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(A^T)^⊥. 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(A^T)^⊥.