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Algebraic Reconstruction Methods
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. A well known example is image reconstruction, which can be modeled using the Radon transform. After expanding the solution into a finite series of basis functions a large, sparse and ill-conditioned linear system occurs. We consider the solution of such systems. In particular we study a new class of iteration methods named DROP (for Diagonal Relaxed Orthogonal Projections) constructed for solving both linear equations and linear inequalities. This class can also be viewed, when applied to linear equations, as a generalized Landweber iteration. The method is compared with other iteration methods using test data from a medical application and from electron microscopy. Our theoretical analysis include convergence proofs of the fully-simultaneous DROP algorithm for linear equations without consistency assumptions, and of block-iterative algorithms both for linear equations and linear inequalities, for the consistent case.

When applying an iterative solver to an ill-posed set of linear equations the error usually initially decreases but after some iterations, depending on the amount of noise in the data, and the degree of ill-posedness, it starts to increase. This phenomenon is called semi-convergence. We study the semi-convergence performance of Landweber-type iteration, and propose new ways to specify the relaxation parameters. These are computed so as to control the propagated error.

We also describe a class of stopping rules for Landweber-type iteration for solving linear inverse problems. The class includes the well known discrepancy principle, and the monotone error rule. We unify the error analysis of these two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using trained rules are demonstrated on examples taken from image reconstruction from projections.

Kaczmarz's method, also called ART (Algebraic Reconstruction Technique) is often used for solving the linear system which appears in image reconstruction. This is a fully sequential method. We examine and compare ART and its symmetric version. It is shown that the cycles of symmetric ART, unlike ART, converge to a weighted least squares solution if and only if the relaxation parameter lies between zero and two. Further we show that ART has faster asymptotic rate of convergence than symmetric ART. Also a stopping criterion is proposed and evaluated for symmetric ART.

We further investigate a class of block-iterative methods used in image reconstruction. The cycles of the iterative sequences are characterized in terms of the original linear system. We define symmetric block-iteration and compare the behavior of symmetric and non-symmetric block-iteration. The results are illustrated using some well-known methods. A stopping criterion is offered and assessed for symmetric block-iteration.

Place, publisher, year, edition, pages
Matematiska institutionen , 2008. , 16 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1186
Keyword [en]
iterative methods; image reconstruction; ART; Cimmino; Kaczmarz; Landweber; sequential iteration; simultaneous iteration; block iteration; semi-convergence; relaxation parameters; stopping rules; discrepancy principle
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-11670ISBN: 978-91-7393-888-4 (print)OAI: oai:DiVA.org:liu-11670DiVA: diva2:18098
Public defence
2008-06-10, Glashuset, Hus B, Ingång 23, Department of Mathematics, Linköping University, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2009-05-08
List of papers
1. On Diagonally Relaxed Orthogonal Projection Methods
Open this publication in new window or tab >>On Diagonally Relaxed Orthogonal Projection Methods
2008 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 30, no 1, 473-504 p.Article in journal (Refereed) Published
Abstract [en]

We propose and studya block-iterative projection method for solving linear equations and/or inequalities.The method allows diagonal componentwise relaxation in conjunction with orthogonalprojections onto the individual hyperplanes of the system, and isthus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation hasproven useful in accelerating the initial convergence of simultaneous andblock-iterative projection algorithms, but until now it was available onlyin conjunction with generalized oblique projections in which there isa special relation between the weighting and the oblique projections.DROP has been used by practitioners, and in this papera contribution to its convergence theory is provided. The mathematicalanalysis is complemented by some experiments in image reconstruction fromprojections which illustrate the performance of DROP.

Place, publisher, year, edition, pages
Philadelphia, PA, United States: Society for Industrial and Applied Mathematics, 2008
Keyword
block iteration, convex feasibility, diagonal relaxation, projection methods, simultaneous algorithms
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-13235 (URN)10.1137/050639399 (DOI)000208048600006 ()
Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2017-12-13Bibliographically approved
2. Stopping Rules for Landweber-type Iteration
Open this publication in new window or tab >>Stopping Rules for Landweber-type Iteration
2007 (English)In: Inverse Problems, ISSN 0266-5611, Vol. 23, no 4, 1417-1432 p.Article in journal (Refereed) Published
Abstract [en]

We describe a class of stopping rules for Landweber-type iterations for solving linear inverse problems. The class includes both the discrepancy principle (DP rule) and the monotone error rule (ME rule). We also unify the error analysis of the two methods. The stopping rules depend critically on a certain parameter whose value needs to be specified. A training procedure is therefore introduced for securing robustness. The advantages of using a trained rule are demonstrated on examples taken from image reconstruction from projections. After training the stopping rules became quite robust and only small differences were observed between, e.g. the DP rule and ME rule.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-13236 (URN)10.1088/0266-5611/23/4/004 (DOI)
Available from: 2008-05-21 Created: 2008-05-21
3. Some Properties of ART-type Reconstruction Algorithms
Open this publication in new window or tab >>Some Properties of ART-type Reconstruction Algorithms
2008 (English)In: Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), / [ed] Yair Censor, Ming Jiang, Alfred K. Louis, 2008, 1, 526- p.Chapter in book (Other academic)
Abstract [en]

This book contains papers presented by leading experts at the "Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT)" held at the Centro di Ricerca Matematica (CRM) Ennio De Giorgi at Pisa, Italy, from October 15 to 19, 2007. The interdisciplinary book consists of research and review papers by leading experts and practitioners in biomedical imaging and intensity-modulated radiation therapy (IMRT).

National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-13237 (URN)8876423141 (ISBN)9788876423147 (ISBN)
Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2013-05-24Bibliographically approved
4. Some Block-Iterative Methods used in Image Reconstruction
Open this publication in new window or tab >>Some Block-Iterative Methods used in Image Reconstruction
2008 (English)Article in journal (Refereed) Submitted
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-13238 (URN)
Available from: 2008-05-21 Created: 2008-05-21
5. Semi-Convergence and Choice of Relaxation Parameters in Landweber-type Algorithms
Open this publication in new window or tab >>Semi-Convergence and Choice of Relaxation Parameters in Landweber-type Algorithms
Manuscript (Other academic)
Identifiers
urn:nbn:se:liu:diva-13239 (URN)
Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2010-01-13

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