The augmented Lagrangian (AL) optimization method has drawn more attention recently in imaging applications due to its decomposable structure for composite cost functions and empirical fast convergence rate under weak conditions. However, for problems, e.g., X-ray computed tomography (CT) image reconstruction, where the inner least-squares problem is challenging, the AL method can be slow due to its iterative inner updates. In this paper, using a linearized AL framework, we propose an ordered-subsets (OS) accelerable linearized AL method, OS-LALM, for solving penalized weighted least-squares (PWLS) X-ray CT image reconstruction problems. To further accelerate the proposed algorithm, we also propose a deterministic downward continuation approach for fast convergence without additional parameter tuning. Experimental results show that the proposed algo- rithm significantly accelerates the “convergence” of X-ray CT image reconstruction with negligible overhead and exhibits excellent gradient error tolerance when using many subsets for OS acceleration.
Variational methods are useful for solving ill-posed inverse imaging problems by minimizing a cost function
with a data fidelity term and a regularization term. For statistical X-ray computed tomography (CT) image
reconstruction, penalized weighted least-squares (PWLS) criteria with edge-preserving regularization can improve
quality of the reconstructed image compared to traditional filtered back-projection (FBP) reconstruction.
Nevertheless, the huge dynamic range of the statistical weights used in PWLS image reconstruction leads to a
highly shift-variant local impulse response, making effective preconditioning difficult. To overcome this problem,
iterative algorithms based on variable splitting were proposed recently. However, existing splitting-based iterative
algorithms do not consider the (unknown) gain fluctuations that can occur between views. This paper
proposes a new variational formulation for splitting-based iterative algorithms where the unknown gain parameter
vector and the image are estimated jointly with just simple changes to the original algorithms. Simulations
show that the proposed algorithm greatly reduces the shading artifacts caused by gain fluctuations yet with
almost unchanged computational complexity per iteration.