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Development and experimental implementation of a physical concept for quality assurance of new CT methods [Elektronische Ressource] / Claudia Christine Brunner

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PHYSIK–DEPARTMENTDevelopment and experimentalimplementation of a physical concept forquality assurance of new CT methodsDoktorarbeitvonClaudiaChristineBrunnerTECHNISCHE UNIVERSITÄTMÜNCHENTECHNISCHE UNIVERSITÄT MÜNCHENFachbereich StrahlenphysikDevelopment and experimental implementation of a physical concept for qualityassurance of new CT methodsClaudia Christine BrunnerVollständiger Abdruck der von der Fakultät für Physik der Technischen Universität Münchenzur Erlangung des akademischen Grades einesDoktors der Naturwissenschaftengenehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. Martin ZachariasPrüfer der Dissertation:1.Hon.-Prof. Dr. Herwig G. Paretzke2.Univ.-Prof. Dr. Franz PfeifferDie Dissertation wurde am 24.11.2010 bei der Technischen Universität München eingereichtund durch die Fakultät für Physik am 21.03.2011 angenommen.ContentsZusammenfassung 11. Introduction 31.1. History of Computed Tomography . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Health risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3. Aims of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82. Theoretical background 112.1. Image reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1. The filtered backprojection algorithm . . . . . . . . . . . . . . . . . . 112.1.2. The reconstruction algorithm OPED . . . . . . . . . . . . . . . . . . . 152.2.

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Published 01 January 2011
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Exrait

PHYSIK–DEPARTMENT

andvelopmentDexperimentale

implementationofaphysicalconceptfor

qualityassuranceofnewCTmethods

arbeitDoktor

onv

ClaudiaunnerBristineChr

TECHNISCHE

UNIVERSITÄT

MÜNCHEN

MÜNCHENTÄUNIVERSITTECHNISCHE

ysikStrahlenphachbereichF

Developmentandexperimentalimplementationofaphysicalconceptforquality
assuranceofnewCTmethods

ChristineClaudiaBrunner

VollständigerAbdruckdervonderFakultätfürPhysikderTechnischenUniversitätMünchen
zurErlangungdesakademischenGradeseines

Dissertation.genehmigten

NaturwissenschaftenderDoktors

Vorsitzender:Univ.-Prof.Dr.MartinZacharias
Dissertation:derPrüfer

1.Hon.-Prof.Dr.HerwigG.Paretzke
2.Univ.-Prof.Dr.FranzPfeiffer

DieDissertationwurdeam24.11.2010beiderTechnischenUniversitätMüncheneingereicht
unddurchdieFakultätfürPhysikam21.03.2011angenommen.

Contents

Zusammenfassung

oductionIntr1.1.1.HistoryofComputedTomography........................
1.2.Healthrisks....................................
1.3.Aimsofthiswork.................................

oundkgrbacTheoretical2.2.1.Imagereconstruction...............................
2.1.1.Thefilteredbackprojectionalgorithm..................
2.1.2.ThereconstructionalgorithmOPED...................
2.2.ThenewscanninggeometryCTdOr.......................
2.3.Imagequalityanalysis..............................
2.3.1.Fourierbasedapproach..........................
2.3.2.Image-spacebasedapproach.......................

vicesDegingIma3.3.1.FDAlaboratoryCTsystem............................
3.2.ClinicalCTscanner................................
3.3.C-armdevice...................................
3.4.NewCTdOrtechnology.............................
3.4.1.CTdOrdemonstrator..........................
3.4.2.CTdOrcombinedwiththeC-armdevice................
3.4.3.ImprovedCTdOrcombinedwiththeclinicalCTscanner.......

4.ocessingprData4.1.FDAlaboratorysystem..............................
4.2.ClinicalCTscanner................................
4.2.1.Analysisoftherawdata.........................
4.2.2.Analysisofthereconstructedimages..................
4.3.CTdOrcombinedwiththeC-armdevice....................
4.4.CTdOrcombinedwiththeclinicalCTsystem.................
4.4.1.Imagereconstruction...........................

1

3368

1111111518202225

3131333637373840

4343464652545858

v

vi

Contents

4.4.2.Analysisofthereconstructedimages..................65

69measurementsDose5.5.1.FDAlaboratorysystem..............................69
5.2.CTdOrcombinedwiththeclinicalCTscanner.................71
5.2.1.CalibrationoftheTLDs.........................71
5.2.2.Dosemeasurement............................72

6.Analysisofimagequality77
6.1.FDAlaboratoryCTsystem............................77
6.2.ClinicalCTsystem................................85
6.2.1.Rawdata.................................85
6.2.2.Reconstructedimages..........................88
6.3.CTdOrcombinedwiththeclinicalCTsystem.................97

OutlookandlusionsConc7.

103

A.MathematicsforthereconstructionoftheCTdOrdata107
A.1.CTdOrcombinedwiththeC-armdevice....................107
A.1.1.DataoftheC-armdevice.........................107
A.1.2.DataoftheCTdOrdemonstrator....................110
A.2.CTdOrcombinedwiththeclinicalCTsystem.................111
A.2.1.DataoftheCTdOrdemonstrator....................112
A.2.2.DataoftheclinicalCT..........................113

yliographBib

115

Zusammenfassung

UmderstetigsteigendenStrahlenbelastungdurchdieComputertomographieentgegenzu-
wirken,werden,unteranderem,neuartigeCT-Systemeentwickelt.ObdieseSystemeaber
tatsächlichmitwenigerDosisdiegleicheBildqualitäterzielen,mussmitgeeignetenVerfahren
untersuchtwerden.DeshalbbeschäftigtsichdieseArbeitmitneuenMethodenzurQualität-
sanalysevonCT-BildernundihrerAnwendungaufdieneuentwickelteCT-GeometrieCT
dOr.StandardmäßigwirddieBildqualitätheutemitaufFouriertransformationbasierenden
Methodenausgewertet,beidenendieAuösungmitderModulationsübertragungsfunktionund
dasRauschenmitdemRauschleistungsspektrumbestimmtwird.DabeiderFouriertransfor-
mationAnnahmengemachtwerden,dievondigitalenSystemennichterfülltwerden,wurdein
dieserArbeitzusätzlicheinpixel-basiertesVerfahrenverwendet,dasdieDatenimBildraum
analysiert.NachdemdieserAnsatzbishernuraufdie2-dimensionaleRadiographieangewen-
detwurde,musstenzuerstentsprechendeMethodenzurMessungundDatenverarbeitung
fürCT-BildermithilfebekannterSystemeentwickeltundgetestetwerden.Diesgeschahin
ZusammenarbeitmitderU.S.FoodandDrugAdministration,anderenLabor-CT-Systemdie
erstenUntersuchungendurchgeführtwurden.Außerdemwurdediebildraum-basierteMethode
auchaneinemkonventionellenCT-GerätdesKlinikumsrechtsderIsarderTechnischen
UniversitätMünchengetestetunddieErgebnissewurdenmitdenenderStandardverfahren
glichen.ervAnschließendwurdenbeideVerfahrenaufdieBilderdesCTdOrSystemsangewendet.Dazu
wurdedervorhandeneCTdOrDemonstratorsoumgebaut,dassernuninKombinationmit
einemklinischenCT-Gerätbetriebenwerdenkann.DieDatenverarbeitungwurdeimVergle-
ichzumVorgängerprojektdeutlichverbessertundfürdieKombinationmitCT-Datenoptimiert.
AußerdemwurdeerstmalsdieDosisimCTdOrgemessenundmitderDosisimherkömm-
glichen.ervCTlichenEszeigtesich,dassdurchdieAbschirmungendesDemonstratorsdieDosisimRingumüber
60%reduziertwird.DieBildqualitätbleibtjedochvorallemaufgrunddergeringenSensitivität
undAnzahlderimDemonstratorverwendetenDetektorenhinterdervonherkömmlichenCT-
zurück.GerätenDieAnwendungdesbildraum-basiertenVerfahrensaufdieverschiedenenSystemeergab,
dassderAnsatzeinendeutlichhöherenMess-undRechenaufwandmitsichbringtalsdie
Standardverfahren.DieEigenschaftendesbildgebendenSystemswerdenaberteilweisebesser
beschriebenalsdurchdieFouriermethoden.

1

oductionIntr1.

1.1.HistoryofComputedTomography

OrdinaryX-rayexaminationsoftheheadhadshowntheskullbones,butthebrainhadre-
mainedagray,undifferentiatedfog.Now,suddenly,thefoghadcleared.Withthesewords
ProfessorTorgnyGreitzawarded1979theNobelPrizeinPhysiologyandMedicinetoAllan
McLeodCormackandSirGodfreyHounsfieldfortheinventionofthecomputedtomography
(CT).TheirworkwasbasedonthefindingsofJohannRadon,anAustrianmathematician,that
thedistributionofamaterialormaterialpropertyinanobjectlayercanbecalculatedifthe
integralvaluesalonganynumberoflinespassingthroughthesamelayerareknown[Rad17].
Inthe1960s,AllanMcLeodCormackfromTuftsUniversityinMassachusetts[Cor73]andSir
GodfreyHounsfieldinHayes,UnitedKingdomatEMICentralResearchLaboratories[Hou73]
appliedthistheorytomedicalapplications.HounsfieldwasluckytoworkforEMI,whichmade
itsmoneythosedaysmostlywiththeenormoussuccessoftheBeatles,providingHounsfield
withalmostunlimitedfundingsforhisresearch[Goe10].Therefore,hewasabletobuildthe
firstprototypeofaCTscannerin1971.AnimageofitisshowninFig1.1aincomparisontoa
modernCTscannerinFig.1.1b.
ThetypeofCTscannerpresentedbyHounsfieldin1971iscalledfirstgenerationCT.It
consistedofasinglepencilbeamandasingledetectorelement,whichwererigidlyconnected.

(a)

(b)

Figure1.1:TheveryfirstCTprototypeinventedbyGodfreyHounsfield(a)andamodernCT
scannerforcomparison(b).(Source:Wikipedia)

3

4

oductionIntr1.

Thesourcewastranslatedacrossthepatienttoobtainasetof160parallelprojectionmeasure-
mentsinonedirectionasshownschematicallyinFig.1.2a[Hsi03].Thesource-detectorpair
wasthenrepeatedlyrotatedabout1°andsubsequentsetsofmeasurementswereobtainedduring
eachtranslationpassingthepatient.Inthiswaythescanofasingleslicetookseveralminutes,
whichledtosevereimagequalityproblemsduetopatientmotion.

Figure1.2:Firstgeneration(a)andsecondgeneration(b)CTscannergeometry.

Inordertoreducethedataacquisitiontime,thesecondgenerationofCTscannersuseda
sourcewithanarrowfanbeamandarowofdetectorstosamplethedata.Aschemeofthe
geometryisprovidedinFig.1.2b.Itallowedthecollectionofseveralprojectionsduringone
translationwhatmadethedataacquisitionsignificantlymoreefficientandwithabout20sec-
ondsperslicealsomuchfasterthanthefirstgenerationscanners.
Thethirdgenerationusesawidefanbeamcombinedwithanarcofdetectorsconcentric
tothesource,whichrotatesaroundthepatient(Fig.1.3).Thefanbeamandthesizeofthe
detectorislargeenoughsothattheentirepatientisinthefieldofviewineachprojection
andthetranslationalmovementisthereforeredundant.Theearlymodelsofthethirdgeneration
scannerssufferedfromproblemswithentangledcables,whichrequiredanalternatingclockwise
andcounterclockwiserotationofthegantry.Thesocalledslip-ringssolvedthisproblemand
clearedthewayforspiralCTdevices[Kal90].Nearlyallstate-of-artscannersonthemarket
todayarethirdgenerationspiralCTs.
Fourthgenerationscannerstriedtoovercomethedrawbacksofthethirdgeneration,suchas
detectorinstabilityandaliasing,withanencloseddetectorring,whichremainsstationaryduring
thescan,whilethex-raysourcerotatesaroundthepatient[Hsi03].Thisgeometryhoweversuf-
fersfromsevereproblemswithscatteredradiation,becauseeachdetectorcellreceivesphotons

1.1.HistoryofComputedTomography

Figure1.3:Thirdgenerationscannergeometry

5

overawideanglesothatnoeffectiveandpracticalscatterrejectioncanbeperformedbypost-
patientcollimation.Furthermore,thehugeamountofdetectorsneededtoobtainaresolution
comparabletoathirdgenerationscannerdrasticallyincreasestheprizeoffourthgeneration
scanners.Therefore,insteadofadoptingthefourthgenerationtechnology,thirdgeneration
scannershavebeenconstantlyimproved.
Thez-yingfocalspottechniqueusesaperiodicmotionofthefocalspotinthelongitudinal
directiontodoublethenumberofsimultaneouslyacquiredsliceswiththegoalofimproved
longitudinalresolutionandeliminationofspiralartifacts.Togetherwiththeimplementationof
multislicedetectors,thescanningtimewasdrasticallyreducedsothatthesetechniquesbecame
standardinmodernCTscanners.Onlinetubecurrentmodulation[Kal99,Gre04]decreasedthe
applieddosetothepatientsignificantlybyadjustingthetubecurrenttotheattenuationineach
projection.Dual-sourceCTsystems(DS-CT),workingwithtwotubesandtwocorrespond-
ingarcdetectorswithanangularoffsetof90°,allowedatemporalresolutiondownto83ms
forcoronaryCTangiographyin2006[Flo06].Flat-detectorcomputedtomography(FD-CT)
promisesahigherspatialresolutionthanconventionalarcdetectors,butstillsuffersfromdraw-
backssuchasalowerdoseefficiency,asmallerfieldofview,alowertemporalresolution,and
[Kal07].scatterTocompletetheoverviewaboutCTsystems,theinverse-geometryvolumetricCT,presented
bySchmidtetal.in2004[Schm04],hastobementioned.Itsuggestsacompletedifferent
geometrywithasourcearrayandmultipleatdetectorarraysofthesameaxialextent.Itis
abletoimagethickvolumesinasinglegantryrotationwithisotropicresolutionandwithout

6

oductionIntr1.

cone-beamartifacts[Maz07].Butthoughitisapromisingapproach,theinverse-geometryCT
isworkinprogressandthereforestillhastoproveitsclinicalfeasibility.

risksHealth1.2.

AccordingtotheBundesamtfürStrahlenschutz[BfS08],doseexposurefromnaturalsourcesis
intherangefrom2to3mSvperyearinGermany.Artificialsourcesaddinaverageanother
2.0mSv/atothisvalue,ofwhich1.8mSv/aoriginatealmostcompletelyfrommedicalexami-
nations.ThecompositionoftheaverageradiationexposureoftheGermanpopulationisshown
1.4a.Fig.in

Figure1.4:StatisticsonradiationexposureofanaverageGermancitizen.

Figures1.4bandcshowthatwhileCTimagingrepresentsonly7%ofthemedicalexamina-
tionsusingx-rays,itcontributes56%tothecollectiveeffectivedosefrommedicine.CTscans
representthereforethelargestarticificalcontributiontotheradiationexposureinGermany.For
otherindustrialnationsthecontributionofCTtotheoveralldoseisinthesameorderofmag-
nitudeorevenhigher,asforexamplefortheUSAandJapan[Bri05].ResearchersatEmory

risksHealth1.2.

7

UniversityinAtlantareportedinAugust2009thatexaminationrecordsofnearly1millionUS
citizenscollectedbyinsurancecompaniesshowedthat68%hadatleastoneCTscaninthree
az09].[FyearsTheInternationalCommissiononRadiologicalProtection(ICRP)[ICRP07]reportsthatCT
isincreasinglybeingusedtoreplaceconventionalx-raystudies.Figure1.5showsthatwhilethe
radiationexposurefromallothermedicalapplicationsexceptCTdecreasedinthelastyears,
thedosefromCTconstantlyincreased.

Figure1.5:StatisticsonradiationexposureofanaverageGermancitizen[BfS08].

However,thevaluesmentionedabovearecollectiveeffectivedoses(i.e.,averagedovera
wholepopulation),thedoseforanindividualpatientcanbeseveraltimeshigher.Forexample,
theadulteffectivedosefromaCTexamoftheheadisabout2mSvandthereforeequivalentto
theadulteffectivedosefromroughly100chestx-rayexaminations.Theeffectivedosefroma
CTexamoftheabdomenwithatypicalvalueintherangeof16mSvequalsabout800chest
A10].[FDxaminationsex-rayStochasticeffectscausedbythisamountofradiation,whichforexampleleadtoanincreased
lifetimeriskforcancer,arenotyetfullyunderstood.Whiletherearetheoriesthatlowdoses
mightevendecreasetheriskofdevelopingcancerthroughtheactivationofanadaptiveresponse
thatmayprotectagainstradiationeffects(hormesis)[Fei05],therearealsotheoriesthatclaim
lowdosestobedisproportionatemoredangerous(supra-linear)[Gof92].Thegeneralscien-
tificconsensus,however,iscurrentlythelinear-no-threshold(LNT)hypothesis[Wal06],which
involvesepidemiologicaldataathigherdosestoestablishananchorpoint,andextrapolates
theexcesscancerrisklinearlydownfromthispointtothelowdoseofinterest[Bre06].This
implicatesthateveryradiationexposureisapossibleriskforthepersonconcerned,whatled
totherecommendationoftheICRPandtheAmericanAssociationofPhysicistsinMedicine
(AAPM)tokeepthepatientdoseaslowasreasonableachievable,theso-calledALARAprin-
RoeV02].AAMP08,[ICRP07,ciple

8

Intr1.oduction

However,sincenoimmediateradiationeffectsappearfromtypicalCTscans,theaware-
nessofthepopulationandevenofthephysiciansandmedicalstaffforhealthrisksduetothe
appliedradiationdoseiscommonlystilllow.Thisattitudechangedslightlywhenseveralac-
cidentsinconnectionwithCTwerereportedinfall2009[Bog09].Oneofthemhappenedin
asmallcommunityhospitalinArcata,NorthCalifornia,whereatwoandahalfyearoldboy,
whocomplainedaboutneckpainafterfallingfromhisbed,wasaccidentallysubjectedtoCT
scansformorethananhour.Skinirritationswereanimmediateconsequenceofthisextreme
overexposure,andradiationexpertspredictthatthechildwilldevelopcataractsinthreetofive
years.Additionally,sincechildrenaremuchmoresensitivetoradiationthanadults,thereisan
unpredictableriseintheboyslifecancerrisk.
AnotherseriesofaccidentswasreportedfromthefamousCedars-SinaiMedicalCenterin
LosAngeles,whereover200potentialstrokepatientswereadministereduptoeighttimesthe
regulardose.TheAmericanCollegeofRadiology(ACR)andAmericanSocietyofNeurora-
diology(ASNR)havethereuponreleasedajointstatementcallingforanestablishedsetofCT
scanandradiationdosageprotocols[Bye09].TheNationalInstitutesofHealth(NIH)startedto
recordthemedicalexaminationswithx-raysandthedosethepatientswereexposedtosothat
thenumberoftestspastandtheestimatedcancerriskforthepatientisavailablefortheattend-
ingphysician[Kri09].Itissupposedthatthisinformationmightinuencehisorherdecision
toorderaparticulartypeofexamandpreventthepatientfromunnecessaryimagingprocedures
thathavealreadybeenconducted.
InFebruary2010,theU.S.FoodandDrugAdministration(FDA)announcedaninitiative
toreduceunnecessaryradiationexposurefromthethreegreatestcontributorstototalradiation
exposurefrommedicalimagingprocedures:CT,nuclearmedicinestudies,anduoroscopy
[FDA10].Thegoalistopromotesafeuseofmedicalimagingdevices,supportinformedclini-
caldecisionmaking,andincreasepatientawareness.Inaddition,theFDAandtheCentersfor
MedicareandMedicaidServicesarecollaboratingtoincorporatekeyqualityassuranceprac-
accreditation.mandatorytheintoticesInGermany,theBundesamtfürStrahlenschutztriestoreduceCTdosesbyintroducingref-
erencevaluesforavariousnumberofscans[RoeV02,BfS10].Thesereferencevaluesprovide
themaximumnecessarydoseforavariousnumberofdiagnosticandinterventionalscans.The
updatedversionincludesforthefirsttimefourexaminationsforchildrenatsixdifferentage
bracketsandweightcategoriesrespectively.Institutionswhichdonotmeetthereferencevalues
areadvisedofhowtooptimizetheirprocedures.

1.3.Aimsofthiswork

SincetheriskduetolowdoseexposuresisstillunknownandCTcontributesmoreandmore
totheeffectiveaveragedoseofthepopulation,dosereductioninCTisaveryurgentgoal
ofefficientradiationprotection.Overall,reducingdosetothepatientinCTdependsmainly

1.3.Aimsofthiswork

9

ontwopoints(Fig.1.6):CTtechnologyandqualityassurance.ExistingCTtechnologycanbe
improvedinhardwareaswellasinsoftware,buttherearelimitations.Inventingnewapproaches
ischallengingandverytime-consuming,buthasthepotentialtoovercometheselimitations.
OneoftheseinventionsisthenewreconstructionalgorithmOPED(OrthogonalPolynomial
ExpansionontheDisc)(Sec.2.1.2),whichisnotbasedontheFourierslicetheoremincontrast
totheconventionallyusedfilteredbackprojectionalgorithm(FBP)(Sec.2.1.1).ThenewCT
geometryCTdOr(CTwithDualOptimalReading)(Sec.2.2)takesadvantageofOPEDs
specialcharacteristicsbyintroducingamask,whichworksbothasashieldingandasasecond
detector.Combined,OPEDandCTdOrallowatheoreticaldosereductionofupto50%
whileprovidingthesameimagequalityasconventionalCTsystemsusingFBP.However,these
theoreticalpredictionsstillhavetobeverified.Takingstepstowardsprovingthemistherefore
oneoftheaimsofthiswork.
ThisleadstothesecondmainpointimportantfordosereductioninCT:thequalityassurance.
Inclinicalroutine,thisimplicatescontrollingthestrictimplementationoftheALARAprinciple
intheinvestigationprotocolsandtheavoidanceofunnecessaryinvestigations.Additionally,
qualityassuranceimplicatestheregularcheckoftheCTscanners.CTdevicesaredefinedby
thequalityoftheproducedimagesandthedosewhichhastobeappliedtoobtaintheseimages.
Onlybothparametersincombinationallowameaningfulestimationofthesystemquality.

Figure1.6:Thevariouswaystoreducedoseincomputedtomographyprovidedthemotivation
ork.wthisfor

10

Intr1.oduction

ImagequalityanalysisisconventionallydoneusingtheFourierbasedmethodswhichquan-
tifythespatialresolutionbythemodulationtransferfunction(MTF)andthenoisebythenoise
powerspectrum(NPS)(Sec.2.3.1).However,theFourierapproachmakessomelimitingas-
sumptionsfordigitalsystemssuchasshiftinvarianceandwidesensestationarity,whicharenot
satisfiedbyrealCTsystems.Therefore,theapplicationofanimage-spacebasedapproachto
CTwasdevelopedadditionally,whichdefinesspatialresolutionandnoiseintheobjectspace
(Sec.2.3.2).Sinceitisnotyetanestablishedmethod,theimage-spaceapproachwasnotonly
comparedtotheconventionalFourierapproach,butitsadaptabilityandpracticabilityforCT
andCTdOrwasverifiedtoo.
Indetail,theimage-spaceapproachwasfirstappliedtoalaboratorycone-beamCT,which
allowedtocontrolallparametersofthemeasurementandthereconstruction.Thecomparison
totheresultsoftheFourierapproachrevealedthesimilaritiesanddifferencesofthetwoap-
proaches.ThesubsequentapplicationtoaclinicalCTtestedtheadaptabilityoftheimage-space
approachtosystemswherenoteveryparametercanbecontrolledorisknown.Itfurthermore
demonstratedthepotentialofthetwoapproachesandtheachievableimagequalityofamodern
.scannerCTSincethehardwareofnewCTscannershastobecharacterizedtoo,butitspropertiesare
maskedbythereconstructionalgorithmandappliedfilter,amethodtoanalyzetherawdata
withtheFourierapproachwasdeveloped.ThiswasdonefortheclinicalCT,butanapplication
tonewCTsystemssuchastheCTdOrwouldbestraightforward,becausethespatialresolution
andthenoiseweredeterminedindependentlyofthenumberofacquiredslices.
InordertoanalyzetheimagequalityoftheCTdOr,firsttheexistingdemonstratorwas
improved.ItcannowbecombinedwithaclinicalCTscannersothatforthefirsttimeitcan
berunintheconfigurationitwasoriginallyinventedfor.Inthecourseofthismodification,a
completelynewconceptforimagereconstructionhadtobedeveloped.Comparisonstoimages
reconstructedwiththeoldalgorithmsdemonstratedtheconsiderableimprovementinimage
quality.ThecombinationwiththeclinicalCTandthenewalgorithmsallowedtoapplythepre-
viouslydevelopedmethodsforimageanalysiswiththeimage-spaceapproachandtheFourier
approach.Inordertothoroughlydescribethiscombinedsystem,additionaldosemeasurements
performed.were

oundkgrbacTheoretical2.

Thischapterprovidesanoverviewofthetheoryessentialfortheworkpresentedhere.First
thetwoappliedreconstructionalgorithms,thefilteredbackprojectionalgorithmandOPED,
areshortlypresentedinSec.2.1.ThenthenewscanninggeometryCTdOr,whichiscom-
binedwithOPED,isdescribedinSec.2.2.Finally,twoapproachesforimageanalysis,the
conventionalFourierbasedandanimage-spacebasedapproach,arepresentedandtheirmain
differencesareemphasizedinSec.2.3.

reconstructionegIma2.1.

(2.1)

2.1.1.Thefilteredbackprojectionalgorithm
Thefilteredbackprojection(FBP)algorithmisthemostcommonlyusedreconstructionalgo-
rithm,implementedinalmosteveryconventionalCTscanner.Likeeveryreconstructionalgo-
rithm,itisbasedontheRadontransform,whichprovidesthebasisforimagereconstruction
from1-dimensionalprojections[Kak01].Theobjectisrepresentedbyatwo-dimensionalfunc-
tionf(x,y)andeachlineintegralthroughitbytheparameters(θ,t),wheretisthedistanceof
thelinetotheoriginandθistheanglebetweenthenormalvectortothelineandthex-axis.A
linejthroughtheobjectisthendescribedby
tj=xcosθj+ysinθj,(2.1)
whichleadstothedefinitionofthelineintegralas
Pθ(t)=(θ,t)linef(x,y)ds.(2.2)
TheRadontransformPθ(t)off(x,y)isdefinedbyrewritingthisequationusingadeltafunction
∞∞Pθ(t)=f(x,y)δ(xcosθ+ysinθ−t)dxdy.(2.3)
−∞−∞AprojectionisformedbycombiningaparallelsetoflineintegralsasgivenbyPθ(t)fora
.θconstantTheFourierSliceTheoremstatesthattheFouriertransformofPθ(t)
∞∞F(u,v)=f(x,y)e−j2π(ux+vy)dxdy(2.4)
−∞−∞

(2.4)11

12

oundbackgreticalTheor2.

Figure2.1:TheFourierSliceTheoremrelatestheFouriertransformofaprojectiontoaline
throughtheorigininthefrequencyspace.

isalinethroughtheoriginoftheFourierdomainrotatedbytheangleθ,whereuandvare
frequenciesintheFourierspace.ThisconceptisdepictedinFig.2.1.
Foraninfinitenumberofprojections,F(u,v)isknownforallpointsinthefrequencydomain
andtheobjectf(x,y)canberecoveredbyusingtheinverseFouriertransform:
∞∞f(x,y)=F(u,v)ej2π(ux+vy)dudv.(2.5)
−∞−∞However,inpracticeonlyafinitenumberofprojectionscanbetakensothatF(u,v)isonly
definedalongafinitenumberofradiallines,whichhavetobeinterpolatedorregriddedtoa
system.coordinateCartesianTheFBPalgorithmsolvesthisproblembyapplyingtheweightingorrampfilterπ|w|/K
toeachprojectioninthefrequencydomain,wherewisthedistancefromtheoriginofthe
coordinatesystemandKisthenumberofprojections.Figure2.2schematicallyshowshow
theweightingfunctionworks.Thedataneededperprojectiontoknowallpointsinfrequency
spacearesupposedtobewedge-shaped(a).Theactualmeasureddataarestripes(b),whichget
wedge-shapedbyapplyingtheweightingfunctiononthem(c).Ateachspatialfrequencyw,
thewedgein(c)hasthesamevolumeasthepie-shapedwedgein(a),butitisperpendicularto
theplane.Thustheweightedprojectionsrepresentanapproximation,whichcanbeimproved
asmuchasdesiredbyusinganappropriatenumberofprojections.

econstructionrImage2.1.

13

Figure2.2:Datainthefrequencydomainneededperprojection(a),theactualmeasureddata
(b),andtheweighteddata(c).
Therampfilter|w|hastobeband-limitedinordertoenableaninverseFouriertransformation.
Inotherwords,therehastobezeroenergycontainedoutsidethefrequencyinterval(−Γ,Γ).
Thisisdonebymultiplyingthefilterwiththewindowfunctionb(w):
b(w)=1|w|<Γ(2.6)
.otherwise0Theresulting,band-limitedrampfilterH(w)=|w|b(w)isshowninFig.2.3.Theimpulse
responseh(t)ofthisfilterisgivenbytheinverseFouriertransformofH(w)independenceof
thesamplingintervalτ=(2Γ)−1:
2∞h(t)=H(w)e+j2πwtdw=2τ12sin2π2t/π2t/τ2τ−4τ12sinπt/π2t/τ2τ(2.7)
−∞

Figure2.3:Frequencyrepresentationoftheband-limitedrampfilterH(w)
Sincetheprojectiondataaremeasuredatdiscreteτ,h(t)isonlydefinedforndiscreteposi-
tions:14τ2,n=0
h(nτ)=0,n=even(2.8)
1−n2π2τ2,n=odd.


14

2.oundbackgreticalTheor

Thediscretizedimpulseresponseh(t)oftheidealbackprojectionfilterisshowninFig.2.4.
Duetothenegativevaluesforoddn,noiseinCTimagesreconstructedwiththeFBPalgorithm
isanti-correlated(compareSec.2.3.2andSec.4.2).

Figure2.4:Impulseresponseoftheidealbackprojectionfilter

ThesecondstepoftheFBPalgorithmisaddingtogetherthe2-dimensionalinverseFourier
transformsofeachweightedprojection.Itiscalledbackprojection,becauseitcanbeperceived
asbackprojectingeachfilteredprojectionovertheimageplane.Sinceinpracticeeachprojec-
tionisonlyoffiniteextent,Pθ(kτ)issettozerooutsidetheindexrangek=0,1,...,N−1.
ThefilteredprojectionQθisthereforedefinedas
1−NQθ(nτ)=τh(nτ−kτ)Pθ(kτ),n=0,1,...,N−1.(2.9)
=0kInordertoimplementthefilteringoperationinafastalgorithm,multiplicationinFourierspace
ispreferredtoconvolutioninobjectspace.This,however,leadstotheimplementationofacir-
cularconvolutioninsteadofatheoreticallycorrectaperiodicconvolution.Thisapproximation
resultsinartifacts,knownastheso-calledwrap-aroundeffectorinterperiodinterference.To
reducetheseartifacts,eachprojectioniszero-paddedbeforeapplyingFouriertransformandfil-
teringoperations,meaningthatthebeginningandtheendofthesequencearefilledwithzeros.
Ifthenumberofzerosisatleastequaltothenumberofsamplesintheoriginalprojectionminus
one(N-1),theartifactscanmostlybeprevented[Hsi03].
Thealgorithmdiscussedaboveisderivedforaparallel-beamgeometrynotgiveninmodern
thirdgenerationCTsystems.Theconversionfromfan-beamprojectiondatatoparallel-beam
datarequiresatwo-stepinterpolationprocesscalledrebinning[Hsi03].Thedifferenceinray
densityatthecenterofthefieldofviewcomparedtotheouterregionsis,however,notac-
countedfor,riskingalossofinformationbyperformingrebinning[Tis10].Therefore,other

econstructionrImage2.1.

15

reconstructionalgorithmsweredeveloped,whichusethedatadirectlywithoutrebinning,such
OPED.e.g.,as

OPEDalgorithmreconstructionThe2.1.2.ThereconstructionalgorithmOPEDwasproposedandprovedtobestableandaccuratebyXu
in2006[Xu06a,Xu06b,Xu07a].OPEDisanacronymforOrthogonalPolynomialExpansion
isc.DtheonTheimplementationoftheOPEDalgorithmreferredtohereapproximatestheobjectfunction
f(x,y)asanexpansionofNChebyshevPolynomialsofthesecondkindUk:
Uk(x)=sin[(k+1)θ],x=cosθ.(2.10)
θsinTheChebyshev√polynomialsofthesecondkindareknowntogenerateanorthogonalsystem
withtheweight1−t2ontheinterval[-1,1][Dun01]
π1√Um(t)Un(t)1−t2dt=sin[(m+1)ϕ]sin[(n+1)ϕ]dϕ=2πδm,n,(2.11)
01−wherem,n≥0.
Forfurtherconsiderations,itisnecessarytointroduceridgepolynomials,whichplayan
importantroleinthestudyofRadontransforms[Xu06b].Theyareusedtodefinebivariate
functionsf(x,y;θ)tobeconstantalongtheirso-calledridgesxcosθ+ysinθ=tatafixed
:tf(x,y;θ)=f(xcosθ+ysinθ).
Forconvenience,theChebyshevpolynomialscanbedenotedintermsoftheridgefunction

Uk(xcosθ+ysinθ)≡Uk(x,y;θ)
DefiningB2=(x,y):x2+y2≤1astheunitdiscandfasafunctionsuchasf:B2→R,
then|f(x,y)|2dxdy<∞⇔f∈L2(B2).
B2ForL2(B2),ithasbeenshownthat[Xu06b]
11πUk(x,y;ϕ)Uk(x,y;ϕ)dxdy=k+1Uk(cos(ϕ−θ)).(2.12)
B2Byreducingtheridgeangleθtokequidistantlydistributedpositionsθk,jonacircumference
jπθk,j=k+1,0≤j≤k,(2.13)

16

oundbackgreticalTheor2.

itcanbefoundthatthetrigonometricpolynomialsUk(x,y;θk,j),j=0,...,kareorthonormal:
π1Uk(x,y;θk,j)Uk(x,y;θk,i)dxdy=k+11Uk(cos(θk,j−θk,i))(2.14)
B21sin[(j−i)π]
=k+1sin[(j−i)kπ+1](2.15)
=δj,i.(2.16)
HilberttheorystatesthatalltrigonometricpolynomialsgenerateadensesetofL2(B2)[Dun01],
therefore∞L2(B2)=Pk,
=0kwherePkdenotesthelinearcombinationsofUk(x,y;θk,j).ThisshowsthatthesetofRidge
Chebyshevpolynomialsofthesecondkind
Uk(x,y;θk,j)=Uk(xcosθk,j+ysinθk,j),
where0≤k≤∞and0≤j≤k,generatesanorthonormalbasisinL2(B2).Thisallowsto
representanyfunctionf∈L2(B2)byanexpansionsuchas
f(x,y)=Uk(x,y;θk,j)π1f(x,y)Uk(x,y;θk,j)dxdy,(2.17)
∞k
B=0j=0kwherethedoubleintegralcorrespondstotheprojectionofthefunctionoverthecorresponding
basisvectorUk(x,y;θk,j).Inordertoapproximatetheobjectfunctionf(x,y),OPEDusesN
RadonProjectionsRf(φν,t)obtainedattheequidistantviewangles
φν=2Nπν,ν=0,...,N−1(2.18)
andatadistancetfromthecenterofthefieldofview.RecallingthedefinitionoftheRadon
transforminSec.(2.1.1),itispossibletorelatetheexpressionofftoitsRadonprojectionsin
polynomials:vChebysheofbasis1f(x,y)Uk(x,y;θk,j)dxdy=Pf(θk,j)Uk(t)dt.(2.19)
1−BTherefore,fcanbewrittenasanexpansionofitsRadonprojections
f(x,y)=Uk(x,y;θk,j)πPf(θk,j,t)Uk(t)dt.(2.20)
∞k11
k=0j=0−1
Xuprovedtwoadditionalproperties,whichareimportantforthereconstruction[Xu06b]:
1.Forany0≤k≤N−1and0≤θ≤2π,
1−NN1Uk(cos(θ−ϕnu))Uk(x,y;ϕν)=Uk(x,y;θ).(2.21)
=0ν

(2.19)

(2.21)

econstructionrImage2.1.

17

2.Foranykand0≤ϕ≤2π,
kUk(cos(θk,j−ϕ)Uk(x,y;θk,j))=(k+1)Uk(x,y;ϕ).(2.22)
=0jByusingGaussianquadrature,whichhasbeenshowntobeexactforpolynomialsinorderof
N[Xu06b],theintegralfromEq.(2.20)canberewrittenas
πPf(θk,j,t)Uk(t)dt=NPf(ϕν,cosψj)sin(k+1)ψj,(2.23)
111N−1
1−=0jwherecosψjdenotesthezerosoftheChebyshevpolynomials.Theresultingformulaforthe
approximationANfoffisobtainedbytruncatingthesummationoverktohaveN−1terms
ANf(x,y)=N1(k+1)Uk(xcosφν+ysinφν)N1Pf(ϕν,cosψj)sin(k+1)ψj
N−1N−1N−1
ν=0k=0j=0
(2.24)IthastobepointedoutthatNisthenumberofprojectionstaken,butitisalsothehighest
gradeofthepolynomialswithwhichtheapproximationcanbedone.Themoreprojectionsare
available,themorepolynomialsofhighergradecanbeusedfortheexpansionandthebetter
getstheapproximationANfoff.
Equation(2.24)resemblesaconventionalvectorialexpansion,whichcanbeintuitivelyex-
plained.ThelasttermisthesummationovertheRadontransformofallbeamswithanangle
ofφnu.ThesecondtermrepresentstheRidgeChebyshevpolynomialsofthesecondkindasthe
correspondingbasisvectors,whicharemultipliedbytheweightingfunction(k+1).Thisfilter
amplifiespolynomialswithhighergradeandisthereforecomparabletotheweightingfunction
oftheFBPalgorithm.Theshapeofthediscretizedimpulseresponsesofthetwoalgorithms
isthusequal,butthescalingisdifferent.Finally,thefirsttermofthereconstructionformula
normalizestheexpressionwiththenumberofprojectionsN.
Equation(2.24)alsoimpliesthatOPEDusesso-calledfan-parallelbeamsineachprojection;
theseareparallelbeamsdistributedaccordingtothezerosofChebyshevpolynomials.Sincethe
zerosofChebyshevpolynomialscorrespondtoequallyangularpointsontheunitcircle[Tis06],
aCTgeometrywhichmakesuseofthesepropertiessuggestsitself.TheresultistheCTdOr
2.2.Sec.indescribedOPEDaswellastheFBPalgorithmwereappliedinthiswork.Therefore,themajorsimi-
laritiesanddifferencesareshortlysummarizedinthefollowing.Bothalgorithmsarebasedon
theRadontransform.However,theFBPalgorithmusestheFourierslicetheoremtorelatethe
objecttoitsRadontransform,whileOPEDusestheexpansionwithChebyshevpolynomials
ofthesecondkind.Thisexpansionisadirectmethod,whichdoesnotneedanyzero-padding
asnecessaryfortheFBPalgorithm.Bothalgorithmsusesomekindofweightingfunctionre-
sultinginsimilarimpulseresponsesandanti-correlatednoiseinthereconstructedimages.The

18

oundbackgreticalTheor2.

FBPalgorithmworkswithequidistantlydistributedparalleldata,whichhavetobecreatedfrom
conventionalfanbeamscannersbyrebinningthedata.OPED,bycontrast,workswithfan-
paralleldata,whichcanbedirectlyobtainedbyreorderingthefanbeamdata.Thecomputing
timeoftheOPEDalgorithmhasbeenshowntobeatleastthesameasfortheFBPalgorithm
[Xu07b]andwithafastimplementationitisuptoabout20-30timesfaster[Xu09].However,
acompleteanddetailedcomparisonbetweenOPEDandtheFBPalgorithmstillhastobedone.
3-dimensionalreconstructionwithOPEDhasnotyetbeenpublished,butinprinciple,theFeld-
kampschemecanbeappliedinasimilarwayasintheimplementationfortheFBPalgorithm.
Itissupposedtobeevenmorestraightforward,becauseanadditionalinterpolationtoobtain
equidistantraysintheconeasdescribedbyGrass[Gra00]isnotnecessaryforOPED.

2.2.ThenewscanninggeometryCTdOr

Inadditiontothehealthrisksduetothedepositeddose(Sec.1.2)andtheinterpolationsnec-
essaryfortheFBPalgorithm(Sec.2.1.1),therearefurtherdrawbacksincurrentCTdevices
[Her08b]:

-Scatteredradiationproducesadditionalnoiseintheimages.

-Duetoaminimalrotationtimedownto300milliseconds[ImPACT10]andagantry
weightofupto2tons[ImPACT09],thecentrifugalforcesinthegantryareenormous.
Thisleadstomechanicinstabilities,whichresultinaslightnutationofthegantryaround
thecenter,therebydecreasingtheaccuracyofthedatacollection.

-Asinglereadingofadetectortakesfinitetimeduringwhichthedetectorslightlymoves,
sothattheeffectivesizeofthedetectorseemstobebroaderthanitreallyis.

AthirdgenerationCTincombinationwithacollimationmaskwithshieldingsonthesideof
thex-raysourcecanovercomethesedrawbacks(Fig.2.5).Thewindowsbetweentheshieldings
aresupposedtohavethesamewidthastheshieldingsitselfinordertoletonlyhalfofthe
radiationcrossthemask.Theamountofradiationappliedtothepatientinawholerotationcan
theoreticallybehalvedbyusingshieldingsmadeofhigh-absorptionmaterialsuchaslead,thick
enoughtopreventthetransmissionofphotonscompletely.Byinstallingdetectorelements
ontheinnersideofthemask,twocomplementarydatasetscanbecollected:Onewiththe
innerdetectorsofthecollimationmask(Fig.2.5)(red),theotherwiththearcdetectorofa
conventionalCTscanner(blue).Thusthesystemhasbeendenoted:CTwithDualOptimal
Reading(CTdOr).Eachofthecomplementarydatasetsproducesaself-containedsinogram
andacorrespondingreconstruction.However,onlythecombinationofbothallowstofully
exploitthepotentialofCTdOrincombinationwithOPED.Inordertocombinethedata
setstherearetwopossibilities:Oneistocombinethefandata,thesecondistoaddthetwo
reconstructedimages.Thefirstmethodismoreaccurateandthereforepromisingtoproduce

2.2.ThenewscanninggeometryCTdOr

19

sharperimagesandsuppressartifactseffectively.However,ifthetwodetectorsetsdonothave

thesamesensitivity,thismethodislikelytoproduceringartifacts.Thesecondmethodismuch

lesspronetodifferentsensitivitiesofthe

ofimagesboth

method.

asw

not

eryv

accurate,

it

detector

is

likely

sets,butifthereconstructionorthealignment

to

produce

images

more

blurred

than

the

first

20

oundbackgreticalTheor2.

(b)(a)Figure2.6:Theseschemesofasimplifiedmaskwith27detectorsandthefanbeamsaftera
rotationabout40°showthateachrayofafanisparalleltoonerayofanotherfanforthemask
detectors(a)andthearcdetector(b).(BycourtesyofHugodelasHeras)

-Inthesameway,thefinitereadingtimeofthedetectorsdoesnotdecreasetheimage
ymore.anresolution

2.3.Imagequalityanalysis
Anintuitivedefinitionofimagequalityratesanimagebywhethertheradiologistcanuseitfor
ameaningfulinterpretation.Thisdefinition,however,isbasedonasubjectiveindividualhuman
judgmentnotappropriatetomeasuretheperformanceofimagingsystemsobjectively.Inorder
toovercomethisproblem,avarietyoftechniqueshasbeendeveloped,whichhaveincommon
thattheyanalyzetheresultingimageofaknowninputsignal.Asafigureofmeritforthesystem
performance,thesignal-to-noiseratio(SNR)hasbeenestablished.Itisbasedontheassumption
thatinordertobedetectableanobjectmusthaveacontrasttoitssurroundingswhichishigher
oratleastequaltotherelativestandarddeviationofthenoise[Ang05].However,theSNRalso
dependsontheshapeandamplitudeofthesignal,thebackgroundandthesuperimposednoise.
Inordertostudythedependenceontheseparameters,differenttaskshavebeenformulated.
Thesimplestoneisthesignal-known-exactly/background-known-exactlytask(SKE/BKE),
whereasignalwithknownparameterssuchassize,shapeandlocationhastobedetected
inadditive,uncorrelatedGaussiannoiseonaknownbackground.TheSKE/BKE-taskis
approximatedbytheRosemodel,whichprovidestherelationshipbetweenthenumberof
imagequantaandtheperceptionofdetail[Bur99].Havingauniformbackgroundwithamean
numberofquantaperunitareaq¯bandanobjectwithameannumberofquantaperunitareaq¯o,
thecontrastCcanbedefinedas:

analysisqualityImage2.3.

21

C=q¯b−q¯o.(2.25)
q¯bTheRosesignaldifferenceΔSRoseisdefinedasthedifferencebetweenthemeannumberof
quantaintheareaconnectedtotheobjectandthemeannumberofquantainanequallysized
areaAinthebackgroundregion:

ΔSRose=(q¯b−q¯o)A.(2.26)
Rosedefinednoiseasthestandarddeviationσbinthenumberofquantaq¯binanareawiththe
samesizeasAinthebackgroundregion[Beu00].Foruncorrelatednoise,thePoissonstatistics
σb=Aqb.(2.27)
describesthestandarddeviationas:
SincetheSNRisdefinedastheratiooftheoutputsignaltothestandarddeviation,theSNRRose
byenvgiisSNRRose=ΔSout=A(√q¯b−q¯o)=CAqb.(2.28)
AqσbbTheSNRprovidesinformationaboutthedetectabilitydoftheobjectandthusaboutits
detectionprobabilityp[Kyp05a].ForaSKE/BKEsituation,dequalstheSNR[ICRU96]
dSKE/BKE=SNR.(2.29)
Foratwo-alternativeforced-choiceexperiment(2-AFC)forexample[Kyp05a],pcanberelated
d1todby
p=21+erf(2),(2.30)
whereerf(d2)denotestheerrorfunction
derf(d)=√22exp[−y2]dy.(2.31)
π20TheresultingsigmoidalcurveisplottedinFig.2.7.Sincethereareonlytwopossibilitiesin
a2-AFCexperiment,thedetectionprobabilityis0.5ford=0.Thedetectionthresholdis
usuallydefinedasthestimulusintensityatwhichp=0.75[Ulr04].
Themeansorstrategybywhichthetaskgetsdoneiscalledtheobserverordecision-maker.
Theobserverofaclinicalimageisnormallyaradiologist,butmuchresearcheffortisbeingex-
pendedonthedevelopmentofcomputeralgorithmsforsuchtasks.Theso-calledidealobserver
usesallstatisticalinformationavailableinordertomaximizethetaskperformance.Therefore,
itprovidesanupperboundagainstwhichallotherobserverscanbecompared.However,ifthe
signalorthebackgroundareonlyknownstatistically,theidealobservermaynotbecalcula-
bleormayrequirelengthynumericalcalculationsusingMonteCarlosimulations.Theideal
observerhasfurthermorethedisadvantagethatitoverestimatestheperformanceofthehuman
observer,wholackstheabilitytoaccountforcorrelationsinimagenoiseeffectively[ICRU96].

22

2.oundbackgreticalTheor

Figure2.7:Thedetectionprobabilitypoftheobjectagainstitsdetectabilitydforatwo-
choice.forcedevalternati

Whenanalyzingclinicalimagingsystemsthatwillhavehumanobserversasend-users,itis
thereforedesirabletousemodelobserversthataremoretractableandthatreectthecapabilities
ofthehumanobserver.TheHotellingobserver[Hot31]isacommonlyusedalternative,because
itdemonstratesthemaximumdiscriminationabilityamongallobserversthatarelimitedto
performonlylinearoperationsondata[Bar85,Bar90].Inessence,theHotellingapproach
modelsthedataasGaussian,regardlessofitstruestatistics.Moreover,ithasbeenfoundto
usefullypredictthehumanperformanceforavarietyofdiscriminationtasks.
Inthiswork,theSNRhasthereforebeencalculatedbasedontheHotelling-observermodel.
ParametersinuencingtheSNRsuchasspatialresolutionandnoiseweredeterminedintwo
differentways:theconventionalFourierbasedapproachandanimage-spacebasedapproach
originatingfromstatisticalinformationtheory.Bothapproachesaredescribedinthefollowing.

2.3.1.Fourierbasedapproach
Inthe1970s,figuresofmeritweredevelopedtospecifythespatialresolutionandthenoiseof
2-dimensionalradiographicsystems.Inordertocomparetheperformanceofdifferentsystems,
themethodshadtobeapplicableonanalogscreen-filmsystemsandprocessablewiththelimited
computationalpowersavailableatthattime.TheFourierbasedconceptsofthemodulation
transferfunction(MTF)andthenoisepowerspectrum(NPS)fulfilledthesecriteriaandbecame
thestandardmethodfordescribingtheimagequalityinradiographicsystems.Widely-usedand
accepted,theseconceptswerealsoadaptedformoderndigitalizedsystemsandCT.

analysisqualityImage2.3.

23

(2.34)

functionertransfModulationTheMTFdescribesthetransferofaninputcontrastthroughtheimagingsystem.Itisdefinedas
theratiooftheoutputmodulationtotheinputmodulationforalinearsystem[Hsi03].Consider
thetransferofasinusoidalsignalh(x)
h(x)=a+bei2πux(2.32)
whereuisthespatialfrequency.h(x)isrealandcorrespondstothemeasurableinputsignal
[Beu00].Themodulationofh(x)isgivenby
|hmax|−|hmin|(a+b)−(a−b)b
Min=|hmax|+|hmin|=(a+b)+(a−b)=a.(2.33)
Theoutputsignald(x)canbederivedfromtheconvolutionofthesignalh(x)withtheimpulse-
responsefunction(IRF)irf(x)
∞d(x)=h(x)∗irf(x)
=h(x)irf(x−x)dx
−∞∞=(a+bei2πux)irf(x−x)dx(2.34)
−∞∞∞=airf(x−x)dx+bei2πuxirf(x−x)dx
−∞−∞=aT(0)+bT(u)ei2πux
whereT(0)equalstheareaundertheIRFandisthereforereal,whileT(u),ingeneral,is
complex.Themodulationoftheoutputisgivenas
Mout=|dmax|−|dmin|=b|T(u)|=Min|T(u)|.(2.35)
|dmax|+|dmin|aT(0)T(0)
TheMTFisdefinedastheratiooftheoutputmodulationtotheinputmodulation
MTF(u)=Mout=|T(u)|.(2.36)
(0)TMinBydefinition,MTF(u)isunityatu=0.
IftheIRFisreal,whichisgenerallythecaseforx-rayimagingsystems,thenT(u)and
MTF(u)areeven.Thismeansthatbothfunctionscanbeexpressedintermsofpositivefre-
quenciesonly.ForsystemswheretheIRFisrealandeven,T(u)isalsorealandevenandno
phase-transferinformationislostbytakingtheabsolutevaluetocalculatetheMTF.
TheMTFofCTsystemsisnormallycalculatedtakingtheabsolutevalueoftheFourier
transformofthepointspreadfunction(PSF).ThePSFistheresponseofthesystemtoanideal
pointobjectoraDiracdeltafunction,δ(x,y).Sinceidealpointobjectsdonotexist,thinwires
orbeadsconsistingofhigh-absorptionmaterialarecommonlyusedtoapproximatethePSF.For

24

Theor2.oundbackgretical

anobjectlargerthanthelimitingspatialresolutionofthesystem,asincoraBesselfunction
canbeusedtocorrectthefiniteextensionoftheobject.
Anideal,analogoussystemhasaconstantMTFequalto1forallfrequencies,becausethe
inputsignalistransferedwithoutanydegradationthroughthesystem.Fordigitalsystems,the
photonsareintegratedoverpixelsofwidthssothateventheidealMTFalwaysdecreaseswith
thesincfunction|sinc(πsu)|[Beu00].Therefore,theMTFofdigitalsystemsdependsstrongly
onthefrequency.Thespatialresolutionofasystemisoftenreferredtoasthefrequencyvalue
wheretheMTFdecreasedto0.1[Kal05].

spectrumwerpoNoiseTherearetwotypesofnoiseintheprojectiondata.Thefirstoneisacontinuouslyvarying
errorduetoelectricalnoise,variationofdetectionefficiencyorroundofferrors.Thesecond
typeofnoiseisafunctionofthenumberofx-rayphotonsthatexittheobject[Kak01].The
noisepowerspectrum(NPS)describesthevarianceofimageintensity(i.e.,imagenoise)for
auniformlyirradiatedimagedividedamongthevariousfrequencycomponentsoftheimage.
Inotherwords,theNPSdefinesthevariance(perfrequencybin)ofagivenspatial-frequency
componentinanensembleofmeasurementsofthatspatialfrequency[Beu00].Themagnitude
oftheNPSreectstherebythedegreeofrandomnessateachspatialfrequencyandtheintegral
ofnoisepoweroverallfrequenciesyieldsthevariance[Rie78].Aconcentrationofnoisepower
inlow-frequencyspaceisduetoacoarsegraininessoftheimages,whileaconcentrationin
high-frequencyspaceiscausedbyafinergraininess.
Homogeneousnoiseimagesinair(2-dimensionalradiography)orinwater(CT)areused
tocalculatetheNPS.TheimagesaredividedinMsquaredregionsofinterest(ROIs),which
areFouriertransformedseparately.ThesizeoftheROIsisnotfixedbutisanoptimumthat
balancestwoopposingeffects.IncreasingthesizeoftheROIsincreasesthenumberofspatial-
frequencypointsatwhichtheNPSiscalculated,butleadstoanincreaseduncertaintyintheNPS
curve.AdecreaseoftheROIsizeresultsinmoreaveraging,butalesswell-definedNPScurve
[Pad05].Inordertobenefitfromtheadvantagesofbothsides,ROIsareoftenchosentooverlap
byuptohalftheirsizeinhorizontalandverticaldirectioninpractice.Apotentialzero-offset
andgeneraltrendsarecorrectedbyfittingandsubtractingasecond-orderpolynomialS(xi,yj)
fromthedata.TheNPSfordigitalizedsystemswithanimageintensityofI(xi,yj)istherefore
asdefinedMNNy2
NPS(un,vk)=ΔxΔyx[I(xi,yj)−S(xi,yj)]e−2πi(unxi+vkyj)(2.37)
MNxNym=1i=1j=1
whereΔx,ΔyisthepixelsizeandNx,NyisthesizeoftheROIsinhorizontalandvertical
[Beu00].elyvrespectidirectionTheNPSinreconstructedCTimagesismainlyinuencedbythedesignofthereconstruction
filterappliedinaparticularalgorithm.Edge-enhancingfiltersincreasethenoiseforfrequencies

analysisqualityImage2.3.

25

relevantforobjectdetection,whilesmoothingfiltersdecreasethenoise,butwithitalsothe
resolution.spatial

SNRbasedFourierTheMTFdescribesthesignaltransferoftheFouriertransformedinputsignal|FT(ΔSin)|
throughtheimagingsystem.IntegratingoverallfrequenciesgivestheoutputsignalΔSout
ΔSout(f)=MTF(f)∙|FT(ΔSin(f))|df.(2.38)
Furthermore,√thevarianceofthebackgroundisdefinedbytheNPSsothatthestandarddeviation
σisgivenbyNPS.FortheHotellingobserver[ICRU96],theFourierbasedSNRF2ourierfora
SKE/BKE-taskcanthereforebederivedfromEq.(2.28)
Fourierσ2NPS(f)
SNR2=ΔS2out=MTF(f)2∙|FT(ΔSin(f))|2df.(2.39)
InherentassumptionsoftheFourierapproach
However,theFourierrepresentationisonlyvalidiftheimagingsystemislinearshiftinvariant
andwidesensestationary[Bar04].Widesensestationaritymeansthatthemeanvalueiscon-
stant,thevarianceisfiniteandtheautocovarianceisonlydependingontherelativedistance
betweentwopixels.However,neitherPoissonnoisenorelectricalnoisefulfillthesecriteria.
Themeanandtheautocovariancefunctionofaprocessconstitutedbyanincidentx-rayphoton
dependthereforeonthelocationonthedetector.Alsothefinitesizeoftheimagingsystem
spoilsthestationaritycriterion,becausetheautocovariancefunctioncouldonlybeinvariant
whenadigitalwrap-aroundwouldbeassumed,meaningthattwopixelsatoppositesides
ofthearraywouldhavethesamecovarianceastwoadjacentpixels.Thisassumptionhasno
physicalbasissothatitcanbestatedthattheassumptionsmadebytheFourierapproacharenot
fulfilledbyrealimagingsystems.Anapproachwhichdoesnotneedtheseassumptionsisthe
image-spacebasedapproachpresentedinthenextsection.

2.3.2.Image-spacebasedapproach
Thechangefromanalogtodigitalsystemsinradiographyandanincreasingcomputational
powerallowedthedevelopmentofalternativeapproachesforimageanalysis.Anew,modern
imageevaluationmethodologyhasbeendevelopedfromconceptsbasedonstatisticaldecision
theory.Itsapplicationtodigitalizedimagesisstraightforward,becauseitisimage-spacebased
anditmodelsthesystempropertiesusingmatrixoperators.Thedeterministicpropertiesofan
imagingsystemaredescribedbytheH-matrix,andthenoiseintheimagesisdescribedbythe
ariancevcomatrix.

262.Theoreticalbackground
H-matrixTheH-matrixHdescribesthetransferofasignalthroughtheimagingsystem[Bar04,Kyp06a,
Kyp08,Kyp09]bymappingadiscreteobjectftoanimageg
g=Hf.(2.40)
ThedimensionalityofHisM×N,whereMisthetotalnumberofimagepixelsingandN
isthetotalnumberofelementsinthediscretizedobjectf.SinceMandNarenormallynot
equal,Hisrectangularingeneralsothatasingularvaluedecompositionhastobeperformed
inordertoanalyzethematrix.Therefore,aHermitiansquarematrixisconstructedusingthe
matrixproductofH†andH.Theeigenanalysisisthenwrittenas
λnun=H†Hun(2.41)
whereλnandunarethesortedeigenvaluesandeigenv√ectorsofH†H.λndefinesthemagnitude
ofeacheigenvectortransferredthroughthesystemandλncorrespondstothesingularvalues
yp08].[KHofEquation(2.41)isofspecialsignificance,becauseitgeneratesasetoforthonormaleigen-
vectorsunthatformthebasisoftheimagingsystem.Sinceunareorthonormalvectors,they
formacompletebasisoflengthN.Consequently,anyobjectfcanbewrittenasanexpansion
intermsoftheseeigenvectors:N
f=αnun,(2.42)
=1nwheretheexpansioncoefficientsαnaregivenbythescalarproduct
αn=un†f.(2.43)
SincetherankRofHisingenerallessthanN,theeigenvaluesλnarerealandpositiveonly
forn<Randzerootherwise.Thismeansthateigenvectorsunwithn>Rarenottrans-
ferredbytheimagingsystem.Suchfunctionsarereferredtoasnullfunctionsoftheimaging
system.Theobjectcanthereforebeseparatedintothemeasurablepartfmeasandthenullpart
[Bar04]:fnullf=fmeas+fnull(2.44)
whereRfmeas=αnun(2.45)
=1nandNfnull=αnunwithHfnull=0(2.46)
+1R=nInpractice,Hiscalculatedfroma2-dimensionalpointspreadfunction(PSF),whichissam-
pledincomputedtomographyfromathinwireorabead(compareSec.2.3.1)[Kyp08,Liu09].

qualityImage2.3.analysis

27

ThemeasuredPSFistheimageg.Thecorrespondingobjectfcanbecreatedasanarraywith
aninfinitesimalthinsignalonexactlyoneentry(thepositionofthewireing)andnosignal
onallotherentries.InordertocalculateH,fandgarereformedto1-dimensionalvectors.
ByshiftingthePSFandthereforeshiftingthepositionofthesignalingandf,eachcolumn
ofHcanbecalculated.ThisisschematicallyshowninFig.2.8,whereeachcolorrepresentsa
differentpositionofthePSF.

Figure2.8:Fromeachpositionoftheinfinitesimalthinsignalinfanditsimageing,one
columnofHcanbecalculated.
Inordertodescribethesystempropertiesaccurately,thePSFhastobeshiftedwithahigher
resolutionthanthepixelsize.Thismeansthatgisshiftedforone-thirdorone-forthofthe
imagepixelsizeatatime,resultinginftobe9to16timeslargerthang.Sincethisgetsvery
computationalintenseforlargeimages,aconvenientsizeforgis32×32pixels,resulting,with
a3-foldresolution,inaH-matrixwithasizeof322×(3∙32)2=1024×9216.
Covariancematrixeigenanalysis
AwaytodescribenoiseinanimagewithoutapplyingFouriertransformationisbycalculating
thecovariancematrixKofthenoise[Bar04].Itquantifieswhetheructuationsinonepixelare
statisticallyrelatedtouctuationsinanotherone.ForanimagegwiththedimensionsNxxNy,
Kisgivenby
Kij=(gi−gi)(gj−gj)∗,(2.47)
wheretheasteriskindicatescomplexconjugate,allowingforthepossibilitythatcomponentsof
x.complebemightgIfgiandgjarestatisticallyindependent,thenKij=0.Thediagonalelementsofthecovari-
ancematrixarethevariancesofthecomponents
Kjj=Var{gj}.(2.48)
Thecovariancematrixcanalsobeexpressedasanouterproduct
K=(g−g)(g−g)†=ΔgΔg†,(2.49)
whereΔg=g−gwithgbeingthemeanvector.

28

oundbackgreticalTheor2.

Tostudythequantumnoiseofanimagingsystem,letgbeonerealizationoftheat-field.
diagonalizedbecanKgKg=ΦMΦ†(2.50)
whereMisthediagonalmatrixoftheeigenvaluesandΦisaunit-lesssquarematrix,whose
columnsaretheeigenvectorsofKg.
SinceKisanHermitianmatrix,i.e.Kij=Kj∗i,ithassomespecialproperties,whichmake
iteasytoworkwith:
-ThesetofeigenvectorsΦspansacompleteorthonormalvectorspace.Φisalsoaunitary
matrix,i.e.Φ−1=Φ†.
-TheeigenvaluesofKarereal,evenifKiscomplex.
-Kisalwayspositive-semidefinite,whichmeansthattheeigenvaluesaregreaterthanor
equaltozero.Ifthecovariancematrixisdefinedappropriately,theeigenvectorsarelin-
earlyindependentandtherankRofK,andaccordinglythenumberofnonzeroeigenval-
ues,equalsitsdimensionsM[Bar04].
SincetheeigenvectorsΦofaHermitianoperatorformacomplete,orthonormalsetofbasis
vectorsintherelevantspace,anyimagegcanbeexpressedas:
g=Φβ,(2.51)
whereβisam×1vectorwithuncorrelatedcomponents{βm}.Thisexpansionofanimagein
eigenvectorsofitscovariancematrixisknownasKarhunen-LoèveorKLexpansion.Itallows
tovisualizethestructureoftheimagenoisebyplottingtheeigenvectorsas2-dimensionalarrays
ofthesamesizeastheROIsthecorrespondingcovariancematrixwascalculatedwith.
Inpractice,thecovariancematrixiscalculatedbydividingnumerous,statisticallyindepen-
dentimagesinquadraticROIs[Bru10].Bydefinition,theseROIshavetobeindependentfrom
eachothersothatthepixelsinonematrixarenotcorrelatedtopixelsinanothermatrix.Theau-
tocovariancespecifieshowfaraparttwopointsmustbefortheiructuationstobeuncorrelated.
by:definedisItK(r1,r2)=[f(r1)−f(r1)][f∗(r2)−f∗(r2)].(2.52)
Whitenoisehasadelta-functioncorrelation,butsincethenoiseinreconstructedimagesis
colored,thecorrelationisadistributionfunctionasshowninFig.2.9a.Duetothefilterapplied
inreconstructionalgorithmssuchasFBPandOPED(compareSec.2.1.1andSec.2.1.2),noise
inCTimagesisanti-correlated,leadingtonegativevaluesoftheautocovariance.
ThecovariancematricesofallROIsareaveragedtoresultinacovariancematrixlikethe
typicaloneshowninFig.2.9b.Thediagonalisbrighterthantherest,becausethecorrelationof
eachpixelwithitself,alsocalledvariance,isofcoursehigherthanthecorrelationofapixelwith
anyotherpixel.Thebrightlinesbesidesthediagonalindicatethecorrelationofeachpixelwith
itsneighbors.Thedarklinesreecttheanti-correlationimplementedbytheFBPalgorithm.

analysisqualityImage2.3.

(a)Typicalautocovariancefunctionobtained
fromimagesreconstructedwithaFBPalgo-
rithm.

(b)Typicalcovariancematrix

Figure2.9:Principalsofcalculatingthecovariancematrix

29

Signal-to-noise-ratioTheimage-spacebasedapproachdefinesthesignaltobedetectedbyanimagingsystemasthe
differencebetweenthemean,ornoise-free,imageg¯b+swithaknownbackgroundthatcontains
thesignaltobedetected,andthemean,ornoise-free,imageoftheknownbackgroundg¯b
yp09]:[KΔSout=g¯b+s−g¯b.(2.53)
UsingEq.(2.40)ΔSoutcanbeexpressedintermsoftheH-matrixandtheobjectf:
ΔSout=H(f¯b+s−f¯b)=HΔf.(2.54)
ForaSKE/BKEtaskandtheHotellingobserver,theSNRforimagescontainingonlyGaus-
siannoiseisdefinedintermsoftheH-matrixHandthecovariancematrixKg[Bar04]:

SNR2Image−based=ΔS†outKg−1ΔSout
=(HΔf)tKg−1HΔf.(2.55)
Iftheimagingsystemwaslinearshiftinvariantandwidesensestationary,KgandHwould
bediagonalizedbyFouriertransformationsothat
Kg←FT→NPS
H←FT→MTF.(2.56)
Therefore,theeigenvaluesofthecovariancematrixandthevaluesoftheNPSwouldbeequal
andtheeigenvectorsofthecovariancematrixwouldbeexponentialwavefunctions.Inthesame

30

oundbackgreticalTheor2.

mannerthesingularvaluesoftheH-matrixwouldcorrespondtothevaluesoftheMTF.Inthis
specialcase,theresultingSNRwouldbethesameforbothapproachespresentedhere.
WithEq.(2.56),theimage-spaceSNRImage−basedofEq.(2.55)caneasilybeconvertedinto
theFourierSNRFourierofEq.(2.39).

Inherentassumptionsoftheimage-spaceapproach

Theimage-spacebasedapproachmakesassumptionstoo.Forcalculatingthecovariancematrix,
itisassumedthatthesubmatricesusedarestatisticallyindependentfromeachotherandthatthe
structureandamplitudeofthenoiseisthesameforall.Byusingalargeenoughpixelspacing
betweenthesubmatricesthefirstpointcanbeaccountedfor,butthesecondoneishardlygiven
inanyreconstructedimage.Duetothecircularnatureofthereconstructionalgorithm,whether
FBPorOPED,thenoiseofimagesdoesnotfulfillthiscriterion.Furthermore,inorderto
accuratelydefinethecovariancematrix,ahugenumberofimagesisnecessary.Forexample,
theresultingcovariancematrixof32×32pixelROIshasasizeof1024×1024.Inorderto
definethismatrixcorrectly,10242≈106ROIswouldbenecessary.InSec.6.1thisproblemis
elaborated.further

vicesDegingIma3.

ThefollowingchapterdescribesthedifferentCTsystemsandexperimentalset-upsusedfor
thiswork.Thetwoapproachesforimageanalysiswerefirstappliedandtestedonalaboratory
cone-beamCTsystem,whichallowedtocontroleverysinglestepfromimageacquisitionto
reconstructionandanalysis(Sec.3.1).ThemethodswerethenappliedtoaclinicalCTdevicein
ordertochecktheirapplicabilitytoconventionalsystemsanddemonstratetheachievableimage
quality(Sec.3.2).InordertobeabletoanalyzetheimagequalityofCTdOr,thedatatreatment
hadtobeoptimized.ThiswasfirstdoneforthecombinationoftheCTdOrdemonstratorwith
theC-armdevice(Sec.3.3,Sec.3.4.2)andthentransferedtothecombinationwiththeclinical
3.4.3).(Sec.CT

3.1.FDAlaboratoryCTsystem

Thestudiesabouttheanalysisofreconstructedimagesusingthepixelbasedapproachweredone
incooperationwiththeU.S.FoodandDrugAdministration(FDA).Thefirstexperimentswere
thereforeperformedattheFDAbench-topat-panel-basedcone-beamCTscanner[Kyp06a,
Kyp06b,Kyp07,Kyp08].Aschematicoftheexperimentalset-upanditsdimensionsisshown
inFig.3.1.Thex-raytubeisaVarianB180(VarianCorp.,SaltLakeCity,UT)withatungsten
anodeandaninherentfiltrationof1mmaluminum.Ithasnominalfocalspotsizesof0.3
and0.6mmofwhichthelargeronewasusedforthemeasurements.Anadditionalfiltrationof
0.25mmytterbium(Yb)wasimplementedtotakeadvantageofthek-edgeemissionsoftungsten
(W).TheYbfiltrationabsorbsmostphotonswithenergieshigherthan59keV,whileallowing
transmissionofphotonsofthek-edgeemissionlinesofthetungstenanode.Theresulting
spectrum,showninFig.3.2,wascalculatedbasedontheoriginalx-rayspectrapublishedby
theInstituteofPhysicsandEngineeringinMedicine[Cra97].Anaperturewasfixedin19.5cm
distancefromthefocalspotofthesourcetocollimatethebeam.Theat-paneldetectorused
intheexperimentalset-up,aVarian4030CB(VarianCorp.SaltLakeCity,UT),wasplaced
142cmawayfromthefocalspotofthesource.Thedetectorhad2048×1536pixelswitha
pixelsizeof195×195µm2anda0.6mmthickcolumnarCsI(Tl)scintillator.Asindicatedin
Fig.3.1,theplaneperpendiculartothedetectorisdefinedasthexy-planeandtheplaneparallel
tothedetectorisdenotedastheyz-plane.
Awater-filledPMMA-cylinder,withadiameterof12.7cmandaheightof20.3cm,was
mountedonarotationaltablein79.5cmdistancefromthesource.Thecylinderhadtwosetsof

31

32

vicesDeImaging3.

Figure3.1:Schematicoftheexperimentalset-up.Theredlinesindicatethexy-andyz-planes.

Figure3.2:Calculatedspectrumfilteredwith1.00mmaluminumand0.25mmytterbium.

12equidistantbuilt-instealball-bearingsformingconcentriccirclesatthetopandatthebottom
asshowninFig.3.3a.Theball-bearingsarecalibrationmarks,whichallownotonlytocorrect
formisalignmentsbutwhichprovidealsoinformationintheimageaboutthescannergeometry
yp06b].[KmethodacquisitionandInordertomeasurethespatialresolutionproperties,aslitwasbuiltconsistingofa
0.025mmmolybdenumfoilbetweentwopiecesofpolystyreneasshowninFig.3.3b.The
polystyrenepieceshadathicknessof2.5cmandanedgedimensionof4.5×4.5cm2.The
molybdenumfoilwasgluedtothepolystyrenewithsuperglueanddirectlyafterwardclenched
firmlytocreateahomogeneous,verythingluelayer.APMMAstageallowedtoplacethe
cubeinthecenterofthewater-filledcylinder,butitwasnotpossibletoplacethecubeonthe
peripheryofthecylinder.Hence,onlythespatialresolutioninthecenteroftheimagescould
determined.be

3.2.scannerCTClinical

(a)

(b)

33

Figure3.3:PMMA-cylinderwithbuilt-instealball-bearingsatthetopandatthebottom(a)
(b).slitmolybdenumtheand

Thescanswereperformedwith100kVptubevoltage,acquiring360projectionimagesper
rotation.Furthermore,100at-fieldand100dark-fieldimagesweretakentoperformtheat-
fieldcorrectionoftheprojectedimages(describedinSec.4.1)[Kyp09].

CTClinical3.2.scanner

ThemethodsdevelopedinthisworkweretestedonaconventionalclinicalCTbeforetheywere
appliedtoCTdOr.Forthispurpose,theSiemensSomatomSensationCardiac64(Siemens
Healthcare,Erlangen,Germany)oftheHospitalrechtsderIsaroftheTechnicalUniversityof
Munichwasused.AnimageofthedeviceisshowninFig.3.4.
TheSomatomSensationisathirdgeneration64-sliceCTwithz-yingspot.Thez-ying
spottechniqueusesaperiodicmotionofthefocalspotinthelongitudinaldirectiontodouble
thenumberofsimultaneouslyacquiredsliceswiththegoalofimprovedlongitudinalresolution
andeliminationofspiralartifacts.Thescannerhasafocus-isocenterdistanceof57.0cmanda
focus-detectordistanceof104.0cm.Thex-raysourceisaSiemensStraton[ImPACT06],which
hasatungstenanodeangledat7°[Kac06].TheImPACTreportdealingwiththisCTsystem
[ImPACT09]specifiesthetotalfiltrationonthecentralaxistobeequalto6.8mmaluminum.
Theresultingspectrum,showninFig.3.5,wascalculatedbasedonthex-rayspectrapublished
bytheInstituteofPhysicsandEngineeringinMedicine[Cra97].Informationaboutwhether
thisalreadyincludesthebowtiefilterandaboutthefiltrationawayfromthecentralaxisisnot
ailable.vapubliclyThedetectorarrayconsistsofultrafastceramicsolidstatedetectorswith672elementsina
rowand40elementsalongthez-axis.Dependingonthescanningmode,differentcombina-

34

vicesDeImaging3.

Figure3.4:SiemensSomatomSensationCardiac64(picturefromofficialSiemensClinical
website)

Figure3.5:CalculatedspectrumonthecentralaxisoftheclinicalCTscannerfor120kVp

tionsofsourcevoltage,sourcecurrentandrotationtimeareadjustable.Themaximalavailable
rotationtimeis1.0s,although2.0scanbesetforcranialscans.Butthisonlyresultsin2scans
witharotationtimeof1.0seach.Furthertechnicaldetailscanbefoundinthecorresponding
ImPACTreports[ImPACT06,ImPACT09].QualityassuranceoftheCTisdoneregularlyby
theclinicalphysicistsofthehospitalaccordingtoDINEN61223-2-6[DIN08].Onephantom
usedfortheanalysisofimagequalityintheHospitalrechtsderIsaristheCatphan®phantom
(PhantomLaboratory,Salem,NY,USA)[Cat06].Itprovidesseveralmoduleswithtestdevices
forhigh-resolution,lowcontrast,slicegeometryandsensitometry,andauniformmoduleto
noise.themeasureForthiswork,thespatialresolutionwasmeasuredusingthewell-establishedmethod[Kwa07,
Har10,Wat10]ofspanningawireinairparalleltotherotationaxis.Inordertoapproximatea
pointsource,thewireconsistedofhigh-absorbingtungstenandhadadiameterof0.08mm.30

scannerCTClinical3.2.

35

imagesweretakenrunningasequentialabdominalscanforadultswiththeminimalavailable
slicethicknessof0.6mm,arotationtimeof1.0s,asourcevoltageof100kVandafield-of-
view(FOV)of250mm.ThescanswereperformedwithfourdifferentmAs-settings,40mAs,
150mAs,250mAsand350mAs.Theimageswerereconstructedwithfourdifferentfilters:
B10s,thesmoothestavailablefilter,B30s,thestandardreconstructionfilterforthisscanpro-
tocol,B40s,aslightlyharderfilteralsooftenusedinclinicalpracticeandB70s,thehardest
.filterailablevaThenoiseoftheclinicalCTscannerwasdeterminedusingthestandardSiemensphantom
showninFig.3.6.Itswater-equivalentregionisawater-filledPMMAcylinderof20cmdiam-
eterand3cmheight.Togetenoughimagesforaproperanalysis,thewaterregionhadtobe
scanned7timestaking30imagesineachcase.Similartothescansofthewire,asequential
abdominalscanforadultswasperformedwiththeminimalavailableslicethicknessof0.6mm,
arotationtimeof1.0s,asourcevoltageof100kVandaFOVof250mm.Theimageswere
reconstructedwiththesamefourdifferentfiltersasdescribedabove.

Figure3.6:StandardSiemensphantom.Thewaterregionwithadiameterof20cmanda
heightof3cmismarkedbyarrows.

Imagequalitywasanalyzedindependenceoftheapplieddose,whichisprovidedbythe
systemwitheveryimageastheweightedcomputedtomographydoseindex(CTDIW)[DIN08].
Table3.1liststhedosedependingonthemAsperscanforthesequentialabdominalscanfor
adultsandtheabovedefinedparameters.Sincethedoseisnotmeasured,butcalculatedbythe
systembasedonthecalibration,theprovideddoseisexactlylineartothemAs.

36

vicesDeImaging3.

Table3.1:CTDIWfordifferentmAs-settingofthemedicalCTscanner
mAsperscanCTDIW[mGy]
1.68406.2915010.4725014.66350

vicedeC-arm3.3.Initially,CTdOrwascombinedwithaC-armdevice.Therefore,thePhilipsMultiDiagnost
ElevaFlatDetector(PhilipsHealthcare,Best,TheNetherlands)oftheHospitalrechtsderIsarof
theTechnicalUniversityofMunichwasemployed[KCA04].ThesystemisshowninFig.3.7.

Figure3.7:PhilipsMultiDiagnostElevaFlatDetector(picturefromofficialPhilipswebsite)
TheC-armdeviceisusedinclinicalroutineasaconvenientdevicetocombinediagnosisand
interventionsonthesamesystem.Ithasapulsedsourcewithtwoavailablefocalspotsizes,
0.6and0.8mm.Sincetheonlyreliableinformationaboutthefiltrationisthatitisthickerthan
2.5mmaluminum,theexpectedspectrumcouldnotbecalculated.Thedistancebetweensource
anddetectorcanbevariedbetween95and125cm.Theat-paneldetectorisaPixium4700
(Trixell)withanactiveareaof381.9×294.1mm2,whichcorrespondstoausefulpixelarray
of2480×1910withapixelsizeof154×154µm2[Dan09].Thesystemcreatestheimagesby
binningandinterpolatingover4.84×4.84pixels,withanimagepixelspacingof0.746mm,in
ordertoobtaina512×512image(personalcommunicationwithatechnicianfromPhilips).

technologydOrCTNew3.4.

3.4.NewCTdOrtechnology

37

demonstratordOrCT3.4.1.ForthisworkademonstratoroftheCTdOrgeometrywasused,whichHugodelasHeras
assembledforhisPhDthesis[Her08b].Thedeviceiscalleddemonstrator,becauseitisonly
intendedtodemonstratetheprinciplesoftheCTdOrconcept,butnottoscanpatients.For
medicalapplication,theringwouldhavetobelargerwithmuchmoredetectorswithahigher
andmorehomogeneoussensitivity.
CsI(Tl)-scintillatorcrystalsof1mm3sizearecoveredbyareectingcoatingofTiO2to
improvelightcaptureefficiency.Figure3.8showshowthedetectorsarewrappedinaprotective
rubbercoverandshieldedatthebacksideby0.5mmlead.197detectorelementswithatotal
widthof5mmareplacedregularlyatadistanceof25cmfromthecenterofthedevice,leaving
windowsof3mmbetweenthem.

Figure3.8:Designofthedemonstratorsmask(BycourtesyofHugodelasHeras)

Opticalfibersconnectthedetectorswithphoto-diodes,whichareverysensitivetoexternal
lightsothattheyhavetobeshieldedbyadditionalcases.Thesoftwareallowsdataacquisition
withsamplingtimesdownto1ms.
Thedetectorringisscrewedtoanaluminumplatetoprovidemechanicalstabilityand
mountedonarotationdesk.ThisdeskiscontrolledbyasoftwarewritteninLabVIEW8.2
(NationalInstruments,Austin,Texas).Probablybecauseofroundingerrorsinthesoftware,the
deskdoesnotrespondprecisely,sothatitrotatessomewhatlessthan360°whenitisorderedto
performafullrotation.Therefore,ascanof365°wasalwaysperformedtoensureacomplete
360°.ofsetdataTheCTdOrgeometryhasthehighestefficiency,whenshieldings,windowsandtheeffective
surfaceofthedetectorshavethesamesize(compareSec.2.2).Duetotheshapeanddesignof
themaskdetectorsthisgeometrycouldnotberealizedexactly.Figure3.9showswhichdata

38

DeImaging3.vices

areactuallycollectedbythedifferentdetectorsystems.Thebluestripes,representingthedata
collectedbythemaskdetectors,areintermittentbystripeswherenodatacanbecollecteddue
tothedifferenceinshieldingsizeandscintillatorcrystalsize.Assumingcompleteabsorption
bytheleadshieldings,only83oftheinitialphotons1passthroughthewindowsandonly31of
themactuallyhitsthescintillator.Therefore,onlyor12.5%ofthetotalamountofphotons
hitsthescintillator.InaperfectCTdOrwithshieldings8ofthesamesizeasthewindowsand
theeffectivescintillatorsurface,50%oftheinitialphotonswouldbedetectedbythemask
detectors.Thisgeometryisamajordrawbackofthedemonstrator,butitdoesnotinuenceits
abilitytodemonstratethepotentialoftheCTdOrtechnology.
ThereddataofthearcdetectorinFig.3.9areadditionallyinterruptedbyalightgreenstripe,
whichconnectstwomaskdetectors.Thesedatacanobviouslynotbecollectedandtherefore
havetobeinterpolated.ThisisinherenttotheCTdOrgeometryindependentofthespecial
drawbacksofthedemonstrator.

3.4.technologydOrCTNew

39

asdepictedinFig.3.11.Separatedpulsescaneasilybeaddedtorecoverafullpulseandzero
samplesaredeleted.Therotationspeedofthedeskwassetto2.5°persecond.However,
sincetheC-armdeviceisnotconstructedwithregardtotakethousandsofimagesinashort
timeperiod,thesourcewouldhaveoverheatedsothat365°-scanshadtobesplitin13series
withabout5minutescoolingtimeinbetween.ForallscansthesourcevoltageoftheC-arm
wassetto96kVpandthesourcecurrentto1mAs.Thelargeroneofthetwoavailablefocal
spotsizes,0.8mm,wasused,becausetheresolutionofthedemonstratorislimitedanyhow
buttheproblemwiththeoverheatingofthesourceissmaller.Thedistancebetweensourceand
detectorwassetto125cmforallmeasurementsinordertohaveareasonablesizedFOVtoscan
thephantom,asliceofanAldersonRandophantom.Itcontainsrealbonesandhumantissue
equivalentmaterial.Thechosensliceisattheheightofthenostrilssothatthereareregionsof
highcontrastaswellasregionsofair.Figure3.12showsapictureofthissliceandaCTimage
ofitobtainedwithaslicethicknessof4mmwiththeSiemensCTscanner(Sec.3.2).
Theatpanelandthemaskhadtobealignedsothatthecentralrayofthesource,marked
inFig.3.10bythedashedline,passedthroughtheisocenteroftheCTdOrring.Therefore,
a2cmthickaluminumcylinderwasplacedinthemiddleoftheringandaseriesofimages
wasrecordedduringarotationof180°.Ifthecylinderdidnotmovebetweentheimagesfor
morethanthewidthofonewindowthealignmentwasassumedtobecorrect,becauseaperfect
alignmentwasvirtuallyimpossiblewiththegivenexperimentalset-up.

Figure3.10:CTdOrplacedbetweenthesourceandthedetectoroftheC-armdevice(By
courtesyofHugodelasHeras)

40

Imaging3.vicesDe

Figure3.11:Thesourcepulseshadanintervalof33.3mswithapulsewidthofapproximately
5ms(red).Bysettingthesamplingtimeofthemaskdetectorsto20ms(blue),itwasensured
thatonlyonepulsewasscannedinasinglesample.(BycourtesyofHugodelasHeras)

(b)(a)Figure3.12:PhotoofthesliceoftheAldersonRandophantomusedforthemeasurements(a)
andtheCT-scanofitobtainedwith4mmslicethickness(b).

3.4.3.ImprovedCTdOrcombinedwiththeclinicalCTscanner
Inthecourseofthiswork,theCTdOrdemonstratorwasimprovedinordertopositionthe
tiltedringinthegantryofaclinicalCTasshowninFig.3.13.
Forstorage,thesensitiveCTdOrringandtheopticalfibersareprotectedbyacasewitha
removablecoverplateonthefrontside.Thestealarm,towhichtheringisnowconnectedbya
joint,canbepulledout.Thejointallowstotilttheringalmostexactlyaround90°.Theheight
ofthearmcanbevariedtopositiontheringintheisocenter.AlthoughinaCTthesourcerotates
aroundthering,theringcanstillberotateditselfincaseitisnecessarytouseitintheC-arm
deviceagain.Allsurfacesaremadeofstainlessstealtoguaranteeeasycleaningandtherefore

CTNew3.4.technologydOr

(a)

(b)

41

Figure3.13:SchemeoftheimprovedCTdOrkeptforstorage(a)andwithextendedarmfor
measurementinaCTgantry(b).

conformtothehighsanitarystandardsintheclinic.TheimprovedCTdOrcombinedwiththe
clinicalCTisshowninFig.3.14.
ThecenteroftheCTdOrringhadtobepositionedasaccuratelyaspossibleintheisocenter
oftheCTscanner.Thispositionwasdeterminedroughlybyensuringthatthedistanceofthe
ringtothecoverofthegantrywasthesameforalldirections.Inanextstep,CTimageswere
takenwithaFOVof50cm.SincetheradiusoftheCTdOrringisexactly50cm,thecenterof
theringwasintheisocenterwhenalldetectorscouldbeseenintheimages.Withthehelpof
theseimages,itcouldalsobeensuredthattheringwasnottilted.
ForthemeasurementswiththeCTdOrring,aheadscanprotocolandaslicethicknessof
18mmwerealwaysusedtoguaranteethatthex-rayconedidnothitthealuminumplatewhere
thedetectorsoftheCTdOraremounted.
ForthenoisemeasurementsaPMMA-diskof20cmdiameterand2.5cmheightwasused.
Thespatialresolutionpropertiesweremeasuredusinga1.0mmsteelwire.

42

eFigur

(b).

3.14:

The

CT

dOr

(a)

ring

positioned

in

the

clinical

CT

from

(b)

the

front

3.

(a)

Imaging

and

the

vicesDe

backside

Data4.ocessingpr

ThischapterdescribesindetailhowthemeasureddataforthedifferentCTsystemsweretreated
andpresentstheresultingreconstructedimages.FortheFDAsystem,thedataprocessingbefore
andafterreconstructionandthestepsperformedforimageanalysisarepresentedinSec.4.1.
FortheclinicalCTsystem,Sec.4.2describestheanalysisofthereconstructedimagesand
additionallyoftherawdatainordertogetinformationaboutthehardwareofthescanner.
Section4.3showsthenewlydevelopeddataprocessingmethod,whichconsiderablyimproved
theimagequalityforthecombinationoftheCTdOrwiththeC-armdevice.Itsapplicationto
theimprovedCTdOrdemonstratorcombinedwiththeclinicalCTandtheperformedimage
analysisisdescribedinSec.4.4.

4.1.FDAlaboratorysystem

WiththeFDAlaboratorysystem,360objectimageswereacquiredinonerotation.Theseim-
ageshadtobecorrectedfortheimagegain,fieldnon-uniformities,theheeleffect,andbad
pixelsandlines[Beu00].Therefore,100atfieldand100darkfieldimagesweretakenad-
ditionally.Flatfieldmeansthatthedetectorwasirradiatedwithoutanobjectintheirradiated
area,whiledarkfieldmeansthatthecountrateofthedetectorwasmeasuredwithoutirradiating
thedetectoratall.Theaverageofthedarkfieldimageswassubtractedfromtheaverageofthe
atfieldimagesanddividedbyitsmeantocalculatetheimagegain.Thecorrectionforthe
imagegainandfieldnon-uniformitieswasdonebysubtractingthemeandarkfieldfromthe
objectimagesanddividingbytheimagegain.Inordertocorrectfortheheeleffect,aplanewas
fittedtothemeanatfieldimageandtheobjectimagesweredividedbytheresultingfunction.
Badpixelsandlinesweredetectedinthemeanatfieldandcorrectedintheobjectimagesby
averagingtheneighboringpixels.Finally,theredundantedgesoftheobjectimageswerecut
andthepixelvalueswerelogarithmized.
Thecorrectedimageswerecalibratedforsmallmisalignmentsbyidentifyingthelocation
ofthestealball-bearings[Kyp06b]toaccommodatetheshift-invariantdetectorpointresponse
andthelocationdependentresponseofthefocalspot.Anexampleforanoriginal(a)anda
correctedandcalibratedimage(b)ofthePMMA-cylinderthatshowsthebeadsatthetopand
atthebottomisgiveninFig.4.1.
Imageswerereconstructedwithapixelcrosssectionof1436×1436andacubicvoxelsize
of100µm,usinganimplementationoftheFeldkampfilteredbackprojectionalgorithmforat

43

44

(a)

(b)

prData4.ocessing

Figure4.1:Original(a)andcorrected(b)imageofthePMMA-cylinder

(a)Cylinderwithawaterregionof877×(b)Molybdenumslitwitharectangularof
877pixels381×400pixels
Figure4.2:Examplesforreconstructedimages

fielddetectors[Tur01].Inordertocorrectforthecuppingeffectinthexy-plane(perpendicular
tothedetector)andtoconvertthepixelvaluestoHounsfieldunits,a2-dimensionalsecond-
orderpolynomialwasfittedtotheimages.Thecorrectionwasperformedbydividingbythe
fittedpolynomial,subtractingoneandmultiplyingby1000,sothattheaveragevaluewaszero,
whichcorrespondstotheHounsfieldunitofwater.AsillustratedinFig.4.2a,asquareof
877×877pixelswascutoutofthewater-phantomregiontoanalyzetheimagenoise.Forthe
evaluationofthespatialresolution,arectangularof381×400pixelswascutoutoftheslit
images(Fig.4.2b).Thesizeoftherectangularwaschosentakingcaretoincludethetailsofthe
pointspreadfunctioninonedirection,butnottheendoftheslitintheotherdirection,where
artifactswouldhavedistortedtheresults.

4.1.systemlaboratoryAFD

45

Fortheestimationofthespatialresolution,100imagesofthemolybdenumslitwererecon-
structedandcorrectedasdescribedabove.Inordertoincreasethesignaltonoiseratio,100
sliceswereaveragedtocreatethelinefunctions[Kyp08].Thepeakofthenarrowestlinefunc-
tionwasfittedwithapolynomial,whilethetailswerefittedwithanexponentialdecayfunction.
Theresultingprofilewasthenrotatedtogeneratea2-dimensionalpointspreadfunction(PSF)
of300×300pixels.The2DPSFwasusedtocalculatetheH-matrixwiththree-foldresolution
2.3.2.Sec.indescribedasTheMTFwascalculatedbyFouriertransformingthe1-dimensionalfittedPSF.Inorderto
obtainasmoothfunction,ahigh-orderpolynomialwasfittedtothe1-dimensionalMTF,which
wasthenrotatedtogeneratethe2-dimensionalMTF.
Fortheestimationofthenoise,thecovariancematrixhastobecalculatedaccordingtoits
definitionfromalargesetofindependentreconstructions.However,forCTthisisverytime
consumingandreducesthelifeexpectancyofthex-raytube.Toovercometheselimitations,
regionsofasingleCTvolumeacquisitionwereassumedtobeindependentofeachotherif
theyaresufficientlyseparatedtoavoidpixelcorrelations.Fortypicalscannersthisdistanceis
supposedtobebetweenfourandtenpixels,dependingonthetypeofscanner,thedirection(xy
oryz)andthereconstructionfilter.Inordertodeterminethenecessarypixeldistanceforthis
scanner,theaxisprofilesofthecentralrowofthecovariancematrixwereanalyzedtoobtainthe
autocovariance.Duetotheanti-correlatednatureofnoiseinimagesreconstructedwiththeFBP
algorithm,theROIspacingwasselectedtobeequaltothepixeldistanceatwhichtheabsolute
valueoftheautocovariancefunctionhaddroppedto1%ofitsmaximum.Forthissystemand
theappliedreconstructionfilter,thepixeldistanceturnedouttobefour.Thewaterregionof
eachofthe200imageswassubdividedinto32×32pixelROIs,resultinginacovariancematrix
of1024×1024pixels.Itseigenvectorswererepartitionedinto32×32pixelarrays,inorderto
visualizethestructureofthenoise.
ForthecalculationoftheNPS,thereconstructedimagesweresubdividedinROIsinthe
samewayasdescribedaboveforthecovariancematrix.The32×32pixelROIswereFourier
transformedandaveragedtoobtainthe2-dimensionalNPS.The1-dimensionalNPS,which
providesinformationaboutthefrequencydistribution,wasobtainedbyaveragingradially.
AlthoughusingROIsfromtheCTvolumeisaconvenientmethod,onehastobeawarethatit
disregardsthatthenoiseinthecenterofanimageisdifferentfromthenoiseintheouterregions.
Therefore,inadditiontoevaluatingthewholecylinderareafornoise,thexy-planesliceswere
dividedintoninesquareregionsandtreatedseparatelyasdescribedabove.Thenumberof
submatricesavailableforthecalculationofthecovariancematrixdecreasedto12800sothat
therespectivecovariancematriceswerenotdescribedasaccuratelyasforthewholeimage.
ThespatialresolutionandthesystemnoisewereusedtocalculatetheSNRfortwosimulated
inputsignals:adiskwithauniformmagnitudeof20HUplacedinsideauniformbackground
of-30HUandatorusofuniformmagnitudeof-30HUplacedinsideauniformbackgroundof
20HU.Thediskmodelsadensemassinsidealowerdensityregion,whilethetorusmodelsa

46

ocessingprData4.

densemasswithafattymargininsideahighdensityregion.Theradiusofthediskwas0.85mm,
theouterradiusofthetoruswas1.1mmandtheinnerradiuswas0.7mm.Thesevalueswere
chosentogeneratetwoobjectswiththesameareasothattheirSNRsweredirectlycomparable.
Furthermore,thedifferenceof0.4mmbetweeninnerandouterradiusistypicalforafattyring
.tumordenseaoutsideInordertocheckwhetherthenumberofROIs,obtainedfromthewholeimagesaswellas
fromthe9regions,wasenoughtoproperlydefinethe1024×1024matrix,abootstrappingcalcu-
lationwasperformed[Efr79].Forthismodern,butcomputer-intenseapproach,newcovariance
matriceswerecalculatedfromROIsamplesofsizen<NumberofavailableROIs,eachof
whichisobtainedbyrandomsamplingwithreplacementfromallavailableROIs.Sincetheco-
variancematrixcanhardlybedefinedbyonlyonevaluesuchasthemeanvalueorthestandard
deviation,thenewcovariancematriceswereusedtocalculatethecorrespondingimage-space
SNRanditsstandarddeviation.Foreachn(n=2∙103,5∙103,1∙104,1.5∙104,5∙104and1∙105),
10runswereperformedrespectivelywiththediskasinputsignalandthenoiseimagestakenat
.mGy9.8

scannerCTClinical4.2.

4.2.1.Analysisoftherawdata
Therawdataarerarelyanalyzed,becausethedatacanbeexported,butnoprogramisavailable
toreadthem.Therefore,thefirststepwastowriteaprogramwhichconvertedtherawdataina
readableformat.FromacooperationoftheHospitalrechtsderIsarwithSiemens,information
wasavailableabouthowthefilesareorganized;forexample,thelengthofthemainheaderand
theframeheaders,thenumberofdetectorsperroworthenumberofviewsperrotationwere
known.Sincethestructureoftherawdataispartofthecooperatesecret,detailscannotbe
presentedhere.Usingthisinformationandaconventionalhex-editorallowedtoreadtheraw
data,theindividualframesandtheheaderinformation.Theexporteddataarepreprocessedby
scalingthelogarithmicattenuationvalues
IData=−clogI0(4.1)
wherecisaconstant,I0istheprimaryintensity,andIistheattenuatedintensity.

resolutionSpatialInCT,thestandardphantomformeasuringtheMTFisawireorabead.Therefore,atungsten
wireinairwasusedforthemeasurementsofthespatialresolution.Thewirehadadiameter
of0.08mmandwasspannedthroughthegantryparalleltotherotationaxis.Thisisawell-
establishedmethod[Kwa07,Har10,Wat10],whichallowsnotonlytoderivetheMTFfromthe
rawdata,butalsofromthereconstructedimages.

scannerCTClinical4.2.

47

WithregardtotheCTdOrgeometrythatcancurrentlyscanonlyonesliceperrotation,the
methodforanalyzingtherawdatawasdevelopedtobeindependentofthenumberofrows
ofthedetector.Therefore,thesinogramoftherawdatawasused,whichiscomposedofthe
centralrowofeachframe,independentofthetotalnumberofrows.Anexemplarysinogramof
thetungstenwireatadistanceofabout4cmfromtheisocenteroftheCTisshowninFig.4.3.
Thesinogramwascreatedrunningasequentialheadroutinewithasourcevoltageof120kVp,
asourcecurrentof300mAs,arotationtimeof1.0sandaslicethicknessof1.2mm.The
amplitudeofthecurveandrespectivelythenumberofpixelsnshadowedbythewiredepends
onthewire-isocenterdistance:Theshorterthedistance,theloweristheamplitude.Whenthe
wireispositionedexactlyintheisocenter,thecurvebecomesastraightline.

Figure4.3:Sinogramofthewireatadistanceofabout4cmfromtheisocenter.Thecolumns
correspondtodifferentviewswhiletherowscorrespondtodifferentdetectorpixels.Thered,
dashedlineindicatesthenumberofdetectorpixelsnwhichareshadowedbythewireina
rotation.complete

Thedark,verticalstripesareartifacts,probablyduetodirt(i.e.,contrastagent)onthegantry
surfaceduringthecalibration.Duringthemeasurementthegantrywasclean,becausethere
arenolightartifactsinthesinogramthatwouldbeproducedbyobjectsinthebeamduetothe
preprocessingoftherawdata.Thediagonallinesoriginatefromasmallslitinthecoversheet
ofthegantry.Thefirst,steeperone,isproducedwhenthesourcepassestheslit,thesecond,
atterone,isproducedwhenthedetectorpassestheslit,becausethedetectorislargerand
thereforeneedsmoreviewstopass.However,bothtypesofartifactsdonotconsiderablydistort
thecalculationoftheMTF,becausetheyareveryfaintintheareaofthewire.
Fujitaetal.[Fuj92]firstmeasuredtheoversampledMTFin2-dimensionalradiographyusing
aslantedleadslit.Sinceawireisnothingbutanegativeslit,asimilarmethodwaschosenin
thiswork.However,severalstepsarenecessarytoconvertthecurvedsinogramofthewireinto
adatasetsuitableforthismethod.UsingtheparametersinTab.4.1,thecalculationoftheMTF
wasperformedinthefollowingsteps:

48

4.ocessingprData

Table4.1:Parametersofthegeometry
ParameterAbbreviationDimension
Distancesource-detectora104cm
Distancesource-isocenterb57cm
Detectorpixelsizes1.41mm
NumberofdetectorpixelsinarowNpix672
NumberofviewsNviews1160
Diameterofthewirew0.08mm

1.Thepositionofthewirewasdeterminedineveryrow.
2.Thenumberofdetectorpixelsnthatareshadowedbythewireinacompleterotationwas
4.3).(Fig.aluatedve3.Thedistanceofthewirefromtheisocenterwascalculatedwiththecoordinatesdefined
inFig.4.4.Themaximumarclengthlmaxwasdeterminedfromthenumberofshadowed
pixelsnofsizesl=n∙s(4.2)
max2Thefactor2isnecessary,becauseonlyhalfofthearcisneededforthefurthercalcula-
tions.Byusingthesource-detectordistancea,themaximumangleαmaxinradianwas
withcalculatedαmax=lmax.(4.3)
aThedistancerofthewiretotheisocenteristhen
r=b∙sinαmax.(4.4)
4.Theheel-effectwascorrectedbyfittingasecondorderpolynomialtoeveryrowandsub-
tractingthefittedcurvefromthisrow.
5.Theslightlyslantedalignmentofthewirewascorrectedbydeterminingtheangleβof
thewireineachframe.Sincetheareacoveredbythetiltedwireisaboutarccosβlarger,
thevalueofthepixelmappingthewirepositioninthecorrespondingrowofthesinogram
ismultipliedwithcosβ.
6.Thechangingdistancebetweenthewireandthedetector,whichcausesprojectionsof
differentsize,wascorrectedusingtheparametersdefinedinFig.4.5.Chordcinevery
pointoftherotationisdefinedfromthepositionjofthewireineveryrow
c(j)=2∙a∙sin(|j−N2pix|−0.5)∙s.(4.5)
a10∙

scannerCTClinical4.2.

Figure4.4:Coordinatesusedtocalculatethedistancewire-isocenterdistance

49

(4.6)

(4.7)

α(j)isthengivenby
sinα(j)=2−→α(j)=arcsinc(j).(4.6)
c(j)
a2a∙Foreveryjtherearetwopossiblepositionsofthewire
γ1(j)=arcsinc(j)∙b
ra2∙∙c(j)∙b(4.7)
γ2(j)=π−arcsin2∙a∙r
Thedistancexbetweensourceandwirethereforecanalsohavetwovalues
b∙sinγ1(j)
x1(j)=sin(π−α(j)−γ1(j))
b∙sinγ2(j)(4.8)
x2(j)=sin(π−α(j)−γ2(j))
Thewireamplitudeofeachviewismultipliedwithoneofthecorrectionfactorsk1,2
k1,2(j)=x1,2(j).(4.9)
bdependingonwhetherthewireisclosertothesourceortothedetector.Thisinformation
isgivenbythefirstderivativeofthecurvethatdefinesthepositionofthewireforeach
view.Ifthederivativeisnegative,thewireisonthesideofthesourceandk1isapplied;
ifthederivativeispositive,thewireisonthesideofthedetectorandk2isapplied.

50

Data4.ocessingpr

Figure4.5:Coordinatesusedtocalculatethedistancebetweensourceandwire

7.Eachrowwasshiftedsothatallpixelsmappingthewirepositionsareonthecentral
el.pixdetector8.Therowsweresortedbythevaluesoftheirmaxima.Thesumofthecentralpixelvalue
pluseithertherightneighboringpixelortheleftneighboringpixelgivesthetendencyof
thewireposition.Therowwiththehighestmaximumispositionedinthecenterandthe
maximumvaluesdecreasetobothsides,wheretherowsbelowthecenterhaveatendency
totheleftwhiletherowsabovethecenterhaveatendencytotheright.Thisway,awire
iscreatedwhichentersthecentralpixelonthefirstrowandleavesitonthelastrow.The
resultingsinogramaftershifting,correctingandresortingthedataisshowninFig.4.6.
9.Theoversampledlinespreadfunction(LSF)wascreatedbyreformingthearrayina1-
dimensionalvectoraddingonecolumnontheendofthenextone[Fuj92,Buh03].The
resultingLSF(Fig.4.7)isthereforeNpix∙Nviews=779520pixelslong.
10.TheMTFwascalculatedbytakingtheabsolutevalueoftheFouriertransformedLSF.In
ordertoemphasizethetrendofthecurvesandtominimizetheuctuationsproducedby
highnoise,theMTFwassmoothedwiththeadjacentaverageover9points.

11.Thecorrespondingspatialfrequencyuwascalculated.Themaximumfrequencyisgiven
bytheNyquistfrequencyof
112∙s=0.355mm(4.10)
12.Thespatialfrequencywascorrectedfortheedgeobliquityby
u2ucorr=cosπ−arctanNviews.(4.11)

(4.11)

Clinical4.2.scannerCT

51

Figure4.6:Sinogramaftershifting,correctingandresortingthedata.Again,thecolumns
correspondtodifferentviewswhiletherowscorrespondtodifferentdetectorpixels.

Figure4.7:LSFsampledfromthesinogram

13.TheMTFwascorrectedforthefinitewidthwofthewirebydividingbythesincfunction
kwire=sin(π∙w∙ucorr).(4.12)
π∙w∙ucorr
14.Valuessmallerthan0.05mm1werecutoff,becausehighnoiseontherawdataledtolarge
uctuationsoftheMTFintherangefrom0to0.05mm1.FortheNPSitiscommonpractice
nottostartthepresentationofthevaluesat0mm1[Han79],butfortheMTFthishasbeen
ag79].[Wtoodone15.TheMTFwasnormalizingto1at0.05mm1.

52

prData4.ocessing

NoiseInprojectionradiography,theconventionalmethodtocalculatetheNPSuses2-dimensional
ROIs.Themethoddevelopedinthisworkincludesasmuchslicesasavailableandaverages
them.WithregardtoitsapplicationtotheCTdOrtechnologyinthefuture,themethodisstill
independentoftheactualnumberofslices.Thecalculationswereperformedinthefollowing
steps:1.Trendcorrectionwasperformedbyfittingasecondorderpolynomialtoeachdetectorrow
ofeveryframeandsubtractingthefittedcurvefromthisrow.
2.A256×NslicespixelROIwastakenfromthemiddleofeveryrow.256pixelsaresufficient
tocoveralsolong-rangecorrelationsaccordingtotheinternationalIECstandard[IEC03].
3.TheFouriertransformedwascalculatedofeachROIofeveryframe.
4.AllrowsofaROIwereaveragedtoproducethe1-dimensionalNPSaccordingtoamethod
developedbyPadgettetal.[Pad05].
5.Thecorrespondingspatialfrequencywascalculated.
6.Thehighuctuationsintherangefrom0to0.05mm1wereremoved.

4.2.2.Analysisofthereconstructedimages
ImagesoftheclinicalCTscannerwerereconstructedwithvariousreconstructionfilters:
-B10s:thesmoothestavailablefilter
-B30s:thestandardfilterforthescanprotocolabdomen
-B40s:anotherfrequentlyusedfilter
-B70s:thesharpestavailablefilter
SincetheCTscanneriscalibratedforthewaterphantom,nocorrectionsforthecuppingeffect
hadtobeapplied.ButsincetheCTdevicesavesthepixelvaluesinHounsfieldunitsplus1024
tooptimizethedatastorage,thisvaluehadtobesubtractedfromthepixelvalues.
Theslightlytiltedtungstenwireallowstogeneratetheoversampledpointspreadfunction
(PSF)accordingtoKwanetal.[Kwa07]byusingall30acquiredslices.Therefore,asmall
regionofinterest(ROI)of17×33pixelsaroundthewirewascutoutofeveryimage.The
centerofmassalongthex-axis(CMx)foreachpointspreadfunction(PSF)wasthencomputed
asxyROI(x,y)
CMx=xyROI(x,y)∙x,(4.13)

scannerCTClinical4.2.

53

wherexandydefinethespatiallocationofthepixelintheROI.TheCMxvaluesforall30
imageswereplottedasafunctionoftheslicenumber.Alinearfunctionwasfittedtothedata
todefinethecentersofthePSFalongtheCTslices.ThePSFofeachslicewasgeneratedby
integratingoverthe17rowsofthecorrespondingROI.EachPSFwasthenshiftedbythefitted
positionofthecenterofmass.AssemblingthePSFsofeverysliceyieldedtheoversampledPSF.
ByanalogywiththeFDAsystem,itspeakwasfittedwithapolynomial,whilethetailswere
fittedwithanexponentialdecayfunction.Theresulting,smoothedPSFwasrotatedtogeneratea
2-dimensionalPSF,whichwasusedtocalculatetheH-matrixwiththree-foldhigherresolution.
TheMTFwascalculatedfromthe1-dimensional,fittedPSFbyFouriertransformation.
Fortheestimationofthenoise,210slicesofthewaterphantomweretaken,whichprovided
amaximalwaterregionof266×266pixels.Thisarraywasagainsubdividedinto32×32pixel
ROIsthathadtobeseparatedbyseveralpixelstoapproximateindependentdatasets.Therefore,
theautocovariancefunctionwascalculatedforeachreconstructionfilter(Fig.4.8).TheROI
spacingwaschosetobethepixeldistanceatwhichtheabsolutevalueoftheautocovariance
functionhaddroppedto1%ofitsmaximum.Asexpected,thesmoothestfilterhadthelongest
rangingcovarianceofupto8pixels.Thisvaluewashenceforwardusedasspacingbetween
theROIsfortheimagesofallfilterstomakesurethatalwaysthesamenumberofROIswas
employedforthecalculationofthecovariancematrix.

Figure4.8:Autocovariancefunctionforthedifferentreconstructionkernels

ForcalculatingtheNPS,thesameROIswereFourier-transformedandaveragedtogetthe
2-dimensionalNPS.The1-dimensionalNPSwasagaincreatedbyradiallyaveraging.
TheSNRforbothapproacheswascalculatedaccordingtoSec.2.3usingasimulateddisk
withanamplitudeof50HUandadiameterof2mm.
Dividingthe266×266pixelwaterregionin32×32ROIswithaspacingof8pixelsresults
in7560ROIsforthewholeimagesandonly840ROIsforeachofthe9squareregions.A

54

prData4.ocessing

bootstrappingcalculationwasperformedtoestimatetheerrorintroducedbyusingsubstantially
lessROIsthanforthelaboratoryCTsystem.Theimage-spaceSNRanditsstandarddeviation
for10runswascalculatedfromsubsamplesofsizen=1500,2500,5000and7500fornoise
imagestakenat10.47mGyandreconstructedwithB30s.

4.3.CTdOrcombinedwiththeC-armdevice

TheexistingprogramsforthereconstructionofimagestakenwiththeCTdOrdemonstratorin
combinationwiththeC-armdevicehadtobeadaptedtoeachdatasetindividually.Sincethis
isaverytimeconsumingprocessandtheresultsdependstronglyontheskillsandpatienceof
theperformer,completelynewprogramswerewrittenfortheimagereconstruction1.Thedata
arenowsampledautomaticallyforeachdatasetbasedonthegeometryoftheset-up.Forthe
developmentoftheimprovedalgorithms,adatasetproducedbyHugodelasHerasforhisPhD
thesiswasused.ThisdatasetwasgainedbyascanoftheAldersonRandophantomatasource
voltageof96kVpandasourcecurrentof1.0mAsperimage.Thesubsequentcomparison
betweentheimagesgeneratedwiththeoldandthenewsoftwareisgoingtodemonstratethat
theimagequalitycouldbeimprovedconsiderably.

vicedeC-armofDataTheC-armdeviceprovidesmorethan4000imagesoftheCTdOrringinonerotation.Inorder
togettherawdatanecessaryforimagereconstruction,alineinthemiddleofthedetectorrow
oftheCTdOriscutoutofeveryimageasshowninFig.4.9a.Anexampleforapartofsucha
rawdatasetacquiredusinganAldersonRandophantomispresentedinFig.4.9b.Eachrhomb
correspondstoonedatapointofthefandatamatrixandthereforehastobeintegrated.The
locationoftherhombsisdeterminedautomaticallybasedonthemathematicsinSec.A.1.1.
Forthisalgorithm,thegeometryoftheset-uphastobeknownexactly.Intheactualmea-
surementset-up,threeparameterscouldhowevernotbedeterminedwithasufficientaccuracy:
thestartingpositionrelativetothering,thesource-to-isocenterdistanceoftheCTdOrandthe
anglebetweenthecentralrayofthefanandtheraywhichpassesthroughtheisocenteroftheCT
dOr.Theprogramcalculatedtheexpectedlocationofthedataforallpossiblecombinationsof
thesethreevaluesandcomparedtheresulttothemeasureddataset.Theparametercombination
withthehighestagreementwiththedatawasthenusedtocreateaso-calledmask.Thisisan
arrayofthesamesizeastheinputdataset,whererelevantpixelsareindicatedby1,whileother
pixelsare0.Thefandataweresampledatthepositionsdefinedbythemask.
Inanextstep,thefandatamatrixwasconvertedintothesinogram.Sincethedetectedsignal
istheoreticallythesame(ignoringscatterradiationandbeamhardening)forrayspassingthe

1TheprogramsweredevelopedtogetherwithDr.OlegTischenkofromtheDepartmentofRadiationPhysicsand
DiagnosticsoftheHelmholtzZentrumMünchen.

4.3.CTdOrcombinedwiththeC-armdevice

55

(b)(a)Figure4.9:ImageoftheC-armdeviceacquiredusinganAldersonRandophantom(a).The
redlineindicatesthelinecutouttogettherawdata.Anextractionofthecorrespondingraw
(b).setdata

windowsfromonesideandthosepassing180°fromtheotherside,thesedatawereaveragedto
decreasethenoise.Thedatawerelogarithmizedaccordingto
DeragedvaDlogged=−logmax(Daveraged).(4.14)
ForthedatasetoftheC-armdevice,onlytherayspassingthroughtwowindowsoftheCT
dOrringcanberecorded.Therayswhichpasstheoreticallyfromdetectortodetectorarealso
requiredforthereconstructionandthereforehavetobeinterpolated.Inthepast,aninterpolation
with1-Dsplineswasused,butforthiswork,severaldifferentmethodsweretested:
-Averagingoverthefouradjacentpixelsonthecornersofthemissingone.
-Interpolatingusingallexistingdatabyweightingeachpixelwithasincfunctionsinxx,
wherexisthedistanceofthepixeltotheinterpolatedone.
-Interpolatingwith1-Dsplines.

-Interpolatingbyweightingwiththesinc2function.
ForthereconstructionofimagesoftheAldersonRandophantom,theinterpolationwiththe
sinc2functionturnedouttoresultinthebestimage.
Thelaststepwastoreorderthefandatamatrixintofan-paralleldataandperformtherecon-
structionusingOPED.Acomparisonbetweenimagesreconstructedfromthesamerawdataset
withtheold(a)andthenewsoftware(b)isshowninFig.4.10.Thenewimageisnoticeably
sharperandshowsmoredetailsthantheoldone.Forexample,theinternaloccipitalprotuber-
ance(redring)isclearlyvisibleintheimagereconstructedwiththenewmethod,whileitis

56

✗✔✖✕

✗✔✖✕

(a)

✗✔✖✕

✗✔✖✕

✗✔✖✕

ocessingprData4.

✗✔✖✕(b)

(c)Figure4.10:ComparisonoftheimagesoftheC-armdevicereconstructedwiththeoldsoftware
(a),thenewsoftwarefortheC-armdevice(b)andthefurtherimprovedsoftwarefortheCT
(c).scanner

blurredintheoldimage.Thesameholdsforthesmallbonesinfrontofthesinusesmaxillaris
(greenring).Thebasilarpartoftheoccipitalboneinthecenterregionoftheskulllooksalso
sharperinthenewimage,butitisdifficulttodecidewhetherthisisreallyduetoahigherimage
qualityoronlyaresultofthesevereartifacts,whichstilldecreasetheimagequalityconsid-
erably.However,thestreakartifactsandtheringaroundthephantomaresupposedtovanish
whenthedatasetiscombinedwiththedataofthemaskdetectors.

detectormaskofDataBesidesthedescribedproblemswiththegeometricaluncertainties,themaskdatahavethead-
ditionaldifficultythattheyareobtainedin13series:12seriesover30°and1seriesover5°,in

4.3.CTdOrcombinedwiththeC-armdevice

57

ordertocorrectfortheinaccuracyoftherotationtable.Figure4.11ashowsanexemplaryraw
datasetforanemptyscanover30°.Thecolumnscorrespondtothe197maskdetectorsand
therowscorrespondtotheviews.Thebroaderstripearetheusabledatameasuredbydetectors
irradiatedfromthefront.Theslimmerstripecorrespondstodetectorswhichwereirradiated
fromthebacksidethroughtheshieldingswhiletheyweredirectlyinfrontofthesource.Each
datasetstartsandendswithviewsduringwhichneitherthesourcewasrunningnorthetable
wasrotating.Thenthesourcestarted,butthetablewasnotyetmoving,therebyproducinga
verticalpatterninthedata.Onlytheviewswhichhaveaslantedpatternareusableforthere-
construction,becausetherethetablewasrotating.Sinceareliablealgorithmtoautomatizethe
selectionofthedatasetscouldnotberealized,itstillhadtobedonemanually,althoughthisis
atime-consuminganderror-proneprocedure.

(d)(c)(b)(a)Figure4.11:RawdataoftheCTdOrintheC-armdevice(a),thecorrespondingmask(b),the
fandataset(c),andtheresultingsinogram(d).Pleasenotethatforbettervisualizationtheraw
datasetshowsanemptyscan,whilethefandataandthesinogramcorrespondtoascanofthe
phantom.

Intheseselecteddata,thepositionsoftherelevantdatapointsweredefinedwiththemathe-
maticspresentedinSec.A.1.2.ByreconstructingthedataoftheC-arm,theexactpositionof
theCTdOrrelativetothesourcewasobtained.Butstilltherewerethreeunknownparameters
left:therelativepositionofthefirstdetectortothesource,thecoordinateontheringandthe
timelagbetweensourcepulseanddetectorreadout.Forallpossiblecombinationsofthesethree
values,theexpectedlocationofthedatawasagaincalculatedandcomparedtothemeasured
dataset.Sincethethreeparametersaredifferentforeachofthe13datasets,thisfittinghadto

58

ocessingprData4.

bedoneforeachdatasetindividually.Therefore,13differentmasksweregeneratedofwhich
oneisshowninFig.4.11b.Inordertocreatethecompletefandataset,12ofthefandatasets
sampledonthedefinedpositionshadtobeadded.The13thfandatasetwasonlyneededpartly
tocompletethe360°rotation.Figure4.11cpresentstheresultingfandatasetofthescanof
theAldersonRandophantom.Itrevealsthattherowscorrespondingtodifferentdetectorshave
largelyvaryingvalues.Bynormalizingeachrowtothesamemeanvalue,thisdrawbackcould
bewidelyovercome.About10lineshadtobesetto0andinterpolatedbycubicsplines,because
theybelongedtobrokendetectors.Thesinogram(Fig.4.11d)wasproducedbyreorderingthe
correctedfandatatofan-paralleldataandlogarithmizingthem.
TheimageswerereconstructedwithOPED.Figure4.12showsthecomparisonbetweenan
imagereconstructedwiththeoldsoftware(a)andwiththenewsoftware(b).Bothimageshave
aboutthesamesharpness,buttheoldimagehasahighercontrastbetweenskullandairregions
thanthenewone.However,theoldimagehasanartificialringaroundthephantom,which
doesnotappearinthenewone.Thestreakartifactsontheedgeoftheskull(redcircle)arealso
fainterinthenewimagethanintheoldone.

CombinationsetsdatabothofForthereconstructionofCTdOrimageswithOPED,therearetwopossibilitiestocombine
thetwodatasets:combiningthefandatabeforethereconstructionoraddingthereconstructed
imagesappropriately.Combiningthefandataeliminatesmoreartifacts,butduetothedifferent
energydependencesofthedetectortypes,concentricringsoccurintheresultingimage.Due
tothehighresolutionofthenewimagefromtheC-armdata,justaddingtheimagesresulted
inabetterimagethancombiningthefandata.Andmostartifactscanceledeachotheroutwith
thismethodtoo.Figure4.13showsthecomparisonbetweenthebestimagesobtainedwiththe
old(a)andthenewsoftware(b).Itisimportanttopointoutthatthebestimagewiththeold
softwarewascreatedbycombiningthefandata,whilethebestimageofthenewsoftwarewas
producedbyaddingtheimagesofthetwodatasets.
Figure4.14showsthattheresultingimagehaslessartifactsthantheimageobtainedwith
theC-armdevice(a)andthatthecontrastisbetterthanintheimageobtainedfromthemask
detectors(b).Consideringthedrawbacksofthedemonstrator(Sec.3.4),theresultingimageis
remarkablygoodanddemonstratesthepotentialoftheCTdOr.However,alotofoptimization
done.betohasstill

4.4.CTdOrcombinedwiththeclinicalCTsystem

reconstructionegIma4.4.1.AscanoftheAldersonRandophantomwasusedtodevelopalgorithmsforanoptimaltreatment
ofthedataobtainedbycombiningtheCTdOrdevicewiththeclinicalCT.Thedataofthemask

4.4.CTdOrcombinedwiththeclinicalCTsystem

(a)

✤✜✣✢

(c)

✤✜✣✢

(b)

✤✜✣✢

59

Figure4.12:Comparisonoftheimagesofthemaskdetectorsreconstructedwiththeoldsoft-
ware(a),thenewsoftwarefortheC-armdevice(b)andtheadaptedsoftwareforthecombination
(c).scannerCTthewith

detectorswerecollectedwiththeminimumavailablesamplingtimeof1ms.Withamaximum
availablerotationtimeoftheCTscannerof1.0s,thisenabled1000viewsandthereforethe
besttimeresolutionwhichcanbeachievedwhencombiningthecurrentCTdOrdevicewith
thisCTscanner.Thesensitivityofthemaskdetectorshoweverissolowthatthissamplingtime
resultedinamaximumcountrateofabout36countspermsinanemptyscanandonly1or
2countspermsbehindanobject.Toreducetheresultingnoise,severalconsecutiverotations
wereaveraged.Sincethedatastoragesystemofthedetectorsislimitedtoonly5000viewsand
duetodelaysatthestartandtheend,somedatagotlost,onlyfourcompleterotationscould
beaveraged.Themeasurementsweredoneintheperfusionmode,becausethisistheonly
modewhichallowscontinuous,repeatedscansonthesameposition.Thescanswererunwith

60

(a)

(c)

(b)

ocessingprData4.

Figure4.13:Comparisonoftheresultingimagesobtainedbycombiningthefandatacreated
withtheoldsoftware(a),byaddingtheimagescreatedwiththenewsoftwarefortheC-arm
device(b)andfortheCTscanner(c).

aFOVof50cm,asourcevoltageof120kVp,asourcecurrentof300mAs,andarotationtime
of1.0s.Theminimalavailableslicethicknessintheperfusionmode,whichis2.4mm,was
used.Theseareparameterstypicalforacranialscanandthereforeoptimalforthephantom.

detectorCTofData

Onecentrallineofeachframewasusedtocreatethesinogramoftherawdata.Inorderto
maximizetheresolution,bothpositionsofthez-yingspotwerecombined.Anexamplefora
rawdatasetoftheCTdevicecombinedwiththeCTdOrringisshowninFig.4.15a.
ThedatawereanalyzedinasimilarwayasthedataoftheC-armdevice,butaccounting
additionallyforthecurvatureofthedetectorandthez-yingfocalspot.Themathematicsused

4.4.CTdOrcombinedwiththeclinicalCTsystem

61

(c)(b)(a)Figure4.14:AddingtheimagesoftheC-armdevice(a)andthemaskdetector(b)resultsinan
imagewithmoredetailsandlessartifacts(c).

(b)(a)Figure4.15:ExemplaryrawdataoftheCTdevicecombinedwiththeCTdOr(a)andthe
createdmask(b)forascanoftheAldersonRandophantomataFOVof50cm.

forthealgorithmcanbefoundinSec.A.2.TherelativepositionandtheangleoftheCTdOr
ringtotheCTwasnotknownexactlyandthereforehadtobemodulatedsimilartotheway
describedinthelastsection.Maskswhichdeterminedthepositionofthedatapointslooked
likeFig.4.15b.ThefandatasetsampledwithitispresentedinFig.4.16a.The1000views
actuallymeasuredhaveagainbeenbinnedto197effectiveviewswhichisequaltothenumber
ofdetectors.Thesinogram(Fig.4.16b)wascreatedinthesamewayasfortheC-armdevice
andtheimageswereagainreconstructedwithOPED.
Figure4.10cshowstheresultingimage.Itscontrastisslightlylowerthanintheimagere-
constructedfromthedataoftheC-armdevice(Fig.4.10b),butthecontoursofthephantom
areclearerandtheartifactsarelessdisturbing.Thisispartlyduetoafurtherimprovementof
thealgorithm,whichavoidsapproximationsatexpenseofalongercalculationtime.Another
reasonaretheoptimizedscanparametersfortheCTscan,whileforthescanwiththeC-arm

62

ocessingprData4.

(b)(a)Figure4.16:Exemplaryfandataset(a)andresultingsinogram(b).

devicestandardparameterswereused.Butstill,artifactssuchasthestreaksonthesideofthe
skull(redcircle)decreasetheimagequality.

detectormaskofDataAnexampleforanaveragedrawdatasetofascanoftheAldersonRandophantomataFOVof
50cmisshowninFig.4.17a.Thelighterstripecorrespondstodetectorswhichwereirradiated
fromthebacksidethroughtheshieldingswhentheyweredirectlyinfrontofthesource.The
exactpositionoftheCTdOrringinrelationtotheCTscannerwasknownfromtherecon-
structionoftheCTdata,buttherelativepositionoftheindividualdetectorstothesourceand
therelativepositionofthesourcetotheshieldingsandwindowshadtobedetermined.Using
theseparametersthepositionsatwhichthedatahavetobesampledwerecalculatedaccording
toSec.A.2.TheresultingmaskfortheexemplarydatasetisshowninFig.4.17b.Thethereof
createdfandataset(Fig.4.17c)wastreatedinthesamewayasforthecombinationwiththe
C-armdevicetoconvertthemintothesinogram(Fig.4.17d)andreconstructwithOPED.
TheresultingimageispresentedinFig.4.12c.Thecontourscanbeseenclearly,butthenoise
degradestheimagequalitydrastically.Thisisontheonehandduetothelowsensitivityofthe
maskdetectors.Astheyarenotbuilttomeasurewithasamplingtimeof1ms,thishighnoise
levelhadtobeexpected.TheimagederivedwiththeC-armdevice(Fig.4.12b)isconsiderably
betterduetothebetterstatisticswithasamplingtimeof20ms.Itisthereforeexpectedthatthe
imagetakenwiththeCTscannerwouldbeatleastasgoodastheonefromtheC-armdeviceif
used.weredetectorsmaskbetterOntheotherhand,thenumberofsamplestakenperrotationdifferedconsiderablyforthetwo
set-ups.WhencombiningtheCTdOrdemonstratorwiththeC-armdevice,morethan7000
samplesperrotationwerecollectedbythemaskdetectors,comparedtoonly1000samplesfor
thecombinationwiththeclinicalCTduetothelimitationsbythemaximumrotationtimeof
theCTsourceof1.0s.
Inordertofurtherincreasethestatistics,severalimagescanbeadded.Thisconsiderably
reducesthenoise,butwithanincreasingnumberofimagesthefinalimagegetsmoreandmore

4.4.CTdOrcombinedwiththeclinicalCTsystem

(a)

(b)

(c)

(d)

63

Figure4.17:ExemplaryrawdataoftheCTdOrringintheCTscanner(a),thepositionsat
whichthedataweresampled(b),thefandataset(c),andtheresultingsinogram(d)forascan
oftheAldersonRandoPhantomataFOVof50cm.

blurred,duetosmallinaccuraciesinthedatasampling.Anotherpossibilityisthenoisereduc-
tionalgorithmdevelopedbyTischenkoetal.[Tis05].Itcomparestwoimagestodistinguish
betweeninformationandnoiseandsubsequentlytocreateanoisereducedimage.Thismethod
worksquitewell,buttheaimofthischapterwastodevelopadataprocessingwhichcanbe
usedfortheanalysisoftheimagequality.Anartificialincreaseoftheimagequalitywould
manipulatetheresultsandthereforebecounterproductive.Moreover,bothmethodsneedmore
thanoneimagetoproducethefinalimage.Sinceitwasalreadydifficulttogetenoughmeasure-
menttimeintheclinictoproducethenecessaryimages,itwouldhavebeenalmostimpossible
toproducetwiceasmuch.Andasthedatahavetobeindependentofeachotherforthenoise
analysis,acorrectionamongeachotherwouldneitherbesuitable.Therefore,theoriginal,but
rathernoisyimage,wasusedforthefurtheranalysis.

databothofCombinationsets

AddingtheimageofthemaskdetectorstotheimagereconstructedfromtheCTrawdatagives
theimageshowninFigure4.18c.Thestreakartifactsonthesideoftheskullarereduced(red
circle),butthecombinedimageissuperimposedbythenoiseofthemaskimage.Thecontrastis

64

prData4.ocessing

slightlyworseforthecombinedimagethanfortheimageobtainedfromtheCTrawdataascan
beseenusingtheexampleoftheoccipitalbone(greencircle).Itisthereforenotclearwhether
thecombinedimageisreallybetterthantheimageobtainedfromtheCTdataforthisdata
set.ComparedtotheimageobtainedwiththeC-armdevice,theimagefromtheCTscanner
(Fig.4.13c)isconsiderablynoisierandhasaworseresolutionsothatlessdetailsarevisible.
Butthiswasalsoreectedinthetimeittooktoacquireoneimage.WhilefortheC-armdevice
ittookaboutthreehours,fortheclinicalCTscanner,theimageacquisition,includingthedata
transfer,wasfinishedinthreeminutes.

✤✜✣✢✤✜✣✢

(a)

✤✜✣✢✤✜✣✢

(c)

(b)

Figure4.18:ReconstructedimagesfromtherawdataoftheCTscanner(a),themaskdetectors
(b)andthecombinationofboth(c).

4.4.CTdOrcombinedwiththeclinicalCTsystem

65

4.4.2.Analysisofthereconstructedimages
Thespatialresolutionwasdeterminedfromcombinedimagesofa1.0mmsteelwire,which
waspositionedatdifferentdistancesfromtheisocenter.Theimagesofthetwodatasetswere
addedbynormalizingthemtothemaximumofthewire.Theresultingimages(Fig.4.19a)show
regularpatterns,whichareduetothedatasamplingwiththecalculatedmask.Thestepsizein
whichthefourunknownparametersarecheckedhastobechosenasanoptimumbetweenaccu-
racyandcomputationaleffort:Thesmallerthestepsize,thehigherthechancetogetanoptimal
fittotherawdata,butthelargerthecomputationtime.Furthermore,thealgorithmassumesa
perfectalignmentoftheshieldings.TherealCTdOrringishoweverslightlyirregularsothat
theagreementbetweentherawdataandthecalculatedmaskcanneverbeperfect.Thepatterns
appearthereforeineveryimagesampledwiththenewalgorithm,butaremaskedpartiallyby
highersignalsforimagesoftheAlderson-Randoandthewaterphantom.
ThePSFcouldnotbeoversampledfromthewireimages,becausetheCTdOrdemonstrator
canonlyproduceoneslice.AccordingtoFuchsetal.[Fuc01],the1-dimensionalPSFwas
insteadsampledbyrotatingthewireimageinstepsof0.5°andaveragingtheresulting720
profiles.Sincerotatingthereconstructedimageinsertspixelinaccuracies,theprofileswere
reconstructedindividuallyforeveryrotationwithanadjustedversionoftheOPEDprogram.
Bydefiningthecenterofthewire,thesamplingintervalandtheangle,anyprofilethroughthe
wirecanbereconstructedwiththehighestavailableaccuracy.
InthesamewayasforthelaboratoryandtheclinicalCTsystem,thepeakoftheresulting
1-dimensionalPSFwasfittedwithapolynomialfunctionandthetailswerefittedwithanex-
ponentialdecayfunction.TheresultingPSFwasrotatedtogeneratethe2-dimensionalPSF,
whichwasagainusedtocalculatetheH-matrixwiththree-foldresolution.TheMTFwascal-
culatedfromthe1-dimensional,fittedPSFbyFouriertransformationandcorrectedforthewire
thicknessdbydividingitthroughthesincfunction:
F(u)=sin(πud),(4.15)
udπwhereuisthespatialfrequency.Forthespatialapproach,nocorrectionforthewirethickness
performed.aswThenoisewasanalyzedinimagesofthePMMA-slice(Fig.4.19b)reconstructedasdescribed
abovewithapixelsizeof0.488×0.488mm.Forthecombinationofthedatasets,theimages
werenormalizedtothemeanvalueofthemaximumhomogeneousregionof285×285pixels.
Inthecombinedimage,theaveragevalueofthewaterregionthereforewas1,whiletheaverage
valueofairwasaround0.ThecalibrationtoHounsfieldunitswasperformedbysubtracting
1andmultiplyingby1000sothattheaveragevalueofthewaterregionbecame0,whilethe
averagevalueofairwasabout-1000.Trendcorrectionwasperformedbyfittinga2-dimensional
second-orderpolynomialtothe285×285pixelROIsandsubtractingthisfunctionfromthe
OIs.R

66

(a)

(b)

ocessingprData4.

Figure4.19:Examplesforreconstructedimagesofthewire(a)andthePMMA-slice(b).

Sincetheacquisitionandreconstructionoftheimageswasverytime-consumingandthemea-
surementtimeintheclinicwaslimited,only30imagesofthewaterphantomweregenerated.
Theresultingautocovariancefunction(Fig.4.20)revealedthattherangeofpixelrelationswas
quitelongandnotthesameforthehorizontalandtheverticaldirection.Thisisprobablydue
totheregularpatternsintroducedintheimagesbythedatasampling.Inordertoaccommodate
therelationsinbothdirections,themaximumnecessarydistanceof10pixelswaschosen.With
aROIsizeof32×32pixelsandaROIspacingof10pixels,30imagesprovidedonly1470
ROIs;byfarnotasmuchasfortheothersystems.Butthisnumberallowedatleasttodetermine
all1024eigenvaluesofthecovariancematrixsothatitwasinvertibleandtheimagespaceSNR
couldbecalculated.Itwashowevernotpossibletocalculatethelocation-specificnoiseforthe
images.dOrCTThespatialresolutionatthethreedifferentdistancesfromtheisocenterandtheaveraged
noiseforthewholeimagewereusedtocalculatetheSNRaccordingtotheimage-spacebased
approachandtheFourierbasedapproach.

4.4.

CT

dOr

eFigur

combined

4.20:

with

the

clinical

ariancevAutoco

CT

function

system

for

the

horizontal

and

the

erticalv

direction

67

measurementsDose5.

Thequalityofanimagingsystemisdefinedbytheimagequalityandthedoseneededtoachieve
it.ThischapterdescribesthedosemeasurementsperformedfortheFDAsystem(Sec.5.1)and
theCTdOrdemonstratorincombinationwiththeclinicalCT(Sec.5.2).

5.1.FDAlaboratorysystem

InordertomaketheresultsofthelaboratoryCTsystemcomparabletoexistingdataofconven-
tionalsystems,thedoseforeachofthefourmAs-settingswasmeasured1.Therefore,alidfor
thecylinderwasconstructed,whichintroducedtwowatertightPMMA-tubesinsidethewater
phantom,oneinthecenterandoneontheperipheryofthelid(Fig.5.1).Toavoidanairgap,the
diameterofthetubescorrespondedpreciselytothediameterofthe0.6ccFarmer-type10×5-
0.6Radcalionizationchamber(RadcalCorp.,Monrovia,USA)usedforthemeasurements.The
tubesallowedtheplacementofthesensitivepartoftheionizationchamberinthecentralplane
ofthewater-filledcylinder.Duringeachdosemeasurement,theemptytubewasfilledwitha
PMMA-rodtoreducetheeffectsoftheairgap.Theelectrometerusedtoreadoutthemeasured
dosesisaRadcal9010(RadcalCorp.,Monrovia,USA).Foreverycurrentintensity,thedose
wasmeasuredinthecenterandattheperipheryofthecylinderthreetimesineachcase.Ta-
ble5.1liststheaveragedresultsofthemeasurementsanditsstandarddeviations.Thevariation
betweenthethreerunsforeverysettingwereapparentlyverysmall.InFig.5.2,themeasured
centerdoseisplottedagainstthemAsperimage.Thelinearfit(dashedline)showsthatdue
tomeasurementinaccuraciesthecenterdosefor1.28mAsperimageisslightlytoolow.This
hastobetakenintoaccountwhenanalyzingthedependenceoftheimagequalityonthedosein
6.1.Sec.

1DosemeasurementswereperformedwiththehelpofSamirAbboudfromtheFDA

69

70

ementsmeasurDose5.

Table5.1:Overviewofthescansandthemeasureddoses
mAsperprojectionTotalmAsCenterdosePeripherydose
[mGy][mGy]0.625225.09.8±0.010.7±0.1
1.28460.817.8±0.119.7±0.1
2.00720.028.4±0.130.9±0.2
2.50900.035.1±0.138.4±0.2

(b)(a)Figure5.1:LidwithtwowatertightPMMA-tubesandadditionalPMMA-rod(a)andthecylin-
(b).lidthiswithder

Figure5.2:RelationbetweenthemAsperimageandthemeasureddoseatthecenter.The
linearfit(dashedline)indicatestheexpecteddeveloping.

5.2.CTdOrcombinedwiththeclinicalCTscanner

71

5.2.CTdOrcombinedwiththeclinicalCTscanner

TLDstheofCalibration5.2.1.ForthedosemeasurementsintheclinicalCT,thermoluminescencedosimeters(TLDs)oftype
TLD-100Hwereused,whichwereprovidedandanalyzedbytheAuswertungsstelleofthe
HelmholtzZentrumMünchen.TheseTLDshaveadiameterof4.5mmandaheightof0.9mm
andarebasedoncrystallinelithiumuoride(LiF),whichhasbeendopedwithsmallquantities
ofmagnesium(Mg),copper(Cu)andphosphor(P).ThecharacteristicsofTLD-100Hthatare
particularlyusefulforradiationdosimetryincludehighsensitivity,almostatphotonenergy
response,alowfadingrateandalineardoseresponse[Mos06].
TheTLDswerecalibratedusingtheradiationemittedby137Cs.Whentheyareirradiatedwith
adifferentspectrum,themeasureddoseshavetobecorrected.Thesecorrectionfactorcanbe
calculatedbydividingthecalibrationfactorsforthedoseequivalentHp(0,07;N)2measuredby
theAuswertungsstellewiththecorrespondingconversionfactorforairkermafromISO4037-
3(Table18)[ISO99].Figure5.3providesthecalculatedcorrectionfactorswithwhichthe
measureddosevalueshavetobemultipliedtogettheactualdoses.Afactorof0.83resultsfor
62.6keV,themeanenergyofthecalculatedCTspectrumat120kVp.SinceN-spectrahavea
completelydifferentenergydistributionthanCTspectra,thisvaluehoweverwasnotusedfor
thecalibration.Theplotismerelysupposedtogiveanimpressionoftheenergydependenceof
TLDs.the

Figure5.3:Correctionfactorfordifferentmeanenergies

2Doseequivalentinadepthof0.07mmforN-spectraandarodphantomconsistingofICRUtissueatareference
m2ofdistance

72

ementsmeasurDose5.

Instead,thecalibrationfactorwasmeasuredintheSecondaryStandardDosimetryLaboratory
(SSDL)oftheHelmholtzZentrumMünchen3.Therefore,50TLDswereirradiatedwitha
spectrumcomparabletotheCTspectrum,whichwasproducedusingtheinformationthatthe
totalfiltrationonthecentralaxisequals6.8mmaluminumfortheheadscan[ImPACT06].
SimilartotheCTscans,asourcevoltageof120kVpwasused.Ashutterrightafterthesource
ensuredthatexactly30mGywereappliedtotheTLDs.Thiswasconfirmedbyirradiatinga
30cm3ionizationchamber(Unidos10001-10003,PTWFreiburg,Germany)witha3mmthick
plexiglassdiskinfrontofittosimulatethefrontsideoftherackwhichheldtheTLDs.The
plexiglassrackwasloadedtwicewith25TLDs,placedatthesamepositionastheionization
chamberbeforeandirradiatedagainwith30mGy.
TheaveragedosemeasuredwiththeTLDswas32.53mGywithastandarddeviationof
0.94mGyor2.9%.ThisresultedinacalibrationfactorkQof
30kQ=32.53=0.92.(5.1)
TogetherwiththecorrectionfactorsinFig.5.3,thisresultdemonstratesthattheTLDsused
forthisworkaresuitedfordosemeasurementsinenergyrangescorrespondingtothosein
scanners.CTclinical

measurementDose5.2.2.DoseintheCTdOrsystemwasmeasuredusingthesliceoftheAldersonRandophantompre-
sentedinSec.3.4.TheAldersonRandophantomhas16holesof4.8mmdiameterfordose
measurements,whicharenormallyfilledwithPMMArods.Theuncertaintyofthemeasure-
mentswasreducedbyplacingtwoTLDsineveryhole.InordertopositiontheTLDsinthe
middleoftheslice,32PMMArodsof4.8mmdiameterand1.2mmheightweremanufactured
tofixtheTLDsfrombothsides.Thephantomandthepositionsoftheholesareshownin
5.4.Fig.FordosemeasurementswiththeCTdOrring,firstthephantomwithoutTLDswasplaced
inthedeviceandsomeimagesweretakenwiththeminimalavailableslicethicknessforhead
scansof18mm.Withthehelpoftheseimages,itwasensuredthattheCTdOrringwasaligned
correctlyandthatthex-rayconedidnothittheplateatthebacksideoftheCTdOrring,while
thewholephantomwasirradiated.ThentheTLDswereplacedinthephantom,whichwas
fixedwithdouble-sidedadhesivetapetotheplateonwhichtheCTdOrringismounted.A
sequentialheadscanwith2.0srotationtime,120kVpand300mAswasperformed.Ithastobe
notedthatthenominalrotationtimeof2.0sfortheheadscanmeansthatthereare2rotations
with1.0srotationtimeeach.Afterthemeasurement,theTLDswerestoredseparatelyinsmall,
numberedplasticbagstobeabletoassignthemtotherightpositions.

3ThesemeasurementswereperformedundertheadviceofWernerPanzerofthesameinstitute.

5.2.CTdOrcombinedwiththeclinicalCTscanner

Figure5.4:PositionsoftheTLDsintheAlderson-Rando-phantom

73

FordosemeasurementswithouttheCTdOrring,thephantomwasstucktothefrontsideof
thewaterphantom,whichisnormallyusedtomeasurethenoiseintheCTscanner(compare
Sec.3.2).Again,itwasensuredthatthephantomwaspositionedaccuratelyandthatthex-
rayconedidnothitthewaterphantombeforetheTLDswereinserted.Theirradiationwas
performedusingthesameparametersasforthemeasurementwiththeCTdOr.Directlyafter
themeasurements,thenumberedbagswheresenttotheAuswertungsstelleforreadout.
ThemeasureddosevaluesweremultipliedwithkQ=0.92toobtainthecalibratedvalues
listedinTab.5.2.ThecalibrationfactorkQmeasuredinSec.5.2.1howevercorrespondstothe
spectrumonthecentralaxisinair,becausetheonlyinformationavailableaboutthefiltration
intheclinicalCTisthatitisequalto6.8mmaluminum[ImPACT06]inthecenter.Usingthis
spectrumisonlyaroughestimationwhichdoesnotallowtodeterminetheexactcalibration
factorfortheTLDsduetoseveralreasons.First,theImPACTreportdoesnotstatedefinitely
whetherthisfiltrationalsoincludesthebowtiefilter.Ifnot,anadditionalfiltrationwouldharden
theactualspectrumonthecentralaxis.Second,thespectrumwillbecomeharderwithincreas-
ingdistancefromthecentralaxis,becauseoftheincreasingthicknessofthebowtiefilter.Since
theexactshape,aswellasmaterialandthickness,ofthebowtiefilterintheclinicalCTscan-
nerispartofthecooperatesecret,thespectraawayfromthecentralaxiscannotbesimulated.
Third,thephantomtissuechangesthespectrumadditionally.Butduetotheinherentfiltration
andthebowtiefilter,thespectrumisalreadysohardthatthesechangesplayaminorrole.

ementsmeasurDose5.74Table5.2:MeasureddoseswithandwithouttheCTdOrringandtheratioofthetwomea-
surementsNumberofholeDosewithCTdOrDosewithoutCTdOrratio
[mGy][mGy]0.3918.967.4210.3719.036.9820.3714.175.2830.3712.474.6240.3614.145.0950.3520.517.1560.3811.924.5370.3610.833.9080.3612.004.2890.3912.774.94100.3611.193.99110.3719.497.29120.4014.906.03130.3815.005.77140.4119.738.17150.4118.657.6816Averageofallratios0.38
Duetothesereasons,theactualcalibrationfactorisestimatedtobeupto0.2smallerthan
themeasuredone,becausethemeanenergiesoftheactualspectraareinarangewherethe
efficiencyoftheTLDsincreasesabruptly.However,thefocusofthisworkliesontherelative
dosecomparisonbetweentheCTscanneronitsownandthecombinationofCTscannerand
CTdOrdevice.SinceithasbeenshowninSec.5.2.1thattheTLDsaresuitablefortheapplied
energies,theactualcalibrationfactorisofminorinterest.
Table5.2showsthattheaveragedratioofthemeasureddosevaluesis0.38.Asdescribedin
Chapter3.4,theleadshieldingsoftheCTdOrringhaveawidthof5mmwhilethewindows
areonly3mmwide.So85=0.625ofthecircumferenceareshieldedand83=0.375are
windows.Theaveragedratioofthedosescorrespondsexactlytothefractionoftheunshielded
partssuggestingthattheshieldingsabsorb100%ofthephotons.
Butsimulatingthespectraonthecentralaxiswithandwithout0.5mmlead(Fig.5.5)reveals
thephotonabsorptiontobeonlyabout93%.Thethinnerpartsonthecornersoftheshieldings
(compareFig.3.8)absorbevenless.Theincompletephotonabsorptioncanalsobeseenin
therawdataoftheCTdOrdevice,wherethecountrateofdetectorsirradiatedfromtheback
sidethroughtheshieldingsisalwaysincreased(Fig.4.11aandFig.4.17a).Thiseffectshould

5.2.CTdOrcombinedwiththeclinicalCTscanner

75

addatleast7%ofthefractionoftheshieldedpart(0.625)tothefractionoftheunshielded
part(0.375),resultinginanexpectedratioofabout0.42betweenthedosewithandwithoutCT
.dOrAnotherissueonehastobeawareofisthatthespectrumishardenedbythemaskandthere-
forethedetectabilityefficiencyoftheTLDsforthatcontributionisfurtherincreased.The
simulatedspectrainFig.5.5showthatthemeanenergyofthespectrumonthecentralaxis
increasesduetothefiltrationwith0.5mmleadfrom62.6keVto74.9keVandthattheshield-
ingsalmostcompletelyabsorbenergieshigherthan88keV,whichcorrespondstotheK-edge
oflead.Butthiseffectissupposedtobemostremarkableinthecenter,sincethespectrumon
theperipheriesisalreadyhardenedbythebowtiefilter.Theharderthespectrum,thelessthe
changecausedbyanadditionalfiltrationsuchastheshieldings.

Figure5.5:CTspectrumatthecenterwherethefiltrationcorrespondsto6.8mmaluminum
(blue)andwithanadditionalattenuationof0.5mmlead(red)

ThereductionofscatteredradiationduetothespecialgeometryoftheCTdOrcanpossibly
counterbalancetheincompleteabsorptionbytheshieldings.Sinceonlyasmallamountof
radiationpassestheshieldings,thescatteredradiationintheshieldedpartsoftheobjectis
small,andthecontributionofscattertodirectlyirradiatedpartsislow.Thiseffectisexpected
tobecomeevenmoreimportantforlargervolumes(e.g.,inmulti-sliceCT),becausethefraction
ofscatteredradiationisincreased.Foraquantitativeestimation,appropriatesimulationswould
havetobeperformed,butthistaskisbeyondthescopeofthiswork.
Ithastobepointedoutthattheresultofthesemeasurementsisonlymeaningfulforthe
demonstratoroftheCTdOr.AclinicalapplicableCTdOrissupposedtohaveatleast1000
detectorsandthereforesmallerandprobablythinnerleadshieldings,resultinginahighertrans-
missionofx-rayphotons.Simulationsshowedthatthisissuecanbesolvedbyusingtungsten
orrheniuminsteadoflead.Theirabsorptionrateishigherintherelevantenergyrangeduetoa
higherdensity,whichmorethancompensatesfortheloweratomicnumbers.Eventhoughmore

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5.ementsmeasurDose

expensivethanlead,tungsten-rheniumalloysarewidelyusedinindustrysothattheyareavail-

ableatareasonableprice.Theresulting

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6.Analysisofimagequality

ThischapterpresentstheresultsobtainedbyanalyzingtheimagesofthedifferentCTsystems
usingbothapproaches,theimage-spacebasedandtheFourierbased.Inordertodecidewhich
methodprovidesamorerealisticdescriptionoftheimagingsystem,theresultswereadditionally
comparedtomeasuredimages.ThesectionsabouttheFDAsystem(Sec.6.1)andtheclinical
system(Sec.6.2)demonstratethepotentialoftheimage-spaceandtheFourierapproachandthe
imagequalityachievablebyconventionalCTgeometries.Section6.3thenpresentstheimage
qualityanalysisfortheCTdOrdemonstratorincombinationwiththeclinicalCT.

6.1.FDAlaboratoryCTsystem

resolutionSpatial

ThespatialresolutionoftheFDAlaboratorysystemwasmeasuredwithamolybdenumfoiland
theimagedatawereprocessedasdescribedinSec.4.1.Figure6.1presentstheprofileofthe
fitted,2-dimensionalPSF,whichwasusedtocalculateboththeH-matrixandtheMTF.

Figure6.1:Profileofthefitted,2-dimensionalPSFusedtocalculatetheH-matrixandtheMTF.

ThesingularvaluesoftheH-matrixsortedindescendingorder(a)andthecorresponding
MTF(b)ofthesystemarepresentedinFig.6.2.Thecurveshavedifferentabsolutevalues,
becausetheMTFisnormalizedto1.Alsotheshapeofthecurvesdiffersconsiderably,as

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theslopeofthesingularvaluesismuchsteeperthantheMTF.Thisindicatesthatthespatial
resolutionofthesystemisworseaccordingtotheimage-spacebasedapproachthanaccording
totheFourierbasedapproach.

Figure6.2:SingularvaluesoftheH-matrix(a)andthecorrespondingMTF(b)ofthelaboratory
CTsystem.Pleasenotethatthescalingonthey-axesisnotthesame.

Thesignaltransferthroughtheimagingsystemwasvisualizedbysimulatingtwoobjectsand
applyingeithertheH-matrixortheMTF:adiskmodelingadensemassinsidealowerdensity
regionandatorusmodelingadensemasswithafattymargininsideahigherdensityregion
(Sec.4.1).Figure6.3andFig.6.4comparethetransferoftheinputsignalsf(a)throughthe
imagingsystemascalculatedwiththeH-matrix(b)andtheMTF(c).Forbothsignalsthe
resultingimageissignificantlyblurredwhenoperatingHonf.Theringseemstobemuch
broaderthanitreallyisandtheholeinthecenterisnotvisible.Theimagesgeneratedwiththe
MTFaremuchsharper,asindicatedbythegentlerslopeoftheMTFcomparedtothesingular
valuesoftheH-matrix.Apossibleexplanationforthisdifferencearetheverylongtailsofthe
scannerPSF,extendingbeyond32×32pixelsor1.6mmrespectively.Thereasonforthese
fairlylongtailsisthelargefocalspot(0.6mm)sothatthefocalspotunsharpnesswidensthe
PSF[Kyp05b].BasedonthePSFimage,ROIsintheorderof100×100pixelswouldbeneeded,
resultinginaH-matrixof104×9∙104pixels.Thissizesmakesitdifficulttomanipulateandto
analyzethematrixbysingularvaluedecomposition.ThepresampledMTFapproachdoesnot
havethisissue,becauseitcapturesthelong-rangetailsatthelowfrequencies.
Inordertoevaluatewhichapproachprovidesamorerealisticdescriptionoftheimaging
system,measuredimagesofanobjectcomparabletoasimulatedsignalarenecessary.Since
therewasnoappropriatephantomavailableforthelaboratoryCTsystem,thisdiscussionhad
tobepostponedtotheclinicalCTsystem(Sec.6.2.2).

6.1.FDAlaboratoryCTsystem

79

(a)(c)(b)Figure6.3:Comparisonofthetransferofthesimulateddisk(a)usingtheimage-spaceapproach
(b)andtheFourierapproach(c).

(c)(b)(a)Figure6.4:Comparisonofthetransferofthetorus(a)usingthespatialapproach(b)andthe
(c).approachourierF

NoiseThecovariancematrixaswellastheNPShave1024eigenvectors,because32×32pixelROIs
wereusedfortheanalysisofthenoise.TheeigenvectorsoftheFouriertransformationareby
definitionexponentialwavefunctions,whiletheimage-spacebasedapproachobtainsoptimized
eigenvectorsfromtheeigenanalysisofthecovariancematrix.Thesamenumberofeigenvectors
ofthecovariancematrixisthereforeexpectedtoprovideabetteroratleastequalrepresentation
oftheimagingsystemastheeigenvectorsoftheFourierapproach.Forabettervisualization,
theeigenvectorsoflength1024havebeenreformedto32×32arrays,whichisjustifiedsince
thecovariancematrixtheydefinewascalculatedfromROIsofthissize.Figure6.5showsthe
first64eigenvectorsofthecovariancematrixforthexy-(a)andtheyz-plane(b)orderedwith
decreasingmagnitudeoftheircorrespondingeigenvalues.Whiletheeigenvectorsofthexy-
planeareirregularwithnopreferreddirection,theeigenvectorsoftheyz-planeshowaclear
verticalpattern,whichindicatesverticalcorrelationsintroducedbythe1-dimensionalnatureof
.filterreconstructiontheInordertovisualizethedifferencebetweentheeigenvectorsets,theexponentialwavefunc-
tionswhichdefinetheFouriertransformationwerecalculatedforthe2-dimensionalNPS.To

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makethemcomparabletotheeigenvectorsofthecovariancematrix,thevaluesofthe2-di-
mensionalNPSweresortedindescendingorderandtherealpartofthefirst64eigenvectorsis
presentedinFigure6.6.Duetothissortingthefirsteigenvectorisnottheonewiththelargest
wavelength,butthedominatingeigenvector.
Thecomparisonrevealsthattheeigenvectorsofthecovariancematricesdonothavethe
regularstructureofexponentialwavefunctions.

(b)(a)Figure6.5:First64eigenvectorsofthecovariancematrixfor35.1mGyinthexy-plane(a)and
in(b).yz-planethe

Figure6.6:Calculatedfirst64eigenvectorsoftheNPS.

Figure6.7presentstheeigenvaluesofthecovariancematrix(solidsymbols)sortedinde-
scendingorderforthexy-plane(a)andtheyz-plane(b).Additionally,thevaluesofthe2-
dimensionalNPSarelistedindescendingorder(opensymbols).Thecomparisonisonlyqual-
itative,becauseeigenvaluesandNPSvaluesarenotbasedonthesamesetofeigenvectors.It
isthereforenotpossibletodirectlyinterpretthelowermaximumandslightlyhigherminimum

6.1.FDAlaboratoryCTsystem

81

Figure6.7:Eigenvaluesofthecovariancematrixandvaluesofthe2-dimensionalNPSsorted
indescendingorderforthexy-plane(a)andtheyz-plane(b).Aseverycurverepresents1024
values,thesymbolsdonotcorrespondtopoints,butareinsertedforbettercurveseparation.

valuesoftheNPS.However,theintegralofthe2-dimensionalNPS,aswellastheintegralof
theeigenvaluesofthecovariancematrix,issupposedtobeequaltothevarianceofthenoise.A
comparisonrevealsthattheintegralsofbothcurvesdifferbylessthan10−6%fromeachother
andarethereforepracticallyequal.Bothapproachesthusagreeonthetotalamountofnoise.
ThisfactwasprovedforCTimagesinthisworkforthefirsttime.
The1-dimensional,radiallyaveragedNPSisshowninFig.6.8forthexy-plane(a)andthe
yz-plane(b).Theanti-correlatednoise,characteristicfortheaxialdirectionofCTscanners,
generatesatypicalcurvefortheNPS,similartotheonesreportedbyWagner[Wag79]and
Hanson[Han79].TheNPSofthexy-planestartsatahighmagnitudeatfrequenciescloseto
zero,hasaslightdropatlowfrequencies,increasesatmid-frequencyrangeanddecreasesto
zeroathigherfrequencies.Forthesamedoselevel,thebumpinmid-frequencyrangeisstrongly
dependentontheutilizedreconstructionfilter,whichmeansthatthemoreedge-enhancingthe
reconstructionfilter,thehigheristhebump.Sincethereconstructionalgorithmusedforthis
workismainlyfocusedonoptimizingtheaxialimages,itdoesnotperformanedgeenhance-
mentfortheyz-planeandthecorrespondingNPSthereforedecreasesmonotonically.

noiseLocation-specificInordertoanalyzethelocation-specificnoise,thexy-cylinderplanewassubdividedintonine
equallysized,squareregionsasshowninFig.6.9aforanimagetakenat35.1mGy.The
covariancematrixandtheNPSofeachregionwerecalculatedtoobtainthelocation-specific
eigenvectorsandeigenvalues.Figure6.9bshowsthe1steigenvectorsatthepositionsofthe
correspondingregions.Theresultingstarlikepatternoftheeigenvectorsillustratestheradially
symmetricnatureoftheFBPreconstructionalgorithm.Therefore,thecentralverticalregion
exhibitsverticallysymmetriceigenvectors,thecentralhorizontalregionexhibitshorizontally

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Figure6.8:NPSforthexy-plane(a)andtheyz-plane(b).

(b)(a)Figure6.9:Anoiseimagetakenat35.1mGywaspartitionedinto9squareregions(a)andthe
1steigenvectorswerepositionedatthecorrespondingregions(b).

symmetriceigenvectors,thediagonalregionsdiagonal,andthecentralregionhasnodirectional
preferencebecausethecontributionsofalldirectionsaresuperimposed.
Thecorrespondingeigenvalues(a)andtheNPS(b)inFig.6.10revealthatthemagnitudeof
noiseishigherinthecentralregionthanintheperipherals,whichisreasonableasthecontrast
isnormallyhigherinthecentralregionofimagesreconstructedwiththeFBPalgorithm.

SNR

SNRtheofComparisonForthesimulateddiskandthetorus,theSNR2isplottedinFig.6.11forbothapproaches.The
image-spaceSNRisforbothsignalssmallerthantheFourierSNR.Thiscanbeexplainedbya

6.1.FDAlaboratoryCTsystem

83

Figure6.10:Eigenvalues(a)andNPS(b)fordifferentregionsofanimagereconstructedfrom
datatakenat35.1mGy.Aseverycurvein(a)represents1024eigenvalues,thesymbolsdonot
correspondtopoints,butareinsertedforbettercurveseparation.

betterspatialresolutionaccordingtotheMTFthanaccordingtotheH-matrix,whilethenoise
wasthesameforbothapproaches.Furthermore,theFourierSNR2islowerforthetorusthanfor
thediskeventhoughtheirsurfaceareaisthesame,whilethespatialSNR2isonlyslightlylower
forthetorusthanforthedisk.TheappliedFouriertransformationseemstobemoredependent
ontheshapeofthesignal,whilethematrixmultiplicationperformedbythespatialapproach
dependsmainlyontheactualsignalarea.
TheSNR2isnotexactlylinearwiththedose,becauseofthenotexactlylinearrelationbe-
tweenthemAs-settingandthedose(compareSec.5.1).Thisexplainsthedeviationforthe
FourierSNR,buttheimage-spaceSNRadditionallyshowsadropinthelastvalue.Sincethis
effectisthesameforbothtasks,itcannotbecausedbythesignalbutprobablybythegeneration
.PSFtheofesholdthrDetectionAsdescribedinSec.2.3,thedetectabilityofanobjectequalstheSNRforaSKE/BKEsituation
andthedetectionthreshold(p=0.75)canberelatedtotheSNRthroughanerrorfunction.In
ordertodemonstratethepracticalbenefitoftheSNR,thedetectionthresholdasafunctionof
theareaandtheamplitudeofadiskisshowninFig.6.12.DuetothehigherSNR,smallerand
lighterdiskscanbedetectedaccordingtotheFourierapproachthanaccordingtotheimage-
spaceapproach.Suchachartcantheoreticallybeusedbytheradiologisttodeterminethe
necessarydoseforanexaminationdependingonthetask.
calculationBootstrappingWithaspacingoffourpixels,105800ROIswerecreatedfromthe877×877pixelwaterregion
ofthe200reconstructedslicestocalculatethecovariancematrixforthewholeimages.For
eachofthe9regions,only12800ROIswereavailabletodeterminethelocation-specificnoise.

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Figure6.11:SNRforthesimulateddisk(a)andthetorus(b).Linearfitsforthedatapointsare
shownwithblack,dashedlines.

Figure6.12:Detectionthresholdfortheimage-space(a)andtheFourier(b)approach.

InordertocheckwhetherthesenumbersofROIsareenoughtoproperlydefinethe1024×1024
matrices,abootstrappingcalculationwasperformedfor9.8mGyasdescribedinSec.4.1.
Asexpected,thestandarddeviationoftheSNRdecreasedwithanincreasingnascanbe
seeninFig.6.13a.Anexponentialdecayfittedthedatapointsverywell.Alsotheactualvalue
oftheSNRdecreasedexponentiallywithanincreasingnumberofROIs(Fig.6.13b).Thefit
convergedtoaSNRvalueof0.675foraverylargenumberofROIswhichisverycloseto0.673,
thevalueobtainedbyusingall105800ROIs.ForthelaboratoryCTsystem,105800ROIswere
thereforeenoughtoaccuratelydefinethecovariancematrixandenabledthecalculationofa
SNR.image-spacereliable

CTClinical6.2.system

85

For12800ROIs,thefitsinFig.6.13gaveastandarddeviationofabout0.7%andaSNR
valueof0.702,whichis4.0%higherthantheactualSNR.SincetheSNRdependsinversely
linearonthecovariancematrixandhenceonitseigenvalues,thelocation-specificeigenvalues
aresupposedtobe4.0%±0.7%lowerthantheactualvalues.Duetothelowstandard
deviation,thecovariancematricesaredefinedaccuratelyenoughforarelativecomparisonof
the9regionssuchasdoneinSec.6.1.

Figure6.13:Resultsofthebootstrappingcalculationsfor9.8mGy.Anexponentialdecayfitted
thestandarddeviation(a)aswellastheactualSNR(b).

systemCTClinical6.2.

FortheclinicalCTsystemnotonlythequalityofthereconstructedimageswasanalyzed,but
alsotheimagequalityoftherawdata.Thefollowingsectionthereforepresentsanewmethod
fortheanalysisoftherawdatawiththeFourierapproach,whichwasdevelopedwithfocuson
newCTgeometriessuchastheCTdOr.Subsequently,imagesoftheclinicalCTreconstructed
withvariousdifferentfilterswereanalyzed.

datawRa6.2.1.resolutionSpatialTheMTFoftherawdatawasdeterminedfordifferentwire-isocenterdistancesaccordingto
Sec.4.2.1.ThedistancesdenotedinFig.6.14werecalculatedfromtheCTrawdatawiththe
proceduredescribedinSec.4.2.1.TheMTFdecreaseswithanincreasingwire-isocenterdis-
tance,becauseofthebowtiefilteroftheCTwhichcompensatesfortheshapeofthehuman

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bodyorhead.Scatteringinthefiltercausesabroadeningoftheeffectivefocalspotsize,which
decreasestheMTF.Sincethefilterisbecomingthickerontheperipheries,thefractionofscat-
teredradiationincreasesandhencetheMTFdecreases.TheMTFcurvesdeterminedfromthe
CTrawdatathereforebehaveaccordingtotheory.

Figure6.14:MTFfromtherawdatafordifferentwire-isocenterdistances

NoiseTheNPSoftherawdataisshowninFig.6.15fordifferentsourcevoltagesandcurrents.As
expected,theNPSdecreaseswithincreasingsourcecurrent.Inordertoquantifythiseffect,the
averageNPSisplottedagainstthesourcecurrentonadouble-logarithmicscaleinFig.6.16a.
Thelinearfitsgivea√slopeofabout-1forallsourcevoltages.Noiseinprojectionradiography
isproportionalto1/N,whereNisthenumberofphotons.ButtheNPSisproportionalto
N2sothatitissupposedtodecreaselinearlywith1/N[Han79].Thiscorrespondsverywellto
slopes.fittedtheFigure6.15alsoillustratesthattheNPSdecreaseswithincreasingsourcevoltage.Thede-
pendenceoftheNPSonthesourcevoltageisplottedinFig.6.16b.Itisduetoanincreasing
outputofphotonswithanincreasingsourcevoltage.Simulatingthespectrafordifferentsource
voltagesUonthecentralaxisandintegratingthenumberofphotonsNgaveaproportionality
ofN∝(1+exp−cU)−1,wherecisaconstant.SincetheNPSisproportionaltoN2,itis
supposedtodecreasewithexp−U/c.Thiscorrespondsverywelltothedata,whichcanbefitted
withanexponentialdecayfunction.
NoticeableinFig.6.15arethepeaksforhighvoltageandhighcurrentsettingsatthespa-
tialfrequencies0.089mm−1,0.177mm−1and0.266mm−1.Thesefrequenciescorrespondto
11.2mm,5.6mmand3.8mmandaccordinglyto8,4and2.7timesthepixelsize.Allowing

systemCTClinical6.2.

Figure6.15:NPSfromtherawdatafordifferentsourcevoltagesandcurrents

87

forsmalldeviationsduetoaveraging,thepeakshaveaboutthesameheight.Forlowervoltages
andcurrents,theyarenotvisibleintheplotsbutmaskedbythequantumnoise.Ifthepeaksare
effectsofsomekindofstructure,firstaveragingoverallrowsandthencalculatingtheFourier
transformshouldincreasetheirheight.Testsshowedhoweverthattherelativeheightofthe
peaksstaysthesame.Furthertestsprovedthatthetrendremovalwithasecondorderpolyno-
mialdoesnothaveanyeffectonthepeaks.Whetherthecorrectionisdoneforthewholerow
oronlyfortheROIandwithwhichorderpolynomialisnotinuencingthem.Theseresults
suggestthatthereasonforthepeaksliesinthereadoutelectronics.Siemenshoweveronlycon-
firmedthattheyknowaboutthereasons,butthattheycannotrevealtheanswerwithreference
secret.corporatethetoThepresentedNPScorrespondstothenoiseinthecenterregionofthefanbeam.Dueto
thebowtiefilter,thenumberofphotonsreachingthedetectorinanairscandecreasesfromthe
centertotheperipheries.ThiscanbeseenintheNPS,ifitiscalculatedfromaROIontheedge
ofthedetector.ThecomparisonshowsthattheNPSattheperipheriesisabout3.7timeshigher
thaninthecenter,independentofthesideofthedetector.

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Figure6.16:DependenceoftheNPSonthesourcecurrent(a)andthesourcevoltage(b).
Linear(a)andexponential(b)fitsareshownasdashedlines.

Theresultspresentedinthissectiondemonstratedthatitispossibletoanalyzethehardware
ofaCTscannerbycalculatingtheMTFandtheNPSfromtherawdata.Theresultscorre-
spondedverywelltothetheoriesforspatialresolutionandnoiseinprojectionradiography.The
presentedalgorithmsarebothdesignedtoworkaswellwithdatafromtheCTdOrtechnology.
Therawdataobtainedwiththedemonstratorarehowevernotsuitableforaproperanalysis.
Thisdependsmainlyonthesmallnumberofdetectorswhichproduceasinogramofonlyabout
67×197pixels.Thesecondreasonistheunequalsizeofwindowsandshieldings,whichresults
indifferenteffectivepixelsizes.
AdrawbackofthismethodishoweverthatthereadoutofCTrawdataisverycomplicated,
sincetheactualstructurevariesnotonlybetweendifferentmanufacturers,butalsobetween
differentmodelsofthesamemanufacturer.Anewreadoutprogramwouldhavetobewritten
foreveryscanner,whichwouldbenearlyimpossiblewithoutinformationthatissubjecttoa
secret.corporate

esgimaReconstructed6.2.2.resolutionSpatialThePSFcurvesoftheclinicalCTsystemwerefittedfollowingtheproceduredescribedin
Sec.4.2.Figure6.17showsthefittedPSFcurvesforthedifferentreconstructionfiltersand
Table6.1liststhecorrespondingparameters.Asexpected,thefull-width-at-half-maximum
(FWHM)ofthepeakscorrelateswiththesharpnessofthefilter:themoreedge-enhancingthe
filter,thenarroweristhePSFandthehigheristhepeakamplitude.

6.2.systemCTClinical

89

ButeventhoughthePSFcurveswerederivedfromimagesofthesameobject,thearea
underthe2-dimensionalPSFisnotequalforthedifferentfiltersbutdecreaseswithincreasing
sharpnessofthefilter.Wecheckedthatthistrendisnotonlyaresultofthefitting,butcanalso
befoundintheoriginalmeasuredimages.

Figure6.17:FittedPSFforthedifferentreconstructionfilters

Table6.1:ParametersofthefittedPSFs.Filtersareorderedwithincreasingsharpness.
FilterFWHM[mm]Maximumvalue[HU]
55617.3B10s74195.6B30s82445.0B40s114304.0B70s

Figure6.18showsthesingularvaluesoftheHmatrix(a)andtheMTF(b)forthedifferent
reconstructionfilters.Thecurveshavedifferentabsolutevalues,becausetheMTFisnormalized
tooneatzerofrequencyaccordingtotheIECnormforprojectionradiography[IEC03],while
thesingularvaluesarenormalizedsuchthatthesingularvectorsareorthonormal.TheMTFat
aspecificfrequencyisthemagnitudewithwhicha2-dimensionalcosinewavefunctionofthe
samefrequencyistransferredthroughtheimagingsystem,whilethesingularvaluescorrespond
tothemagnitudewithwhichthesystemssingularvectorsaretransferredthroughtheimaging
system.DuetothenormalizationoftheMTFat0frequency,theMTFofB10s,B30sandB40s
havevalueshigherthan1.Thisdoesnotmeanthattheamountofinformationishigheronthe
outputofthesystemthanontheinput.Thesmoothkernelsseemtosuppresstheveryhigher
frequencies,whilethesharpestkernelhashigherMTFatthelargerfrequencies.
Forbothapproachesholdsthatthesmootherthefilterthelowertheintegralofthecorre-
spondingcurve.Butrankingthemaximumvaluesgivesadifferentpicture:B40shasthehigh-

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Figure6.18:SingularvaluesoftheH-matrix(a)andthecorrespondingMTF(b)oftheclinical
scanner.Pleasenotethatthescalingonthey-axesisnotequal.

estmaximumvalue(0.43,1.07)forbothapproaches,followedbyB30s(0.37,1.05),whilethe
maximumvaluesofthesmoothestandthehardestfilter,B10s(0.33,1.03)andB70s(0.34,1),
equal.almostareThespatialresolutionishigheraccordingtotheFourierapproachthantothespatialapproach,
becauseofthestrongdecreaseofthesingularvaluesoftheHmatrixcomparedtotheMTF.
Thesimulatedsignaltransferofadiskwithadiameterof2mmandanamplitudeof50HU
provesthisassumption.TheimageinthetopleftcornerofFig.6.19correspondstoB10s,top
righttoB30s,bottomlefttoB40sandbottomrighttoB70s.Allimagesofoneapproachare
normalizedtothemaximumofthefourimages.ThesignalstransferedbytheH-matrixshow
aclearrelationwiththesmoothnessofthefilter.ThesignalstransferedbytheMTFaremuch
moresimilarforthedifferentfilters,thoughatrendfromasmoothertoasharpersignalisstill
visible.TheimagesobtainedfromB30sandB40sarehardlydistinguishableduetothevery
similarMTFcurves.Forallfilters,theresolutionpredictedbytheFourierapproachishigher
thanthatpredictedbytheimage-spaceapproach.
Inordertoevaluatewhichapproachgivesabetterdescriptionoftheimagingsystem,images
ofa3mmTeonpinwereused,gainedfromtheslicegeometryandsensitometrymoduleof
theCatphan®phantom[Cat06].A32×32pixelregionwascutoutofthereconstructedimage
aroundthepin.Adiskwiththesamediameterandanamplitudeof990HUcorresponding
toTeonwassimulated.TheH-matrixandtheMTFwereappliedtoitandmeasurednoise
wasadded.TheresultingimageswerethencomparedtothemeasuredimageoftheTeonpin
(Fig.6.20).TheexampleoftheB30sfilterat10.47mGyshowsthattherealimageisbetween
theimagesgeneratedwiththeH-matrixandtheMTF:Itislessblurredthantheimagegenerated

systemCTClinical6.2.

91

(a)(b)Figure6.19:Signaltransferaspredictedbytheimage-basedapproach(a)andtheFourier
approach(b).ThetopleftimagerespectivelycorrespondstoB10s,toprighttoB30s,bottom
lefttoB40sandbottomrighttoB70s.

withtheH-matrix,butitisnotassharpastheimagegeneratedwiththeMTF.Theprofilesofthe
transferedsignalsinFigure6.21confirmthisstatement.Forallfilters,theamplitudepredicted
bytheFourierapproachconsiderablyexceedstheamplitudeofthemeasuredsignalandthe
FWHMissmallerthanthemeasuredone.TheMTFthereforesystematicallyoverestimatesthe
performanceoftheimagingsystem.ForB10sandB30s,theamplitudeestimatedbythespatial
approachislowerandtheFWHMislargerthantheoriginal,resultinginanunderestimation
ofthesystemperformance.ForB40sandB70s,however,theamplitudecalculatedwiththeH
matrixisveryclosetothemeasuredamplitude,providingaconsiderablybetterdescriptionof
thesignaltransferthantheFourierapproach.

(c)(b)(a)Figure6.20:Measuredimageofa3mmTeonpinoftheCatphan®phantomacquiredwith
10.47mGyandreconstructedwithB30s(a).Thecorrespondingsimulatedsignalgenerated
withtheH-matrix(b)andwiththeMTF(c).

92

Noise

imageofAnalysis6.quality

Figure6.21:Profilesofthetransfered3mmTeonpin

Thefirst25eigenvectorsofthecovariancematricesforthedifferentfiltersat10.47mGyare
showninFig.6.22sortedwithdecreasingmagnitudeoftheircorrespondingeigenvalues.Since
thestructureofthenoisedependsonthefilterandonlytheheightofthenoisedependsonthe
dose,theeigenvectorslookrathersimilarfordifferentdosesettingssothatonly10.47mGyis
shownhere.Thepatternislargerforsmootherfilters,whileitisverysmall-structuredforB70s.
TheeigenvaluesofthecovariancematrixarepresentedinFig.6.23byusingagaintheex-
ampleof10.47mGy.Foraquantitativecomparisonalsothevaluesofthe2-dimensionalNPS
sortedindescendingorderareinserted.Asexpected,thesmootherthefilter,theloweristhe
noiseintheimages.ThesharpestfilterB70s,however,producesconsiderablymorenoisein
theimagesthantheotherfilters.Although,particularly,thehighernumbersarequitedifferent
for−3thetw2o2approaches,thedifferenceoftheintegralswasforallcalculatedcurveslessthan
10HUmm.Therefore,thetotalamountofnoiseisdescribedsimilarbybothapproaches.
Thedependenceofthe1-dimensionalNPSonthefilterandonthedoseisshowninFig.6.24
usingtheexampleof10.47mGy(a)andB30s(b)respectively.Foranincreasingsharpnessof
thefilternotonlytheheightofthecurveincreases,butalsotheshapechangestowardabumpin

systemCTClinical6.2.

93

Figure6.22:First25eigenvectorsforthedifferentreconstructionfiltersat10.47mGy:topleft
B10s,toprightB30s,bottomleftB40sandbottomrightB70s.

themid-frequencyrange.Fordifferentdosesettings,theshapeisthesame,butthemagnitude
ofthenoisedecreaseswithincreasingdose.

noiseLocation-specificInthesamewayasfortheFDAsystem,theimagesoftheclinicalCTscannerweredivided
into9squareregionsandtheirindividualcovariancematriceswerecalculatedandanalyzed.
The1steigenvectorsatthepositionoftheircorrespondingregionsareshowninFig.6.25.The
eigenvectorsofB10shaveabroadandsmoothstructure,whichgetsfinerwithanincreasing
sharpnessofthefilter.B30s,B40sandB70shaveanindistinctstarlikepattern,whichisvisible
butbyfarnotasclearasfortheFDAsystem.Bycontrast,B10sshowsnostarlikepatternatall,
butthestructureofthenoiseseemstobethesameforthewholeimage.
ThedependenceofthemagnitudeofnoiseisshownfortheextremefiltersB10sandB70s
inFig.6.26.InaccordancewiththesimilareigenvectorpatternsforB10s,thecorresponding
eigenvaluesdependlittleonthelocation.Themagnitudeofnoiseissensitivetothelocation
whensharperfiltersareapplied,showingthelargestdifferenceforB70s.Sincedividingthe

94

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Figure6.23:Eigenvaluesandthevaluesofthe2-dimensionalNPSsortedindescendingorder
.mGy10.47for

Figure6.24:VariationoftheNPSindependenceofthefilterat10.47mGy(a)andofthedose
(b).B30sat

266×266pixelwaterregionin9squareregionsresultedinonly840ROIsforall210slices,
only840eigenvaluesofthecovariancematrixcouldbedefined.Theremainingeigenvalues
wereessentiallyzerosothethecovariancematrixdidnothavefullrankandwasthereforenot
invertible(compareSec.6.2.2).

6.2.systemCTClinical

(a)

(c)

(b)

(d)

95

Figure6.25:1steigenvectorsatthepositionofthecorrespondingregionsfor10.47mGy:top
leftB10s,toprightB30s,bottomleftB40sandbottomrightB70s.

SNR

SNRtheofComparisonTheSNRforthesimulateddisk(diameter2mm,amplitude20HU,background-30HU)is
showninFigure6.27forthespatial(a)andtheFourier(b)approach.ThespatialSNRis
smallerduetoaworsespatialresolutioncalculatedfromtheHmatrixthanfromtheMTF.The
FourierSNRhasaboutthesamevaluesforthefiltersB30sandB40s.Almostequallysuited
isB10s,butB70sisconsiderablyworse.Forthespatialapproach,B40sisclearlybetterthan
B30s.NoticethatB70sexceedsB10s.Thisisaveryinterestingexampleofhowthefiltersare
describedbythedifferentapproaches:whilethespatialSNRclearlyfavorsB40sforthistask,
theFourierSNRratesB30sandB40sequallyandB10sisevaluatedtobenotmuchworse.
ComparingtheseresultswithFigure6.20andFigure6.21leadstotheconclusionthatforthe
harderfilters,B40sandB70s,thespatialSNRisexpectedtobemorerealisticthantheFourier

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Figure6.26:Location-specificeigenvaluesforB10s(a)andB70s(b).

SNR.Forthesmootherfilters,B10sandB30s,therealSNRliesprobablyinbetweenthetwo
approaches.

Figure6.27:SNRforthesimulateddiskfortheimage-space(a)andtheFourier(b)approach.
Linearfitsforthedataareshownwithblack,dashedlines.

ComparingtheseresultswithFig.6.20andFig.6.21leadstotheconclusionthattherealSNR
liesinbetweenthetwoapproaches,butisforB40sprobablyclosertotheimage-spaceSNR.

calculationBootstrappingThewaterregionofthe210slicestakenwiththeclinicalCTscannerhadasizeof266×266
pixels.Dividingthisareainto32×32ROIswithaspacingof8pixels(Sec.4.2)gave7560

6.3.CTdOrcombinedwiththeclinicalCTsystem

97

ROIsintotal.Abootstrappingcalculationwasperformed(n=1500,2500,5000and7500)for
noiseimagestakenat10.47mGyandreconstructedwithB30stocheckwhetherthisnumberof
ROIswaslargeenoughtoappropriatelydefinethecovariancematrix.
ThestandarddeviationaswellastheSNRdecreaseexponentially,approaching0.034and
4.21respectively(Fig.6.28).Theimage-spaceSNR,forwhichall7560availableROIswere
used,was3.97.Thedifferenceof0.24isproducedbytherandomsamplingwithreplacement,
whichisperformedbydefinitionforthebootstrappingcalculation.Itsuggeststhatthenumber
ofROIswasnotenoughforanappropriatedefinitionofthecovariancematrixsothatthere-
peatedcontributionofsomeROIsartificiallydecreasedthenoise.IftherewereenoughROIs,
thereplacementwouldnotchangetheresultandtheSNRcalculatedwiththebootstrapping
methodwouldconvergetotheimage-spaceSNRasitdidforthelaboratoryCTsystem.The
image-spaceSNRissupposedtobeclosertotherealvalue,becauseofthegoodagreement
oftheintegraloftheeigenvalueswiththeintegraloftheNPS.ThemagnitudeoftheNPSis
independentofthenumberofROIsusedforthecalculation,butthemoreindependentpixels
areused,thelessnoisyistheNPS.TheIECnorm[IEC03]requiresatleast4millionindepen-
dentimagepixelsforthecalculationoftheNPS.Since7560ROIscorrespondtomorethan7.7
millionimagepixels,theNPSisexpectedtobeaccuratelydetermined.
Whenpartitioningtheimagesin9regions,only840ROIscouldbecreated.Therefore,
only840eigenvaluesofthe1024×1024covariancematrixweredefined,whiletheremaining
184eigenvalueswerezero.Theresultingmatrixdidnothavefullrankandwasthereforenot
invertibleasitwouldbenecessaryforthecalculationoftheSNR.Thefits(Fig.6.28)suggesta
standarddeviationof27%andaSNRofabout20;476%oftheactualvalue.Thecorresponding
eigenvaluesaresupposedtobeaboutthisfactorlowerthantherealeigenvalues.Duetothe
highstandarddeviation,thecomparisonofthelocation-specificnoiseinSec.6.2.2hastobe
care.withinterpreted

6.3.CTdOrcombinedwiththeclinicalCTsystem

Sincethemethodselaboratedinthepreviouschaptersrequirealargenumberofimagesfor
theanalysis,theycouldnotbeappliedtotheCTdOrincombinationwiththeC-armdevice.
Instead,theanalysishadtobeperformedusingimagesfromthecombinationoftheCTdOr
demonstratorwiththeclinicalCTscanner,thoughSec.4.4showedthattheimagequalityhas
tobemuchlower.Therefore,theresultspresentedinthischapterarenotsuitedtodemonstrate
potentialadvantagesoftheCTdOrgeometryaboveconventionalsystems.Thischapteris
rathersupposedtostudytheapplicabilityoftheimage-spaceandtheFourierapproachtonew
CTgeometriessuchastheCTdOr.

98

qualityimageofAnalysis6.

Figure6.28:Resultsofthebootstrappingcalculations.Thestandarddeviation(a)aswellas
theactualSNR(b)decreaseexponentially.

resolutionSpatial

Thespatialresolutionwasmeasuredwithasteelwireof1.0mmdiameteratdifferentdistances
fromtheisocenter.Figure6.29presentsthefittedPSFcurvesforthethreesettings.Thecurves
werenormalizedandonlythevaluesbetween-50and+50pixelareshownsothatthevery
smalldifferencesbetweenthepeakshapescanbeseen.TheFWHMisthesameforallcurves,
butthelargerthedistancetotheisocenter,thesmootherthetransitionfromthepeaktothetails.
ThisisalsoreectedinthelowersingularvaluesoftheH-matrix(a)andintheMTF(b)in
Figure6.30.Thecloserthepeakistotheisocenter,thehigheristheMTF.Duetodifferentnoise

Figure6.29:NormalizedPSFcurvesatdifferentdistancesfromtheisocenter.

6.3.CTdOrcombinedwiththeclinicalCTsystem

99

patternsindifferentregionsoftheimages(Fig.4.19a),itcanhowevernotbeexcludedthatthis
effectisjustanartifact.Furthermore,thehighersingularvaluesbehaveinadifferentway.For
0.9cmdistancefromtheisocenter,theslopeofthecurveissteeperthanfor4.1cmand7.2cm,
whichdecreaseinaboutthesameway.Itisthereforeexpectedthattheimage-spaceSNRisbest
for4.1cm,followedby7.2cmand0.9cm.
Notablyistheabruptdrop-offafterthefirstvalueoftheMTF,whichisduetothesizeof
thewire.Whilethe0.08mmwireusedfortheclinicalCTwasapproximatelyapointobject,
the1.0mmwireusedintheCTdOrdoesnotfulfillthiscriteria.Amuchthinnerobjectcan
howevernotbedetectedbytheCTdOrdemonstrator.

Figure6.30:SingularvaluesoftheH-matrix(a)andcorrespondingMTF(b)

Thesizeofthewirehashoweveralsoitsadvantages,becauseitallowstousetheimage
directlyforacomparisonofthesignaltransferdescribedbythetwoapproaches.Thewire
wassimulatedasadiskwith1.0mmdiameterandtheH-matrixandtheMTFwereappliedto
itrespectively.SincethereconstructedimageofthewirewasnotcalibratedtoHU,onlythe
shapeofthepeaknottheamplitudewascompared.Forthisreason,nonoisewasaddedtothe
images.simulatedFigure6.31showsthemeasuredimageofthewire(a),thesimulatedsignalgeneratedwith
theH-matrix(b)andwiththeMTF(c).Therealimageisblurredquitesimilartotheimage
predictedbytheimage-spaceapproach,whiletheFourierapproachpredictsamuchsharper
image.Thisbecomesevenclearerwhencomparingtheprofilesoftheimagesasithasbeen
doneinFig.6.32.Themeasuredimageisnotsymmetric,buttheFWHMoftheprofilepredicted
bytheimage-spaceapproachisonlyslightlylargerintheoriginalimage,whiletheFWHM
obtainedwiththeFourierapproachismuchsmaller.Thisisunlikelytobeduetoaninaccurately
definedMTF,sincetheMTFissupposedtogetevenbetterforthinnerobjectsasitsdrop-
offwoulddisappear.FortheCTdOr,theimage-spaceapproachprovidesthereforeamuch

100

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betterdescriptionofthesystemthantheFourierapproach,whichconsiderablyoverestimates
performance.its

(c)(b)(a)Figure6.31:Measuredimageofthe1.0mmsteelwire(a),thecorrespondingsimulatedsignal
generatedwiththeH-matrix(b)andwiththeMTF(c).

Figure6.32:Normalizedprofilesofthetransfered1.0mmsteelwire.

NoiseNoiseinCTdOrimageswasdeterminedfrom30imagesofaPMMA-slice,whichwereused
tocalculatethecovariancematrixandtheNPS.Figure6.33presentsthefirst64ofthe1024
eigenvectorsofthecovariancematrixsortedwithdecreasingmagnitudeoftheircorresponding
eigenvalues.Thedominatingeigenvectorsdifferconsiderablyfromtheeigenvectorsoftheother
systemsaswellasfromexponentialwavefunctions,sincetheyhaveverylarge,butquiteregular
structures.Theyprobablydescribethenoiseproducedbytheartifactpatternsintheimages.
Startingatabouteigenvectornumber34,thestructurebecomessimilartotheeigenvectorsof
theothersystems.Thisexampleshowsthattheimage-spaceapproachprovidesaconvenient
methodforthecharacterizationofthenoisestructureinnewCTgeometriessuchastheCT
.dOr

6.3.CTdOrcombinedwiththeclinicalCTsystem

Figure6.33:First64eigenvectorsofthecovariancematrix

101

Figure6.34ashowstheeigenvaluesofthecovariancematrixandthevaluesofthe2-dimen-
sionalNPSsortedindescendingorder.Thecurvesdifferconsiderablyfromeachother,and,in
contrasttotheothersystems,theintegralsofthecurvesdotoo.ThesumoftheNPSvaluesis
morethandoublethesumoftheeigenvalues.Thiscorrespondstotheresultsofthebootstrap-
pingcalculationswhichrevealedthatthenoiseisunderestimatedbytheimage-spacebased
approach,whennotenoughROIswereusedtocalculatethecovariancematrix.
The1-dimensionalNPSinFigure6.34bhasitsmaximumatlowfrequenciesandsubse-
quentlydecreaseswithoutabumpinmid-frequencyrange,becausenoedge-enhancingfilter
wasusedforthereconstruction.

Figure6.34:ResultsofthenoiseanalysisoftheCTdOrimages:(a)Eigenvaluesofthe
covariancematrixandvaluesofthe2-dimensionalNPSsortedindescendingorderand(b)the
NPS.1-dimensional

102

SNR

qualityimageofAnalysis6.

Figure6.35givestheSNRfortheCTdOratdifferentdistancesfromtheisocenter.Asexpected
fromtheresultsforthespatialresolution,theFourierSNRdecreaseswithincreasingdistance
fromtheisocenter,whiletheimage-spaceSNRdoesnothaveatrend,butfollowstheorder
predictedbythesingularvaluecurvesoftheH-matrix.
Inthisspecificcase,themeasuredimage-spaceSNRislargerthantheFourierSNR.The
resultsofthebootstrappingcalculationsfortheothersystemsdemonstratedhoweverthatthisis
onlyduetothesmallnumberofROIsusedtocalculatethecovariancematrix.Inordertocorrect
forthiseffectbasedonthebootstrappingcalculationsfortheclinicalCTsystem(Sec.6.2.2),
themeasuredimage-spaceSNRhastobedividedby1.9resultinginacorrectedimage-space
SNRthatisagainsmallerthantheFourierSNR(Fig.6.35).

Figure6.35:SNRfortheCTdOrdemonstratorincombinationwiththeclinicalCT.Besides
themeasuredSNRforbothapproaches,alsothecorrectedimage-spaceSNRisshown.

TheanalysisofthenoiseandtheSNRcanthereforeonlybeaproof-of-principlethatthe
image-spaceapproachcanbeappliedtotheimagesobtainedwiththeCTdOrdevice.Butsince
theirquantitativeinformationisknowntobebiased,theabsolutevalueshavetobeinterpreted
withcare.

OutlookandlusionsConc7.

Thisworkdealtwiththedevelopmentandexperimentalimplementationofaphysicalconcept
forqualityassuranceofnewCTmethodswithaspecialfocusontheCTdOrtechnique.Besides
dosemeasurementsthiswasachievedbyapplyingtheconventionalFouriermethodsaswellas
byintroducingacompletelydifferentimage-spacebasedapproach.
TheFourierapproachisawell-establishedmethodwhichiscommonlyappliedtoCTimages.
Thedetailsindatatreatmenthoweverstillvarywidelybetweendifferentstudies.Thedata
processingdevelopedforthisworkhasbeenproposedtotheIECinSeptember2010tobecome
commonstandard1.Afinaldecisionhasnotbeenmadetillthecompletionofthiswork,but
sinceitpassedtheinitialsurvey,thereisagoodchancethatthepresentedmethodswillbepart
ofthenexteditionsoftheIECnormsaboutimagequalityinCTscanners.
Althoughnormallyonlyappliedtoreconstructedimages,inthisworktheFouriermethods
werealsousedfortheanalysisofCTrawdatainordertoseparatethequalityofthehard-
warefromthereconstruction.Incontrasttoformerstudies,thedataprocessingwasespecially
developedwithregardtotheapplicationtonewCTmethods,suchastheCTdOrconcept.
ThemeasurementmethodsweredesignedtobesimpleandsuitableforeveryCTgeometry.
TheresultingMTFandNPSprovidedinformationaboutthescannersystemadditionaltothe
informationobtainedfromcommonanalysisofthereconstructedimages.
EventhoughtheFouriermethodsarewell-establishedwithdigitalizedsystems,theymake
assumptions,suchasshift-invarianceandwidesensestationarity,whicharenotfulfilledbyreal
CTscanners.Inordertodeterminetheerrorsintroducedbyneverthelessapplyingthem,an
image-spacebasedapproachwaspresentedinthiswork,whichispixelbasedandtherefore
well-suitedforreconstructedimages.Asthefirstcomprehensiveapplicationofthisapproach
toCT,theexperimentalimplementationandthedatatreatmenthadfirsttobedevelopedusing
knownscanners.Onceaccomplished,acomparisonoftheresultstotheFourierbasedapproach
revealedthesimilaritiesanddifferences.Despitetheeigenvectorsandeigenvaluesnotbeing
equal,themagnitudeofnoiseturnedouttobethesameforbothapproaches,forthelaboratory
andtheclinicalCTsystem.Thespatialresolutionwashoweverconsiderablydifferentforthe
twoapproaches,becausetheH-matrixactuallyblursasimulatedinputsignalmuchmorethan
theMTF.AcomparisonwithrealimagesrevealedthattheMTFtendstooverestimatetheper-
formanceofanimagingsystem,whiletheH-matrixtendstounderestimateit.Butforsome

1ProposedAmendmentforIEC61223-3-5Ed.1[IEC04]andIEC61223-2-6Ed.2[IEC06]byIacovosS.Kypri-
anouandStanleyH.Stern

103

104

OutlookandConclusions7.

cases,theimage-spaceapproachactuallyprovidesareasonabledescriptionofthesignaltrans-
fer.SincetheSNRisbasedonthenoiseandthespatialresolution,itisalwayshigherforthe
Fourierapproachthanfortheimage-spaceapproach.Fromthecomparisonwithrealimages,
theactualSNRofthesystemissupposedtolieinbetweentheimage-spaceSNRandtheFourier
SNR.Theapplicationoftheimage-spaceapproachtovariousCTsystemsrevealedalsoitsad-
vantagesanddisadvantages.Itsmajoradvantageisthatitisdirectlyapplicabletodigitalized
systems,becauseitispixelbasedanditmodelsthesystempropertieswithmatrixoperators.
TheassumptionsnecessaryforFouriertransformationdonothavetobemade.Theoptimized
eigenvectorsgiveadditionalinformationaboutthestructureofthenoiseandthespatialreso-
lutionissometimespredictedmorerealisticallybytheH-matrixthanbytheMTF.Therefore,
thisapproachhasthepotentialtoprovideabetterdescriptionoftheimagingsystemthanitis
possiblewiththeFourierapproach.
Buttheanalysisinimage-spaceinvolvesalsodisadvantages,becauseneitherspatialresolu-
tionnornoisecanbeassignedtotheircorrespondingspatialfrequencies.Thisishoweveran
importantinformation,especiallyasbothMTFandNPSdonotnecessarilydecreaselinearly.
Anotherdrawbackisthehighmeasurementandcomputationaleffort,whichisrequiredtoget
enoughdataandanalyzethemappropriately.Insteadof4∙106independentpixelsneededfor
theNPSaccordingtotheIEC[IEC03],thebootstrappingcalculationsinthisworkshowedthat
atleast107independentpixelsarenecessaryforaproperdescriptionofthecovariancematrix
andthatlesspixelsleadtoahighlybiasedSNR.Takingenoughimagesisatime-consuming
processevenforcompletelyautomatedsystemssuchastheclinicalCT,butfortheCTdOr
demonstratoritwasimpossible.Furthermore,thecomputationaleffortfortheanalysisaccord-
ingtotheimage-spaceapproachisenormous.AlthoughROIsoftheminimumreasonablesize
of32×32pixelswereused,thecalculationofthecovariancematrixtookseveralhoursand
theanalysisoftheresulting1024×9216H-matrixoverranthecapacityofastandardcomputer.
Forcomparison,theMTFandtheNPScanbecalculatedinsecondsonthesamecomputer.
TohandlelargerROIsizes,whichwouldprovideabetterresolution,ahigh-capacitycomputer
needed.beouldwThesedrawbacksmakeitimpossibleatpresenttoincludetheimage-spaceapproachinto
dailyroutineofqualityassurance.Nevertheless,analyzingeachsystematleastoncewiththe
image-spaceapproachcangiveimportantadditionalinformation.Theerrorimplementedbythe
assumptionsoftheFouriermethodscanbeshowntobenegligibleasforthenoiseinthelabo-
ratoryandtheclinicalCTsystem.Butalsodiscrepanciescanberevealedbetweenthespatial
resolutionpredictedbytheMTF,theH-matrixandtheactualvalue.Detectingthelimitations
oftheconventionalmethodscanthenhelptoaccountforthemindailyroutine.
ThiswasfirstdemonstratedwiththeimprovedversionoftheCTdOrdemonstratorincom-
binationwiththeclinicalCTsystem.Inordertogetasgoodimagesaspossible,newalgorithms
fordatatreatmenthadtobedevelopedfirst.Theapplicationofthenewalgorithmsonadataset

105

sampledwiththeC-armdeviceandthecomparisonwithimagesprocessedwiththeformersoft-
waredemonstratedtheimprovementofimagequalityachieved.ForthecombinationoftheCT
dOrwiththeC-armdevice,remarkablygoodimagescouldbeproduced,consideringthedraw-
backsofthedemonstrator.Duetoadrasticdecreaseofsamplingtimeandnumberofviews,the
imagequalitywasmuchworsewhencombiningtheCTdOrringwiththeclinicalCT.Theres-
olutionwasdecreased,thenoisewasconsiderablyhigher,andthedatasamplingimplementeda
regularpatternintheimages.Theresultingnoisewasbiasedlowfortheimage-spaceapproach,
becausenotenoughindependentpixelswereavailableforanappropriateanalysis.Sinceonly
onesliceisacquiredbytheCTdOrdemonstrator,nooversampledPSFwasmeasuredforthe
determinationofthespatialresolution.Furthermore,thehighnoiseandthepatternsintheim-
agesintroducedsevereuncertaintiesintheresultsforthespatialresolution.Independentofthe
approachused,theanalysisoftheimagequalityhasemphasizedthemajordisadvantagesofthe
.demonstratordOrCTHowever,themeasureddosereductionofmorethan60%justifiestofollowuptheconcept
ofCTdOr.Butbeforeaclinicalprototypecanbemanufactured,severalissueshavetobe
considered.Theringmusthaveadiameterofabout70cmtobelargeenoughforahumanbody
andthenumberofdetectorshastobeincreasedtoatleast1000inordertoachieveanadequate
resolution.Thisgivesamaximumwidthofthemaskdetectorsof1.1mm.Furthermore,fora
straightforwardcombinationofthedatasets,themaskdetectorsshouldhaveasensitivitysimilar
tothearcdetector.Thisdefinesforagivenmaterialaminimumthicknessofthedetector.
Theshieldingsmustaccountforthis,buttheymustnotbemuchbroaderthanthedetectors.
Aprobablyresultingdecreaseofthethicknesscouldbecountervailedbyreplacingleadby
tungstenorrhenium,whichhavehigherdensitiesthanleadandhenceahigherabsorption.
TheseconsiderationsrecentlyledtotheintroductionofamodifiedversionoftheCTdOr
geometry.Insteadofthemaskdetectorsontheinnersideoftheshieldingsandthearcdetector
outsidethering,itworkswithonlyonearcdetectorinsidethering,whichcollectsbothdata
setssimultaneously.Thissolvespotentialproblemswiththethicknessofthemaskdetector
andthesmallerthicknessoftheshieldings.ThemodifiedCTdOrisnotanadditionalring
inanotherwiseconventionalCTsystemanymore,butanindependentnewCTtechnology.A
prototypeforamicro-CTsystemisunderconstructioninordertoinvestigateitspotentialand
finallyclarifyifthetheoreticallypredictedadvantagesoftheCTdOrgeometrycanbeachieved
.realityin

A.Mathematicsforthereconstructionofthe
datadOrCT

A.1.CTdOrcombinedwiththeC-armdevice

A.1.1.DataoftheC-armdevice

ThecalculationsweredonebasedonthecoordinatesystemdefinedinFig.A.1.

FigureA.1:CoordinatesysteminC-armdevice

ThereferencesystemOxyzisdefinedbytheoriginO,whichcoincideswiththesource,a
vectorOxnormaltothedetector,avectorOyparalleltothedetector,thecentralplaneOxyand
avectorOznormaltothecentralplane(notshowninthefigure).
LetO1x1y1z1beacoordinatesystemrotatinginthereferencesystemtogetherwiththering
aroundO1z1.ItsoriginO1isthecenteroftheringandO1z1isthesymmetryaxisofthecylinder.
Theangleαisdefinedastheangleofrotation.

107

108

A.MathematicsforthereconstructionoftheCTdOrdata

ThelengthoftheradiusvectorconnectingOandO1isR=Rx2+Ry2,whereRxandRy
as:definedareRx=−Rcosζ,(A.1)
Ry=Rsinζ.
Ideally,ζissupposedtobezero,however,adeviationof±1◦intheactualexperimentalset-up
madeitnecessarytoincludeζasanadditionalparameterinthecalculations.
DefiningrastheradiusoftheCTdOrring,thecoordinatesoftheringinOxyzare:
x=Rx+rcos(γ+α),(A.2)
y=Ry+rsin(γ+α).
Theequationofalineconnectingtwodistinctpointsγ1,γ1+kontheringis:
xcosα+γ1+k+γ1+ysinα+γ1+k+γ1=
22(A.3)
Rxcosα+γ1+k+γ1+Rysinα+γ1+k+γ1+rcosγ1+k−γ1.
222Inthesystem,raysaredefinedaslines,whichgothroughγ1,γ1+kandtheoriginO.This
conditionholdsfortheangleαthatsatisfiestheequation:
Rxcosα+γ1+k+γ1+Rysinα+γ1+k+γ1=−rcosγ1+k−γ1.(A.4)
222Substitutingequation(A.1)intoequation(A.4)resultsin:
cosζ+α+γ1+γ1+k=rcosγ1+k−γ1.(A.5)
R22Asanyrayischaracterizedwithtwoparameters,αandγ,theangleβbetweenray(α,γ)and
thenormaltothedetectoris:
tanβ=−Ry+rsin(γ+α).(A.6)
Rx+rcos(γ+α)
Therefore,they-coordinateofthepointwhereray(α,γ)hitsthedetectorinadistanceD
fromOare:Ry+rsin(γ+α)
y=−DRx+rcos(γ+α).(A.7)
Thestripe(j,k)isdefinedasthecollectivityofrays,whichpassthewindowsjandj+k.In
ordertodescribeasinglewindowtowindowstripe,thepointsontheCTdOrringaredefined
A.2:Fig.toaccordingγ1=j∙2π+2π∙5,(A.8)
8197197γ2=γ1+2π∙3,(A.9)
8197γ1+k=γ1+k2197π,(A.10)
γ2+k=γ2+k2π,(A.11)
197

A.1.CTdOrcombinedwiththeC-armdevice

109

takingintoaccountthatthewidthoftheshieldingsis5mmandthewidthofthewindowsis
mm.3

FigureA.2:Angleswhichdefinearay

Foranystripe(j,k),therearefourpositionsα1,α2,α3andα4ofthering.Thefirstangle,
α1,denotesthesourceanglewhereexactlyonerayrunsfromγ1tothepointγ1+k:
α1=−ζ−γ1+γ1+k+arccosrcosγ1+k−γ1.(A.12)
2R2Therefore,thelastangle,α2,describesthesourceanglewhereexactlyonerayrunsfromγ2to
thepointγ2+k:
α2=−ζ−γ2+γ2+k+arccosrcosγ2+k−γ2.(A.13)
2R2α3andα4denotethepointswherearayrunsfromγ1toγ2+korγ2toγ1+krespectively:
α3=−ζ−γ1+γ2+k+arccosrcosγ2+k−γ1,(A.14)
2R2α4=−ζ−γ2+γ1+k+arccosrcosγ1+k−γ2.(A.15)
2R2Thenforeachsourceimpulsefromthecorrespondinginterval,allpixelsbetweenyminand
ymaxhavetobeadded.Forallimpulsesbetweenα1andα3,yminandymaxaredefinedas:
Ry+rsin(γ1+α)
ymin=−DRx+rcos(γ1+α),
(A.16)Ry+rsin(γ1+k)+α)
ymax=−DRx+rcos(γ1+k+α).

110

A.MathematicsforthereconstructionoftheCTdOrdata

(A.17)

Forallimpulsesbetweenα4andα2:
ymin=−DRy+rsin(γ2+α),
Rx+rcos(γ2+α)
Ry+rsin(γ2+k+α)(A.17)
ymax=−DRx+rcos(γ2+k+α).
Andforallimpulsesbetweenα3andα4:
ymin=−DRy+rsin(γ1+k+α),
Rx+rcos(γ1+k+α)(A.18)
Ry+rsin(γ2+k+α)
ymax=−DRx+rcos(γ2+k+α).
Usingtheseformulas,anotherprogramdeterminedwhetheradatapointbelongedtoarayand
one.whichtosoif

A.1.2.DataoftheCTdOrdemonstrator
ThecoordinatesystemisdepictedinFig.A.3.

FigureA.3:CoordinatesystemintheCTdOr

AnypointoftheringisrepresentedinOxyas
x=R+rcos(γ+ωt),(A.19)
y=rsin(γ+ωt).
Foragivenpointonthering,γ+dγ(t)issupposedtobethecurrentpositionwherelineOγ
intersectstheringatthemomentt.Thesevaluesofγaregivenbytherelation
cosγ+ωt+dγ+rcosdγ=0.(A.20)
2R2

A.2.CTdOrcombinedwiththeclinicalCTsystem

111

SolvingEq.(A.20)relativetotgives
t=π−γ−dγ−arccosrcosdγ1.(A.21)
ω2R2Hence,forthedetectorlocatedatpointγoftheringandforanygivenvaluedγamomentt
canbefoundsuchthatatthismomenttherayhitsthedetectorthroughtheringpointγ+dγ.
Therefore,foranywindowintheringitispossibletofindthetimeintervalduringwhichthe
detectorisirradiatedthroughthiswindow.
Additionally,alldetectorsarereadoutsimultaneouslywithfrequencyq1,whilethepulse
frequencyofthesourceoftheC-armdeviceisq2.Therefore,thepositionofthesourceinthe
momentofpulseνisν
αpν=α0p+ωq,(A.22)
2whereωistherotationvelocityofthering.Equivalently,thepositionofthesourceinthe
isireadoutofmomentαir=α0r+ωi.(A.23)
q1Onlythoseαirareinterestingwhichsatisfythefollowingconditionsforagivenvalueofindex

1.thedetectorisopentothesourcebeinginpositionανp
2.αir−1<αpν<αirsothateventαirhappenseitherdirectlyaftereventανporbothevents
coincide.

A.2.CTdOrcombinedwiththeclinicalCTsystem
OxyzisthereferencesystemrelatedtotheCTdOrring,whereOisthecenterofthering.The
referencesystemrelatedtotheCTscannerisO1x1y1z1,whereO1correspondstotheisocenter.
LetδxδyδzbecoordinatesofO1inOxyz.Thetransformationfromonereferencesystemtothe
isother

x1cosθ20sinθ2100x−δx
y1=0100cosθ1sinθ1y−δy,(A.24)
z1−sinθ20cosθ20−sinθ1cosθ1z−δz
whereθ1andθ2areparametersofangulardiscrepancybetweenxyzandx1y1z1.Thiscanbe
aswrittenx1=(x−δx)cosθ2−(y−δy)sinθ1sinθ2+(z−δz)sinθ2cosθ1,
y1=(y−δy)cosθ1+(z−δz)sinθ1,(A.25)
z1=−(x−δx)sinθ2−(y−δy)sinθ1cosθ2+(z−δz)cosθ1cosθ2.

112

A.MathematicsforthereconstructionoftheCTdOrdata

(A.26)

(A.27)

Inx1y1z1themotionofthesourceisdescribedwithR
Rcos(ωt)
R(t)=Rsin(ωt).(A.26)
0Thedetectorringisdescribedasacylinder,thepointsofwhichinxyzarer
βcosrr(β,z)=rsinβ.(A.27)
zA.2.1.DataoftheCTdOrdemonstrator
LetR0bethevectordirectedfromOtothesource.Lettherayemittedbythesourcehitpoint
r(β,0)throughpointr(β+Δβ,z)oftheringcylinder.Forthis,therayequality
Δr×RO=Δr×r(A.28)
isvalid,whereΔr=r(β+Δβ)−r(β).Inordertodescribetheorthogonalprojectiononto
planeOxy,onlythez-coordinateofEq.(A.28)isneeded.Theexpandedformofthisequation
is
Rcos(ωt)cosβ+Δ2βcosθ2−sinβ+Δ2βsinθ1sinθ2+
+Rsin(ωt)sinβ+Δ2βcosθ1=(A.29)
=rcosΔβ−δxcosβ+Δβ+δysinβ+Δβ.
222Inordertosolvethisequationrelativetot,itisrewrittenas
Ω1cos(ωt)+Ω2sin(ωt)=rcosΔβ−1δxcosβ+Δβ+δysinβ+Δβ,

R2R22(A.30)
whereΩ1andΩ2aredefinedas
Ω1=cosβ+Δβcosθ2−sinβ+Δβsinθ1sinθ2,
22Ω2=sinβ+Δβcosθ1.
2Byintroducingangleφsuchthat
ΩΩ21cosφ=Ω12+Ω22,sinφ=Ω12+Ω22,
Eq.(A.30)canbewrittenas
βΔrcos(ωt−φ)=RΩ12+Ω22cos2−,(A.31)

(A.31)

A.2.CTdOrcombinedwiththeclinicalCTsystem

113

asdefinediswhererΔβΔβ
21=RΩ2+Ω2δxcosβ+2+δysinβ+2.
EquationA.31hastwosolutions
ωt=φ±arccosRΩ2r+Ω2cosΔ2β−,(A.32)
21ofwhich+istheneededone.ForthespecialcasewhentheCTdOrdeviceisperfectly
alignedinthegantry(θ1,θ2=0),φ=β+Δ2βand=0sothatEq.(A.32)simplifiesto
ωt=β+Δβ+arccosrcosΔβ.(A.33)
2R2LetΔα=2197πbethesegmentoftheringwhichcorrespondstooneshieldingandonewindow.
Themomentoffirstreadoutisreferredtoasinitialmoment.Theangleβjisdescribedin
,jdetectorofdependenceβj=(λ1+j)Δα,(A.34)
whereλ1,0≤λ1<1,isafreeparameter.
AccordingtoEq.(A.32),thetimeinterval,duringwhichdetectorjcanbeseenfromthe
sourcethroughwindowk,isΔt=t2−t1
t1=φ(βj,Δβ1)+arccosc1cosΔ2β1−1(βj,Δβ1)ω1,
t2=φ(βj,Δβ2)+arccosc2cosΔβ2−2(βj,Δβ2)1,
2ωwhereΔβ1=kΔα+Δ2S,Δβ2=(k+1)Δα−Δ2S(A.35)
dependsontheangularsizeofoneshieldingΔS.

A.2.2.DataoftheclinicalCT
UsingEq.(A.25),alineofintersectionoftheiso-planewiththecylindercanbefoundeasily.
Sincefortheiso-planez1=0,itfollows
θsinθsinz−δz=(x−δx)cosθ1cos2θ2+(y−δy)cosθ11.(A.36)
ByinsertingthisequationintothefirsttwoequationsofEq.(A.25),oneobtainstheparametric
representationofthelineinO1x1y1:
x1(β)=(rcosβ−δx)cosθ1,
cosθ1cosθ2(A.37)
x2(β)=(rsinβ−δy)cosθ2+(rcosβ−δx)sinθ2sinθ1.
cosθ1cosθ2

114

A.MathematicsforthereconstructionoftheCTdOrdata

Forarayemittedfromthesourceatpositionαandpassingthroughpointsβ1andβ2defined
byEq.(A.37),thefollowingrelationmusthold:
(x1(β+Δβ)−x1(β))(Rsinα−y1(β))=(y1(β+Δβ)−y1(β))(Rcosα−x1(β)).(A.38)
InsertingEq.(A.37)intoEq.(A.38)andsimplifyingtheexpression,resultsinEq.(A.29),which
canbeusedforthedeterminationofpositionαgivenβ1andΔβ.
Inordertodeterminethepointthedetectorishitbyagivenray,thecoordinateofthisrayin
thefanneedstobeknown.Theequationoftherayemittedbythesourcefrompositionαto
pointβ1onthelinedefinedbyEq.(A.37)is
x1(y1(β1)−Rsinα)+y1(Rcosα−x1(β1))=R(y1(β1)cosα−x1(β1)sinα).(A.39)
Thisexpressioncanbereformedtotheequationofviewx1cosφ+y1sinφ=t,where
t=y1(β1)Rcosα−x1(β1)Rsinα(A.40)
(y1(β1)−Rsinα)2+(Rcosα−x1(β1))2
isthedistanceofthelinetotheisocenter.Thedesiredcoordinateoftherayisψ=arccost.
LetΔlbethesizeofthedetectorbin,Dthesourcetodetectordistance,andNthenumber
ofdetectorelements.Takingintoaccounttheyingfocustechniqueandthequarterdetector
offset,rayψhitsdetectorbini
i=ψ−ψ0,
ψΔwheretheresolutionofthefanbeamΔψis
Δψ=1Δγ
D2and1πψ0=2−N+4Δψ.
Again,theinitialmomentisthemomentoffirstreadout.Theuncertaintyoftheringposition
ismodeledviaparameterλ2,0≤λ2<1,sothatcoordinatesofpointsβ1(j),β2(j)that
arejwwindocharacterizeβ1(j)=(λ2+j)Δαβ2(j)=β1(j)+ΔW,(A.41)
whereΔWisthesizeofthewindows.Thisequationmakesclearthatλ2hastobehandled
.λfromindependently1

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wledgmentsknoAc

ThisworkwouldnotexistwithoutthehelpofnumerouspeoplewhoIwanttothankhere:

-Prof.Dr.Dr.HerwigG.Paretzkeforgivingmethepossibilitytoworkonthistopicunder
hissupervisionandprovidingexperiencedadviceonmyproblems.

-Prof.Dr.ChristophHoeschenforgivingmethechancetoworkinhisdepartmentinthis
veryinterestingfieldofresearch.Hissupport,motivationandscientificguidancewere
essentialforthiswork.Especially,IwanttothankhimforarrangingmystayattheFDA.

-IacovosJakeKyprianou,Ph.D.,forgivingmethechancetospendthreemonthsatthe
FDA.UnderhispatientsupervisionIlearnedtonsoftheoryaboutimagingandhissupport
andnumerousideaswereessentialforthiswork.Especially,Iwanttothankhimforthe
timehestillfindstosupervisemeeventhoughIlonglefttheFDA.

-Dr.HelmutSchlattlforalwaysbeingavailableformeandfortakingcareofallmysmall
andlargescientific,administrativeandcomputerproblems.Hissensibleandcarefulcor-
rectionsthoroughlyimprovedthiswork.

-Dr.OlegTischenkoforhelpingmewiththereconstructionoftheCTdOrimages.Espe-
cially,Iwanttothankhimforalwayshavingtimeformeandpatientlyansweringallmy
detail.inquestions

-Dr.HugodelasHerasforteachingmeeverythingabouttheCTdOr,providingmewith
hisprogramsandallowingmetousehisfiguresinmywork.Especially,Iwanttothank
himforprovidingmeshelterandtakingcareofmeduringmytimeinWashington.

-BernhardRengerfromtheHospitalrechtsderIsarforspendingnumerouseveningsinthe
clinicwithme,takingimagesandhelpingmetodevelopthemostappropriateset-upand
procedure.measurement

-Dr.MatthiasGreiterandDr.FelixSchöferforalwaysbeingavailableforfruitfuldiscus-
sionsandansweringallmycountlessquestions.

-TilmanJanzenforextensivelyproofreadingmyworkandprovidingmoralsupportwhen-
needed.erve

123

124

yBibliograph

-KyleMyers,Ph.D.,andAldoBadano,Ph.D.,forallowingmetoworkattheFDAand
givinghelpfulinputtomywork.

-SamirAbboudformeasuringthedoseintheFDAsystemandsendingmeallthead-
ditionalinformationIneededtoevaluatethedata.Especially,Iwanttothankhimfor
spendinghourswithmeinthemachineshophelpingmetobuildmyphantoms.

-StefanieHurowitzforacquiringtheimagesattheFDAforme,evenifitsometimestook
severalattemptstillitworkedthewaywewanteditto.

-WernerPanzerforhelpingmetocalibratetheTLDsandansweringallmynumerous
topic.thisaboutquestions

-TheAuswertungsstelleoftheHelmholtzZentrumMünchen,andespeciallyMarkusFigel,
forprovidingtheTLDs,readingthemout,andpatientlyansweringallmyquestions.

-ThemachineshopoftheHelmholtzZentrumMünchen,andespeciallyMartinScherer,
foralwaysprocessingmystuffasfastaspossible.

-ThewholeDepartmentofMedicalRadiationPhysicsandDiagnosticsfortheverykind
workingatmosphereandforscientificandmoralsupportwheneverneeded.

-Myfamilyandfriendsforalwayssupportingandhelpingme,fortheirleniencyandpa-
tience.

FurthermoreIwanttothank

-TheGermanAcademicExchangeService,DAAD,forfundingtwomonthsofmystayat
A.FDthe

-TheLifeScienceFoundationforunbureaucraticallyfinancingthethirdmonthofmystay
USA.thein