Constrained block-iterative Landweber scheme for image reconstruction

Abstract

The Landweber method provides a framework to formulate iterative algorithms for image reconstruction problems with large, sparse and unstructured system matrices. In a previous study, the authors established the convergence conditions for a general Landweber scheme in both simultaneous and block-iterative [or ordered-subset (OS)] formats with either consistent or inconsistent data, without constraints. Constrained iterative algorithms provide a mechanism for incorporating prior knowledge such nonnegativity, bounds, finite spatial or spectral supports, <i>etc</i>. Hence, they have been widely used in practice. Although the simultaneous constrained (or projected) Landweber scheme was well studied, the convergence of the constrained block-iterative Landweber scheme is unknown. Block-iterative schemes are recently intensively studied theoretically and applied widely. In this paper, we report convergence conditions of a constrained block-iterative Landweber scheme. Prior knowledge is represented as convex sets in which an image of interest must stay. The constrained block-iterative Landweber scheme is constructed by alternatively performing a projection onto convex sets (POCS) and a conventional block-iterative Landweber iteration. The POCS method has been used before for constrained image reconstruction to satisfy both imaging equations and convex constraints. Our approach is different from the conventional application of the POCS method in that we use Landweber iteration for the imaging equations and perform POCS only for the convex constraints. While the conventionally applied POCS method requires Moore-Penrose inverses of matrix blocks, our constrained block-iterative method only takes transposes of such matrix blocks, and improves the computational complexity greatly.