Statistical tomographic image reconstruction metho

StatisticalTomographicImageReconstructionMethodsforRandoms-PrecorrectedPETMeasurementsbyMehmetYavuzAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophy(ElectricalEngineering:Systems)inTheUniversityofMichigan2000DoctoralCommittee:AssociateProfessorJereyA.Fessler,ChairProfessorAlfredHeroProfessorW.LeslieRogersProfessorAndrewE.YagleThisversionisformattedsinglespacedtosavepaperwhenprinting.Itisnottheocialarchivedversion.ABSTRACTStatisticalTomographicImageReconstructionMethodsforRandoms-PrecorrectedPETMeasurementsbyMehmetYavuzChair:JereyA.FesslerMedicalimagingsystemssuchaspositronemissiontomography(PET)andelectron-icallycollimatedsinglepositronemissiontomography(SPECT)recordparticleemissioneventsbasedontimingcoincidences.Thesesystemsrecordaccidentalcoincidence(AC)eventssimultaneouslywiththetruecoincidenceevents.Similarlyinlowlight-levelimag-ing,thermoelectronsgeneratedbyphotodetectorareindistinguishablefromphotoelectronsgeneratedbyphoto-conversion,andtheireectissimilartotheACevents.DuringPETemissionscans,accidentalcoincidence(AC)eventsoccurwhenphotonsthatoriginatefromseparatepositron-electronannihilationsaremistakenlyrecordedashavingarisenfromthesameannihilation.InPET,generallyasignicantportionofthecollecteddataconsistsofACeventsthatareaprimarysourceofbackgroundnoise.Also,duringPETtransmissionscans,photonsthatoriginatefromdierenttransmissionsourcescauseACevents.InPET,themeasurementsareusuallypre-correctedforACeventsbyreal-timesubtractionofthedelayedwindowcoincidences.Randomssubtractioncompensatesinmeanforaccidentalcoincidences,butdestroysthePoissonstatistics.Wedevelopstatisticalimagereconstructionmethodsforrandomspre-correctedPETmeasurementsusingpenalizedmaximumlikelihood(ML)estimation.Weintroducetwonewapproximationstothecomplicatedexactlog-likelihoodofthepre-correctedmeasurements:onebasedona\shiftedPoissonmodel,andtheotherbasedonsaddle-pointapproxima-tionstothemeasurementprobabilitymassfunction(pmf).Wecompareestimatorsbasedonthenewmodelstotheconventionaldata-weightedleastsquares(WLS)andconven-tionalmaximumlikelihood(basedontheordinaryPoisson(OP)model)usingexperiments,simulationsandanalyticapproximations.Fortransmissionscans,wedemonstratethattheproposedmethodsavoidthesystematicbiasoftheWLSmethod,andleadtosignicantlylowervariancethantheconventionalOPmethod.Wealsoinvestigatethepropagationofnoisefromthereconstructedattenuationmapsintotheemissionimages.Interestingly,thenoiseimprovementsintheemissionimageswiththenewmethodsareevengreaterthantheimprovementsintheattenuationmapsthemselves.Tocorroboratetheempiricalstudies,wedevelopanalyticalapproximationstothereconstructedimagecovarianceandwealsodevelopanalyticalapproximationsforthepropagationofnoisefromattenuationmapsintothereconstructedemissionimages.Theresultsoftheanalyticapproximationsareshowntobeingoodagreementwiththeexperimentalresultssupportingtheimprovementswiththenewmethods.Similarly,fortheemissionreconstructions,wedemonstratethattheproposedmethodsleadtosignicantlylowervariancethantheconventionalOPmethodandalsoavoidsys-tematicpositivebiasoftheOPmethod.AlthoughtheSPmodelisshowntobeslightlybiasedforemissionscanswithverylowcountrates,thesaddle-pointmodelisfreeofanysystematicbiasandperformsalmostidenticallytotheexactlog-likelihood.Also,weinves-tigatethebias-variancetrade-osofthemodelsin1-Dbyanalyzinghowclosetheyperformtothe\uniformCramer-Raobounds.ThenewmethodsoerimprovedimagereconstructioninPETthroughmorerealisticstatisticalmodeling,yetwithnegligibleincreaseincomputationovertheconventionalOPmethod.cMehmetYavuz2000AllRightsReservedTomywifeSemaiiACKNOWLEDGEMENTSIwouldliketoexpressmydeepestgratitudetomyadvisorProfessorJereyA.Fesslerforhisenlighteningandconstructiveguidancethroughoutmygraduatestudy.Hisunder-standing,encouragementandmoralsupporthelpedmeatallstagesofmygraduatework,andmademyPh.D.researchalivelylearningexperience.IwouldliketothankTUBITAKfortheirnancialsupportwithscholarshipfortherstyearofmygraduatestudy.IwouldalsoliketothanktomyadvisorProfessorJereyFessler,ProfessorLesRogersandNationalInstituteofHealthforsupportingmenanciallywithresearchassistantship.IwouldalsoliketoexpressmygratitudetoProfessorAlfredHero,ProfessorLesRogersandProfessorAndrewYagleforservinginmycommitteeandhelpingmewiththeirideas,NealClinthorneforhishelpfulsuggestions,WebStaymanforhishelpwiththemodiedquadraticpenaltyandmycolleaguesHakanErdogan,WebStayman,SteveTitusandmanyothersforsharingideasandfriendship.Finally,Iwishtothanktomyparents,mybrother,andmydearwifeSemafortheirlovingsupportandencouragement.iiiTABLEOFCONTENTSDEDICATION......................................iiACKNOWLEDGEMENTS..............................iiiLISTOFTABLES...................................viiLISTOFFIGURES..................................viiiLISTOFAPPENDICES...............................xiiiCHAPTERS1Introduction...................................11.1BackgroundandMotivation......................11.2OrganizationoftheThesis.......................41.3OriginalContributio