A generalized Kaczmarz algorithm with projection adjustment

Abstract

Kaczmarz algorithm (KA), also known as algebraic reconstruction technique (ART) in medical imaging area, is a commonly used iterative method for solving large-scale linear system of equations. It has found wide applications in many fields such as medical imaging, electromagnetic inverse scattering and geophysics. Various improved KAs have been developed to promote the efficiency of the algorithm, e.g., Block KA, Simultaneous Algebraic Reconstruction Technique (SART), and so on. The author has also presented an accelerated Kaczmarc algorithm with projection adjustment (KAPA), which remarkably promotes the convergence speed of KA. However, KAPA has to additionally store the previous vector for each hyperplane defined by an equation. For some very large-scale practical problems, e.g., 3D CT imaging problem, the additional storage load of KAPA may become unaffordable. To alleviate this problem, a generalized KAPA (denoted as GKAPA1) is proposed in this paper. In the generalized KAPA (GKAPA1), additional acceleration adjustment is only implemented for a subset of hyperplanes, instead of all hyperplanes in KAPA. By using a very small subset, the additional storage are remarkably reduced, and a good balance of speed acceleration and additional storage could be achieved. A randomized version of GKAPA1 (GRKAPA1) has also been proposed in this paper to further speed up the convergence. Numerical simulations of solving linear equations have been conducted to verify the efficiency of the proposed algorithms. GRKAPA1 is also applied to the computed tomography (CT) image reconstruction problem and compared with other algorithms. The simulation results show that the performance of generalized KAPA is much better than KA and close to KAPA. Considering its much less additional storage and smaller computation load, it seems that generalized KAPA is more attractive for practical applications.