Abstract
Kaczmarz's alternating projection method has been widely used for solving mostly over-determined linear system of equations A x = b 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 A 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.
Highlights
Kaczmarz introduced an iterative algorithm for solving a consistent linear system of equations Ax = b with A ∈ RM×N
The Kaczmarz method is a method of alternating projection (MAP) and it has been widely used in medical imaging as an algebraic reconstruction technique (ART) [2, 3] due to its simplicity and light computation
The simulations were computed in parallel for each of the methods: Kaczmarz (K), randomized Kaczmarz hyperplane angles (RKHA), and randomized Kaczmarz (RK)
Summary
Kaczmarz (in [1]) introduced an iterative algorithm for solving a consistent linear system of equations Ax = b with A ∈ RM×N. This method projects the estimate xj onto a subspace normal to the row ai at step j + 1 cyclically with i = j(modM) + 1. Needell (in [5]) extended the work of [4] for noisy linear systems and developed a bound for convergence to the least square solution for Ax = b. Needell developed a randomized Kaczmarz method that addresses coherence effects [6], and she analyzed the convergence of randomized block Kaczmarz method [7]. Brezinski and Redivo-Zaglia (in [12]) utilizes the work of Galantai for accelerating convergence of regular Kaczmarz method
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