning in \! II . . 65
5.2.1 Temporal Clustering Analysis (TCA) . . ..... 65
5.2.2 Nonnegative Matrix Factorization (NMF) . .... 66
5.2.3 Autoassociative Network for Subspace Projection . 68
5.2.4 Optimally Integrated Adaptive Learning (OIAL) ...... ..69
5.2.5 Competitive Subspace Projection (CSP) . ... 70
5.2.5.1 hard competition .................. .. 71
5.2.5.2 soft competition. ................. .72
5.2.6 Algorithm Analysis .................. .. .. 75
5.2.7 I\! lI Application with Competitive Subspace Projection 76
5.3 Results ................... ... ....... 78
5.4 D discussion . . . . . . .. . ... 85

6 CONCLUSIONS AND FUTURE WORK ................. .. 88

6.1 Conclusions . . . . . . . .. 88
6.2 Future W ork . . . . . . . .. 89

APPENDIX

A MRI BIRDCAGE COIL .................. ........ .. 94

B MEASURING THE SIGNAL-TO-NOISE RATIO IN MAGNETIC RES-
ONANCE IMAGING: ACAVEAT ............. .... .. 100

B.1 Introduction ............. . . . ...... 100
B.2 The Signal-to-Noise Ratio (SNR) .................. .. 101
B.3 Measuring the Signal-to-Noise Ratio ................ 104
B.4 Illustration .................. ............ 106


. . . ..... 101
B.3 Measuring the Signal-to-Noise Ratio . . . ..... 104
B.4 Illustration . . . . . . . 106










B.5 Concluding Remarks . . . . . . 109

C QUALITY MEASURE FOR RECONSTRUCTION METHODS IN PHASED-
ARRAY MR IMAGES ................... ......... 111

C.1 Image Quality Measure Review ................... 111
C.2 Methods ................... .............. 112
C.2.1 Traditional SNR measures ................... 112
C.2.2 Local nonparametric SNR measure ............. .. 113

D MRI IMAGE RECONSTRUCTION VIA HOMOMORPHIC SIGNAL PROCESS-
ING ................... .... ............... 115

D.1 Data Model ................... ........... 115
D.2 Homomorphic signal processing ................... .115
D.3 Numerical Results . . . . . . . 117
D.4 Concluding Remarks . . . . . . 121

E HOMOSENSE: A FILTER DESIGN CRITERION ON VARIABLE DEN-
SITY SENSE RECONSTRUCTION ................... 124

E.1 Introduction ................... ............ 124
E.2 Method ................... .............. 124
E.3 Results and Discussion .. . ............. ..... 126
E.4 Conlusion ................... ............ 127

F HYBRID1DSENSE, A GENERALIZED SENSE RECONSTRUCTION 129

F.1 Introduction ................... ........... 129
F.2 Method ................... .............. 129
F.3 Results and Discussion ..... ............. ..... 130
F.4 Conclusion ................... ............ 131

G TRAJECTORY OPTIMIZATION IN K-T GRAPPA . . 33

G.1 Introduction . . . ... .......... 133
G.2 Method . . . . . . . . 33
G.3 Results and Discussion ... . . ..... .... 135
G.4 Conclusions . . . . . . . 37

REFERENCES . . . . . . . . 138

BIOGRAPHICAL SKETCH . . . . ........ 147















LIST OF TABLES


Table page

A-1 8 leg birdcage coil parameters ............... .... 95

D-1 N. ii i. I1 entropy of (a) SoS, (b) homomorphic signal processing, and
(c) contrast-enhanced homomorphic signal processing. . . ... 121

G-1 k-t pattern comparison in k-t GRAPPA in reduction factor 4 cardiac images. 135















LIST OF FIGURES


Figure page

1-1 The principle of magnetic moment, (a)Proton spin, (b) Angular procession
of a proton spin in an external magnetic field .... . . 4

1-2 Block diagram of magnetization detection by a receiver coil. . 5

2-1 The four element phased-array coil. .............. .. 16

2-2 Performance of the four algorithms, SVD (circle), ML (square), LS (star),
SoS (triangle), shown in terms of image reconstruction SER (dB) versus
measurement SNR (dB). Clearly, ML and LS perform almost identically
outperforming SVD and SoS, which also perform identically. ...... ..25

2-3 The vivio image obtained from a) Coil 1 b) Coil 2 c) Coil 3 d) Coil 4. The
coil sensitivity estimates for f) Coil 1 g) Coil 2 h) Coil 3 i) Coil 4, and j)
the reconstructed image obtained using the SoS reconstruction method. .27

2-4 The ratio of the maximum singular value to the average of the smaller
three singular values of the measurement matrices for 5x5 non-overlapping
regions a) summarized in a histogram and b) depicted as a spatial distrib-
ution over the image with .-i i,-' 1,. values assigned in logl0 scale, brighter
values representing higher ratios. .............. ...... 28

2-5 The reconstructed images using a) SVD b) ML c) LS d) SoS approaches. 29

2-6 The estimated local SNR levels of the reconstructed images using a) SVD
b) ML c) LS d) SoS approaches, where the top left region is the noise
reference. Notice that in (a)-(d) the SNR levels are overlaid on the recon-
structed image of the corresponding method. To prevent the numbers from
,l. in.-r these images are stretched horizontally. The top left corner of
each image is used as the noise power reference. . . ..... 32

3-1 Block diagram of the linear multiple model mixture and learning scheme. 37

3-2 Block diagram of the nonlinear multiple model mixture and learning scheme. 40

3-3 Transverse crossections of a human neck as measured by the four coils
from one training sample. .................. ..... 46

3-4 Coronal crossections of a human neck as measured by the four coils used
as the testing sample. ................... ........ 47









3-5 Desired reconstructed image, (a) estimated by averaging the SoS recon-
struction for each coil image sample, (b) SNR performance of the estimated
desire. . . . . . . . . . 48

3-6 Adaptive learning performance, (a) Learning curve of winner models for
the model number 4,8,16, (b) Learning curve of the linear mixture of
competitive linear models system for the model number 4,8,16. . 49

3-7 Learning curve of the nonlinear mixture of competitive linear models sys-
tem for the model number 4. . . . . . . 50

3-8 The reconstruction image, (a) From one transverse training sample by
nonlinear mixture of local linear experts, (b) The SNR performance of the
reconstruction. . . . . . . ...... 50

3-9 Pixel classification for the model number 2, 4, 8, 16. . ... 51

3-10 Reconstructed images and their SNR performances from the mixture of
competitive linear models system with the model number 16 and the coil
number 4, 36. . . . . . . . ...... 52

3-11 Reconstructed test images for a coronal crossection from a human neck,
(a) SoS without whitening (b) SoS with whitening (c) Linear mixture of
models, (d) Nonlinear mixture of models. . . . ....... 53

3-12 SNR performances of the reconstructed test images for a coronal crossec-
tion from a human neck, (a) SoS without whitening (b) SoS with whitening
(c) Linear mixture of models, (d) Nonlinear mixture of models. . 54

3-13 Image quality measure, (a)-(b) The two reconstructions by nonlinear mix-
tures of models using two near idential 4 coil samples, (c) The noise power
from the subtration of the two reconstruction images in (a). . 55

4-1 Block diagrom of the nonlinear multiple model mixture and learning scheme. 58

4-2 Histogram of output error from the well-trained MLP network by MSE. 59

4-3 Adaptive learning performance, (a) The information potential learning
curve, (b) The kernel variance anealing curve. . . . 60

4-4 The reconstruction images of the coronal image by (a) ITL training and
(b) MSE training. . . . . . . ...... 61

4-5 The SNR performance of the reconstruction images of the coronal image
by (a) ITL training and (b) MSE training. . . . ....... 62
5-1 Block diagram of autoassociative network. . . . 69

5-2 The block diagram of competitive subspace projection methodology. 71









5-3 Three dimensional synthetic data, (a) projected to its first and second
dimension, where the third dimension is insignificant in classification (b)
clustering data in (a) by k-means, (c) clustering data in (a) by optimally
integrated adaptive learning (OIAL), (d) clustering data in (a) by com-
petitive subspace projection (CSP). The intersected lines in (c) and (d)
represent the two projection axes for each cluster. . ..... 77

5-4 The learning curve in the second phase of training from competitive sub-
space projection for M 1, 2, 3 (The mean square error (S\!S1) is normal-
ized by the input signal power). .................. .... 79

5-5 The projection axes for the number of the projection axes M = 2 and
model number K = 3 after the second phase training of competitive sub-
space projection is completed. ............... .... 80

5-6 The cluster centroids for model number K = 4 and projection axes M = 2. 82

5-7 The cluster centroids for model number K = 3 and projection axes M = 2. 83

5-8 The four basis images (1-2 upper row and 3-4 lower row from left to right)
are determined by NMF using real fMRI data. ............. 84

5-9 The encoding time series corresponds to four basis images by NMF using
real \ I data. .................. ............. ..85

5-10 The temporal maxima plot for temporal clustering analysis (TCA) method. 86

5-11 Functional region localization by (a) temporal clustering analysis (b) non-
negative matrix factorization and (c) competitive subspace projection. 87

A-1 Transmit only birdcage coil flow chart. ................. 96

A-2 Receiver coil flow chart, with C1, C2 are the parallel combination of a
20pF capacitor and a 1-15pF adjustable capacitor; C3,C8 are the parallel
combination of a 4.7pF capacitor, a 91pF capacitor, and a 39pF capacitor;
C4 is the parallel combination of a 3.9pF capacitor and a 1-15pF adjustable
capacitor; C5, C7 are the parallel combination of a 91pF capacitor and a
39pF capacitor; C6 is the parallel combination of a 18pF capacitor and a
1-15pF adjustable capacitor .............. .. .... 97

A-3 Schematic representation of a single transmit/receive switching circuit for
protection of the receiving preamplifier. ................. 98

A-4 Block diagram of the quadrature transmit coil, and receive- only phased
array setup ............... ............. .. 99

B-1 Synthetic data example. (a) Original noisy step function signal x, (b)
transformed (squared) signal y, and (c) the true and the measured SNR
levels .................. .................. .. 107


. . . . . . . . . . 107









B-2 Reconstruction images and their SNR performance. (a) SoS, (b) SNR of
SoS . .. . . . .. . . . 109

B-3 Reconstruction images and their SNR performance. (a) logrithm of SoS,
(b) SNR of logrithm of SoS. ............... .. ..... 110

B-4 Reconstruction images and their SNR performance. (a) median filtered
SoS, (b) SNR of median filtered SoS. ................ . 110

D-1 Canonic form for homomorphic signal processor. ............. .115

D-2 Photograph of the phased array coil, transmit coil, and cabling. . 117

D-3 Vivo sagittal images of cat spinal cord from coil 1-4 and the spectral
estimate of SoS. ............... ............ .. 118

D-4 (Upper row) Spatial distribution of the coil sensitivities for four coil sig-
nals. (Lower row) Spectral distribution of the coil sensitivities for four coil
signals . .................... ............. 119

D-5 The reconstruction image contrast versus the high-pass filter cutoff fre-
quency and the stopband magnitude. .................. 120

D-6 High-pass filter to eliminate coil sensitivities. .............. ..121

D-7 Reconstructed images. (a) Sum-of-squares (sos), (b) homomorphic signal
processing, (c) contrast-enhanced homomorphic signal processing, and (d)
reconstruction from the filtered coil sensitivities. ............. ..122

D-8 The pdf distribution of the reconstructed images. ........... .123

E-1 SoS of axial phantom data. ................ .... 126

E-2 High-pass and low-pass filter with order 4 and cutoff frequency at 64. .. 127

E-3 Central PE line from Reconstructions of homoSENSE of MSE = 0.2 :',
SENSE of MSE-2.19'. compared with SoS. .. . . .... 128

F-1 Reconstruction of variable density imaging with 64 ACS lines and R = 4;
(a) SENSE, MSE 1.',n.'; (b) HybridldSENSE, MSE 1.71 .. (c) SoS 131

G-1 k-t trajectory in k-t GRAPPA. ................ .... 136

G-2 k-t pattern comparison in k-t GRAPPA in R = 5 cardiac images. ... 137















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COMPETITIVE MIXTURE OF LOCAL LINEAR EXPERTS FOR MAGNETIC
RESONANCE IMAGING

By

Rui Yan

May 2006

C'!I iir: Jos6 C. Principe
M, i i Department: Electrical and Computer Engineering

Magnetic resonance imaging (\! RI) is an important contemporary research field

propelled by expected clinical gains. MRI includes many interesting specialties. Re-

cently the data acquisition time in scanning patients became a critical issue. The

collection time of MRI images can be reduced at a cost of device complexity by using

multiple phased-array coils, which bring the problem of adequately combining multi-

ple coil images. In this dissertation, the problem of combining images obtained from

multiple MRI coils is investigated from a statistical signal processing point-of-view

with the goal of improving signal-to-noise-ratio (SNR) in the reconstructed images.

A new adaptive learning strategy using competitive learning as well as local linear ex-

perts is developed by treating the problem as function approximation. The proposed

method has the ability to train on a set of images and generalize its performance to

previously unseen images.

To validate the effectiveness of the adaptive method in MRI iin ,-i,- the com-

petitive mixture of experts was also tested in the extraction of information from

functional MRI (:i\!ll) images. The problem is to localize the functional pattern









corresponding to an external stimulus. Although this problem has been widely inves-

tigated using a block paradigm (i.e. processing synchronized with the external stimu-

lus), the proposed competitive mixtures model provides a self-organizing method that

can be especially useful in \ IRI experiments when the response time is unknown.

To our knowledge, it is the first time that competitive learning is included into i!lRI

signal analysis with good results.















CHAPTER 1
INTRODUCTION

1.1 Literature Review of Magnetic Resonance Imaging

Magnetic resonance imaging (\! I ) is an imaging technique using radiofrequency

waves in a strong magnetic field mostly for inner human body examination. This

method can provide images with better quality than regular x-rays and CAT scans

for soft-tissues inside the body. Widely used as a noninvasive diagnostic tool in the

medical community, MRI is used to detect the early evidence of many ailments of

soft tissues such as brain abnormalities, coronary artery diseases and disorders of the

ligaments, etc.

1.1.1 History of MRI

The phenomenon of magnetic resonance imaging was independently discovered

by Bloch [1] and Purcell et al. [2] in 1946, which led to their Nobel Prize in 1952.

The relaxation times of tissues and tumors were found to be different by Dama-

dian in 1971 [3]. This discovery opened a promising application area for MRI. In

1973 Lauterbur [4] proposed magnetic resonance imaging using the back projection

method, for which he shared the Nobel prize in 2003. Ernst et al. [5] introduced the

Fourier Transform of the k-space sampling into 2D iin ,-ii,- resulting in the modern

MRI technique and a shared Nobel prize in 1991.

1.1.2 fMRI

In the last decade, MRI imaging has been subdivided into two main categories.

One technique images time-varying processes within an image series and is called

functional MRI ( -\ IRI) [6]. The purpose of this technique is to understand how func-

tional regions inside the brain respond to external stimuli. The information between









the functional regions of the brain and the cognitive operations has been investi-

gated [7]. The temporal partitioned activity demonstrates functional independence

with respect to the localized spatiality inside the brain [8]. The challenges remain

in localizing brain function when there is no priori knowledge available about a time

window in which a stimulus may elicit response. Thus there is no timing for the

brain's response to align. The spatial active regions can still be located according to

the temporal response activated by a single stimulus [9]. Therefore \! I I provides a

method to understand the mapping between brain structures and their functions.

1.1.3 Image Reconstruction in Phased-Array MRI

Another research aspect focuses on fast imaging with multiple receiver coils.

The increased equipment complexity increases the signal-to-noise-ratio (SNR) by ap-

proximately combining the coil images from the different coils. Thus for a certain

SNR image quality level, phased-array imaging techniques can dramatically reduce

the scanning time, which has the benefits of reducing the motion artifacts of the

image. Roemer et al. proposed a pixel by pixel reconstruction method, named sum-

of-squares (SoS), to reconstruct coil images [10]. They showed that this method loses

only 10' of the maximum possible signal-to-noise-ratio (SNR) with no priori infor-

mation of the coils' positions or RF field maps. This result sets the foundation of

phased-array image reconstruction and demonstrates its prevalence in the industry.

Based on SoS, a substantial body of research has focused on sophisticated techniques

for phase encoding together with the use of gradient coils. This work includes the sen-

sitivity encoding for fast MRI (SENSE) technique [11] and simultaneous acquisition

of spatial harmonics (SMASH) imaging [12]. Both methods reduce the scanning time

by undersampling along the gradient-echo direction from k-space in parallel data col-

lection. Debbins et al. [13] si --l. -. I1 adding the images coher' i ;,l after their relative

phases were properly adjusted by another calibration scan. This method increased

the imaging rate by reducing demands, such as bandwidth and memory, while it kept









much of the SNR performance compared to SoS. Walsh et al. used adaptive filters to

improve SNR in the image [14]. Kellman and McVeigh proposed a method that can

use the degrees of freedom inherent to the phased array for ghost artifact cancellation

by a constrained SNR optimization [15]. This method also needs a priori informa-

tion on reference images without distortion to estimate coil sensitivities. Bydder et

al. proposed a reconstruction method that estimated the coil sensitivities from the

smoothened coil images to reduce noise effects [16]. A B i,, ,i method using itera-

tive maximum likelihood with a priori information in coil sensitivities was presented

recently by Yan et al. [17]. Recently image reconstruction methods incorporating lo-

cal coil sensitivity features have been proposed such as parallel imaging with localized

sensitivities (PILS) [18], local reconstruction [19], etc.

1.2 Magnetic Resonance Imaging Basics

1.2.1 Interaction of a Proton Spin with a Magnetic Field

Magnetic resonance imaging originates from understanding the nature of a proton

spin. It is the proton spin rather than the electron spin which is applicable in MRI

due to its field homogeneity as well as being noninvasive to the human body [20, 21].

Proton spin expresses processing a positive charge. This angular procession creates

an effective current loop, which generates its own field, called a magnetic moment

tt (Fig. 1 -(a)). The interaction of the magnetic moment with an external magnetic

field B tends to align tt to B. This alignment is an angular procession considering

B is the axis, determined by the Bloch equation


=t x B (1-1)


The geometrical representation in Fig. 1 l(b) demonstrates that the proton spin

rotates left-handed around B with the magnitude of pt fixed. From Eqn. (1-1), the



























(a) (b)

Figure 1-1. The principle of magnetic moment, (a)Proton spin, (b) Angular proces-
sion of a proton spin in an external magnetic field.

Larmor procession formula is derived


S= 7B (1-2)


where 7 is the gyromagnetic ratio of the proton and u is named the Larmor frequency.

It is shown that the rotation frequency of the proton magnetic moment is determined

both by the external field B and proton nature 7. Based on the biological abundance

of hydrogen (,' .'-), this proton is taken as the measured nuclei with gyromagnetic

ratio equal to 42.58MHz/T.

1.2.2 Magnetization Detection and Relaxation Times

Inside a given macroscopic volume, protons finally align with the external field

either by parallel alignment or anti-parallel alignment. The number of protons par-

allel to the external field is larger than that of protons anti-parallel aligned due to

the Boltzmann distribution. The energy difference between these two states is called

spin excess. Spin excess generates a net equilibrium magnetization M0 proportional

to the spin density. However, this quantum spin energy is much smaller compared








5




z BO



M 0



-------
Con

x y



Figure 1-2. Block diagram of magnetization detection by a receiver coil.

with the thermal energy kT, where k is Boltzmann's constant and T absolute tem-

perature. Thus, net magnetization M0 cannot provide detectable signals. Therefore,

another 1 radiofrequency pulse B1 is required to flip the magnetization orthogonal

to the external field direction for procession (Fig. 1-2). The flux change due to the

magnetization procession can be detected by the electromotive force (emf) induced

in the vicinity receiver coil given by reciprocity principle


emf = (M B,)d3r (1-3)

where Brf is the magnetic field from the receiver coil.

However, spinning magnetization is affected by the relaxation times including

longitude relaxation time Ti and transversal relaxation time T2. The longitude re-

laxation time TI determines the speed of alignment back to direction of the static

field B, which is due to the interaction from magnetization M and external field B;

while the transversal relaxation time T2 measures the dephasing effect of the spin-spin

decay caused by the interaction among spins.









1.2.3 Magnetic Resonance Imaging

The key for imaging is to map the measured signals from the receiver coil with

the spatial locations. This can be achieved by applying a linearly spatial varying field

B + Gz, taking z direction as an example. So the received signal proportional to the

Larmor frequency c(z) = 7(B + Gz) gives spatial slice information in z direction.

The relationship between the received signal and the spin density is given by the

Fourier Transform


x(k) p(z)e-i27kdz (14)


where p(z) is the one dimensional spin density and k(t) = fo G(r)dr represents the

spatial frequency in k-space. The p(z) reflects image intensity and can be resolved

by an inverse Fourier Transform


p(z) x(k -' dk (1-5)

Instead of the above 1D imaging, two dimensional spatial extension is easy to

accomplish by adding one more encoding direction. Suppose that phased-array coils

consisting of n, coils are used for the parallel data collection and let (xk,yk),k =

1, nc be the coordinate of the kth coil. Let x, y, z be orthogonal unit vectors

that span the Cartesian coordinate system under consideration, and suppose that a

suitable gradient magnetic field is applied to enable selective excitation of a thin slice

parallel to the (x, y) plane, -w z = o. At a given coordinate x, y, zo and time t, let

G,(t) and G,(t) be the strength of the external magnetic field and define
/0

Jt (16)
ky(t) = Gy(r)dr









Then, for a given kth receiver coil, the received time domain signal can be written as


Xk(t) = e- iwt f p(x,y)Ck(x, y)e i(kx(t)(x-xk)+ky(t)(Y-Yk))dxdy + ek(t) (1 7)


where A is a constant, and p(x,y) is proportional to the "transverse magnetiza-

tion" (which is essentially the quantity of interest in the imaging), Ck(x,y) is the

sensitivity and ek(t) is noise both from the kth coil.

Equation (1-7) shows that the received signal x(t) is equal to the 2D Fourier

transform of the multiplication of the true pixel value p(x, y) and the coil sensitivity

Ck(x,y) sampled at kx(t) and k,(t).
After the inverse Fourier Transform is applied to the received k-space signal xk,

the resulting spatial signal Sk(i,j) from coil k at coordinate (i,j) is the observed by


Sk(i) = pCk (i) + nk(i), k = 1, 2, (1-8)

where nk(i,j) is complex-valued (Gaussian), wide sense stationary (WSS), zero-mean,

spatially white noise, which is possibly correlated across coils with covariance matrix

Q (spatiotemporally constant due to the WSS assumption [10, 22, 23]). Note that
the noise correlation, if properly compensated for, does not pose a limitation to the

achievable image quality [24]. In this signal model, the specific values of the coil

sensitivities are, in general, not known. However, some a priori knowledge in the

form of statistical distributions or structural constraints (such as spatial smoothness)

may be available.

1.3 Main contribution and introduction to appendix

The follow-up dissertation starts in chapter 2 with the optimal reconstruction

with known coil sensitivities. The maximum likelihood estimation gives the best

reconstruction with the known coil senstitivities. However, coil sensitivities is not

known a prior information in practice. The conventional sum-of-squares (SoS) method

solves the problem by estimating the coil sensitivities from pixel based data itself.









The dissertation demonstrates that SoS is an optimal linear combination base on

the signal-to-noise-ratio (SNR) analysis while the optimality is hard to satisfy in

practice. This disadvantage drives us to research novel reconstruction methods. By

incorporating the local smoothness property of coil sensitivities, three statistical im-

age reconstructions are proposed, named as singular value decomposition method,

B li-,i ,in maximum-Likelihood reconstruction, and least squares with smoothness

penalty. These methods gains a 1-2dB SNR improvement compared with SoS. Al-

though the statistical methods give analytical solutions, they don't have the capabil-

ity to manipulate historical data. Therefore, chapter 3 switches to adaptive learning

methods to extract features from historical scanning images. Once the adaptive net-

work is well trained, it can be generalized to other unknown scanning images. Com-

petitive learning combined with local linear experts is proposed in this dissertation

to implement the divide-and-conquer strategy in this function approximation case.

Such competitive learning topology incorporates intelligence into the adaptive net-

work to decouple the subtasks which has weak correlations in between. By training

a considerable amount of samples, the SNR improvement in the test image is signifi-

cant. Chapter 4 further this idea to information theoretical 1l. iiiir where the error

criterion changes from mean square error to Renyi's entropy. This is an extension

from second order statistics to higher order statistics. This competitive learning idea

is extended from supervised learning to unsupervised learning by proposing compet-

itive subspace projection in chapter 5. It is applied in functional MRI area and helps

to locate the activated spatial and temporal patterns inside brain. As a summary,

conclusions consisting of discussion and some proposed future work are described in

chapter 6.

Besides the main body of the dissertation, some complimentary work is worth

to mention. As we know, the MRI scanning with phased array coils specify different

coil configuration to different parts of patients or phantoms. Thus the coil design









needs careful consideration in a certain scanning case. Appendix A briefly describes

the four element birdcage coil used in the data collection.

Medical image quality is al-i-b a tough but interesting topic which measures the

amount of true object information extracted. The difficulty is due to lack of knowledge

of the true visual system, the noise and the blurring effect. Normally Signal-to-noise

ratio (SNR) measures the image noise while contrast-to-noise ratio (CNR) measures

the blurring effect. In case of pixel based image reconstruction, blurring effect is

ignorable and SNR is normally used as the image quality measure in full sampled

data case. However, nonlinear transformation changes the second order statistics.

Thus SNR measurement may give a fake image quality evaluation. Appendix B

describes the problem in detail. In order to conquer this problem, appendix C gives

a image quality speculation using nonparametric pdf estimation.

Except for the proposed statistical image reconstruction methods all in image

space, appendix D gives another perspective in modeling this problem in spectral

domain. Homomorphic signal processing helps bridges the filtering process between

spectral domain and the image domain. The final quantitative entropy in image

quality is also an interesting measurement.

The following three Appendix chapters describes my intern work in Invivo Cor-

poration. In partial parallel acquisition (PPA), it attracts much interest to sample the

k-space using variable density. Naquist sampling is usually at low frequency and un-

dersampling is usually at high frequency. Thus Naquist sampling conserves the image

energy and leads to high SNR in final reconstruction; the undersampling reduces the

scanning time by the acceleration factor. However, the combination from the recon-

structions of the two parts separately is a challenge. The ring effect is obvious in final

reconstruction if the two parts are naturally separately; on the other hand, filtering

the two parts may incorporate bias into the final reconstruction as well. Appendix

E gives an optimal filter design strategy to minimize the bias effect with smoothing







10


filter. Appendix F extends the k-space sampling to an arbitrary trajectory. Thus the

partial parallel image reconstruction is generalized by an inverse problem in hybrid

space. Dynamic imaging with undersampling is a hotspot. People are interested in

different reconstruction methods, such as k-t BLAST, k-t SENSE and k-t GRAPPA,

etc. Little work is done on how the k-t sampling trajectory affects the reconstruction

performance. Appendix G gives an optimal search criterion in finding the optimal

k-t trajectory related to k-t GRAPPA method.














CHAPTER 2
STATISTICAL IMAGE RECONSTRUCTION METHODS

2.1 Optimal Reconstruction with Known Coil Sensitivities

It is well-known in the statistical signal processing literature that for complex-

valued received signals, assuming that the coil sensitivities are known, the SNR-

optimal linear combination of the measurements for estimating r(i,j) is given by

pj cH(ij)Q-1s('j)
H(,j)) -t= (2-1)

where H denotes the conjugate-transpose (Hermitian) operation, c(i,j) is the vector

of coil sensitivities and s(i,j) is the vector of measurements for pixel (i,j). The SNR-

optimality of this reconstruction method among all linear combiners can be proved,

for example, by applying the Cauchy-Schwartz inequality [25]. The SNR for this

reconstruction method can be determined to be Ip21 C1 2/a2, where a2 is the noise

power (of both real and imaginary parts).

2.2 Sum-of-squares (SoS)

The sum-of-squares (SoS) method, proposed by Roemer et al. as a pixel by pixel

reconstruction method [10], is extensively implemented in the industry due to its high

image reconstruction quality and simple math calculation. This method estimates the

coil sensitivity Ck at the kt coil as ck

N
6k Sk Sk 2 (2-2)
k=l









Based on the coil sensitivity estimated in Eqn. (2-10), The SoS reconstruction p can

be interpreted as an optimal linear combination

N N
P -~- -cSk -2 2 (23)
k=1 Ckl2 k=1

where s = [sl,- Snc] contains all signal elements in n, coils. In most practi-

cal cases, the noise across coils is correlated, assuming spatial wide sense stationar-

ity (WSS). The coil vector s needs to be prewhitened by the noise covariance matrix

Q before using the basic SoS reconstruction. Thus, a whitened SoS is written as


p = sTQ-s (2-4)

2.2.1 SNR Analysis of SoS

Maximum-ratio combining (optimal combining): If the coil sensitivities Ck

are known, the optimal estimate of p can be shown to be

SEk CkSk Ek C= k (-5)
N 1 P+ N (25)
EC1 ICk 2 E l CI 2

where (.)* stands for the complex conjugate. A neat and self-contained derivation of

this result can be found in, for example [10, 26], although it also follows directly by

using some standard results on minimum variance estimation theory [27]. We can

easily establish that p is unbiased, i.e., E[p] = p, where E[.] stands for statistical

expectation. Then the SNR in p is equal to [10, 26, 27]

Ip12 p12 2(2 1C 2)2
SNRopt k1 I 1
E [1p- p12] E 2 c21 ICk 2.2) (2 6)
Y:N 12









Sum-of-squares (SoS) Reconstruction: The SoS method is applicable when

{ck} are unknown. The reconstructed pixel is obtained via


pi = F 2k (2-7)
\k=l

(This SoS estimate can be interpreted as an optimal linear combination according to

Eq. (2-5) but with ck replaced by Sk! N =1 Sk 2 [16].) Clearly, if the noise level

goes to zero the SoS estimate converges to p -+ p/]k ck l2 which is in general

not equal to p. Therefore, SoS reconstruction typically yields severely biased images,

even in the noise-free case. Unless Ck is constant for all coils (which is certainly not

the case in practice), this bias depends on the coil number k and hence it cannot

be corrected for if Ck is unknown. Also, Ck are typically not constant over an entire

image, and therefore the bias will be location-dependent, which may imply serious

artifacts in the image.

We next analyze the statistical properties of the SoS method. For a high input

SNR, the expression for p in Eq. (B-15) can be written:

N N
S= pk + k 2 [p2 k 2 + 2pR(c*ek) + 12]
k=1 k=1

p IC2 V1 R(c ek)
k1=k 1 ICk2
(2 8)
N [ Zk-IR(Ck*ek)
P ICk 2 =+ C2
k=1 k=E 1 Ck2

= F 2Ck 2 Nk1 R(CCk)
kl kN 1 Ck 2

where R denotes the real part. In the first approximation, the higher order term is

discarded, while a first order Taylor series expansion is used in the second approx-

imation. Clearly, E[p] / p in general and thus we see again that SoS gives biased









images. The SNR of p is obtained as:


P k1 = Ck2 2( 1CN 2)
SNRsos = ( C (2-9)



which is equal to the same as the SNR for optimal combining with known coil sen-

sitivities (see Eq. (2-6)). Therefore, from a pure SNR point of view, SoS is optimal

for high input SNR.

2.2.2 Conclusion

SoS reconstruction method possesses many advantages. First, the SoS method

... ,i -i i I ically approaches reconstruction optimality as all measurement (coil) signal-

to-noise-ratio (SNR) levels increase [28]. This high SNR performance ensures the final

reconstruction image quality, which is the most important virtue of SoS. Second, it

gives an unbiased estimate in the noise-free case. As we can see, if the noise level
N 2
goes to zero the SoS estimate converges to p p ik 1 ICk With ck estimated in

Eqn. (2-10), k= l1Ck 2 is one. Thus this SoS estimate p approaches the true pixel

value p, which also explains the reason why choosing such coil sensitivity estimator in

Eqn. (2-10). Besides, SoS doesn't need any prior information. On one hand, with no

need for prescan or other information about the magnetic field, data collection is sim-

plified. On the other hand, no statistical assumption concerning the coil sensitivities

is predetermined and thus reduces the modeling error.

However, the widely-used sum-of-squares method has its own disadvantages.

Though it has the .i.- 1!, I ical SNR optimality property, the condition for this op-

timality, which is the high measurement SNR condition, is not ahv--, satisfied in

practice [28], especially in phased-arrays, where the coils measure only a portion of

the image. This creates the problem of considering pure noise pixels equally weighted

to pixels with actual signal. Another potential disadvantage for the SoS method

and other SoS based methods (e.g., SENSE & SMASH) lies within the statistical









assumption of spatial wide sense stationarity (WSS) of the noise. Since, in general,

the noise covariance matrix Q is not known a priori, a region consisting of pure noise

pixels must be used to estimate it empirically. This often requires a manual selec-

tion of the v...:,;, pixels or another reference scan containing only noise, under the

additional assumptions that the noise statistics are stationary within each imaging

trial and are independent from the object being imaged. If the noise exhibits local

properties in the spatial domain (e.g., the noise statistics differs from the signal region

to the background region), the noise covariance estimated from the global space or a

certain local space distorts or ignores some effective information and thus hurts the

reconstruction.

2.3 Reconstruction Methods Using Prior Information on Coil
Sensitivities

In this section, I present three image reconstruction methods for phased array

MRI that are optimal in the least-squares or maximum-likelihood sense. To this end,

one of the following two assumptions will be made:

Al. The coil sensitivities remain 'j'l.'.i:,,,' ,, 1.; constant over a small region Q

consisting of N pixels, i.e., c(i,j) = cfor(i,j) E .

B1. The coil ., ,;.:..- ,,. .1 1,1. .I a .;, -in.... il, i with the spatial location, within the

regions of interest.

In order to justify these assumptions, consider the images of a cat spinal cord shown

in Fig. D-3(a)-D-3(d) taken using the 4-coil phased array shown in Fig. 2-1 (4.7T,

TR OOO1ms, TE=15ms, FOV 10 5cm, matrix 256128, slice thickness 2mm, sweep

width 26khz, 1 average). Regarding the SoS as a linear combination methodology,

the equivalent coil sensitivity estimates produced by this algorithm are found in

Eqn. (2-10). These estimated coil sensitivity profiles generated by the SoS are also











,.'i "*:
A
Li .
000


Figure 2-1. The four element phased-array coil.


shown in Fig. D-4(a)-D-4(d), as well as the reconstructed image estimate (Fig. D
7(a)). Notice in Fig. D-4(a)-D-4(d) that the four spatial coil sensitivity profiles
exhibit a smooth behavior as a function of the spatial coordinates.
A similar structural behavior of the coil sensitivity profiles has also been ob-
served in images of various other objects, including phantoms and human tissues.
This observation is the main motivation behind the two assumptions stated above.
The three reconstruction methods that are proposed below take advantage of this
structural quality of the coil sensitivities over space to generate optimal results in a
statistical array signal processing framework under the assumptions stated.









2.3.1 Singular Value Decomposition (SVD)

For a phased array imaging system consisting of n, coils, under assumption Al

the data model for some small region f simplifies to the following vector-matrix

equation


mS pcT + N (2-10)


where p is the vector of pixel values in the region,S = [si, sc,] is the measurement

matrix of sizeN x nc, and N is the noise matrix (of the same size as S) consisting of

independent samples across pixels, but possibly correlated across coils.

In the ideal noise-free case, S has rank one, and the left and right singular

vectors of S are p and c, respectively. However, the presence of noise increases the

rank of S; hence the left singular vector and the right singular vector corresponding

to the maximum singular value will yield the least squares estimates of p and c [25].

Specifically, if

A1 0 0 v1

S= u O 0 = U.VT (2-11)

0 0 \A, v

is the singular value decomposition (SVD) of S, then ul and vi minimize I|S -

uiAilvl 2 (in Eqn. (2-11), U and V are orthonormal singular vector matrices and

E is a diagonal matrix that contains the singular values in descending order). The

estimate of the image in region 2 is therefore p = Alul and the corresponding coil

sensitivity vector estimate for this region is c = v given the unit energy constraint

on c. The procedure must be repeated for all regions in the whole image. Using

eigenvalue perturbation theory, the .-i-ii!ilill ic SNR of this method can be found to

be identical to that of optimal linear combining. The second assumption used in this

approach (besides Al) is that











A3. The measurement matrix has an effective rank of one. Effectively, this is equiv-

alent to i--i,',,:i,:. that the coil measurement SNR levels are suaff. :.. il high. In the

noise-free measurement case, Al implies A3.

In order to demonstrate the validity of this assumption, we resort to the same cat

spinal cord image example shown in Fig. D-3. Fig. 2-4(a) shows the histogram of the

ratio between the largest singular value of the local measurement matrix to the mean

of the other three singular values (there are four singular values since there are four

coils). Since there are very few small singular value ratios, we conclude that in most

local regions the rank-one measurement matrix assumption accurately holds. In fact,

the noise-only regions dominantly contribute to the small singular-value-ratios. To

illustrate this fact, in Fig. 2-4(b) we also present the singular-value-ratio as a function

of spatial coordinate for the cat-spine image, using 5 x 5 square local regions.

2.3.2 Bayesian Maximum-Likelihood (ML) Reconstruction

The B i, i ,I ML reconstruction approach also relies on assumption Al; there-

fore, it operates on a set of small regions that constitute a partitioning for the whole

image. In addition, any available statistical information about the coil sensitivities

and noise in the form of probability distribution functions (pdf) are incorporated in

the formulation. This is stated formally in the following assumption.

A4. Suff :.. ill; accurate a priori information ,.j,,:i,:.i. the p, 1r'.al.,;11./ distribution

function of the coil sensitivities and the additive measurement noise is available.

The principle behind ML reconstruction is to maximize the a posteriori probability

of the observed data given the image pixel values, and is formulated in the following

optimization problem


p= ar.iiu .i pp(SIp) = p(S,cfp)dc I p(SIc, p)p(c)dc (2-12)









Here p(Slp) is the conditional pdf of the measurement matrix given the image,

p(S, clp) is the joint pdf of the measurement matrix and the coil sensitivity vec-
tor conditioned on the image, p(Slc, p) is the conditional pdf of the measurement
matrix given the coil sensitivity vector and the image, and finally,p(c) is the pdf of
the coil sensitivity vector. 1 Assuming that the noise in the measurements is jointly
Gaussian, we have


p(Sc, p) -= N-" QI-Q N [-p I (S pcT)Q-1/212] (2-13)

A Gaussian noise distribution can often be justified by invoking the central limit
theorem [29]. In addition, ML formulations with Gaussian disturbance terms tend
to give rise to mathematically convenient expressions, often in a least-squares form,
which are often intuitively appealing. (For instance, it is not hard to show that the
max-SNR reconstruction of Eqn. 2-1 is equivalent to ML if the noise is Gaussian.) If
we further assume that the density of c is also Gaussian with mean p, and covariance
A, the conditional pdf of the observed data becomes 2

p(Slp) p(S, c p) de = p(Sp, c)p(c) de
f 2
= -" IAI| exp- A-1/2(c _t) 2-Mnc Q exp- (S pcT)Q-12 2 dc

=7-(M+1)nc Q-nc Al exp (S- pcT)Q- 1/ 2- A- 1/2c 2 dc

(2-14)



1 Note that if a priori information about p is available (which is however unlikely)
in the form of a pdf, p(p), it can be incorporated in the optimization problem in
Egn. (2-12) by multiplying it with p(Slp) to result in a reconstruction that is optimal
in the maximum a posteriori (1\ AP) sense.
2 The randomness assumption for c emanates from the fact that it is a spatially
varying unknown parameter. In B ,., -i n estimation theory, unknown deterministic
parameters are typically treated as random variables.









The incorporation of a priori knowledge about model parameters via B i, -i iI

statistics has the advantage that the uncertainty in the value can be controlled by

adjusting the covariance matrix A. For example, a situation with little initial knowl-

edge about the value of c can be represented by a matrix A with large eigenvalues.

On the other hand, setting A = 0 results in a least-square optimal estimation of p

corresponding to c = gt.

The above integral result is a product of an exponential function multiplied with
T
an determinant, where both have a Q- & p part. It appears not to be directly

straightforward to maximize the total p.d.f. with respect to p, and as an approxima-

tion we simply minimize the sum of the two norms inside the integral in (2-14) with

respect to both the parameters p and c. For this purpose, we use a cyclic algorithm:

1. begin initialize po,T, i = 0

2. compute co : co <-- arg mine F(c; po)

3. do i <- i + 1

4. compute p : pi+ <- arg minp F(p; c)

4. cpi+ <-- arg mine F(c; pi)

5. until F(p1+i, cis+) F(pi, c) < T

6. return p pi+, c ci+-

7. end

where the cost function F, the pixel vector p and the coil sensitivity c are related by


F A-12( ) + (S pc)Q- 12 (2-15)


c+1 [TT, + A- A- [TS + A- (2-16)


(2-17)


pi+ = [BfB]- 'B S


+ -H !


(2-17)


Pi+ [BHBi]-1B S









where 0 stands for the Kronecker product and

S1

B, = (Q- c) 0 I, S = (Q- 0 I)
(2-18)
Sn"

T,= (Q-) 0pi, g, A 2p,

In an MRI application, we may obtain p and c either via analytical modeling

of the electromagnetic fields associated with the coils, or via calibration scans of a

phantom with known contrasts; by modulating the parameter A, we can directly

influence the accuracy of the prior knowledge of c. Such efforts to compute the

coil sensitivity patterns must use the finite-difference time-domain (FDTD) method,

which is a computational method to solve Maxwells equations. FDTD divides the

problem space into rectangular cells, called Yee-cells, and uses discrete time-steps [30,

31]. This approach has been successfully employ, .1 to compute the sensitivity patterns

of transmit and receive coils for MRI [32]. The noise covariance, on the other hand,

can be estimated from the coil images using portions of the frame that do not have

any signal.

Since a closed-form expression for the solution of this reconstruction algorithm

is not available, it is difficult to obtain an .i-i~piiih ll ic SNR expression. Nevertheless,

since the solution is the fixed-point of the iterations, perturbation methods could be

used to obtain an SNR expression, possibly after tedious calculations.

2.3.3 Least Squares (LS) with Smoothness Penalty

Given the measurement model in Eqn. (1-8) and assumption A2, a simple and

intuitive approach is to solve a penalized least-squares (LS) problem to reconstruct

the image from the coil measurements. Recall that LS methods coincide with ML if

the error is Gaussian. A natural smoothness penalty function is one that attempts

to minimize the first and second order spatial derivatives of the coil sensitivities.










However, such an approach alone does not solve the problem, because the optimal

solution of a penalized LS criterion tends to yield images with large intensity. This is

so because decreasing the amplitude of the coil sensitivity profile decreases its deriv-

atives as well, causing the reconstructed image to be scaled up by the same amount.

Therefore, it appears necessary to also impose a penalty on the total energy of the

image. The resulting penalized least squares criterion, which has to be minimized to

obtain the optimal reconstructed image, is given in Eqn. (2-19)


J(p, ci,...




Jo(p, C,...


Ji(ci,.. .


en) =(1 A A2 A3)Jo(P, c1, Cnj + AIJI(Cl,... ,Cn

+ A2 2(l,..., ,c) + 3J (p)
nc M N
,Cnj) -= [S (1,J)- P(1,J)ck (,)]2 + [Sk(,j)- 2P(,j)Ck(,j)]2
k=1 i l j 1
nc M N nc M N

k=1 i=2 j 1 k=1 i 1 j 2

( 2 A ckl2 A CII 2)
k= 1
nc M N
, nc ) [Ck(i,j) 2Ck(I- 1,j) + Ck( 2, j)]2
k=1 i=3 j 1
nc M N
+ Z [ck j) 2k(, j ) + Ck(i, 2)]2
ki= i-l j=3

(BIck 2 +IB2cT 2)
k= 1
M N
J3(P) 2 I 2)]2 1
i 1 j 1


(2-19)


where p now denotes the vector of pixel values for the whole image; hence no parti-

tioning is required here. Note that the penalty term in this LS formulation can be


- I)]2


J2 (C, .










interpreted as a B li, -~i i prior. 3 The gradient G of the cost function in (2-19) with

respect to the optimization variables W = [pT, T, ... nTiT is

dJ
Op
9J
G a = (1 A1 A2 A3)Go + AIGI + A2G2 + A3G3


9J
Benc .
__Jo n O J1
j 2 Z i [p O Ck )Ckl] ai

DJ0 [ci 0p0 p- s* 0 p |J1
G
C1 GiI -C1 (2-20)


a.J L [nc Op o p sc o) p] _a J
DJ2 0 aJ3
Op Op

G2 2(BTBIc1 + c1BT B2) G
0J2 2(B TBc, + cBT B2) J3
G2 cn 1 3 cl




where 0 denotes element-wise vector product. In (2-20), Ai and Bi are non-symmetric

sparse Toeplitz matrices that arise from the matrix formulation of the first and second

order differences. In particular, Al and A2 are ( -1l)xM and (N-l)xN matrices with

Is on the main diagonal and -Is in the first upper diagonal, and B1 and B2 are

(\1-2)xM and (N-2)xN matrices with Is on the main diagonal, -2s in the first upper

diagonal, and Is in the second upper diagonal. All other entries of these matrices

are zeros. Similar to the case of the B li, -i i, reconstruction algorithm, obtaining

an ..i-mptotic SNR expression for this algorithm should be possible although it is

algebraically complicated.




3 More details on the relation between smoothness constraints and a priori infor-
mation via B ,i, -i in statistics can be found in [32].









In general, least squares criteria can be shown to be equivalent to the maximum

likelihood principle if the probability distributions under consideration are Gaussian,

or perhaps other symmetric unimodal functions where the peak of the distribution

corresponds to its mean value as well [27]. Besides the three statistical reconstruc-

tions, a image reconstruction method based on spectral decomposition is worth to

mention in Appendix D.

2.4 Results and Discussion

The performances of the proposed algorithms are first evaluated using synthetic

data. The data model for this data is as follows. A random image consisting of 9

pixels whose jth pixel value is drawn from Uniform [j, j + 1] and then normalized

such that the norm of the intensity vector is unity, pTp = 1. The measurement

vector in each coil is obtained by Sk = (rho + ek)ck, where c = [cl, 4]T is the

coil sensitivity vector (with each entry selected from Uniform [0, 1]), ek is zero-mean,

unit-covariance Gaussian noise (also independent across coils), and a is the standard

deviation of the additive noise determined by the specific measurement SNR that is

being simulated. 4 All four algorithms (SVD, ML, LS, and SoS) are applied to this

synthetic data in 20000 Monte Carlo simulations for each measurement SNR level,

where all parameters are randomized as described above in every trial.

The image intensity estimate vectors of all four algorithms are normalized to

unity such that in the comparison with the ground truth (which is available in this

setup) using signal-to-error ratio (SER) without considering scaling errors. The SER



4 Note that this is not a very realistic situation, since in an actual MRI, the mea-
surement SNR in a coil is also determined by its sensitivity coefficient. In this ex-
ample, however, the noise is added to the image before coil sensitivity scaling is
applied, merely for convenience in representing results (such that a single SNR value
describes the data quality). In fact, it will become evident in the application to real
data that statistical signal processing approaches benefit more from this variability
in measurement SNR of coils.













28 -





m 22

1 20 --L-

18 -G -( .SVD

16 4-' -SoS
4 O"'Bayesian
14 --- -

12- I I
0 5 10. 15 20
SNR (dB)
FiNure 2-2. Performance of the four algorithms, SVD (circle), ML (square), LS (star),
SoS (triangle), shown in terms of image reconstruction SER (dB) versus
measurement SNR (dB). Clearly, ML and LS perform almost identically
outperforming SVD and SoS, which also perform identically.

is defined as SER(dB) = 10log 0 (1 p 2/1 p _- ||2), where p is the normalized esti-
mate obtained using the corresponding algorithm. The results of this Monte-Carlo
experiment on the described synthetic data are presented in Fig. 2-2 in terms of av-
erage reconstruction SER versus measurement SNR for all algorithms. These experi-
ments show that all four algorithms .-i~-li i11 l. ically (as the SNR approaches infinity)
achieve equivalent reconstruction SER levels. For low SNR, however, although the
SVD and SoS yield the same level of SER performance, the ML and LS algorithms
provide a slight (about 0.6 dB) gain in SER.












As a second case study, all four algorithms are applied to the multiple coil images

presented in Fig. D-3(a)-D-3(d), which are collected by the coil array shown in Fig. 2

1 with the previously specified measurement parameters (In Appendix A, a detailed

phased array coil is introduced). For the two iterative methods (\ l, and LS), the

SoS estimate of the coil sensitivity profiles and image intensities are utilized as initial

conditions. In addition, for both SVD and ML algorithms, 5 x 5 non-overlapping

regions in which the coil sensitivity is assumed to be constant are used, and the scale

ambiguity for the solution of each region is resolved by normalizing the power of

the reconstructed signal for that region to that of the SoS reconstruction. The ML

algorithm uses a noise covariance estimate Q obtained from a purely noise region

of the coil images, and in an ad-hoc manner, the covariance of the coil sensitivity

distribution is assumed to be A = I. Also quite heuristically, in the LS algorithm,

all three weight parameters are set to Ai 0.1. 5 In phased array MRI, the quality

of reconstructed images is often quantified by SNR, as the true image is usually

unknown. 6 The reconstructed images obtained by these four methods, as well



5 Experiments performed to establish an understanding of how these parameters
affect the reconstruction performance demonstrated that extreme values (both in
smaller and larger directions) degrade the quality of the image. In general, the
authors observed that for all three coefficients values in the interval [0.05,0.1] are
reasonable. Values greater than 0.1 tend to overemphasize the penalty functions,
while values smaller than 0.05 do not provide sufficient smoothing.

6 The SNR calculated here (given in dB scale) is the ratio of the power of the re-
constructed image intensity in the region of interest to the power of the reconstructed
image intensity in a reference region, which presumably consists only of noise. Un-
der the spatially WSS noise assumption, the SNR calculated using this method is on
average equal to the SNR+1 (in linear scale), where the latter is the conventional def-
inition common in the signal processing literature. In the examples shown in Fig. 2-6,
a rectangular region at the top left corner, which consists of pure noise, is selected as
the reference noise power region. The SNR in the other rectangular regions, as shown
in Fig. 2-6, are calculated by dividing the signal power in the selected region by the
noise power estimated from the reference region. The values are then converted to
decibels using the 10 log1io() formula.





















(a) (b) (c)


(f) (g) (h) (i)

Figure 2-3. The vivio image obtained from a) Coil 1 b) Coil 2 c) Coil 3 d) Coil 4. The
coil sensitivity estimates for f) Coil 1 g) Coil 2 h) Coil 3 i) Coil 4, and j)
the reconstructed image obtained using the SoS reconstruction method.

as the estimated local SNR levels of these reconstructed images are presented in

Fig. 2-5& 2-6. By comparing the SNR estimates in Fig. 2-6(a)-2-6(d), we observe

that the SVD and SoS methods, in general, produce images with equal SNR levels

(although SVD is observed to be more sensitive to noise and measurement artifacts as

discussed below), whereas the ML approach improves the SNR by up to 2dB and the

LS approach improves the SNR by up to 3dB over the performance of SoS. However,

the correlation between SNR and image quality will be explained in Appendix B and

C.

At first look, a clear artifact in the SVD reconstructed image shown in Fig. 2

5(a) is visible. Although this artifact is not as visible in the other three reconstructed


'I '










250


200


150
50

100 40

30
50 20

10

0 20 40 60 80 100 0
(a) (b)

Figure 2-4. The ratio of the maximum singular value to the average of the
smaller three singular values of the measurement matrices for 5x5 non-
overlapping regions a) summarized in a histogram and b) depicted as a
spatial distribution over the image with -.i i,-' i1. values assigned in logo1
scale, brighter values representing higher ratios.


images (Fig. 2-5(b)-D-7(a)) due to the small size of the figures, upon closer exami-

nation, we see that this horizontal artifact also exists in these images. The reason for

this artifact is identified as a horizontal measurement artifact that exists in all four

coil measurements at that location (most strongly seen in the first coil). This artifact,

along with measurement noise, is amplified in the SVD reconstruction method to the

highly visible level in Fig. 2-6(a). The reason for this amplification of noise and

outliers can be understood by investigating Fig. 2-4(b). The ratios of the maximum

singular values to minimum ones are not as large in the top half of the coil measure-

ment image as the same ratios in the bottom half of the image. Consequently, A3

is not as strongly satisfied in the top half as the bottom half. This causes the SVD

algorithm to pass the existing measurement noise to the reconstructed image with

some amplification. The artifact in the measurements is also amplified in the process.

























(a) (b)


Figure 2-5. The reconstructed images using a) SVD b) ML c) LS d) SoS approaches.


Phased-array MRI research has experienced an increased interest in the last

decade due to the potential gains in both imaging quality and acquisition speed.

Although many algorithms have been proposed for phased-array MR image recon-

struction, in addition to the perhaps most commonly used sum-of-squares algorithm,

these approaches are not based on a statistical or optimal signal processing frame-

work.

In this chapter, the problem of combining images obtained from multiple MRI

coils is studied from a statistical signal processing point-of-view with the goal of

improving SNR in the reconstructed images. In order to pursue this approach, cer-

tain model assumptions must be made. I developed a set of assumptions that were

observed to hold on data collected from real measurements, and three alternative al-

gorithms, stemming from well-established statistical signal processing techniques, and

founded on these assumptions were proposed. The new proposed methods, namely

singular value decomposition, maximum-likelihood, and least-squares with smooth-

ness penalty, were evaluated on synthetic and real data collected from a four-coil

phased array using a 4.7T scanner for small animals. A quantitative analysis of the

reconstructed images obtained using measurements of a cat spinal cord revealed that


() (d)









it is possible to improve the quality of the final images (in terms of local SNR) by up

to 2dB using the maximum-likelihood approach and up to 3dB using the least-squares

approach.

The SNR is a convenient and widely used quality assessment instrument for MR

images. The use of the singular value decomposition and least-squares methods statis-

tically make sense when this second order quantity is utilized for quality assessment.

On the other hand, other quantitative measures such as signal-to-contrast ratio might

be more representative of image quality as perceived by a human observer. In that

case, alternative optimization criteria for optimal reconstruction of the coil measure-

ments must be derived. These alternative criteria must be consistent with the desired

quality measure, as well as being sufficiently simple.

There are still unsolved issues, however. For example, if the original measure-

ments already have high SNR, then the reconstructed image using SoS performs close

to maximum ratio combining; therefore a few dB of gain in reconstruction SNR may

not be visible to the human eye. With the maximum-likelihood approach, I used the

standard circular-Gaussian noise model; yet I ended up with a relatively complicated

expression that needs to be maximized. More accurate statistical signal models might

improve the performance of the approach; nevertheless, computational complexity is

alv-i-, a concern for MRI.

Therefore, the disadvantages of SoS reconstruction and other statistical image

reconstruction methods drive me to research further this topic. All these methods,

without exception, relied on building algorithms based on statistical or structural

assumptions about the signal model. These approaches were either heuristic or sta-

tistical in nature. An adaptive signal processing framework has not yet been studied

for phased array MRI. In the next chapter, I propose to tackle the image reconstruc-

tion problem in multiple-coil MRI scenarios by a competitive mixture of experts. The

expected gains from this approach include the following: there is no need to propose









or discover signal models that describe the measurements well (a must in statistical

signal processing approaches) and the local structure of the input space is naturally

extracted from the data. Thus the key difficulty to estimate the coil sensitivities is

avoided. Moreover, adaptive systems are more flexible and robust to inconsisten-

cies and nonstationarities in the data as they can be updated on-line while in use.

With a meaningful adaptation paradigm adaptive systems are able to approximate

optimal statistical signal processing approaches (to the limits set by the topology)

while requiring less design effort. However, the adaptive framework requires a desired

response for adaptation operation, as will be discussed below.





















n nn n eSn n E




(a)






Sl m em nnn n
re1
M M MMM m m em
iBEMl:]
eMM mlM MM m l
i 31:]
eMMMM m
ME lml MM m l
iiMMl:]


MMMMM
e ine e e m

n nn I II i






(b)

ESz mBEE
SE1aiam m E DI EfIEa
EE amE3m EEam E3EB
iE m niE~EmniEm S]
iEa nim BE nRiEnl
Em aBElmE EZ amEB
ME~aEnam EDl~ m
mE aE am EniE a
mE imE3]lBEaDlEEBIEB


Figure 2-6. The estimated local SNR levels of the reconstructed images using a)
SVD b) ML c) LS d) SoS approaches, where the top left region is the
noise reference. Notice that in (a)-(d) the SNR levels are overlaid on
the reconstructed image of the corresponding method. To prevent the
numbers from squeezing, these images are stretched horizontally. The
top left corner of each image is used as the noise power reference.















CHAPTER 3
SUPERVISED LEARNING IN ADAPTIVE IMAGE RECONSTRUCTION
METHODS, PART A: MIXTURE OF LOCAL LINEAR EXPERTS

3.1 Local Patterns in Coil Profile

As we reviewed, fast MRI imaging using a phased-array of multiple coils has to

cope with an implicit inhomogeneous reception profile in each coil [33]. This feature is

described by coil sensitivity profiles, and explains the B1 field map generated from the

coil geometry. Due to ehe spatial configuration of phased-array coils, the sensitivities

of the coils are restricted to a finite region of space. This local coil sensitivity feature

is used in recent MRI image reconstruction such as parallel imaging with localized

sensitivities (PILS) [18], local reconstruction [19], etc.

Besides the sensitivity map locality, it is of interest whether the thermal noise

generated in the receiver coils possesses local property. The thermal noise Voise,, in

the coils is excited by the imaged lossy body in the coil vicinity, where the rms voltage

of the noise is given by Nyquist's formula [34, 35]


4Voise = VkTBAf RL (3-1)

where k is the Boltzmann's constant, TB the temperature of the body, Af the band-

width of the preamplifier attached to the coil and RL the equivalent loss resistance

of the coil. For a given designed coil system, the thermal noise Voise should solely

depend on RL. The loss resistance RL is affected by many factors, i.e., the geome-

tries of the coil and body, their positions relative to each other, the conductivity and

complex permittivity of the dielectric and the coil coupling. The load affects RL by

transferring the uncoupled coils into coupled and finally influence the generated noise.

This effect produces the local noise property distinct in the desired image region and









background image region. However, due to complex local structure inside the image

region, the noise property is hard to estimate there. Based on these local patterns

in coil profiles, an adaptive signal processing strategy is proposed to extract local

features and incorporate them into image reconstruction.

3.2 Competitive Learning

Competitive learning algorithms are widely used in pattern classification [36],

vector quantization [37], and time series prediction [38], etc. They employ competi-

tion among the processing elements (PEs) by lateral connection or a certain training

rule. The simple competitive learning with the winner-take-all (WTA) activation rule

leads to a PE underutilization [39]. Two other schemes, named as fre'-1ii. l-v- -I itive

competitive learning (FSCL) and the self-organizing map (SOM), are addressed to

solve this problem. FSCL incorporates the conscience term into the training to drive

all the PEs inside the network into competition [40]. The SOM method proposed by

Kohonen uses a soft competition scheme to adapt not only the activation PE but also

its neighborhood [41]. Competitive learning has also been used in image processing as

image compression [42], image segmentation [43] and color image quantization [44].

3.3 Multiple Local Models

The idea of using multiple simple and local models to represent complicated non-

linear systems has gained interest in recent years. The multiple model idea comes

from the following reasoning: if any nonlinear mapping or space representation can be

subdivided into different subset, the utilization of multiple models reduces the cou-

pling of different subsets and gives a better representation and mapping performance.

If each model is identically linear, this multiple model method is usually called local

linear model or local linear expert. In most cases, the parameter for learning multiple

models is based on competitive learning while the competition strategy is either com-

petitive or cooperative. However, it was stated that the cooperative and competitive









mixtures of local linear models in classification generalize better performance than

radial basis function (RBF) and MLP with a comparable number of parameters [45].

Function approximation is an interesting area to use local models in representing

a nonlinear input-output mapping. The idea of local linear modeling was applied in

predicting chaotic time series, where the nonlinear dynamics is learned by the local

approximation [46, 47], as well as in the nonlinear autoregressive model parameter

estimation given a Markov structure [48]. Jacobs et al. proposed a mixture of experts

network followed by a gated network, used in multispeaker vowel recognition [49].

This divide-and-conquer strategy first provides a new view of either modular version

of rimultilli. r supervised network or an associative version of competitive learning.

The authors control the coupling among the multiple models by using a negative

log probability cost function. However, the gating weights are input-based and not

adapted to optimality. Fancourt introduced a cooperative fashion to the mixture of

experts network, where both the model parameters and gating weights are trained

by the Expectation-Maximization (EM) algorithm [50]. This method determines the

proportion of the datum which belongs to a single linear model according to its

posterior probability and combines the linear wiener solution in each model. The soft

competition actually provides a new data set strictly following the assumed Gaussian

distribution for each model, it may incorporate the modeling error into the final

estimation.

Another example in modular networks is applied to the local training of the

radial basis function (RBF). It shows that the k-nearest neighbor method (KNN) or

RBF network can be generalized to a local learning model based on different kernel

selection [51]. A local model networks which incorporates local learning to a radial

basis function (RBF) is proposed also in the divide-and-conquer strategy [52]. A

Growing Multi-Experts method is proposed as a novel modular network which adds

a local linear model to the RBF at the gating network stage [53]. In spite of the









network topology, it deploys a redundant experts removal algorithm to remove the

redundant models in order to exploit the optimal network architecture. Among these

methods, the parameter choices of the kernel (usually Gaussian kernel) needs careful

consideration. Based on the smooth tails of the Gaussian kernel, the decomposition

of the input space is somewhat overlapped, which means that the local properties

are not fully satisfied. Besides, the kernel parameter is adapted by the pure input

r llppiii which doesn't take the input-output mapping into account. Now I propose

to implement the competitive mixture of local linear experts method into image

reconstruction in phased-array MRI. It achieves simple local properties for each model

while the training is based on the input-output mapping while the gated network has

the universe approximation ability.

3.4 The Linear Mixture of Local Linear Experts for Phased-Array MRI
Reconstruction

In order to circumvent the difficulties associated with SoS, and to improve the

quality of the reconstructed image in terms of SNR, an adaptive training approach

with a mixture of experts can be employ, -l [50]. In the case of MRI, it is possible

to obtain a sequence of training images from a phantom statistically representative

of the actual objects to be imaged, or equivalently, a training set of images from a

subject in the beginning of the session. The supervised adaptive training process

learns these sample properties and stores them in system weights. Once the image

reconstruction system is trained (calibrated) with this set of images, it can then be

utilized for reconstructing images from scans of other subjects (e.g., tissues). The

desired high quality output is formed by utilizing a standard image reconstruction

algorithm (such as SoS) on multiple scans of the training image set and averaging

the reconstructed images to generate a high SNR desired output image with dij as

the desired output for pixel (i,j). An alternative to multiple images is to increase

the scan time to improve the quality of a single scan image. In training, the network




















Sij *



Model M -V 4
Competitive
LMS Training


Figure 3-1. Block diagram of the linear multiple model mixture and learning scheme.


is expected to map the inputs sij obtained from single scan images to the clean

desired output dij. Note that the training is not necessarily related to the image to

be detected later (i.e., it can come from a phantom placed on the MRI) because the

goal is to determine the spatial profile of the coils, which is largely unrelated to the

object being imaged.

A schematic diagram of the proposed image reconstruction topology is depicted

in Fig. D 1. This topology consists of multiple linear models operating on the coil

measurement vectors that specialize in different regions of the measurement vector

space. For pixel (i,j), model m produces an output xm that is the linear combination

of the input vector si;j: x = wT sij, where Wm are the model weights for m =

1, ... M, M being the number of linear models. The input vector sij may consist

of only the coil measurements for pixel (i,j) in a training input image or it may

include the coil measurements for pixel (i,j) and its neighbors (in which case the

neighborhood radius must be specified). The neighborhood is typically a small q x q

square region centered at pixel (i, j).


__



__









These linear models are trained competitively using the LMS algorithm [54] (in

a winner-take-all fashion), where the criterion of the competition is the output mean

square error (\!1S;) evaluated over a neighborhood of pixels (which is an r x r region

centered at the (i,j) pixel [50]. For the nth epoch

Wm(n+ + 1)= wm(n) + pisei(n)
(3-2)
ei(n) d w(n)Ts
2

where m = arg minm k ,(d(k) xm(k))2 is the winning model index chosen from

[1, ... M], r2 is the number of pixels in the local region, /1 is the step size and only
the model with the smallest MSE is updated. The r2 local region has the effect of

noise suppression in case the current pixel is very noisy, and prevents the wrong model

selection. This procedure is repeated for multiple epochs until the weight vectors of

all models converge. As a result, the competitive learning phase maps the gray scale

coil amplitude images to multiple local experts based on the spatial clustering of coil

vectors and their projection to the desired response.

The multiple model outputs which capture the spatial local features of the coil

images, are then combined to produce an estimate of the image intensity at the (i,j)

pixel using


j gXij (33)

where xij = WTsij is the vector of outputs from the multiple models (W =

[wl, w u]), and the mixing weights are also linear combinations of the input,
i.e., gi, = Vsij. Once the multiple linear models are trained with competitive

LMS, the mixing matrix parameters V can be trained with LMS using the outputs

of the competitive models as the input and the same desired output, as illustrated in









Fig. D-1.

V(n + 1) = V(n) + p2xsTe2(n)
(3-4)
e2(n) = d- sTV(n)Tx

where P2 is the step size. Alternatively, V can be determined using the analytical

linear least square solution.

Since, it is assumed that both the model outputs and the mixture weights are

linear functions of the inputs, in the test phase, the output of the proposed mixture

of linear experts can be written as


A,? = sVTWTsJ = sfGsi, (3-5)

We note that Eqn. (3-5) implicitly used to reconstruct a testing images is similar to

the whitened SoS reconstruction given in (2-4) 1 except that the weighting matrix

is trained using the MSE criterion and the multiple model concept over a training

(calibration) set. For spatially stationary noise characteristics and perfect training,

the two procedures should be equivalent since the competition is based on the noise

power. The adaptive approach has the advantage that if the noise is not spatially

stationary, the local models will specialize to different modalities of the noise and

the adaptive mixture model will still be able to produce high SNR reconstructions

reliably.

3.5 The Nonlinear Mixture of Local Linear Experts for Phased-Array
MRI Reconstruction

The image reconstruction system described above can be improved by replacing

the linear combination stage by a nonlinear combination of local linear outputs, here

implemented as a rmiltil ivr perception (\ I.P). It is sufficient for the MLP to have



1 The square root in the SoS reconstruction is ignored so as to establish a compa-
rable equation with the proposed method.

















Sij


-= Model M XM L
-- ICompetitive
LMS Training MLP Training


Figure 3-2. Block diagram of the nonlinear multiple model mixture and learning
scheme.

a single hidden li. r and a linear output 1I rV. Due to the universal approximation

capabilities of MLPs, it is expected that this new topology will improve the final

SNR by better emphasizing the outputs of the linear models that are relevant and

deemphasizing the outputs of those models that are not relevant for the current pixel.

This nonlinear mixture model (with M inputs, L hidden processing elements, and

one linear output) and the adaptation strategy are shown in Fig. 3-2.

The output of the MLP is given by pi,j;


yj f x(VxI + bi)
(3-6)
pi,j v= y2, + b2

where f(.) is the sigmoid shaped nonlinear function of the hidden lI, r. The MLP

weights V1, v2, bi, b2 are trained with error backpropagation according to the MSE

criterion [55]. The inputs to the MLP are the outputs of the linear models and the

desired output is the same dij that is used to train the linear models competitively.

3.6 Results

In this section, the performance of the proposed mixture model approach in

phased-array MRI reconstruction is demonstrated using transverse (45 measurements)









and coronal (9 measurements) fast collection human neck images in a 4-coil MRI sys-

tem (fast spin echo(FSE), TE=15ms, TR=150ms, ETL = 2, FOV = 40cm, slice

thickness = 5mm, matrix = 160x128, NEX=1). Sample images from the four coils

for both cross-sections are shown in Fig. 3-3 & 3-4, which is noisy due to the short

scanning time for each sample. All transverse cross-section measurements are used

for training and one of the coronal cross-section measurements (the one that is shown

in Fig. 3-4) is used for testing the resulting network. The desired reconstructed

image is estimated by averaging the SoS reconstruction for each training coil im-

age sample (Fig. 3-5(a)). Its high SNR performance demonstrates a clean and low

noise desired response (Fig. 3-5(b)). Both training and testing data sets consist of

magnitude images normalized to [-1, 1] before processing.

The training procedure, made in two stages, needs some special discussion. In

the first stage, the weights in the local linear experts are trained competitively by

LMS. The number of competitive models is selected to be M = 4 (as will be explained

below). The input vector sij corresponds to only one pixel (i, j). The training of

the local models stops after 20 epochs with the step size /i = 0.01 demonstrated in

Fig. 3-6(a). After the weights in the first stage are well trained, the multiple expert

outputs are taken as the input to the second combination stage and dij used in the

first stage is again taken as the desired response. The linear mixture finishes training

in 5 epochs with 2 = 0.01 by LMS algorithm shown in Fig. 3-6(b). Alternatively,

the nonlinear mixture network is a 3-1 i-r MLP network with one linear output PE,

5 hidden PEs and M input PEs corresponding to M multiple experts. The training

stops in 30 epochs with the step size 2 = 0.005 by the backpropagation algorithm

shown in Fig. 3-7. The reconstructed image calculated from one training sample of

the transverse cross-sections by the trained nonlinear mixture of local linear experts

shows a peak SNR of 33 dB (Fig. 3-8), which is still 12 dB lower than the peak SNR









of the desired response image (Fig. 3-5). This SNR gap means that there is still room

for improvement in future research.

The coil measurements of the test image (coronal cross-section) are combined

using SoS (without and with whitening) as well as the proposed mixture model net-

work. Since a reference (a ground truth) is not available in MRI, typically the image

quality is measured by the empirical SNR measure, which in fact does not conform to

the traditional definition of SNR in signal processing. The procedure for computing

the SNR is as follows:

1. Find a reference region in the reconstructed image where there is no signal (i.e., a

pure noise region).

2. Compute the variance of the noise in this reference region.

3. For all other regions, compute the signal power (which includes both the actual

signal and the remaining noise in that region).

4. Calculate SNR in a region as the ratio of the power of the signal in that region

to the variance of the noise in the reference region. Convert SNR to dB. In order

to optimally configure the proposed method, parameter a n ,!i1,-; is addressed now.

First, I demonstrate the specialization of the local linear models of the first stage dur-

ing training. The important question concerning the number of local linear models

is also addressed. Fig. 3-9 shows the spatial distribution of the pixels for a sample

image using trained multiple local models for the cases where M = 2, 4, 8, 16. One

can observe that the 2-model system basically segments the image into noise and

signal regions. As M is increased, the additional models help segment the signal and

noise regions to smaller partitions depending on their local statistics. As expected,

as the number of models is increased, the MSE of the winning models in training

converges to progressively lower value as shown in Fig. 3-6(a). However, the overall

MSE of the final output (after the combination of the multiple model outputs) does

not decrease significantly when the number of local linear models is increased above









M = 4(shown in Fig. 3-6(b)). Due to additional computational load and generaliza-

tion considerations, one should in practice select the smallest number of models that

yield satisfactory performance. In measuring the performance, the modeling MSE

and the SNR of the reconstructed image can be monitored simultaneously to make

a decision. In our MRI system, for example, M = 4 is a logical choice, and will be

used in the rest of the experiments.

Next, I study the effect of increasing the spatial filter order q (the neighborhood)

for each coil image in reconstructing the center pixel value. In general, the input

vector sij for reconstructing pixel (i, j) can consist of all pixels in a neighborhood of

pixel (i,j) from all coils. For example, if a 3 x 3 region centered around pixel (i,j)

is selected, then the competing linear combiners become 9-tap 2-dimensional spatial

FIR filters on each coil, yielding a total of p = 4 x 3 x 3 = 36 input values, assuming

n, = 4 coils. This extension of the input vector to include neighboring pixels in the

reconstruction of the center pixel allows designing competing minimum-MSE spatial

filters for each coil. Increasing the size of these spatial filters will introduce additional

smoothing capabilities that help increase SNR. As an illustration, the performance

of the linear mixture of local linear experts approach is demonstrated on the coronal

cross-section reconstruction using noisy measurements from 4 coils. Compared to

the SNR performance of the competitive linear experts using only the center pixel (4-

dimensional input vector) shown in Fig. 3-10(a)&3-10(c), the SNR obtained by using

a 36-dimensional input (i.e., a 3 x 3 neighborhood for the FIR filters) shown in Fig. 3-

10(b)&3-10(d) is up to 4dB higher in the signal regions. This noise suppression is

achieved at the cost of some blurring of the sharp details in the reconstructed image,

due to the low-pass filtering effect of the increased-length spatial filters;so smaller

masks or multiple input multiple output combination method are advisable.

Now I analyze the advantage of nonlinear combination of experts. Since both the

linear and nonlinear mixtures of local linear experts use the same driving models, the









MSE from the competitive winner models in training can be regarded as a lower bound

for the MSE of the overall system for both paradigms, which is around 295 for M = 4

case shown in Fig. 3-6(a). However, the selection of the winner requires the knowledge

of the desired response, which is not available during testing. Therefore, I resort to a

linear or nonlinear combination adapted in the training set. The combination phase

combines not only the winner expert corresponding to the current pixel but also

the other M 1 experts, where the winner changes within the M models when the

adaptation goes from one pixel to another. Thus, the MSE of the overall network

should be worse than that of the winners. The final MSE computed in nonlinear

mixtures demonstrates a closer value to the lower bound (\r l; of around 350) than

the MSE in the linear mixture (around 530) in Fig. 3-7 & 3-6(b), which responds to

higher SNR in reconstruction.

Finally, the nonlinear mixture of local linear experts is compared to SoS, whitened

SoS, and linear mixture of local linear experts. The comparison is based on the re-

construction of the same coronal cross-section image from 4 coil measurements. For

whitened SoS, the whitening covariance matrix is estimated from the coronal cross-

section measurements. The reconstructed images and the estimated SNR levels are

presented in Fig. 3-11&312. Focusing only on the SNR levels in the signal regions,

we observe that the nonlinear mixture of local linear experts approach improved the

performance up to 4dB, 5dB, and 15dB over that of the linear mixture of local linear

experts, SoS with whitening, and SoS methods, respectively. The light white region

at the upper-left corner is used as the noise reference in computing the SNR levels.

As we know, the definition of SNR as a measure for image quality is not ap-

propriate. Furthermore, one might argue that the noise is only suppressed in the

background region (i.e. not affecting the signal region). In spite of the absence of

any theoretical justification, this was practically demonstrated wrong. Two nearly

identical samples of the human spinal cord image collected by four coils are collected.






45


The two reconstructed images and the noise given by the subtraction are shown in

Fig. 3-13. It can be seen that the noise is evenly distributed with no correlation

neither with the background nor with the signal region; on the contrary, they are

spatial evenly distributed despite of the residue structure. Thus the SNR measure

accesses both the noisy and signal regions.
































(a) Coil 1 (b) Coil 2


(c) Coil 3 (d) Coil 4


Figure 3-3. Transverse crossections of a human neck as measured by the four coils
from one training sample.

































(a) Coil 1 (b) Coil 2


(c) Coil 3 (d) Coil 4


Figure 3-4. Coronal crossections of a human neck as measured by the four coils used
as the testing sample.





































(a) Desire reconstructed image (b) SNR of the desire reconstructed




mated desire.






























Model#: 4
-*- Model#: 8
-- Model #: 16
















5 10 15 2
Number of epochs


-- Model #: 4
-- Model #: 8
- Model#: 16


1.5 2 2.5 3 3.5
Number of epochs


4 4.5 5


Figure 3-6. Adaptive learning performance, (a) Learning curve of winner models for

the model number 4,8,16, (b) Learning curve of the linear mixture of

competitive linear models system for the model number 4,8,16.















3000


2500-

E
w 2000-


o 1500-


C 1000-


500


0
0 5 10 15 20 25 30
Number of epochs


Figure 3-7. Learning curve of the nonlinear mixture of competitive linear models
system for the model number 4.


(a) (b)

Figure 3-8. The reconstruction image, (a) From one transverse training sample by
nonlinear mixture of local linear experts, (b) The SNR performance of
the reconstruction.




51








IUI




(a) M-2 (b) M-4

I I



I M- M
(c) M 8 (d) M 16


Figure 3-9. Pixel classification for the model number 2, 4, 8, 16.


ac-- *
.




























(b) M-4, p-36


(c) M 4, p 4
1.4dB 4

241 2;4g 2 B


E (d) M=4, p=36



(d) M-4, p-36


Figure 3-10. Reconstructed images and their SNR performances from the mixture of
competitive linear models system with the model number 16 and the
coil number 4, 36.


(a) M=4, p=4





























(a) (b)


(c) (d)

Figure 3-11. Reconstructed test images for a coronal crossection from a human neck,
(a) SoS without whitening (b) SoS with whitening (c) Linear mixture
of models, (d) Nonlinear mixture of models.

















^ EE^ m


(a)

E Em~me~

E~mm ~mm


_--U.-_^^,^,---
E M 7"'J-1 Vm AM





(b)



MM: .11 UME
-3 r 1 ? L '6"M -





(d)
~em'~
sm m
~


Figure 3-12. SNR performances of the reconstructed test images for a coronal crossec-
tion from a human neck, (a) SoS without whitening (b) SoS with whiten-
ing (c) Linear mixture of models, (d) Nonlinear mixture of models.





























(a) (b)


Figure 3-13. Image quality measure, (a)-(b) The two reconstructions by nonlinear
mixtures of models using two near idential 4 coil samples, (c) The noise
power from the subtration of the two reconstruction images in (a).















CHAPTER 4
SUPERVISED LEARNING IN ADAPTIVE IMAGE RECONSTRUCTION
METHODS, PART B: INFORMATION THEORETIC LEARNING (ITL) OF
MIXTURE OF LOCAL LINEAR EXPERTS

4.1 Brief Review of Information Theoretic Learning (ITL)

In the last chapter, the nonlinear competitive local linear experts network esti-

mates the reconstruction image by combining the outputs from the multiple linear

model by a muiltil i-r perception ( I1I.). This nonlinear combination is superior to

the linear combination due to its nonlinear compression to the models which don't

win the adaptations. The training for the MLP weights is based on minimizing mean

square error (\ SE) criterion. This criterion is extensively applied in the training of

linear or nonlinear systems due to its mathematical simplicity and practicability. It is

based on the assumption that the second order statistics is sufficient to represent the

data distribution. This is often true since the probability density distributions (pdfs)

of many systems exhibits Gaussianity, where their pdfs are exactly determined by

mean and variance. Besides, Gaussianity is also supported by the central limit theo-

rem for a large amount of sample number. However, neural networks don't necessarily

confine the moments of their output error to the first and second order statistics due

to the implicit nonlinear processing elements (PEs), which is true for generalized

nonlinear networks. Thus, minimizing MSE might not be sufficient to capture all the

information used to train the network.

As we know, entropy is used to measure the uncertainty of a random variable due

to a its pdf. Shannon firstly defined the average information of a random variable,

named Shannon entropy, which is formulated in


H,(x)= -E[log f(x)] (4-1)










where E[.] and f(.) are the expection and pdf of a random variable x, respectively.

With the Taylor series theory at the point x = 0, Shannon entropy can be expanded

as

H() E[x 1 aO[log f(x)] (4 2)
H,(x) -Ely, O (4-2)
n=O

Thus Eqn. (4-2) shows that the Shannon entropy is a expectation of the weighted sum

of all order moments while the weights depends on the higher order derivatives of pdf.

Thus entropy contains the combined information of all the moments. If higher order

moments make sense in some cases, the entropy measure as a adaptation criterion is

more suitable than MSE. Instead of Shannon entropy, Renyi entropy is more widely

used due to its mathematical attractivity and generality where Shannon entropy is

only one of its special case. The research of incorporating the information theoretic

quantities into the adaptive training guided by Dr. Principe in our CNEL lab has

been for nearly a decade. The up-to-date contribution to incorporate minimizing

Renyi entropy criterion in supervised learning strategy is shown in [56].

4.2 ITL Bridged to MRI Reconstruction

I are interested in applying this ITL training method to our combination strategy.

A schematic diagram of the proposed image reconstruction topology is depicted in

Fig. 4-1. The topology is the same as in Fig. D 1 except that the optimization

strategy in the second stage is alternated by information theoretic learning instead

of minimizing MSE.

The first question to ask is whether or not this training method compassing

higher order moments is needed. The pdf of the error distribution of the well-trained

MLP network by minimizing MSE gives the detail of validity. If the pdf of the error

demonstrates Gaussianity or can be simply described by the first and second order

statistics, there is no reason we should incorporate this ITL training idea into this

network. However, Fig. 4-2 shows a super-Gaussian distribution with kurtosis of

















sil -


Model M XM L
SCompetitive
LMS Training ITL Training


Figure 4-1. Block diagrom of the nonlinear multiple model mixture and learning
scheme.

27.4. The pdf shows slim tail and dominant main lobe with two peaks inside with

one dominant and another having 17'. peak value of the first one. Since the pdf

is not unimodel, it can't be exactly described by the lower order moments and the

application of ITL strategy to this problem is suitable.

4.3 ITL and Recursive ITL Training

As we know, Renyi's entropy of order a is defined as

1 +
H(x) log f (x) dx (4-3)
1 a J_.

Special interest is focused on Renyi's quadratic entropy (a = 2) here for simplicity.

Parzen window is used to estimate the pdf with Gaussian kernels, and simplifies the

Renyi's quadratic entropy as


H2(x) log[- G(x x, a2) (44)
p=l q=l

where the information potential V(x) is defined as V(x) = P1 1 CZ, G(xp

x,, a72). To simplify the cost function, the minimization of entropy is equivalent to

the maximization of the information potential due to the monotone logarithm. Thus

the adaptation is driven by the gradient of the information potential to the MLP







59



histogram of error
5000

4500-

4000-

3500-

3000-

2500-

2000-

1 500-

1 000-

500

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Figure 4-2. Histogram of output error from the well-trained MLP network by MSE.



weight w given by

9V
Aw = -av
9w
1N N (4-5)
~N2 ( 2j2 OW OW
= 9- G. ^pP p, 2ow2
p=l q= 1

However, since the probabilistic density function estimated at each sample point

needs to utilize the whole data set, the computational load of ITL training is of o(NV2).

In our case of MRI training image case of N = 45 x 160 x 128 = 921600 samples,

o(NV2) is not applicable. Thus a recursive entropy estimator, which greatly reduces

the computational load, is defined as



fk+1(x) (1 )f- k(x) + AG(x k1, 2) (4-6)


Then the information potential and its derivative are provided to drive the adaptation


-- 1 k
Vk+1 = (1 A)Vk + A- G(xp Xk+1,a2


p=k-L+l
(47)
1 (1 5) +f A G(x, -Zk+1, 02)(

p= k-L+1


G(x, Xk+1, 2)( W

p= k-L+l






60



Leading curve kernel variance
07 01
009
06
008




> 4 005
03 0045


02 003
002
001

0 50 100 150 0 50 100 150
# of epoch Number of epoch
(a) (b)

Figure 4-3. Adaptive learning performance, (a) The information potential learning
curve, (b) The kernel variance anealing curve.


Thus the computational load of the recursive ITL training greatly reduce to o(NL),

where

In the MRI reconstruction problem, suppose forgetting length L = 20, the com-

putational load is o(NL) = o(921600 x 20) = o(18432000). Considering that the

algorithm needs a large number of epochs (for example, 150 epochs), the matlab code

still takes a long time. Fortunately, matlab provides inferfaces to external routines

written in other languages, called MATLAB Application Program Interface. Specifi-

cally, the C engine routine allows calling the components of a C MEX-file. Thus the

recursive ITL code was written in C and based on an self-defined interface function,

matlab can call this core code in C to greatly save computational time. Though the

ITL code consists lots of matrix computation, this C'\ I I code uses only approxi-

mately 1/20 execution time, which makes the code feasible.

4.4 Results

The performance of the proposed ITL training method is also demonstrated in

phased-array MRI reconstruction using transverse (45 measurements) and coronal (9



























(a) (b)


Figure 4-4. The reconstruction images of the coronal image by (a) ITL training and
(b) MSE training.


measurements) fast collection human neck images collected in a 4-coil MRI system

used in the previous chapter. All the training, testing and desired samples are exactly

the same as being used before.

Fig. 4-3(a) shows the learning process for the normalized information potential

V of the output error vs. number of epochs. The normalized information potential

V remains between (0, 1). It shows how much error remains in the output during

training, where the higher information potential, the lower error. V = 1 denotes a

zero output error ideally which means that the output perfect matches the desire.

In our case, the final V = 0.58 shows a still quite large error after convergence. At

the same time, the kernel size annealing from 0.1 -> 0.005 is shown in Fig. 4-3(b).

Since the Parzen window pdf estimator can be considered as a convolution between

the true pdf and the kernel function, the kernel size annealing can avoid the local

minima trap in the processing of training and finally achieve a solution close to global

optima.










t, it. AM






(a)
BBiZifHIfl
/ ar^MafliD
1 1r 1r^ff t f
B||'I3BII II


D5mmmm


(b)


Figure 4-5. The SNR performance of the reconstruction images of the coronal image
by (a) ITL training and (b) MSE training.
The final reconstruction testing image given from the ITL learning shows a peak
2 dB higher SNR than that of the MLP training using minimizing MSE criterion,
shown in Fig. 4-4&4-5. Then I can conclude that the information of the higher order
moments in the nonlinear training needs to be considered. Actually the ITL training
takes this point into consideration.















CHAPTER 5
UNSUPERVISED LEARNING IN i\llRI TEMPORAL ACTIVATION PATTERN
CLASSIFICATION

5.1 Brief Review of fMRI

The interest in understanding brain functions dates back to several centuries

ago. But it was Gall et. al who argued for the first time that functional modules are

localized at specified regions and correlated to particular tasks [57]. However, it was

only in the last decade that the rapid development of functional magnetic resonance

imaging ( \ I RI) techniques allowed dynamic mapping of the brain processes with fine

spatial resolution. The information between functional brain regions and cognitive

processes has being investigated [7], and the temporal segmented activity demon-

strates functional independence with respect to the localized brain .,111 i nlr [8]. So

far the main methodology in -\ \ II is to segment the activation region in terms of the

temporal response given by the external periodic stimulus. Such stimulus alternates

between task and control conditions giving a supervised baseline for the temporal

response. Plenty of methods have been proposed to address this problem, which can

be roughly categorized into model-based and model-independent. Correlation analy-

sis (CA), as a model-based method [58, 59], combines the subspace modeling of the

hemodynamic response and the use of the spatial information to analyze :\ I RI series.

However, the model-based methods are not effective in neuronal pattern analysis when

the temporal information is not available. Thus various model-independent methods

were proposed, including principal component analysis (PCA) [60, 61], independent

component analysis (ICA) [62, 63] and clustering methods [64, 65, 66] to quantify the

S\! RI responses.









However the challenge remains in localizing brain function when there is no a

priori knowledge available about the time window in which a stimulus may elicit a re-

sponse [67, 68]. In such cases there's no timing for the brain response, so conventional

segmentation with stimulus is impossible. In addition, the \ I RI signal is subject to

high level of noise, especially for non-repeatable physiological events or relatively long

events (compared to cognitive processes) in the brain, such as those following eating

and drinking. A temporal clustering analysis (TCA) method was proposed to reveal

the brain response following eating [9]. This is a space-time methodology that tries to

bridge the gap between spatial localizations and temporal responses. However, TCA

can still be improved and its performance is hampered by several assumptions that

are not necessarily satisfied by cognitive signals measured by \ IllI. New methods are

required for dealing with these challenges.

Subspace projection method as used in the image deconvolution seems to be suit-

able for this task. They have the advantage of data compression and noise cancelation

and are widely implemented in image processing, such as image compression [69], hy-

perspectral image classification [70], etc. The optimal linear subspace projection in

terms of preserving energy is the well known principal component analysis (PCA) [71].

However, in many cases the global PCA is not optimal in particular when the data dis-

tributions are far from Gaussian. Competitive learning is known for its powerful local

feature extraction as demonstrated in the later chapters of this dissertation. It can

also be applied in unsupervised mode as in vector quantization combined with PCA

[72, 73]. Haykin et al. proposed the OIAL (optimally integrated adaptive learning)

method, which gives smaller MSE and higher compression ratio [74]. However, OIAL

doesn't take the bias among models into consideration, which leads to sub-optimal

results. Fancourt et al. combined the mixture of experts and PCA into a cooperative

network to segment time series and images [75]. Both methods are sensitive to the

initial condition especially when input is in a high dimensional space. The SOM










method proposed by Kohonen addressed a soft competition scheme to adapt not only

the activation PE but also its neighborhood [41], which is a good way to solve the

initial condition problem. We incorporate this idea into subspace projections, named

competitive subspace projection (CSP) method, to represent data optimally not only

in terms of local projection axes but also local cluster centroids. This methodology

introduces for the first time the competitive learning into \ I RI image processing. The

advantage of this method lies in the fact that it doesn't need any prior information of

time course segmentation, since it is self-organizing. The unsupervised vector space

representation optimally clusters vector of time series, which gives optimal spatial

task-oriented segmentation. This segmentation is uncorrelated with image content

and has a good noise rejection performance.

5.2 Unsupervised Competitive Learning in fMRI

5.2.1 Temporal Clustering Analysis (TCA)

The CSP methodology will be compared with temporal clustering analysis (TCA)

[9]. TCA effectively extracts the statistical properties from a 3-dimensional data

space (the 2-dimensional spatial image plus the time dimension) and forms a prob-

abilistic sequence over time where each element Nmax (t) of the sequence represents

the number of pixels which reach maximum value throughout the time series. Given

the \ I RI image of size M x N at discrete time t, where t = 1, L, and the pixel

value pij(t) at instant t with i =1,. M and j = 1,..., N, the temporal maxima

response Nmax(t) can be written as
M N
NIx((t)=f>p(t)) (5-1)
i=1 j=1

where f(pi,j(t)) = 1 if p~j((t) > pi,j(t*),Vt*,t* / t; 0 otherwise. This method im-

plicitly assigns probability P(i,j,t) = 1 to pixel (i,j) at the peak time of t, while

it assigns probability P(i,j,t) = 0 for all other time instants. Next, f(p~,(t))


f(pij(t))









at each pixel at each time instant is summed to obtain the temporal maxima re-

sponse Nmax (t) -= i I,, P(i,j,t). This quantity is a measure of grouping ac-

tivation(possibly due to a common cause) since it assumes that functional response

happens not in a separate voxel but in a group of voxels. Such group of voxels can be

distinguished by the temporal maxima response due to their similar temporal peaks.

This method has been successfully applied to mapping the brain activities fol-

lowing glucose ingestion. It provides a deterministic analytical solution with straight-

forward computations. However, it has limitation. Firstly, it is suitable for event-

related \ I RI function localization where the response is demonstrated only in short

time peaks. It is not suitable for other task-related \! RI problems. Secondly, it is

affected by impulsive noise and outliers modeling false temporal maxima, thus yield

wrong estimates for the response time and region.

5.2.2 Nonnegative Matrix Factorization (NMF)

NMF is a procedure to decompose a non-negative data matrix into the product

of two non-negative matrices: bases and encoding coefficients. The nonnegativity

constraint leads to a sparse representation, since only additive, not subtractive, com-

binations of the bases are allowed [76]. A K x L nonnegative data matrix S, where

each column is a sample vector, can be approximated by NMF as


S =WH + E (5-2)

where E is the error and W and H have dimensions K x R and R x L, respectively.

W consists of a set of R basis vectors, while each row of H contains the encoding

coefficients for each basis. The number of bases is selected to satisfy Rx (K+L) < KL

so that the number of equations exceeds that of the unknowns.

The key point of applying NMF to \ Il I is to map the i\!RI images to the

factorization matrix S for which the product W and H corresponds to spatial and

temporal events that make sense hypersiologically. S is a huge matrix where each










column of S is a vectorized 2D spatial \ll RI image of dimension K = MN, and

the number of columns represents the number of image samples along the discrete

time axis. Given the factorization in Eqn. (5-2), each basis function w,, which is

the rth column of W where r = 1, R, is considered to be a vectorized 2D local

feature image of dimension MN; the corresponding vector hr, which is the rth row of

H, codes the intensity and the timing of the activation for the corresponding basis

image wr in the reconstruction of the NMF approximation. If the encoding vector hr

demonstrates sparsity, i.e., if it peaks occasionally, these peaks might be correlated

with the response time (to the stimulus). In addition, the corresponding basis images

will also highlight the spatial details of the response of the brain to the particular

stimuli. Thus, the decomposition of S into W and H jointly provides the answer to

when and where functional regions act.

The decomposition of S into W and H can be determined by optimizing an

error function between the original data matrix and the decomposition. Two possible

cost functions used in the literature are the Frobenius norm of the error matrix

I S- WH| 1| and the Kullback-Leibler divergence DKL (SI I WH). The nonnegativity

constraint can be satisfied by using multiplicative update rules discussed in [77] to

minimize these cost functions. In this dissertation, we will employ the Frobenius norm

measure, for which the multiplicative update rules that converge are given below,

H ,j(k + 1) = H, (k) (WTS)j
(WTWH) (53)
(SHT)i,,
Wi,k(k + 1) W,,(k) (SHT)i
(WHH T) i,

where Aa,b denotes the element of matrix A at ath row and bth column. It has been

proven in [77] that the Frobenius norm cost function is nonincreasing under this

update rule.









5.2.3 Autoassociative Network for Subspace Projection

Subspace projection is widely implemented in signal processing applications such

as data compression and noise cancellation. The goal of this method is to map data

from a higher dimensional space to a lower dimensional space while the in i" features

of the data are preserved. Given a N-dimensional vector x and a projection matrix

W = [wi, i v w r] (M < N for data compression), the projected vector y is written

as


y = WT (5-4)

where the basis vector wi is orthonormal to each other. The optimal linear sub-

space projection in terms of the second order moment is principal component analy-

sis (PCA). It preserves maximum variance of the projected random variable (named

principal components) with the constraint of orthogonal axes. PCA solution can be

achieved by singular value decomposition (SVD). The orthonormal weight matrix

W can be also estimated by an unsupervised Hebbian learning strategy known as

generalized Hebbian algorithm (GHA) and adaptive principal components extrac-

tion (APEX) [71].

From another perspective, PCA can be considered as minimizing the reconstruc-

tion mean square error (\!SE;) by a constrained linear projection. Thus Hebbian

learning is equivalent to an autoassociative network, as shown in Fig. 5-1 [78]. The

output of the hidden -1v-r is the projected random variable y and the desired re-

sponse is nothing but the original data itself x. Minimizing MSE between x and the

reconstruction WWTx allows LMS (least mean square) adaptation for weight ma-

trix W. Thus an unsupervised model is equivalently sovled by a supervised learning

scheme, which is computationally attractive.
























Figure 5-1. Block diagram of autoassociative network.


5.2.4 Optimally Integrated Adaptive Learning (OIAL)

As we know, principal component n i, lJi ; (PCA) maximizes the variance in the

rotated space. If the data is naturally modeled by a Gaussian distribution, PCA

optimally represents the data structure in terms of minimum MSE between input

and reconstructed projections. However, most real data, such as images, is not well

modeled by a single Gaussian distribution. PCA is not optimal in this case. A

productive alternative is to project each cluster by a projection network where the

network parameters are determined locally by the clustered data. This subspace pro-

jection is superior to other classification methods based on minimizing the distance

between input and cluster centers, such as LBG and k-means algorithm, because it

preserves the input structure better and has less effect of the vector scale ambiguity

problem. To achieve this classification goal, competition or cooperation among mul-

tiple expert networks is needed. Haykin proposed an optimally integrated adaptive

learning (OIAL) method incorporating competition among each PCA network using

generalized Hebbian adaptation [78]. The routine of this method is listed as the fol-

lowing:

1. Initialize K transform matrices W1, WK.

2. For each training input vector x,









a) classify the vector based on the subspace classifier


x E Ci, if Pixll = max>,11PjXzl (5-5)

where Pi = WfWi,and

b) update transform matrix Wi according to


Wi Wi + aZ(x, Wi) (5-6)


where a is a learning parameter, and Z(x, Wi) is a learning rule that converges to

the M principal components of {x x E CE}.

3. Repeat for each training vector until the transformation converges.


5.2.5 Competitive Subspace Projection (CSP)

The OIAL method is optimal in terms of MSE only if each cluster approxi-

mates the same cluster centroid. However, complex data structure generally doesn't

conforms to this strict condition. In order to conquer this difficulty, we propose

a strategy, competitive subspace projection (CSP), to cluster data using subspace

projection while explaining the different centroids in each clusters. The adaptation

inside the network switches from initial soft competition to final hard competition.

The block diagram of this CSP network is shown in Fig. 5-2. It consists of multi-

ple (K) autoassociative networks corresponding to K patterns to be classified. When

a input vector x enters the system, the K experts compete in terms of MSE between

input x and the reconstruction x. The winning expert is chosen based on a specific

minimum MSE criterion. The winning expert and its neighborhood are adapted us-

ing LMS with the reconstruction x as the desired response given each x. After the

adaptation for the whole CSP network converges, the input data is classified to K

patterns corresponding to K autoassociative networks.































Figure 5-2. The block diagram of competitive subspace projection methodology.


5.2.5.1 hard competition

The competition strategy is denoted as hard competition if only a single ex-

pert with the least MSE is chosen as the winner. The network architecture and

its optimization methodology with hard competition are described as follows. Each

autoassociative network explains one cluster centroid by introducing bias either to

the hidden l1-,-r or output lv.-r, while the bias is simultaneously adapted with the

projection matrices. The bias on the output l-1v,-r is preferred due to its mathe-

matical simplicity. Thus the cost function J(W, b) for each expert consists of two

partsJ(W, b) = J1(W, b) AJ2(W), where J1(W, b) defines the cost function sur-

face and J2(W) is the orthogonality constraint with weighting factor A. The two









items J1(W, b) and J2(W) are given by

J1(W, b) =-\ x 2

-(||aX||2 2xTWWTx + xTWWTWWTx
2
+ 2xTWWTb 2xTb + b b) (5-7)
M M
J2(W) = 2w [ww|]
i= 1 j 1, i

where x is the input vector containing neighborhood pixels, x = Wy+b = WWTx+

b is the reconstruction, y = WTx is the projection vector which has less dimension

than that of x and wi is the ith column vector of matrix W. Based on the ma-

trix lemma of Oaaa/aw = 2J(a, w)a where Jij(a, w) = aajl/wi, the adaptation

criterion at the nth iteration is written as

Aw,(n) =r][yi(n)(x(n) _x(n)) + x(n)(w(n)x(n) y (n))
M
+ A(2wi(n) P(w, wi)wj(n))] (5-8)


Ab(n) =ri[x(n) x(n)]

where P(wi,wj) is M x M matrix with Pr,s(wi,wj) = 0 if r / s;

Pr,s(wi, wj) = sign(iwi(r) wj(r)) if r = s. sign(.) is a sign function and

r,s=0,--- ,M -1.

We notice that the length of the projection vector y is less than the vector length

of x due to its subspace dimension compression. This dimension reduction results in

the error between the input x and the reconstruction x, where it is the error which

drives the adaptation. A full space projection in y causes error zero and makes no

sense for reconstruction.

5.2.5.2 soft competition

The initial condition for hard competition is a tough issue. Several options in

choosing the initial W and b are the following:


nitial W and b are the following:









1. Choose small random variables.

2. Use global eigenvectors plus small random perturbations added to each class.

3. Arbitrarily divide the data into K classes, estimate the largest L eigenvectors.

4. Arbitrarily divide the data into K classes, estimate the smallest L eigenvectors.

However, in high dimensional space, none of the above estimation for the initial

condition assures convergence for all patterns.

In the competitive strategy, the performance of hard competition depends on

how well data fits the input space. If data only covers part of space with a certain

structure, initial W and b for some models may stay far from the data structure. Thus

these model weights will not be able to win adaptation and lead to null models. In an

extreme case where a specific model alv--, wins adaptation, no competition strategy

is applied. Since the number of samples needed to fit input space exponentially

increases with dimension, high dimensional data is likely to have an initial condition

problem.

Soft competition is an alternative to solve the initial condition problem. In

soft competition, not only the winning model but also its neighboring models are

adapted. Here we are not interested in preserving the topology mapping like SOM

since subspace projection is able to preserve the complex structure of data while the

interest lies at the adaptation robustness soft competition provides. The adaptation

methodology consists of two independent phases in soft competition. Robustness is

achieved in the first phase, which deals with the topological ordering of the weights

and drive all model weights spatially close to data. The second phase is a convergence

phase. It finally tunes the model weights to the local structure of input with a much

smaller step size compared with that in the first phase.

Each model adaptation is modified by a Gaussian weighting function

Aw,(n) -= rAin)[yi (n)(x(n) x(n)) + x(n)(wi(n)Tx(n) yi(n))]
(5 9)
Ab(n) = nAi(n)[x(n} x(n})









where Ai(n) is the weighting function of the ith model, which has a general form of


An() = exp(- ) (5-10)
(510)

where di(n) is a distance measure showing how close the ith model fits the local

cluster and o(n)2 is the kernel width. In order to derive the proper di(n) and o(n)2

in competitive subspace projection, a few criterions needs to be satisfied:

1. In each adaptation, the winning model with least MSE should get the largest

adaptation step while the other model adaptations depend on how well they fit the

input.

2. In the first phase of training, the weighting function should be controlled in a

given dynamic range such that all models are robustly adapted independent of data

structure.

3. In the second phase of training, the weighting function should finally approximately

shrink to a delta function centered at the winning model to achieve the final winner-

take-all fashion.

Based on the listed criteria, di(n) and o(n)2 are such that the weighting function is

given by

(ei(n) e .(u))>
Ai(n) = exp(- ((n) C ))) (5-11)
I ko0 f ((ek( ) Ci*(n))2, (eCk() ei**(n))2)

where the reconstruction error in model i, the winning model i* and nearest neighbor

to the winning model i** are ei(n) = i(n) x(n), e.(n) = xi.(n) x(n) and

ei**(n) = xi*.(n)-x(n), respectively; f(.) is a nonlinear truncation function to control

the extreme large ei(n) for stable convergence; 1 represents a scalar proportional to

the epoch index. If i = i*, Ai(n) = 1 gives the largest step size; otherwise in the

first epoch where k = 1, Ai(n) is ah--v- in the range of [0.3679, 1], where assures

the tuning of the neighboring in the first phase of training. In the second phase of

training, a large number of epochs are needed for fine-tuning the input. Thus I will









finally approach a large integer and exponentially shrink the weighting function to

approximate a delta function.

5.2.6 Algorithm Analysis

The advantage of the competitive subspace projection method lies in three as-

pects. First, adapting bias simultaneously with the projection axis gives optimal

data representation. Without the bias in the multiple autoassociators, the com-

petition gives the same clustering as OIAL does. This kind of clustering actually

separates the space into multiple cones with vertices at the origin. This kind of

clustering neglects varied cluster locations while the proposed CSP methodology re-

gards the cluster as a combination of its spatial location and its shape linearly repre-

sented by the projected axes. Furthermore, unlike some other methods as local PCA

[73] which treats finding spatial locations and shapes of clusters as two independent

processes, CSP couples the two aspects of cluster representation and adapts them

simultaneously to optimally represent data space. Secondly, this subspace projec-

tion method perform noise suppression. Finally, this method trains the competitive

system by supervised training instead of unsupervised training. This is done by es-

timating the desired response with the autoassociator. Therefore the computational

load is greatly reduced.

There exist three issues which needs further discussion. The first states that ex-

plicit orthogonality is constrained to the projection weight W. Without an orthogo-

nality constraint, the data can still be projected into a subspace while the projection

efficiency is not guaranteed. Take W for one expert for example, any new weight

matrix W = WR rotated by R where R = R-1 satisfies the same projection error

for that expert. Thus this is equivalent for data inside this cluster. However, some

data outside this expert may follow W other than W to cause misclassification.

The second issue is the weighting function Ai. This weighting function is derived

by the three basic soft competition criterions logically. Although it is specified to CSP,









Ai itself can be generalized to other unsupervised learning problems. It is a comprise

between the softmax activation function in fuzzy clustering and the usually used

Gaussian kernel in self-organizing map (SOM). On one hand, the proposed weighting

function shares the advantage of fuzzy clustering where all the soft competition is

determined by the whole cluster center statistics. This information is more accurate

than the Gaussian kernel in SOM. On the other hand, it incorporates the truncation

nonlinear function into Ai which maintains the flexibility of hard competition as SOM

does after shrinking.

Another issue is the scale ambiguity for inputs. As is known, subspace projection

is independent of the norm while any Euclidean distance clustering methods, e.g.

LBG and K-means, take norm into account. This proposed competitive subspace

projection method is a mixture of projection method and Euclidean distance methods

and thus has scale ambiguity for inputs. A cluster is defined by its local structure,

which means how the local data is grouped. Its center, projection axes and projected

variance are the linear representation of cluster itself which linearly determine the

structure and shape of cluster. Thus, a scalar multiplication can be considered as

a center shift and expanded or shrinking projected variance with the same axes. If

the center is shifted (original center is nonzero), this scalar multiplication should

generate a new cluster while it is arguing if the center is unchanged (original center

is zero). Thus the competitive subspace projection which takes the norm jointly with

the subspace projection is reasonable.

5.2.7 fMRI Application with Competitive Subspace Projection

We are interested in using this proposed method to detect the functional re-

gions in I\ RI brain images. The purpose is to detect when and where the response

takes effect inside the brain after stimulus. There are multiple sampling time instants

where each sampling time corresponds to a 2D brain image with the same size. From

another aspect, each pixel in the 2D brain image has its time response, which reflects

















01 A Oz.





-02 .
S*.


-02-

-03-

04 -03 -02 -01 0 01 02 03 04 05


(a)


.03
02 o


0 &a o 0


C luster
X02X A Cluster
-0 3 \ -----


-04 -03 -02 -01 0 01
Dimesion 1

(c)


02 03 04 05


02
01


0
-02-1
-2 cluster

-03- Cluster4

-04 -03 -02 -0.1 0 01 02 03 04 0:
Dimension 1

(b)





01



01


jCO + Cltr3


04 -03 02 01 0 01 02 03 04 05


(d)


Figure 5-3. Three dimensional synthetic data, (a) projected to its first and second

dimension, where the third dimension is insignificant in classification (b)

clustering data in (a) by k-means, (c) clustering data in (a) by optimally

integrated adaptive learning (OIAL), (d) clustering data in (a) by com-

petitive subspace projection (CSP). The intersected lines in (c) and (d)

represent the two projection axes for each cluster.



the change of its pixel intensity through time. Some pixels in the background noise


and part inside the brain may not respond to the stimulus and thus their intensities


fluctuates in a small dynamic range due to scanning noise. Other pixels may re-


spond to it with delay and demonstrate a certain time course structure in time. The


classification using the proposed competitive subspace projection can segment the


different patterns based on this time structure difference among pixels. The patterns


which have specific time structures give information of spatial location and temporal


response peak of a given functional effect.









5.3 Results

Clustering Performance Comparison: The synthetic data used to demon-

strate the clustering performance is generated with complex structure. The data

which is three dimensional is projected to its first and second dimensions for classi-

fication since the third dimension is insignificant in segmentation (Fig. 5-3(a)). The

data structure consists of four natural clusters where their shapes are two approxi-

mate rectangles, one circle and one ellipse. The two rectangle clusters approximate

the circle cluster from different directions while the ellipse cluster t i,-; compara-

tively far from the other three. Three clustering methods are compared based on

the synthetic data, which are kmeans, optimally integrated adaptive learning (OIAL)

and competitive subspace projection (CSP). It is shown in Fig. 5-3(b) that kmeans

groups data well except that it misclassifies some samples inside its neighboring end

side of the two rectangle clusters. The reason is that k-means cannot reflect complex

clustering boundary in classification. Fig. 5-3(c) reveals that the segmentation which

OIAL does conforms to separating the input space into multiple cones with the ver-

tex at the origin due to the fact that the in-between cluster centroid distances are

not considered. We can see that CSP gives a reasonable clustering fitting its natural

structure in terms of subspace projection (Fig. 5-3(d)). Here the preserved subspace

dimension is two while the projection axes to the first and second dimensions of the

synthetic input are demonstrated.

Task Detection: The I\! RI brain images detecting task-related effects are

collected on eight human volunteers using a 3 Tesla MRI scanner at the UF. A

gradient-echo echo-planar imaging (EPI) pulse sequence was used with the following

scan parameters: TR/TE/FA = 6s/30ms/900, Field of View = 240mm, matrix size

= 64x64 with an in-plane resolution of 1.875 x 1.875 mm2 and a single slice (3.5 mm

thick). The functional images consist of 750 samples in total.



















LU -
U) 0.74- --


0.72-


0.7 -


0.68 1 1
0 50 100 150 200 250 300
Number of epochs

Figure 5-4. The learning curve in the second phase of training from competitive
subspace projection for M = 1,2, 3 (The mean square error (\MS I) is
normalized by the input signal power).


The functional images form a 3D matrix of size 64 x 64 x 750. Then each pixel

vector is preprocessed with mean-removal from its pixel intensity with normalized

standard deviation (mean-removed along time for each pixel independently). The

mean-removal along time series for each pixel eliminates the in-between correlation

due to image statistics, which leads to an independent time responses for pixel inten-

sities. The normalization process reduces the effect of the scales of the vector norm

and further avoids divergence in training phase.

The preprocessed pixel vector xij at spatial location [i, j] is taken as the input of

competitive subspace network with unsupervised learning with A = 0.01 and the trun-

cation function f(a, b) = 5b if a > 5b and a otherwise. The soft competition adapts

the winning model and its neighborhood models with different step size. There's two

phases in training. The first phase uses a large step size (T71 = 0.02) and small number

of epochs (epoch number = 10) to train the model weights with MSE criterion. In










The first projection axis
0.04-..... I
Model 1
0.02 Model 2
***** Model 3

0-

-0.02-

-0.04
0 100 200 300 400 500 600 700 800

The second projection axis
0.3
Model 1
0.2-- Model 2
****** Model 3





-0.1
0 100 200 300 400 500 600 700 800
Number of samples

Figure 5-5. The projection axes for the number of the projection axes M = 2 and
model number K = 3 after the second phase training of competitive
subspace projection is completed.


the first few epochs, the weighting function Aj(n) in soft competition assures every

model to be adapted to avoid null models. Thus the cluster weights are topologically

ordered around the range of input vector localization. This soft competition stage

gives an approximate clustering estimate which is softly specified by the data itself.

After the training in the first stage converged, a smooth competition strategy is used

with a small step size (T2 = 0.005) and a much larger epoch number 300 are used to

slowly train the weights to precisely capture the input structure. The desired pattern

in spatial and temporal domain is achieved after final convergence.

The first problem in training is to determine how to choose the minimum num-

ber of hidden 1-.r M in autoassociator without sacrificing performance. This is

equivalent to find how many minimum projection axes are needed to extract the time









structure. As we know, the \IllI image series xij form a vector space of high di-

mension T. C'!IL-I ii3g xij is dividing the space into multiple patterns, where each

pattern resembles a stimulated temporal response plus noise or only noise in inactive

regions (the preprocessing excludes image content interference). This is determined

by the additive noise level of data. If the noise level is pretty high, noise may be dom-

inant in main axes and the effective time structure may have to be extracted from

the second, or even more insignificant axis. Fig. 5-4 demonstrates how the learning

curve in the precise second phase of training is affected by the number of projection

axis M. It shows that the final mean-square error (\! I1;) reduces 5.7'. and only 1.5'.

when M increases from 1 to 2 and from 2 to 3 respectively. We can conclude that the

clustered time information can be well represented by using the first two axes. This

is also illustrated in Fig. 5-5. The useful time structure is exhibited in the second

axes while the first axis is chosen to represent noise. Thus only two projection axis

M = 2 is enough for competition in this case. Besides, in cases of the time structure

is dramatically dominated by noise, the two projection axes are still preferred for

algorithm stability.

Another important question is how to determine the number of models K needed

for competition. One pattern is needed for inactive pixels and at least one more

pattern for the task activated pixels. However, we don't know how many patterns

are stimulated in advance. We have to predetermine a model number K, K = 4,

to check if any models inside are actually subdivided from natural one cluster. The

cluster centroids of K = 4 models are shown in Fig. 5-6. It is demonstrated that

the cluster centroids from model 1 and 2 overlap somewhat, which means that the

model 1 and 2 should come from one natural cluster and could be combined. Thus

only three models are necessary for this task.

The purpose of the task used in our \ IRI study is to identify the neural correlates

underlying eye-blinking and apply our new analysis approaches to dissociate neuronal










Cluster centroids
0.25
... Model 1
Model 2
I -- Model3
0.2 ,.,, Model 4
I t
!t
0.15 -
11 1
ii I ii

0.1 ii
II !









-0.1 I
0 100 200 300 400 500 600 700 800
Number of samples

Figure 5-6. The cluster centroids for model number K = 4 and projection axes M =
2.


response from the response induced by motion (i.e. the eye-blinking per se) at the

eyes and by the physiological background (i.e, the periodic cardiovascular effects).

From a methodological point of view, the purpose of the task design and the analysis

procedure (i.e., I was blind to the timing of the protocol, in other words, I am not

aware of the exact time when the subject was performing the eye-blinking task) is to

identify a time window for the response when there is no a priori knowledge about

the timing of task on-set.

First, according to our results, there are two peaks found, which are consistent

with the actually recorded protocol (blind to me). So all thses methods seem to be

able to detect time windows but our methods are more sensitive than TCA. Finally,

the localization detection derived from the temporal clustering analysis (TCA), non-

negative matrix factorization (NMF), competitive subspace projection (CSP) meth-

ods are compared. The nonnegative matrix factorization method with four bases









Cluster centroids


-0.1'
0 100 200 300 400 500 600 700 800
Number of samples


Figure 5-7. The cluster centroids for model number K = 3 and projection axes M =
2.

gives the time series classification and their corresponding spatial classification in

Fig. 5-8& 5-9 while the competitive subspace projection method with K = 3 and

M = 2 gives the time series classification in Fig. 5-7 and their corresponding spatial

classification in Fig. a-11. It shows that both methods localizes two activated re-

gions. One region is around eyes where the task of blinking at two separate instants

corresponds to the two peaks in its cluster centroid at sample 136 and 366. This is

close to the result given by the TCA method (sample index 126 and 367) shown in

Fig 5-10. However, TCA can't localize the blinking task focusing on the eye region

as NMF and CSP does, shown in Fig. 5-11. These outliners in TCA are due to the

noise inference. It demonstrates the proposed method has more robust to noise rejec-

tion performance. Another region located by NMF and CSP is at the center cortical

region with periodic oscillation while TCA completely ignores it.


completely ignores it.































Figure 5-8. The four basis images (1-2 upper row and 3-4 lower row from left to
right) are determined by NMF using real -l\ li data.


Our results further s-l--:- -1 that the localization based on TCA may not be sen-

sitive enough, even for detecting the strong motion-related changes in \ I RI signals.

However, we failed to detect any cortical activation using either method but instead

our new methods detected both the response induced by blinking per se at the eyes

and response induced by physiological background noise, i.e., the periodic cardiovas-

cular effects in the central spinal fluid (CSF). But our methods are still better in the

localization of changes in i\ IRI signal than TCA. The reason why we did not detect

any blinking-related cortical activation may be lying in the fact that our methods

are not sensitive enough to detect small BOLD response overshadowed by the larger

non-BOLD effects induced by both motion and periodic physiological noises. A fur-

ther step is to refine our method based on a filtering procedure, so as to remove noise

components first defined and then to detect the BOLD response.










0.1

0.00

0 100 200 300 400 500 600 700 800
0.1

0.05


300 400 500
Sampling time


Figure 5-9. The encoding time series corresponds to four basis images by NMF using
real \ I RI data.


5.4 Discussion

The proposed competitive subspace projection provides an optimal space repre-

sentation in terms of MSE. The contribution of this proposed CSP method is three-

fold. First of all, CSP gives a more comprehensive view of dat gilustering features,

where a cluster is determined by its location (cluster centroid) and its shape (mul-

tiple orthogonal projection axes). Such view in nonparametric vector quantization

leads to the proposed CSP. It is equivalent to merge the traditional K-means and

OIAL method together. Secondly, The two sides of cluster features are not separate

in adaptation. They are dynamically coupled and adjusted in the adaptation to en-

sure the feature integrity. Finally, the proposed soft competition strategy shares the

advantages of both fuzzy clustering and SOM which uses the global cluster statistics

for local learning while assure final convergence to hard partition.

Functional MRI analysis provides a valuable tool for understanding brain activ-

ity in response to external stimuli. In this dissertation, we firstly incorporate soft

competition as a tool for extracting temporal and spatial activations in sequences


spatial activations in sequences