AIR Tools - A MATLAB Package for Algebraic Iterative ...

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xii LIST OF FIGURES 4.6 Relative error histories for an ART method . . . . . . . . . . . . 43 4.7 The minimum relative errors for different λ-values for an ART method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.8 Optimal number of iterations for an ART method . . . . . . . . . 44 4.9 Relative error histories for a SIRT method with maximum number of iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.10 Minimum relative error for a SIRT method with maximum number of iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.11 Optimal number of iterations for a SIRT method with maximum number of iterations . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.12 Illustration of line search . . . . . . . . . . . . . . . . . . . . . . . 48 6.1 Parallel beam illustration . . . . . . . . . . . . . . . . . . . . . . 62 6.2 Fan beam illustration . . . . . . . . . . . . . . . . . . . . . . . . 63 6.3 Seismic tomography illustration . . . . . . . . . . . . . . . . . . . 64 6.4 The two exact phantoms . . . . . . . . . . . . . . . . . . . . . . . 65 7.1 Relative error histories for test of DROP . . . . . . . . . . . . . . 68 7.2 Relative error histories for test of DROP using weighting . . . . 69 7.3 Ψ-based relaxations for symmetric Kaczmarz . . . . . . . . . . . 71 7.4 Training of relaxation parameter using Cimmino’s projection method 72 7.5 Training of relaxation parameter using Kaczmarz’s method . . . 72 7.6 Training of relaxation parameter using randomized Kaczmarz . . 73 7.7 Relative errors for the SIRT methods with trained λ . . . . . . . 73 7.8 Relative errors for the ART methods with trained λ . . . . . . . 74
LIST OF FIGURES xiii 7.9 Training of relaxation parameter using Cimmino’s projection method with maximum number of iterations . . . . . . . . . . . . . . . . 76 7.10 Training of relaxation parameter using Kaczmarz’s method with maximum number of iterations . . . . . . . . . . . . . . . . . . . 76 7.11 Training of relaxation parameter using randomized Kaczmarz method with maximum number of iterations . . . . . . . . . . . . 77 7.12 Relative error for the SIRT methods using line search . . . . . . 78 7.13 Relative error using the Ψ-based relaxations . . . . . . . . . . . . 79 7.14 Relative error using the modified Ψ-based relaxations . . . . . . 79 7.15 Relative errors for the SNARK test problem with different relaxation strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.16 Training of stopping rule for Cimmino’s projection method . . . 84 7.17 Training of stopping rule for DROP . . . . . . . . . . . . . . . . 84 7.18 Training of stopping rule for Kaczmarz’s method . . . . . . . . . 85 7.19 Illustration of the stopping rules for the SIRT methods . . . . . . 86 7.20 Illustration of the stopping rules for the ART methods . . . . . . 87 7.21 Ψ-based relaxation with stopping rules . . . . . . . . . . . . . . . 90 7.22 Line search with stopping rules . . . . . . . . . . . . . . . . . . . 91 7.23 Training λ with stopping rules for SIRT methods . . . . . . . . . 93 7.24 Training λ with stopping rules for ART methods . . . . . . . . . 94 A.1 Illustration of projection on hyperplane where origo is in the hyperplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A.2 Illustration of projection on the hyperplane wheer origo is not in the hyperplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 A.3 Illustration of the roots . . . . . . . . . . . . . . . . . . . . . . . 152
Page 1 and 2: AIR Tools - A MATLAB Package for Al
Page 3: Summary In this master thesis a MAT
Page 7: Preface This master thesis is prepa
Page 10 and 11: viii Contents ̟ average number of
Page 12 and 13: x CONTENTS 5 Stopping Rules 53 5.1
Page 16 and 17: xiv A.4 Zoom of the roots . . . . .
Page 18 and 19: 2 Introduction 1.1 Sturcture of the
Page 20 and 21: 4 Introduction
Page 22 and 23: 6 Theory of Inverse Problems and Re
Page 24 and 25: 8 Theory of Inverse Problems and Re
Page 26 and 27: 10 Theory of Inverse Problems and R
Page 28 and 29: 12 Theory of Inverse Problems and R
Page 30 and 31: 14 Iterative Methods for Reconstruc
Page 50 and 51: 34 Semi-Convergence and Choice of R
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48 Semi-Convergence and Choice of R
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54 Stopping Rules assume that the n
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56 Stopping Rules 〈r k , r k + r
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58 Stopping Rules 5.2 Normalized Cu
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60 Stopping Rules
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62 Test Problems ray i x 1 x 2 x 3
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64 Test Problems x 1 x 2 x 3 x 4 x
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66 Test Problems
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68 Testing the Methods relative err
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70 Testing the Methods the simplifi
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78 Testing the Methods Relative Err
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80 Testing the Methods Minimum rela
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82 Testing the Methods The minimum
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84 Testing the Methods τ 55 50 45
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88 Testing the Methods Testing the
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92 Testing the Methods Total Work U
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96 Testing the Methods
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98 Manual Pages TRAINING ROUTINES t
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100 Manual Pages calczeta Purpose:
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102 Manual Pages cav Purpose: Synop
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104 Manual Pages - options.restart
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106 Manual Pages cimminoProj Purpos
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108 Manual Pages - options.restart.
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110 Manual Pages cimminoRefl Purpos
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114 Manual Pages drop Purpose: Syno
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118 Manual Pages fanbeamtomo Purpos
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120 Manual Pages kaczmarz Purpose:
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122 Manual Pages See also: X = kacz
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124 Manual Pages that the iteration
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126 Manual Pages paralleltomo Purpo
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128 Manual Pages randkaczmarz Purpo
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130 Manual Pages See also: imagesc(
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132 Manual Pages lambda. As default
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134 Manual Pages References: 1. A.
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136 Manual Pages Algorithm: The ele
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138 Manual Pages The second output
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140 Manual Pages trainDPME Purpose:
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142 Manual Pages trainLambdaART Pur
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144 Manual Pages trainLambdaSIRT Pu
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146 Manual Pages
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148 Conclusion and Future Work rela
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150 Appendix bi = 0 ⇒ O ∈ Hi: a
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152 Appendix imaginary 0.8 0.6 0.4
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154 Appendix To use Newton-Raphson
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156 BIBLIOGRAPHY [9] A. Dax, Line s
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