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