Acceleration of Landweber-type algorithms by suppression of projection on the maximum singular vector

A procedure is developed that speeds up the convergence during the initial stage (the first 100 forward and backward projections) of Landweber-type algorithms for iterative image reconstruction, which include the Landweber, generalized Landweber, and steepest descent algorithms. In this procedure, the singular vector associated with the maximum singular value of the PET (positron emission tomography) system matrix is identified, and then projection of the data on this singular vector is suppressed after a single Landweber iteration. Typical PET system matrices have a significant gap between their two largest singular values; hence this suppression allows larger gains in subsequent iterations, speeding up convergence by roughly a factor of three. The present work includes: (1) a study of the singular value spectra of typical PET system matrices; (2) a study of the effect on convergence of projection on the maximum singular vector; and (3) a study of the convergence behavior of the procedure applied to the Landweber, generalized Landweber, steepest descent, conjugate gradient, and algebraic reconstruction technique algorithms. A comparison is made with the maximum-likelihood expectation-maximization algorithm.<>