1;Preface;5 2;Contents;8 3;1 Introduction;13 3.1;1.1 Computed Tomography - State of the Art;13 3.2;1.2 Inverse Problems;14 3.3;1.3 Historical Perspective;16 3.4;1.4 Some Examples;19 3.5;1.5 Structure of the Book;23 4;2 FundamentalsofX-rayPhysics;26 4.1;2.1 Introduction;26 4.2;2.2 X- ray Generation;26 4.3;2.3 Photon-Matter Interaction;42 4.4;2.4 Problems with Lambert-Beer's Law;57 4.5;2.5 X- ray Detection;59 4.6;2.6 X- ray Photon Statistics;70 5;3 Milestones of Computed Tomography;85 5.1;3.1 Introduction;85 5.2;3.2 Tomosynthesis;86 5.3;3.3 Rotation- Translation of a Pencil Beam(First Generation);89 5.4;3.4 Rotation- Translation of a Narrow Fan Beam (Second Generation);93 5.5;3.5 Rotation of aWide Aperture Fan Beam ( Third Generation);94 5.6;3.6 Rotation-Fix with Closed Detector Ring (Fourth Generation);97 5.7;3.7 Electron BeamComputerized Tomography;99 5.8;3.8 Rotation in Spiral Path;100 5.9;3.9 Rotation in Cone-BeamGeometry;101 5.10;3.10 Micro- CT;103 5.11;3.11 PET- CT Combined Scanners;106 5.12;3.12 Optical Reconstruction Techniques;108 6;4 Fundamentals of Signal Processing;110 6.1;4.1 Introduction;111 6.2;4.2 Signals;111 6.3;4.3 Fundamental Signals;111 6.4;4.4 Systems;113 6.5;4.5 Signal Transmission;115 6.6;4.6 Dirac's Delta Distribution;118 6.7;4.7 Dirac Comb;121 6.8;4.8 Impulse Response;124 6.9;4.9 Transfer Function;125 6.10;4.10 Fourier Transform;127 6.11;4.11 Convolution Theorem;133 6.12;4.12 Rayleigh's Theorem;134 6.13;4.13 Power Theorem;134 6.14;4.14 Filtering in the Frequency Domain;135 6.15;4.15 Hankel Transform;137 6.16;4.16 Abel Transform;141 6.17;4.17 Hilbert Transform;142 6.18;4.18 Sampling Theoremand Nyquist Criterion;144 6.19;4.19 Wiener-Khintchine Theorem;150 6.20;4.20 Fourier Transform of Discrete Signals;153 6.21;4.21 Finite Discrete Fourier Transform;154 6.22;4.22 z- Transform;156 6.23;4.23 Chirp z- Transform;157 7;5 Two-Dimensional Fourier-Based ReconstructionMethods;160 7.1;5.1 Introduction;160 7.2;5.2 Radon Transformation;162 7.3;5.3 Inverse Radon Transformation and Fourier Slice Theorem;172 7.4;5.4 Implementation of the Direct Inverse Radon Transform;176 7.5;5.5 LinogramMethod;179 7.6;5.6 Simple Backprojection;184 7.7;5.7 Filtered Backprojection;188 7.8;5.8 Comparison Between Backprojection and Filtered Backprojection;192 7.9;5.9 Filtered Layergram: Deconvolution of the Simple Backprojection;196 7.10;5.10 Filtered Backprojection and Radon's Solution;200 7.11;5.11 Cormack Transform;203 8;6 Algebraic and Statistical ReconstructionMethods;210 8.1;6.1 Introduction;210 8.2;6.2 Solution with Singular Value Decomposition;216 8.3;6.3 Iterative Reconstruction with ART;220 8.4;6.4 Pixel Basis Functions and Calculation of the SystemMatrix;227 8.5;6.5 Maximum Likelihood Method;232 9;7 Technical Implementation;250 9.1;7.1 Introduction;250 9.2;7.2 Reconstruction with Real Signals;251 9.3;7.3 Practical Implementation of the Filtered Backprojection;264 9.4;7.4 Minimum Number of Detector Elements;267 9.5;7.5 Minimum Number of Projections;268 9.6;7.6 Geometry of the Fan-Beam System;270 9.7;7.7 Image Reconstruction for Fan-Beam Geometry;271 9.8;7.8 Quarter-Detector Offset and Sampling Theorem;302 10;8 Three-Dimensional Fourier-Based ReconstructionMethods;311 10.1;8.1 Introduction;311 10.2;8.2 Secondary Reconstruction Based on 2D Stacks of Tomographic Slices;312 10.3;8.3 Spiral CT;317 10.4;8.4 Exact 3D Reconstruction in Parallel-Beam Geometry;329 10.5;8.5 Exact 3D Reconstruction in Cone-Beam Geometry;344 10.6;8.6 Approximate 3D Reconstructions in Cone-BeamGeometry;374 10.7;8.7 Helical Cone-Beam Reconstruction Methods;402 11;9 Image Quality and Artifacts;410 11.1;9.1 Introduction;410 11.2;9.2 Modulation Transfer Function of the Imaging Process;411 11.3;9.3 Modulation Transfer Function and Point Spread Function;417 11.4;9.4 Modulation Transfer Function in Computed Tomography;419 11.5;9.5 SNR, DQE, and ROC;428 11.6;9.6 2D Artifacts;430 11.7;9.7 3D Artifacts;452 11.8;9.8 Noise in Reconstructed Images;469 12;10 Practical Aspect