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 A ^T Ax = A ^T 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 A ^T 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 $\left[0,\frac{\pi }{3}\right]\sim \left[0,\frac{\pi }{2}\right]$ .

METHODS:

We propose a reweighted method of improving the condition number of A ^T Ax = A ^T b and the corresponding preconditioned Landweber iteration scheme. The weight means multiplying A ^T Ax = A ^T b by a matrix related to A ^T 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.