Table of Contents
Journal of Medical Engineering
Volume 2014 (2014), Article ID 908984, 15 pages
http://dx.doi.org/10.1155/2014/908984
Research Article

Kaczmarz Iterative Projection and Nonuniform Sampling with Complexity Estimates

Department of Computer Science, Tennessee State University, 3500 John A. Merritt Boulevard, Nashville, TN 37209-1500, USA

Received 21 August 2014; Revised 26 October 2014; Accepted 27 October 2014; Published 15 December 2014

Academic Editor: Hengyong Yu

Copyright © 2014 Tim Wallace and Ali Sekmen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Kaczmarz’s alternating projection method has been widely used for solving mostly over-determined linear system of equations in various fields of engineering, medical imaging, and computational science. Because of its simple iterative nature with light computation, this method was successfully applied in computerized tomography. Since tomography generates a matrix with highly coherent rows, randomized Kaczmarz algorithm is expected to provide faster convergence as it picks a row for each iteration at random, based on a certain probability distribution. Since Kaczmarz’s method is a subspace projection method, the convergence rate for simple Kaczmarz algorithm was developed in terms of subspace angles. This paper provides analyses of simple and randomized Kaczmarz algorithms and explains the link between them. New versions of randomization are proposed that may speed up convergence in the presence of nonuniform sampling, which is common in tomography applications. It is anticipated that proper understanding of sampling and coherence with respect to convergence and noise can improve future systems to reduce the cumulative radiation exposures to the patient. Quantitative simulations of convergence rates and relative algorithm benchmarks have been produced to illustrate the effects of measurement coherency and algorithm performance, respectively, under various conditions in a real-time kernel.