Preconditioned block Kaczmarz methods for linear equations with an application to computed tomography.

Abstract

Background: Preconditioned Kaczmarz methods play a pivotal role in image reconstruction. A fundamental theoretical question lies in establishing the convergence conditions for these methods. Practically, devising an efficient block strategy to accelerate the reconstruction process is also critical.

Objective: This paper aims to introduce the convergence conditions for the preconditioned Kaczmarz methods and design the block strategy with corresponding preconditioners for these methods in computed tomography (CT).

Methods: We establish a kind of useful convergence conditions for the preconditioned block Kaczmarz methods and prove the dependence of the convergence limit on the initial guess. Tailored for the CT problem, we also propose a new method with a novel block strategy and specific preconditioners, which ensure accelerated convergence.

Results: Numerical experiments with the Shepp-Logan phantom and a real chest CT image demonstrate that our proposed block strategy and preconditioners effectively accelerate the reconstruction process by the preconditioned block Kaczmarz methods while maintaining satisfactory image quality.

Conclusions: Our proposed method, which incorporates the designed block strategy and specific preconditioners, has superior performance compared to the traditional Landweber iteration and the block Kaczmarz iteration without preconditioners.