This book provides an overview of X-ray technology, the historic developmental milestones ofmodern CT systems, and gives a comprehensive insight into themain reconstruction methods used in computed tomography. The basis of reconstruction is, undoubtedly, mathematics. However, the beauty of computed tomography cannot be understood without a detailed knowledge of X-ray generation, photon– matter interaction, X-ray detection, photon statistics, as well as fundamental signal processing concepts and dedicated measurement systems.Therefore, the reader will find a number of references to these basic disciplines together with a brief introduction to the underlying principles of CT.
1 Introduction 1
Computed Tomography – State of the Art 1
Inverse Problems 2
Historical Perspective 4
Some Examples 7
Structure of the Book 11
2 Fundamentals of X-ray Physics 15
Introduction 15
X-ray Generation 15
X-ray Cathode 16
Electron–Matter Interaction 19
Temperature Load 23
X-ray Focus and Beam Quality 24
Beam Filtering 28
Special Tube Designs 30
Photon–Matter Interaction 31
Lambert–Beer's Law 32
Mechanisms of Attenuation 34
Problems with Lambert–Beer's Law 46
X-ray Detection 48
Gas Detectors 48
Solid-State Scintillator Detectors 50
Solid-State Flat-Panel Detectors 52
X-ray Photon Statistics 59
Statistical Properties of the X-ray Source 60
Statistical Properties of the X-ray Detector 64
Statistical Law of Attenuation 66
Moments of the Poisson Distribution 68
Distribution for a High Number of X-ray Quanta 70
Non-Poisson Statistics 72
X Contents
3 Milestones of Computed Tomography 75
Introduction 75
Tomosynthesis 76
Rotation–Translation of a Pencil Beam (First Generation) 79
Rotation–Translation of a Narrow Fan Beam (Second Generation) 83
Rotation of aWide Aperture Fan Beam (Third Generation) 84
Rotation–Fix with Closed Detector Ring (Fourth Generation) 87
Electron Beam Computerized Tomography 89
Rotation in Spiral Path 90
Rotation in Cone-Beam Geometry 91
Micro-CT 93
PET-CT Combined Scanners 96
Optical Reconstruction Techniques 98
4 Fundamentals of Signal Processing 101
Introduction 102
Signals 102
Fundamental Signals 102
Systems 104
Linearity 104
Position or Translation Invariance 105
Isotropy and Rotation Invariance 105
Causality 106
Stability 106
Signal Transmission 106
Dirac's Delta Distribution 109
Dirac Comb 112
Impulse Response 115
Transfer Function 116
Fourier Transform 118
ConvolutionTheorem 124
Rayleigh'sTheorem125
PowerTheorem 125
Filtering in the Frequency Domain 126
Hankel Transform 128
Abel Transform 132
Hilbert Transform 133
SamplingTheorem and Nyquist Criterion 135
Wiener–KhintchineTheorem141
Fourier Transform of Discrete Signals 144
Finite Discrete Fourier Transform 145
z-Transform 147
Chirp z-Transform 148
Contents XI
5 Two-Dimensional Fourier-Based Reconstruction Methods 151
Introduction 151
Radon Transformation 153
Inverse Radon Transformation and Fourier SliceTheorem 163
Implementation of the Direct Inverse Radon Transform 167
Linogram Method 170
Simple Backprojection 175
Filtered Backprojection 179
Comparison Between Backprojection and Filtered Backprojection 183
Filtered Layergram: Deconvolution of the Simple Backprojection 187
Filtered Backprojection and Radon's Solution 191
Cormack Transform 194
6 Algebraic and Statistical Reconstruction Methods 201
Introduction 201
Solution with Singular Value Decomposition 207
Iterative Reconstruction with ART 211
Pixel Basis Functions and Calculation of the SystemMatrix 218
Discretization of the Image: Pixels and Blobs 219
Approximation of the SystemMatrix in the Case of Pixels 221
Approximation of the SystemMatrix in the Case of Blobs 222
Maximum Likelihood Method 223
Maximum Likelihood Method for Emission Tomography 224
Maximum Likelihood Method for Transmission CT 230
Regularization of the Inverse Problem 235
ApproximationThroughWeighted Least Squares 238
7 Technical Implementation 241
Introduction 241
Reconstruction with Real Signals 242
Frequency DomainWindowing 244
Convolution in the Spatial Domain 247
Discretization of the Kernels 252
Practical Implementation of the Filtered Backprojection 255
Filtering of the Projection Signal 255
Implementation of the Backprojection 258
Minimum Number of Detector Elements 258
Minimum Number of Projections 259
Geometry of the Fan-Beam System261
Image Reconstruction for Fan-Beam Geometry 262
Rebinning of the Fan Beams 265
Complementary Rebinning 270
XII Contents
Filtered Backprojection for Curved Detector Arrays 272
Filtered Backprojection for Linear Detector Arrays 280
Discretization of Backprojection for Fan-Beam Geometry 286
Quarter-Detector Offset and SamplingTheorem 293
8 Three-Dimensional Fourier-Based Reconstruction Methods 303
Introduction 303
Secondary Reconstruction Based on 2D Stacks of Tomographic Slices 304
Spiral CT 309
Exact 3D Reconstruction in Parallel-Beam Geometry 321
3D Radon Transform and the Fourier Slice Theorem321
Three-Dimensional Filtered Backprojection 326
Filtered Backprojection and Radon's Solution 327
Central SectionTheorem 329
Orlov's Sufficiency Condition 335
Exact 3D Reconstruction in Cone-Beam Geometry 336
Key Problem of Cone-Beam Geometry 339
Method of Grangeat 341
Computation of the First Derivative on the Detector 347
Reconstruction with the Derivative of the Radon Transform 348
Central SectionTheorem and Grangeat's Solution 350
Direct 3D Fourier Reconstruction with the Cone-Beam Geometry 354
Exact Reconstruction using Filtered Backprojection 357
Approximate 3D Reconstructions in Cone-Beam Geometry 366
Missing Data in the 3D Radon Space 366
FDK Cone-Beam Reconstruction for Planar Detectors 371
FDK Cone-Beam Reconstruction for Cylindrical Detectors 388
Variations of the FDK Cone-Beam Reconstruction 390
Helical Cone-Beam Reconstruction Methods 394
9 Image Quality and Artifacts 403
Introduction 403
Modulation Transfer Function of the Imaging Process 404
Modulation Transfer Function and Point Spread Function 410
Modulation Transfer Function in Computed Tomography 412
SNR, DQE, and ROC421
2D Artifacts 423
Partial Volume Artifacts 423
Beam-Hardening Artifacts 425
Motion Artifacts 432
Sampling Artifacts 435
Electronic Artifacts 435
Detector Afterglow 437
Metal Artifacts 438
Contents XIII
Scattered Radiation Artifacts 443
3D Artifacts 445
Partial Volume Artifacts 446
Staircasing in Slice Stacks 448
Motion Artifacts 450
Shearing in Slice Stacks Due to Gantry Tilt 451
Sampling Artifacts in Secondary Reconstruction 454
Metal Artifacts in Slice Stacks 455
Spiral CT Artifacts 456
Cone-Beam Artifacts 458
Segmentation and Triangulation Inaccuracies 459
Noise in Reconstructed Images 462
Variance of the Radon Transform 462
Variance of the Reconstruction 464
Dose, Contrast, and Variance 467
10 Practical Aspects of Computed Tomography 471
Introduction 471
Scan Planning 471
Data Representation 475
Hounsfield Units 475
Window Width and Window Level 476
Three-Dimensional Representation 479
Some Applications in Medicine 482
11 Dose 485
Introduction 485
Energy Dose, Equivalent Dose, and Effective Dose 486
Definition of Specific CT Dose Measures 487
Device-RelatedMeasures for Dose Reduction 493
User-RelatedMeasures for Dose Reduction 499
