A multilevel approach to the algebraic image reconstruction problem

Abstract

The problem of reconstructing an image from its Radon transform profiles is outlined. This problem has medical, industrial and military applications. Using the computer assisted tomography (CAT) scan as an example, a discretization of the problem based on natural pixels is described, leading to a symmetric linear system that is in general smaller than that resulting from the conventional discretization. The linear algebraic properties of the system matrix are examined, and the convergence of the Gauss-Seidel iteration applied to the linear system is established. Next, multilevel technology is successfully incorporated through a multilevel projection method (PML) formulation of the problem. This results in a V-cycle algorithm, the convergence of which is established. Finally, the problem of spotlight computed tomography, where high quality reconstructions for only a portion of the image are required, is outlined. We establish the formalism necessary to apply fast adaptive composite (FAC) grids in this setting, and formulate the problem in a block Gauss-Seidel form. Numerical results and reconstructed images are presented which demonstrate the usefulness of these two multilevel approaches