## Fundamentals of Computerized Tomography: Image Reconstruction from ProjectionsThis revised and updated text presents the computational and mathematical procedures underlying data collection, image reconstruction, and image display in computerized tomography. New topics: the fast calculation of a ray sum for a digitized picture, the task-oriented comparison of reconstruction algorithm performance, blob basis functions and the linogram method for image reconstruction. Features: Describes how projection data are obtained and the resulting reconstructions are used; Presents a comparative evaluation of reconstruction methods; Investigates reconstruction algorithms; Explores basis functions, functions to be optimized, norms, generalized inverses, least squares solutions, maximum entropy solutions, and most likely estimates; Discusses SNARK09, a large programming system for image reconstruction; Concludes each chapter with helpful Notes and References sections. An excellent guide for practitioners, it can also serve as a textbook for an introductory graduate course. |

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### Contents

1 | |

12 Probability and Random Variables | 18 |

An Overview of the Process of CT | 27 |

23 Data Collection for CT | 30 |

24 Voxels Pixels and CT Numbers | 31 |

25 The Problem of Polychromaticity | 32 |

26 Reconstruction Algorithms | 34 |

Physical Problems Associated with Data Collection in CT | 37 |

87 Why So Popular? | 157 |

Other Transform Methods for Parallel Beams | 159 |

92 The Fourier Method of Reconstruction | 161 |

93 Linograms | 166 |

94 RhoFiltered Layergram | 173 |

Filtered Backprojection for Divergent Beams | 177 |

102 Choice of the Window Function | 181 |

103 Point Response Function | 182 |

32 Beam Hardening | 42 |

33 Other Sources of Error | 44 |

34 Scanning Modes | 47 |

Computer Simulation of Data Collection in CT | 53 |

42 Creation of a Phantom | 54 |

43 A PiecewiseHomogeneous Head Phantom | 56 |

44 Head Phantom with a Large Tumor and Local Inhomogeneities | 59 |

45 Creation of the Ray Sums | 60 |

46 Fast Calculation of a Ray Sum for a Digitized Picture | 63 |

Data Collection and Reconstruction of the Head Phantom | 66 |

52 TaskOriented Comparison of Algorithm Performance | 69 |

53 An Illustration Using Selective Smoothing | 73 |

54 Reconstruction from Perfect Data | 78 |

55 Effects of Photons Statistics | 83 |

56 Effect of Beam Hardening | 86 |

57 The Effects of Detector Width and Scatter | 91 |

58 Simulation of Different Scanning Modes | 95 |

Basic Concepts of Reconstruction Algorithms | 100 |

62 Transform Methods | 106 |

63 Series Expansion Methods | 108 |

64 Optimization Criteria | 111 |

65 Blob Basis Functions | 119 |

66 Computational Efficiency | 122 |

Backprojection | 125 |

72 Implementation of the Backprojection Operator | 127 |

73 Discrete Backprojection | 131 |

Filtered Backprojection for Parallel Beams | 135 |

82 Derivation of the FBP Method | 139 |

83 Implementation of the FBP Method | 140 |

84 Fourier Transforms | 143 |

85 Sampling and Interpolation | 146 |

86 The Choice of Convolving and Interpolating Functions | 147 |

104 Noise Reconstruction | 186 |

105 Comparison of Algorithms Based on Reconstructions | 188 |

Algebraic Reconstruction Techniques | 193 |

112 Relaxation Methods for Solving Systems of Inequalities and Equalities | 196 |

113 Additive ART | 201 |

114 Tricks | 205 |

115 Efficacy of ART | 210 |

Quadratic Optimization Methods | 217 |

122 Richardsons Method for Solving Systems of Equations | 221 |

123 Smoothing Matrices | 224 |

124 Implementation of Richardsons Methods for Image Reconstruction | 226 |

125 A Demonstration of Quadratic Optimization | 227 |

Truly ThreeDimensional Reconstruction | 234 |

131 ThreeDimensional Series Expansion | 236 |

132 Dynamically Changing 3D Phantoms and Their Projections | 237 |

133 ThreeDimensional Reconstructions of the Dynamic Phantom | 240 |

ThreeDimensional Display of Organs | 243 |

142 Boundary Detection | 246 |

143 Hidden Surface Removal | 251 |

144 Shading | 253 |

Mathematical Background | 259 |

152 The Line Integral of the Relative Linear Attenuation | 260 |

153 The Radon Inversion Formula | 261 |

154 A Picture Is Not Uniquely Determined by a Finite Number of Its Views | 265 |

155 Analysis of the Photon Statistics | 267 |

156 The Integral Expression for Polychromatic Ray Sums | 269 |

157 Proof of the Regularization Theorem | 270 |

158 Convergence of the Relaxation Method for Inequalities | 273 |

277 | |

292 | |

### Common terms and phrases

approximation assume attenuation coefficient backprojection bandlimiting basis functions blobs calibration measurement computerized tomography converges convolving cross section data collection defined denote density detector dimensionality discrete random variable display divergent beam elemental object energy estimate evaluation example FBP method Fourier transform Gaussian random variable grid Hamming window head phantom illustrated image reconstruction image vector implementation IROI iterative step line integrals linear attenuation coefficient linear interpolation mathematical matrix minimizes monochromatic nearest neighbor interpolation nonnegative number of photons obtained optimization P-value parameters photon statistics picture distance measures picture function picture region pixel Poisson random variable polar variables polychromatic problem produced Radon transform ray sum real numbers reconstruction algorithms reconstruction from projections reconstruction region referred relative linear attenuation relaxation method Richardson's method sample scanning mode scatter selective smoothing shown in Fig SNARK09 standard projection data three-dimensional tomography tumor two-dimensional values voxels window function x-ray x-ray source zero