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

Area of Science:

• Image reconstruction
• Computational imaging
• Applied mathematics

Background:

• Limited-angle reconstruction is crucial but challenging due to severe ill-posedness.
• The linear system's normal equation (AT Ax = AT b) is ill-posed with a large condition number.
• Existing methods struggle to produce valid results from limited projection data.

Purpose of the Study:

• Develop and validate a new algorithm for improved limited-angle image reconstruction.
• Address the challenge of reconstructing images from small angle ranges ([0,π/3]∼[0,π/2]).
• Enhance the condition number of the linear system for better reconstruction.

Main Methods:

• Propose a reweighted method to improve the condition number of AT Ax = AT b.
• Implement a preconditioned Landweber iteration scheme.
• Apply iterative weighting to the linear system, observing monotonic decrease in condition number.

Main Results:

• The reweighted algorithm significantly improves image reconstruction from limited angles.
• Numerical experiments demonstrate superiority over Landweber, Cimmino, NWL-a, and AEDS algorithms.
• The condition number of the reweighted system approaches 1 with increased weighting iterations.

Conclusions:

• The proposed reweighted algorithm is effective for limited-angle reconstruction.
• It successfully reconstructs valid images even with small angle ranges.
• This method offers a significant advancement for solving ill-posed inverse problems in imaging.