The generalized Landweber iteration in positron emission tomography.

Abstract

This research makes three contributions to the field of image reconstruction in positron emission tomography (PET). First, we study the convergence behavior of multigrid implementations of the following iterative algorithms: Landweber; generalized Landweber; algebraic reconstruction technique (ART) and maximum likelihood expectation maximization (MLEM). The results suggest that if the image has a "local smoothness property" (the values of four neighboring fine-grid pixels to be grouped as a coarse-grid pixel are close to each other), then the reconstruction of high-frequency components (associated with the singular vectors corresponding to small singular values in the PET system geometry matrix) is accelerated. If the local smoothness property does not hold, or if there are no high-frequency components in the image, then a multigrid implementation does not speed up the convergence. Second, we study the effect of the lowest-frequency component, defined as the DC component, on the speed of convergence of the generalized Landweber iteration. We have shown there is a large ratio between the two largest singular values of the system matrix of a typical PET system. A new "DC-suppression" procedure has been designed, which removes the DC component after a single Landweber iteration in order to speed up the reconstruction of higher-frequency image components for the Landweber, generalized Landweber and steepest descent iterative algorithms. Reconstruction of high-frequency image components has been accelerated by a factor of three when the ratio of the two largest singular values is 1.68. The performance of the generalized Landweber iteration with DC-suppression is comparable to ART, and faster than the conjugate gradient iteration, during the first 100 forward and backward projections. Third, a new variable shaping matrix method has been proposed for the generalized Landweber iteration, in order to either accelerate the reconstruction of high-frequency image components, or to attenuate high-frequency image components for regularization. This method can also be applied to finding the truncated singular value decomposition solution for a system of linear equations.