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

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Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800 Kongens Lyngby, Denmark Phone +45 45253351, Fax +45 45882673 reception@imm.dtu.dk www.imm.dtu.dk
Summary In this master thesis a MATLAB package AIR Tools with implementations of several iterative algebraic reconstruction methods for discretizations of tomography problems is developed. The focus is mainly on the two classes of methods: Simultaneous Iterative Reconstruction Technique (SIRT) and Algebraic Reconstruction Techniques (ART). The package also includes three simplified test problems from medical and seismic tomography. For each iterative method a number of strategies for choosing the relaxation parameter and the stopping rule are presented and implemented. The relaxation parameter can be chosen as a fixed parameter or chosen adaptively in each iteration. For the fixed case a training strategy is developed for finding the optimal parameter for a given test problem. The stopping rules provided in the package is the Discrepancy Principle, the Monotone Error Rule and the NCP criterion. For the first two methods a training strategy is also provided for finding an optimal stopping parameter. In addition simulation studies and comparisons of performance of the available methods and strategies are presented and discussed. This thesis also includes manual pages for each implemented routine that describes the use of the implemented routines. KEYWORDS: ART methods, SIRT methods, iterative methods, semi-convergence, relaxation parameters, stopping rules, tomography.
Page 1: AIR Tools - A MATLAB Package for Al
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
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Page 18 and 19: 2 Introduction 1.1 Sturcture of the
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36 Semi-Convergence and Choice of R
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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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