Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction

We study the convergence of a class of accelerated perturbation-resilient block-iterative projection methods for solving systems of linear equations. We prove convergence to a fixed point of an operator even in the presence of summable perturbations of the iterates, irrespective of the consistency o...

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Bibliographic Details
Published in:Inverse problems 2012-03, Vol.28 (3), p.35005-19
Main Authors: Nikazad, T, Davidi, R, Herman, G T
Format: Article
Language:English
Subjects:
Algorithms
block-iterative algorithms
Blocking
Convergence
Exact sciences and technology
Image reconstruction
image reconstruction from projections
Mathematical analysis
Perturbation
perturbation resilience
Physics
Projection
Resilience
superiorization
total variation
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Summary:We study the convergence of a class of accelerated perturbation-resilient block-iterative projection methods for solving systems of linear equations. We prove convergence to a fixed point of an operator even in the presence of summable perturbations of the iterates, irrespective of the consistency of the linear system. For a consistent system, the limit point is a solution of the system. In the inconsistent case, the symmetric version of our method converges to a weighted least-squares solution. Perturbation resilience is utilized to approximate the minimum of a convex functional subject to the equations. A main contribution, as compared to previously published approaches to achieving similar aims, is a more than an order of magnitude speed-up, as demonstrated by applying the methods to problems of image reconstruction from projections. In addition, the accelerated algorithms are illustrated to be better, in a strict sense provided by the method of statistical hypothesis testing, than their unaccelerated versions for the task of detecting small tumors in the brain from x-ray CT projection data.
ISSN:0266-5611
1361-6420
DOI:10.1088/0266-5611/28/3/035005