A preconditioned landweber iteration scheme for the limited-angle image reconstruction

Abstract

BACKGROUND: The limited-angle reconstruction problem is of both theoretical and practical importance. Due to the severe ill-posedness of the problem, it is very challenging to get a valid reconstructed result from the known small limited-angle projection data. The theoretical ill-posedness leads the normal equation AT Ax = AT b of the linear system derived by discretizing the Radon transform to be severely ill-posed, which is quantified as the large condition number of AT A. OBJECTIVE: To develop and test a new valid algorithm for improving the limited-angle image reconstruction with the known appropriately small angle range from [ 0 , π 3 ] ∼ [ 0 , π 2 ] . METHODS: We propose a reweighted method of improving the condition number of AT Ax = AT b and the corresponding preconditioned Landweber iteration scheme. The weight means multiplying AT Ax = AT b by a matrix related to AT A, and the weighting process is repeated multiple times. In the experiment, the condition number of the coefficient matrix in the reweighted linear system decreases monotonically to 1 as the weighting times approaches infinity. RESULTS: The numerical experiments showed that the proposed algorithm is significantly superior to other iterative algorithms (Landweber, Cimmino, NWL-a and AEDS) and can reconstruct a valid image from the known appropriately small angle range. CONCLUSIONS: The proposed algorithm is effective for the limited-angle reconstruction problem with the known appropriately small angle range.