Computed Tomography: From Photon Statistics to Modern Cone-Beam CT

525 Pages·2008·18.713 MB·English

by  Thorsten Buzug Prof. Dr. (auth.)

ThorstenM.Buzug
ComputedTomography
ThorstenM.Buzug
Computed Tomography
From Photon Statistics
to Modern Cone-Beam CT
With475Figuresand10Tables
123
ThorstenM.Buzug,Prof.Dr.
InstitutfürMedizintechnik
UniversitätzuLübeck
RatzeburgerAllee
Lübeck
Germany
E-mail:[email protected]
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Preface
This book provides an overview of X-ray technology, the historic developmental
milestonesofmodernCTsystems,andgivesacomprehensiveinsightintothemain
reconstruction methods used in computed tomography. The basis of reconstruc-
tionis,undoubtedly,mathematics. However,thebeautyofcomputedtomography
cannotbeunderstoodwithoutadetailedknowledgeofX-raygeneration,photon–
matterinteraction,X-raydetection,photonstatistics,aswellasfundamentalsignal
processingconceptsanddedicatedmeasurementsystems.Therefore,thereaderwill
findanumberofreferencestothesebasicdisciplinestogetherwithabriefintroduc-
tiontotheunderlyingprinciplesofCT.
ThisbookisstructuredtocoverthebasicsofCT:fromphotonstatisticstomod-
erncone-beamsystems.However,themainfocusofthebookisconcernedwithde-
tailed derivations ofreconstruction algorithms inDandmodernDcone-beam
systems. A thorough analysis of CT artifacts and a discussion of practical issues,
suchasdoseconsiderations,providefurtherinsightintomodernCTsystems.While
mainlywrittenforgraduatestudentsinbiomedicalengineering,medicalengineer-
ingscience,medicalphysics,medicine(radiology),mathematics, electricalengin-
eering,andphysics,experiencedpractitionersinthesefieldswillbenefitfromthis
bookaswell.
Thedidacticapproachisbasedonconsistentnotation. Forexample,thenota-
tionofcomputedtomographyisusedinthesignalprocessingchapter.Therefore,
contrarytomanyothersignalprocessingbooks,whichusetime-dependentvalues,
this book uses spatial variables in one, two or three dimensions. This facilitates
the application of the mathematics and physics learned fromthe earlier chapters
todetectorarraysignalprocessing,whichisdescribedinthelaterchapters.Add-
itionally,specialattentionhasbeenpaidtocreatingatextwithdetailedandrichly
discussedalgorithm derivations ratherthancompactmathematical presentations.
Theconcepts should give evenundergraduate students the chancetounderstand
theprincipalreconstructiontheoryofcomputedtomography.Thetextissupported
byalargenumberofillustrations representingthegeometryoftheprojectionsit-
uation. Sincetheimpactofcone-beam CTwillundeniably increaseinthefuture,
three-dimensionalreconstructionalgorithmsareillustratedandderivedindetail.
Thisbookattemptstocloseagap.Thereareseveralexcellentbooksonmedical
imagingtechnologythatgiveacomprehensiveoverviewofmodernX-raytechnol-
ogy, computed tomography, magnetic resonance imaging, ultrasound, or nuclear
VI Preface
medicinemodalities like PETand SPECT.However, these books oftendonot go
intothemathematical detailofsignalprocessingtheory.Ontheotherhand,there
areanumber of in-depthmathematical books oncomputed tomography that do
notdiscusspractical issues. Thepresentbook isbased ontheGerman book Ein-
führungindieComputertomographie, whichfirstappeared during thesummerof
.Fortunately,sincethebookwasusedbymanyofmystudentsinlectureson
EngineeringinRadiology,MedicalEngineering,SignalsandSystemsinMedicine,and
TomographicMethods,Ireceivedalotoffeedbackregardingimprovementsonthe
firstedition. Therefore, the idea arose to publish an English version of the book,
whichisacorrectedandextendedfollow-up.
IwouldliketothankSiemensMedicalSolutions,GeneralElectricMedicalSys-
tems,andPhilipsMedical Systems,whogenerouslysupportedmylaboratories in
thefield ofcomputed tomography. Inparticular, Iwould liketothank myfriend
Dr. Michael Kuhn, former Director of PhilipsResearch Hamburg. It was his ini-
tiativethatmadepossiblethefirstinstallation ofCTinmylabsin.Addition-
ally, I have to thank Mrs. Annette Halstrick and Dr. Hans-Dieter Nagel (Philips
MedicalSystemsHamburg),LeondeVries(PhilipsMedicalSystemsBest),Doris
Pischitz,JürgenGreimandRobbyRokkitta(SiemensMedicalSolutionsErlangen),
DieterKeidelandJanLiedtke(GeneralElectricMedicalSystems)formanyphotos
inthisbook.IwouldliketothankWolfgangHärer(AXICC,SiemensMedicalSolu-
tions),Dr.GerhardBrunst,(GeneralElectricMedicalSystems),Dr.ArminH.Pfoh,
DirectorofGeneralElectric ResearchMunich,Dr.Wolfgang Niederlag(Hospital
Dresden-Friedrichstadt), Prof.Dr.HeinzU.Lemke(TechnicalUniversity Berlin),
Dr. Henrik Turbell (Institute of Technology, Linköpings Universitet), and Prof.
Dr.Dr.JürgenRuhlmann(MedicalCenterBonn)forthecourtesytoallowmetouse
theirillustrations andphotos.Further,IhavetothanktheDigitalCollections and
ArchivesofTuftsUniversity,theCollectionofPortraitsoftheAustrianAcademyof
Sciences,andtheRöntgen-KuratoriumWürzburge.V.forthecourtesytoallowme
tousetheirphotos.
Additionally, I have to thank my friends, colleagues, and students for proof-
reading and translating parts ofthebook. Inalphabetical orderIappreciated the
helpof:
Dr.BerndDavid(PhilipsResearchLaboratoriesHamburg)
KatieDechambre(MilwaukeeSchoolofEngineering)
ErinFredericks(CaliforniaPolytechnicStateUniversity,SanLuisObispo)
SebastianGollmer(UniversityofLübeck)
Dr.FrankoGreiner(UniversityofKiel)
TobiasKnopp(UniversityofLübeck)
DieterLukhaup(Schriesheim-Altenbach)
AndreasMang(UniversityofLübeck)
Prof.Dr.-Ing.AlfredMertins(UniversityofLübeck)
JanMüller(UniversityofLübeck)
Dr.Hans-DieterNagel(PhilipsMedicalSystemsHamburg)
SusannePreissler(RheinAhrCampusRemagen)
Preface VII
TonyShepherd(UniversityCollegeLondon)
VyaraTonkova(RheinAhrCampusRemagen)
ManyspecialthanksgotoSebastianGollmer,AndreasMang,andJanMüller,who
didthecopyediting ofthecompletemanuscript. However,fortheerrorsthatre-
main,Ialoneamresponsibleandapologizeinadvance.
Iwouldliketothanktheproductionteamatle-texaswellasPaulaFrancisfor
copyediting.Further,IhavetothankSpringerPublishing,especiallyDr.UteHeil-
mannandWilmaMcHughfortheirexcellentcooperationoverthelastfewyears.
Finally,IwouldliketothankmywifeKerstin,whohassupportedandsustained
mywritingeffortsoverthelastfewyears.Withoutherhelp,patience,andencour-
agementthisbookwouldnothavebeencompleted.
Lübeck,June ThorstenM.Buzug
Contents
1 Introduction........................................................ 1
. ComputedTomography–StateoftheArt......................... 1
. InverseProblems .............................................. 2
. HistoricalPerspective .......................................... 4
. SomeExamples................................................ 7
. StructureoftheBook........................................... 11
2 FundamentalsofX-rayPhysics........................................ 15
. Introduction .................................................. 15
. X-rayGeneration .............................................. 15
.. X-rayCathode.......................................... 16
.. Electron–MatterInteraction.............................. 19
.. TemperatureLoad....................................... 23
.. X-rayFocusandBeamQuality............................ 24
.. BeamFiltering.......................................... 28
.. SpecialTubeDesigns .................................... 30
. Photon–MatterInteraction...................................... 31
.. Lambert–Beer’sLaw..................................... 32
.. MechanismsofAttenuation .............................. 34
. ProblemswithLambert–Beer’sLaw .............................. 46
. X-rayDetection ............................................... 48
.. GasDetectors........................................... 48
.. Solid-StateScintillatorDetectors .......................... 50
.. Solid-StateFlat-PanelDetectors........................... 52
. X-rayPhotonStatistics ......................................... 59
.. StatisticalPropertiesoftheX-raySource ................... 60
.. StatisticalPropertiesoftheX-rayDetector ................. 64
.. StatisticalLawofAttenuation............................. 66
.. MomentsofthePoissonDistribution ...................... 68
.. DistributionforaHighNumberofX-rayQuanta ........... 70
.. Non-PoissonStatistics ................................... 72
X Contents
3 MilestonesofComputedTomography ................................. 75
. Introduction .................................................. 75
. Tomosynthesis ................................................ 76
. Rotation–TranslationofaPencilBeam(FirstGeneration)........... 79
. Rotation–TranslationofaNarrowFanBeam(SecondGeneration) ... 83
. RotationofaWideApertureFanBeam(ThirdGeneration) ......... 84
. Rotation–FixwithClosedDetectorRing(FourthGeneration)....... 87
. ElectronBeamComputerizedTomography ....................... 89
. RotationinSpiralPath.......................................... 90
. RotationinCone-BeamGeometry............................... 91
. Micro-CT..................................................... 93
. PET-CTCombinedScanners.................................... 96
. OpticalReconstructionTechniques .............................. 98
4 FundamentalsofSignalProcessing .................................... 101
. Introduction .................................................. 102
. Signals........................................................ 102
. FundamentalSignals ........................................... 102
. Systems....................................................... 104
.. Linearity ............................................... 104
.. PositionorTranslationInvariance......................... 105
.. IsotropyandRotationInvariance.......................... 105
.. Causality............................................... 106
.. Stability................................................ 106
. SignalTransmission............................................ 106
. Dirac’sDeltaDistribution....................................... 109
. DiracComb................................................... 112
. ImpulseResponse.............................................. 115
. TransferFunction.............................................. 116
. FourierTransform ............................................. 118
. ConvolutionTheorem .......................................... 124
. Rayleigh’sTheorem............................................. 125
. PowerTheorem................................................ 125
. FilteringintheFrequencyDomain............................... 126
. HankelTransform ............................................. 128
. AbelTransform................................................ 132
. HilbertTransform ............................................. 133
. SamplingTheoremandNyquistCriterion......................... 135
. Wiener–KhintchineTheorem.................................... 141
. FourierTransformofDiscreteSignals ............................ 144
. FiniteDiscreteFourierTransform................................ 145
. z-Transform................................................... 147
. Chirpz-Transform............................................. 148
Contents XI
5 Two-DimensionalFourier-BasedReconstructionMethods ............... 151
. Introduction .................................................. 151
. RadonTransformation ......................................... 153
. InverseRadonTransformationandFourierSliceTheorem .......... 163
. ImplementationoftheDirectInverseRadonTransform ............ 167
. LinogramMethod ............................................. 170
. SimpleBackprojection.......................................... 175
. FilteredBackprojection ......................................... 179
. ComparisonBetweenBackprojectionandFilteredBackprojection ... 183
. FilteredLayergram:DeconvolutionoftheSimpleBackprojection .... 187
. FilteredBackprojectionandRadon’sSolution...................... 191
. CormackTransform............................................ 194
6 AlgebraicandStatisticalReconstructionMethods ....................... 201
. Introduction .................................................. 201
. SolutionwithSingularValueDecomposition ...................... 207
. IterativeReconstructionwithART ............................... 211
. PixelBasisFunctionsandCalculationoftheSystemMatrix ......... 218
.. DiscretizationoftheImage:PixelsandBlobs ............... 219
.. ApproximationoftheSystemMatrixintheCaseofPixels .... 221
.. ApproximationoftheSystemMatrixintheCaseofBlobs .... 222
. MaximumLikelihoodMethod................................... 223
.. MaximumLikelihoodMethodforEmissionTomography .... 224
.. MaximumLikelihoodMethodforTransmissionCT ......... 230
.. RegularizationoftheInverseProblem ..................... 235
.. ApproximationThroughWeightedLeastSquares............ 238
7 TechnicalImplementation............................................ 241
. Introduction .................................................. 241
. ReconstructionwithRealSignals ................................ 242
.. FrequencyDomainWindowing........................... 244
.. ConvolutionintheSpatialDomain........................ 247
.. DiscretizationoftheKernels.............................. 252
. PracticalImplementationoftheFilteredBackprojection ............ 255
.. FilteringoftheProjectionSignal .......................... 255
.. ImplementationoftheBackprojection ..................... 258
. MinimumNumberofDetectorElements ......................... 258
. MinimumNumberofProjections................................ 259
. GeometryoftheFan-BeamSystem............................... 261
. ImageReconstructionforFan-BeamGeometry.................... 262
.. RebinningoftheFanBeams.............................. 265
.. ComplementaryRebinning............................... 270
XII Contents
.. FilteredBackprojectionforCurvedDetectorArrays ......... 272
.. FilteredBackprojectionforLinearDetectorArrays .......... 280
.. DiscretizationofBackprojectionforFan-BeamGeometry.... 286
. Quarter-DetectorOffsetandSamplingTheorem ................... 293
8 Three-DimensionalFourier-BasedReconstructionMethods.............. 303
. Introduction .................................................. 303
. SecondaryReconstructionBasedon2DStacksofTomographicSlices 304
. SpiralCT ..................................................... 309
. Exact3DReconstructioninParallel-BeamGeometry .............. 321
.. 3DRadonTransformandtheFourierSliceTheorem......... 321
.. Three-DimensionalFilteredBackprojection ................ 326
.. FilteredBackprojectionandRadon’sSolution ............... 327
.. CentralSectionTheorem................................. 329
.. Orlov’sSufficiencyCondition............................. 335
. Exact3DReconstructioninCone-BeamGeometry ................ 336
.. KeyProblemofCone-BeamGeometry .................... 339
.. MethodofGrangeat ..................................... 341
.. ComputationoftheFirstDerivativeontheDetector......... 347
.. ReconstructionwiththeDerivativeoftheRadonTransform.. 348
.. CentralSectionTheoremandGrangeat’sSolution ........... 350
.. Direct3DFourierReconstructionwiththeCone-Beam
Geometry .............................................. 354
.. ExactReconstructionusingFilteredBackprojection ......... 357
. Approximate3DReconstructionsinCone-BeamGeometry......... 366
.. MissingDatainthe3DRadonSpace ...................... 366
.. FDKCone-BeamReconstructionforPlanarDetectors....... 371
.. FDKCone-BeamReconstructionforCylindricalDetectors... 388
.. VariationsoftheFDKCone-BeamReconstruction .......... 390
. HelicalCone-BeamReconstructionMethods...................... 394
9 ImageQualityandArtifacts .......................................... 403
. Introduction .................................................. 403
. ModulationTransferFunctionoftheImagingProcess.............. 404
. ModulationTransferFunctionandPointSpreadFunction .......... 410
. ModulationTransferFunctioninComputedTomography .......... 412
. SNR,DQE,andROC........................................... 421
. 2DArtifacts................................................... 423
.. PartialVolumeArtifacts ................................. 423
.. Beam-HardeningArtifacts ............................... 425
.. MotionArtifacts ........................................ 432
.. SamplingArtifacts ...................................... 435
.. ElectronicArtifacts...................................... 435
.. DetectorAfterglow...................................... 437
.. MetalArtifacts.......................................... 438