Kaczmarz算法收敛解的性态分析

Kaczmarz算法作为一种重要的代数重建技术(ART)在医学成像及诊断研究中起着很重要的作用。随着计算机硬件技术的发展,诸如ART、SIRT等迭代算法由于其良好的抗干扰性能及数据缺失情况下良好的成像能力逐渐受到人们的重视。本文主要基于矩阵广义逆的定义和性质证明,当x⁽⁰⁾∈R(A^T)^⊥时Kaczmarz算法迭代序列的极限为Moore-Penrose广义解的性质。理论表明Kaczmarz方法求解相容性和不相容性问题都是适定方法,本文从数值实验的角度验证了Kaczmarz方法的“适定”性和求解扰动问题时的“半收敛”性。另外,Kaczmarz方法当x⁽⁰⁾∈R(A^T)^⊥时还是一类正则化方法。

The Property Analysis of the Convergent Solution to Kaczmarz Method

Kaczmarz method is an important algebraic reconstruction techniques(ART) and play an important role in medical imaging and diagnosis. With the development of computer hardware, these iterative algorithms, such as ART, SIRT, attract people's attention due to their excellent performance in image reconstruction problems with anti-interference and absent data. In this paper, on the basis of the definition and the properties of the generalized inverse, we prove that the limit of the iterative sequence from Kaczmarz method is Moore-Penrose generalized solution as x(0)∈R(AT)⊥. The theoretical results show that Kaczmarz method is 'well-posed' method for consistent and inconsistent problems. In this paper, we verify the 'well-posed' of Kaczmarz method and its 'semi-convergence' for perturbed problems by numerical test. In additional, Kaczmarz method is also a regularization method as x(0)∈R(AT)⊥.