Augmented Lagrangian (AL) methods for solving convex optimization problems with linear constraints are attractive for imaging applications with composite cost functions due to the empirical fast convergence rate under weak conditions. However, for problems such as X-ray computed tomography (CT) image reconstruction, where the inner least-squares problem is challenging and requires iterations, AL methods can be slow. This paper focuses on solving regularized (weighted) least-squares problems using a linearized variant of AL methods that replaces the quadratic AL penalty term in the scaled augmented Lagrangian with its separable quadratic surrogate (SQS) function, leading to a simpler ordered-subsets (OS) accelerable splitting-based algorithm, OS-LALM. To further accelerate the proposed algorithm, we use a second-order recursive system analysis to design a deterministic downward continuation approach that avoids tedious parameter tuning and provides fast convergence. Experimental results show that the proposed algorithm significantly accelerates the convergence of X-ray CT image reconstruction with negligible overhead and can reduce OS artifacts when using many subsets.
Statistical image reconstruction (SIR) methods are studied extensively for X-ray computed tomography (CT) due to the potential of acquiring CT scans with reduced X-ray dose while maintaining image quality. However, the longer reconstruction time of SIR methods hinders their use in X-ray CT in practice. To accelerate statistical methods, many optimization techniques have been investigated. Over-relaxation is a common technique to speed up convergence of iterative algorithms. For instance, using a relaxation parameter that is close to two in alternating direction method of multipliers (ADMM) has been shown to speed up convergence significantly. This paper proposes a relaxed linearized augmented Lagrangian (AL) method that shows theoretical faster convergence rate with over-relaxation and applies the proposed relaxed linearized AL method to X-ray CT image reconstruction problems. Experimental results with both simulated and real CT scan data show that the proposed relaxed algorithm (with ordered-subsets [OS] acceleration) is about twice as fast as the existing unrelaxed fast algorithms, with negligible computation and memory overhead.
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