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Inverse Probl. 2012 Mar 1;28(3). pii: 035005. Epub 2012 Feb 10.

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

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1
Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran.

Abstract

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.

KEYWORDS:

block-iterative algorithms; image reconstruction from projections; perturbation resilience; superiorization; total variation

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