Algebraic reconstruction techniques can be made computationally efficient (positron emission tomography application)

Abstract

Algebraic reconstruction techniques (ART) are iterative procedures for recovering objects from their projections. It is claimed that by a careful adjustment of the order in which the collected data are accessed during the reconstruction procedure and of the so-called relaxation parameters that are to be chosen in an algebraic reconstruction technique, ART can produce high-quality reconstructions with excellent computational efficiency. This is demonstrated by an example based on a particular (but realistic) medical imaging task, showing that ART can match the performance of the standard expectation-maximization approach for maximizing likelihood (from the point of view of that particular medical task), but at an order of magnitude less computational cost.

A Sampling Kaczmarz--Motzkin Algorithm for Linear Feasibility

“…The original Kaczmarz method simply cycles through the equations sequentially, so its convergence rate depends on the order of the rows. One way to overcome this is to use the equations in a random order, rather than sequentially [HS78,HM93,Nat01]. More precisely, we begin with Ax ≤ b, a linear system of inequalities where A is an m × n matrix with rows a i and x 0 an initial guess.…”

Introduction: a brief overview of iterative algorithms in X-ray computed tomography

“…There is growing interest in iterative reconstruction algorithms mainly due to recent surge in computational power. In this paper, a brief overview of iterative algorithm in X-ray CT is presented [2][3][4][5][6][7].…”

Randomized block Kaczmarz method with projection for solving least squares

“…A poor ordering may lead to very slow convergence. To overcome this obstacle, one can select the row a i at random to improve the convergence rate [HM93,Nat01]. Strohmer and Vershynin proposed and analyzed a method which selects a given row with probability proportional to its 2 norm [SV09,SV06].…”