Image reconstruction algorithms for achieving high-resolution positron emission tomography images

2004·6.9 MB·English

by  Chang, Ji-Ho

IMAGERECONSTRUCTIONALGORITHMS
FORACHIEVINGHIGH-RESOLUTION
POSITRONEMISSIONTOMOGRAPHYIMAGES
By
JI-HOCHANG
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
2004
Copyright2004
by
Ji-HoChang
Idedicatethisworktomyfamily.
ACKNOWLEDGMENTS
Firstofall.Iwouldliketothankmyacademicadvisor,Dr.JohnM.M.Anderson,
for hisgreat guidanceofmyresearch. I alsowould liketothankmycommittee
membersDr.JianLi,Dr.FredJ.Taylor,andDr.BernardA.Mairfortheirprecious
comments and corrections on my dissertation. Especially, I would like to thank
Dr.BernardA.Mairforhiscommentsandcorrectionsonconvergenceproofofthe
proposedalgorithms. IthankYoonchulKimandKerkilChoiwhohavesharedwith
mevaluablediscussionsrelatedtomyresearch. Finally,Iamgratefultomyfamily,
whohavealwaysprayedformyhealth.
IV
TABLEOFCONTENTS
page
ACKNOWLEDGMENTS
iv
ABSTRACT
vii
CHAPTER
1 BACKGROUND 1
1.1 OverviewofPositronEmissionTomography(PET) 2
1.2 PETScanner 3
1.3 SourcesofError 6
1.4 SystemModelforEmissionData 10
2 LITERATUREREVIEW 16
2.1 PenalizedMaximumLikelihoodAlgorithms 17
2.2 ScatterCorrectionMethods 19
2.3 SummaryoftheProposedAlgorithms 20
3 PENALIZEDMAXIMUMLIKELIHOODALGORITHM 23
3.1 PenalizedMaximumLikelihood(PML)Algorithm 24
3.2 ConvergenceProof 33
3.3 PropertiesofthePMLAlgorithm 37
4 ACCELERATEDPENALIZEDMAXIMUMLIKELIHOODALGORITHM 39
4.1 ConvergenceProof 40
4.2 AcceleratedPenalizedMaximumLikelihood(APML)Algorithm . 44
4.3 DirectionVectors 52
4.4 PropertiesoftheAPMLAlgorithm 55
5 QUADRATICEDGEPRESERVINGALGORITHM 57
6 JOINTEMISSIONMEANANDPROBABILITYMATRIXESTIMATION 63
6.1 ScatterMatrixModel 65
6.2 JointMinimumKullback-LeiblerdistanceMethod 68
6.3 ProbabilityCorrectioninProjectionSpace(PCiPS)Algorithm . . 69
v
7 SIMULATIONSANDEXPERIMENTALSTUDY 79
7.1 RegularizedImageReconstructionAlgorithms 80
7.1.1 SyntheticData 82
7.1.2 RealData 87
7.2 APMLalgorithm 112
7.3 PCiPSAlgorithm 116
8 CONCLUSIONSANDFUTUREWORK 138
8.1 Conclusions 138
8.2 FutureWork 142
APPENDIX
A HUBER’SSURROGATEFUNCTIONS 144
B SURROGATEFUNCTIONSFORPENALTYFUNCTION 145
C STRICTCONVEXITYOFPMLOBJECTIVEFUNCTION 147
D SOLUTIONTOUNCONSTRAINEDOPTIMIZATIONPROBLEM
INMODIFIEDAPMLLINESEARCH 149
E CONVEXITYOFSURROGATEFUNCTIONS
FOROBJECTIVEFUNCTIONSINAPMLLINESEARCH 151
REFERENCES 154
BIOGRAPHICALSKETCH 160
vi
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
IMAGERECONSTRUCTIONALGORITHMS
FORACHIEVINGHIGH-RESOLUTION
POSITRONEMISSIONTOMOGRAPHYIMAGES
By
Ji-HoChang
August2004
Chair: JohnM.M.Anderson
MajorDepartment: ElectricalandComputerEngineering
Inthisdissertation,wepresentfouralgorithmsforreconstructinghigh-resolution
imagesinPET.Thefirstalgorithm,referredtoasthepenalizedmaximumlikelihood
(PML)algorithm,iterativelyminimizesaPMLobjectivefunction.Ateachiteration,
thePMLalgorithmgeneratesafunction,calledasurrogatefunction,thatsatisfies
certainconditions. Thenextiterateisdefinedtobethenonnegativeminimizerof
thesurrogatefunction. ThePMLalgorithmutilizesstandardde-coupledsurrogate
functionsforthemaximumlikelihoodobjectivefunctionofthedataandde-coupled
surrogatefunctionsforacertainclassofpenaltyfunctions. Asdesired, thePML
algorithmguaranteesnonnegativeestimatesandmonotonicallydecreasesthePML
objectivefunctionwithincreasingiterations. ForthecasewherethePMLobjective
functionisstrictlyconvex, which is truefortheclassofpenaltyfunctionsunder
consideration,thePMLalgorithmhasbeenshowntoconvergetotheminimizerof
thePMLobjectivefunction.
ThedrawbackofthePMLalgorithmisthatitconvergesslowly. Thus,a“fast”
versionofthePMLalgorithm,referredtoastheacceleratedPML(APML)algorithm,
vii
wasdevelopedwhereanadditionalsearchstep,calledapatternsearchstep,isper-
formedaltereachstandardPMLiteration. Inthepatternsearchstep,anaccelerated
iterate,whichhaslowercostthanthestandardPMLiterate,isfoundbysolvinga
certainconstrainedoptimizationproblemthatarisesateachpatternsearchstep. The
APMLalgorithmretainsthenicepropertiesofthePMLalgorithm.
Thethirdalgorithm,referredtoasthequadraticedgepreserving(QEP)algo-
rithm,aimstopreserveedgesinthereconstructedimagessothatfinedetails,such
assmalltumors,aremoreresolvable. TheQEPalgorithmisbasedonaniteration
dependent,de-coupledpenaltyfunctionthatintroducessmoothingwhilepreserving
edges. Thepenaltyfunctionwasdevelopedbymodifyingthesurrogatefunctionsof
thepenaltyfunctionforthePMLmethod.
In PET, thereareseveralerrors that havetheneteffectofintroducingblur
intothereconstructedimages. Weproposeamethodthataimstoreduceblurin
PETimages. Themethodisbasedontheassumptionthatthe “true” probability
matrixfortheobservedemissiondataisaproductofanunknownnonnegativematrix,
calledascattermatrix,anda“conventional”probabilitymatrix. Underthesuggested
framework,theproblemistojointlyestimatethescattermatrixandemissionmeans.
Weproposeanalternatingminimizationalgorithmtoestimatethembyminimizing
acertaindistance.
The algorithms are qualitatively and quantitatively assessed using synthetic
phantomdataandrealphantomdata.
viii
CHAPTER 1
BACKGROUND
Medicalimagingmodalities,suchasX-raycomputedtomographyandmagnetic
resonanceimaging, areusedtoobtainimagesofanatomicalstructureswithinthe
humanbody. However,incertainmedicalapplications,itisalsoimportanttoget
physiologicalinformation. Thereasonisbecausephysiologicalchangescanindicate
diseasestatesearlierthananatomicalchanges [1], Positronemissiontomography
(PET)andsingle-photonemissioncomputedtomography(SPECT)arewidelyused
medicalimagingmodalitiesthatacquirephysiologicalinformationonbothhuman
andanimalsubjects.
In SPECT, physiologicalinformationisobtainedbyimagingthedistribution
ofgamma-rayorX-rayemittingradio-isotopeswithinthehumanbody [1], After
theradio-isotopesareintroducedintothehumanbody,theradio-isotopesdecayand
emitgamma-rayorX-raysinglephotons. ASPECTscannerdetectsthesephotons
withhelpofcollimatorsthataremadeoflead. Sincealargepercentagephotonsare
absorbedbytheleadcollimators,thesensitivityandaccuracyofSPECTislimited.
InPET,thereisnoneedforleadcollimatorsbecausecollimationisperformedby
electroniccircuitsthatareconnectedtothedetectors. Consequently,PETpossesses
relativelyhighsensitivityandaccuracy,whencomparedtoSPECT. Althoughcost
isthemajorlimitationofPET,recentresearchadvances,suchasalessexpensive
materialsfordetectorsandscannerconfigurationsthat needasmaller numberof
detectorsthanconventionalscanners, havehelpeddecreaseitscost [1]. Themain
advantageofSPECToverPETisitssubstantiallylowercost.
1
2
WepresentabriefoverviewofPET.PETscanner,andcertainerrorsinPET
inSection1.1,1.2,and1.3,respectively. Then,weprovidesomebackgroundonthe
imagereconstructionprobleminPET.
1.1 OverviewofPositronEmissionTomography(PET)
MostclinicalapplicationsofPETareinoncologicalcasesinvolvingthediagnosis
andstagingofcancer, treatmentplanningandmonitoringoftumors, detectionof
recurrenttumors,andlocalizationofbiopsysitesincaseswhentherearetumorsin
theheadorneck[2,3]. PETisalsousedinthediagnosisofcoronaryarterydisease
andotherheartdiseases[2,3].
In PET, physiological informationisacquired byimagingthedistribution of
positron-emittingisotopes,suchas13N,150,18F,andnC,withinthehumanbody[3].
Thepositron-emittingisotopesareboundtocompoundswith knownbiochemical
properties. Compoundsthatarelabeledwithpositron-emittingisotopesarecalled
radiopharmaceuticals. Thechoiceoftheradiopharmaceuticaldependsonitsappli-
cation. Forexample,2-[18F]fluoro-2-deoxy-d-glucose([18F]FDG)isusedforimaging
braintumorswhile [13N]ammoniaisusedforthedetectionofcoronaryarterydis-
ease[3]. Aftertheradiopharmaceuticalisintroducedintothesubject(injectionor
inhalation),positronsareemittedasthepositron-emittingisotopesdecay.Anemitted
positronannihilateswithanearbyelectronwithinthebodycausingthegeneration
oftwohighenergy (511keV) photons. Thetwophotons,whichcanpenetratethe
subject,travelinalmostoppositedirections. Ideally,photonpairsthataregenerated
byapositronannihilatingwithanelectronwillbedetectedbyapairofdetectors.
Iftwoelectronicallyconnecteddetectorsdetectapairofphotonswithinashort
timeinterval(e.g.,<10ns),thenthedetectionisrecordedalongthelineconnecting
thetwodetectors, whichiscalled aline-of-response. Intheabsenceofanerror,
thedetectionindicatesthat thereis an annihilationsomewherealongtheline-of-
response.Thedetectionofaphotonpairisreferredtoasacoincidence. Inaddition,