Computed radiation imaging : physics and mathematics of forward and inverse problems

303 Pages·2011·3.698 MB·English

by  E. Hussein

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Computed Radiation Imaging
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Computed Radiation
Imaging
Physics and Mathematics of Forward
and Inverse Problems
Esam M.A. Hussein
Department of Mechanical Engineering
University of New Brunswick
Fredericton, N.B. Canada
AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•OXFORD
PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
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Elsevier
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Firstedition2011
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Contents
Preface xi
1 RadiationImaging 1
1.1 WhyRadiation? 1
1.2 ImagingModalities 2
1.3 DirectandReconstructedImaging 6
1.4 TheForwardandInverseProblems 7
1.5 ForwardandInverseMapping 10
PartI TheForwardProblem 13
2 RadiationTransport 15
2.1 Introduction 15
2.2 Variables 15
2.3 CrossSections 17
2.4 BoltzmannTransportEquation 20
2.5 Source-FreeSteady-StateProblem 21
2.6 Steady-StateProbleminVoid 23
2.7 Point-KernelMethod 24
2.8 ChargedParticles 25
3 MeasurementModels 27
3.1 Formulation 27
3.2 Scaling 27
3.3 MeasuredResponse 28
3.4 Sensitivity 29
3.5 Variability 30
3.6 Components 30
3.6.1 Source 31
3.6.2 Detector 32
3.7 ImageGrid 33
3.8 Idealization 34
3.9 ComputerCoding 35
3.9.1 Verification 35
3.9.2 Validation 35
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4 Transmission 37
4.1 BasicModel 37
4.2 PhysicalRelevance 40
4.2.1 Photons 40
4.2.2 Neutrons 42
4.3 Discretization 42
4.4 NatureofRadiationSource 43
4.4.1 GammaRays 43
4.4.2 X-Rays 45
4.4.3 Neutrons 47
4.5 SecondaryRadiation 48
4.6 Scattering 49
4.7 Sensitivity 50
4.8 Variability 51
5 Emission 53
5.1 EmbeddedRadiation 53
5.2 InducedEmission 55
5.3 Discretization 58
5.4 Sensitivity 59
5.5 Sources 59
5.6 InterferingEffects 61
6 Scattering 63
6.1 Introduction 63
6.2 Single-ScatteringModel 64
6.3 MultipleScattering 65
6.4 ComptonScattering 66
6.5 NeutronElasticScattering 68
6.6 Discretization 70
6.7 Sensitivity 71
PartII TheInverseProblem 73
7 Features 75
7.1 Discretization 75
7.2 Well-PosedProblem 77
7.3 Existence 78
7.4 Uniqueness 78
7.5 Continuity 79
7.6 Ill-PosedProblem 79
7.7 Ill-Conditioning 80
7.A BasicsofFunctionalAnalysis 82
7.A.1 WithinaSpace 82
7.A.2 MappingbetweenSpaces 85
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Contents vii
8 Formulation 87
8.1 Matrix 87
8.2 Functional 88
8.3 Analytic 89
8.3.1 FourierTransform 89
8.3.2 Backprojection 91
8.4 Probabilistic 92
8.A ProbabilisticBasisofMaximum-Likelihoodand
Least-SquaresMethods 94
9 PreprocessingofMeasurements 97
9.1 NumberofMeasurements 98
9.1.1 Sampling 98
9.1.2 ErrorMinimization 100
9.2 FrequencyAnalysis 104
9.2.1 FrequencyFiltering 105
9.2.2 ImpulseResponse 107
9.2.3 DetectorAperture 108
9.3 SpatialFilteringofNoise 109
9.3.1 Moving-AverageFilter 109
9.3.2 GaussianFilter 110
9.3.3 RecursiveWeightedMovingAverageFilter 111
9.3.4 MatchedFilter 112
9.3.5 MultipoleFilters 113
9.4 ConsistencyandSmoothing 113
9.4.1 RegularizedLeast-Squares 114
9.4.2 DynamicProgramming 115
9.4.3 SplineSmoothing 119
9.4.4 CrossValidation 120
9.4.5 Mollification 120
10 Matrix-BasedMethods 125
10.1 ErrorPropagation 125
10.2 SingularValueDecomposition 126
10.3 LeastSquares 128
10.4 RegularizationMethods 128
10.4.1 Approach 128
10.4.2 MathematicalSignificance 130
10.4.3 AnOptimizationPerspective 132
10.4.4 MinimumInformation 133
10.4.5 DoublyRelaxed 134
10.4.6 EstimateofSolution 134
10.4.7 GradientofSolution 134
10.4.8 CovarianceMatrix 136
10.4.9 SpatialCorrelation 137
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viii Contents
10.4.10 Modeling-ErrorCompensation 138
10.4.11 Piecewise 139
10.4.12 Variational 146
10.4.13 MaximumEntropy 147
10.4.14 SolutionBounding 149
10.5 Regularization-ParameterDetermination 150
10.5.1 ConvergenceRegularization 151
10.5.2 DiscrepancyPrinciple 152
10.5.3 L-Curve 152
10.5.4 MinimumPredictiveError 153
10.5.5 GeneralizedCrossValidation 154
10.5.6 MinimumBound 155
10.5.7 StatisticalHypothesisTesting 155
10.5.8 RegularizationbyIteration 157
10.6 IterativeMethods 157
10.6.1 GeometricApproach 158
10.6.2 SuccessiveApproximation 159
10.6.3 SteepestDescent 161
10.6.4 ConjugateGradient 162
10.6.5 ConvergenceMetricsandStoppingCriteria 163
10.7 NonlinearProblems 165
10.7.1 Quasi-Linearization 166
10.7.2 SuccessiveApproximation 166
10.7.3 Newton-Raphson 167
10.7.4 Gauss-Newton 168
10.7.5 Levenberg-Marquardt 168
10.7.6 ConjugateGradient 169
10.8 Software 169
11 FunctionalOptimization 173
11.1 Formulation 173
11.2 EffectofNumberofMeasurements 175
11.3 SensitivitytoMeasurementUncertainty 176
11.4 Minimization 177
11.5 SearchMethods 177
11.5.1 SequentialLinearization 177
11.5.2 GradientDriven 178
11.6 GeneticEvolution 178
11.7 SimulatedAnnealing 180
11.8 NeuralNetworks 182
12 AnalyticMethods 185
12.1 RadonTransform 185
12.2 Two-DimensionalFourierTransforms 187
12.2.1 FundamentalEquations 187
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Contents ix
12.2.2 CompletenessandContinuity 190
12.2.3 Discretization 190
12.2.4 Interpolation 193
12.2.5 DirectFourierInversion 194
12.2.6 Fourier-FrequencyFiltering 195
12.3 Backprojection 195
12.3.1 Direct 196
12.3.2 FourierFiltered 197
12.3.3 RadonFiltered 198
12.3.4 Shepp-LoganFiltered 198
12.3.5 ConvolutionFiltered 199
12.3.6 NoiseFiltering 201
12.3.7 ErrorPropagation 201
12.4 Fan-BeamTransmissionTomography 202
12.4.1 Rebinning 202
12.4.2 DirectBackprojection 204
12.5 Cone-BeamTransmissionTomography 206
12.6 EmissionImaging 208
12.7 ScatterImaging 213
12.8 ComputerCodes 214
12.9 WaveletTransforms 214
13 ProbabilisticMethods 219
13.1 Bayesian-MinimumInformation 219
13.2 PoissonDistribution 221
13.3 NormalDistribution 223
13.4 Maximumaposteriori(map) 224
13.5 TheMonteCarloMethod 228
14 IncompleteProblems 231
14.1 Incompleteness 231
14.2 GeneralSolutionMethods 232
14.2.1 Least-SquaresSolution 232
14.2.2 Twomey-PhillipsSolution 232
14.2.3 AlgebraicGeometricSolution 233
14.2.4 Pseudoinversion 233
14.2.5 OptimizationMethods 233
14.2.6 AnalyticSolutions 233
14.2.7 ProbabilisticSolution 234
14.3 EstimationMaximization 234
14.3.1 PoissonDistributionLikelihood 236
14.3.2 ScatterImaging 240
14.3.3 OrderedSubsetEstimationMaximization 241
14.4 MarkovRandomFields 241