Convergence of iterative algorithms for image reconstruction

We introduce a general iterative scheme for image reconstruction based on Landweber's method. Within our configuration, the block-iterative (BI) version can be formulated from its simultaneous version easily, and vice versa. This provides the mechanism to formulate new algorithms from known algorithms. It can be shown that some of the widely used iterative algorithms, such as the ART, SART, Cimmino's Algorithm and recently designed DWE and CAV, are special examples of the general scheme or its BI version. By applying the convergence results of the general scheme, the corresponding convergence in the consistent or inconsistent cases, for the block-iterative or simultaneous versions, of those specific algorithms can be established under mild conditions on the relaxation parameters.

1 Introduction

Image reconstruction arises in applications. Most of the imaging systems can be modeled by the following equation $$Ax=b \eqno{\hbox{(1)}}$$ where the observed data the original image is a non-zero matrix. The problem is to reconstruct the image x from the observed data b. A direct solution is not feasible because of the ill-posedness, the noisy corrupt data b and huge data dimension in practice.