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

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Resumé I dette eksamensprojekt udvikles en MATLAB programpakke, AIR Tools, med implementeringer af flere iterative algebraiske rekonstruktions metoder til diskretiseret tomografi problemer. Det primære fokus er p˚a to klasser af metoder: Simulatan Iterative Rekonstruktion Teknik (SIRT) og Algebraiske Rekonstruktions Teknikker (ART). Programpakken indeholder ligeledes tre simple testproblemer fra medicinsk og seismisk tomografi. For hver iterative metode præsenteres og implementeres en række strategier til at vælge relaxations parameteren samt stopkriterier. Relaxations parameteren kan enten vælges som en konstant parameter eller den kan vælges adaptivt i hver iteration. For det konstante tilfælde er der udviklet en træningsstrategi til at finde den optimale værdi for et givent testproblem. Stopkriterierne, der er tilgængelige i denne pakke, er discrepancy princippet, monotone error reglen samt NCP kriteriet. For de to første metoder er der givet en træningsstrategi til at finde den optimale værdi for stopparamerten. Yderligere er studier og sammenligninger af metodernes og strategiernes opførsel ogs˚a præsenteret of diskuteret. Eksamensprojektet indeholder ogs˚a manual sider til hver implementeret funktion, som beskriver benyttelsen heraf. STIKORD: ART metoder, SIRT metoder, iterative metoder, semi-konvergens, relaxation parameter, stopkriterier, tomografi.
Page 1 and 2: AIR Tools - A MATLAB Package for Al
Page 3: Summary In this master thesis a MAT
Page 9 and 10: List of Symbols The following is a
Page 11 and 12: Contents Summary i Resumé iii Pref
Page 13 and 14: List of Figures 2.1 SVD basis . . .
Page 15 and 16: LIST OF FIGURES xiii 7.9 Training o
Page 17 and 18: Chapter 1 Introduction In the begin
Page 19 and 20: 1.1 Sturcture of the Thesis 3 • C
Page 21 and 22: Chapter 2 Theory of Inverse Problem
Page 23 and 24: 2.2 SVD and Picard Condition 7 U(:,
Page 25 and 26: 2.3 Spectral Filtering 9 2.3 Spectr
Page 27 and 28: 2.5 Resolution Limit 11 2.5 Resolut
Page 29 and 30: Chapter 3 Iterative Methods for Rec
Page 31 and 32: 3.1 Simultaneous Iterative Reconstr
Page 37 and 38: 3.2 Algebraic Reconstruction Techni
Page 43 and 44: 3.4 Block-Iterative Methods 27 we d
Page 45 and 46: 3.4 Block-Iterative Methods 29 cons
Page 47 and 48: 3.4 Block-Iterative Methods 31 wher
Page 49 and 50: Chapter 4 Semi-Convergence and Choi
Page 51 and 52: 4.1 Semi-Convergence for SIRT Metho
Page 53 and 54: 4.1 Semi-Convergence for SIRT Metho
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4.2 Choice of Relaxation Parameter
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Chapter 5 Stopping Rules In the pre
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5.1 Stopping Rules with Training 55
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5.1 Stopping Rules with Training 57
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5.2 Normalized Cumulative Periodogr
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Chapter 6 Test Problems This softwa
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x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9
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Shepp−Logan Phantom, N = 100 (a)
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Chapter 7 Testing the Methods In th
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7.1 Convergence of DROP 69 relative
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7.3 Test of the Choice of Relaxatio
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7.4 Stopping Rules 83 after 2 itera
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7.4 Stopping Rules 85 τ 820 810 80
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7.4 Stopping Rules 87 Stopping inde
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7.5 Relaxation Strategies Combined
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Chapter 8 Manual Pages ITERATIVE SI
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The Demo functions This MATLAB pack
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See also: cav,cimminoProj,cimminoRe
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103 lambda. As default lambda is se
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See also: cimminoProj, cimminoRefl,
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107 The second output info is a vec
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References: 109 1. G. Cimmino, Calc
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References: 113 1. G. Cimmino, Calc
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115 lambda. As default lambda is se
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See also: cav, cimminoProj, cimmino
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Algorithm: 119 The element aij is d
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landweber Purpose: Synopsis: The Cl
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Examples: - options.stoprule.type 1
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127 phantom is reshaped as a vector
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sart Purpose: The Simultaneous Alge
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- options.restart.M = a vector cont
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seismictomo Purpose: Synopsis: Crea
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symkaczmarz Purpose: Synopsis: The
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See also: [A b x] = paralleltomo(50
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141 - options.restart: Only availib
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Examples: 143 Generate a “noisy
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Use of options: The following field
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Chapter 9 Conclusion and Future Wor
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Appendix A Appendix A.1 Orthogonal
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A.1 Orthogonal Projection on a Hype
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A.2 Investigation of the Roots 153
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Bibliography [1] A. H. Andersen and
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BIBLIOGRAPHY 157 [24] G. N. Hounsfi
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