Nien Hung, Fessler Jeffrey A
IEEE Trans Med Imaging. 2015 Feb;34(2):388-99. doi: 10.1109/TMI.2014.2358499. Epub 2014 Sep 16.
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 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.
用于求解具有线性约束的凸优化问题的增广拉格朗日（AL）方法，由于在弱条件下具有经验快速收敛速率，对于具有复合成本函数的成像应用很有吸引力。然而，对于诸如X射线计算机断层扫描（CT）图像重建等问题，其中内部最小二乘问题具有挑战性且需要迭代，AL方法可能会很慢。本文重点研究使用AL方法的线性化变体来求解正则化（加权）最小二乘问题，该变体用其可分离的二次替代函数代替缩放增广拉格朗日中的二次AL惩罚项，从而得到一种更简单的基于有序子集（OS）可加速分裂的算法，即OS-LALM。为了进一步加速所提出的算法，我们使用二阶递归系统分析来设计一种确定性向下延拓方法，该方法避免了繁琐的参数调整并提供快速收敛。实验结果表明，所提出的算法显著加速了X射线CT图像重建的收敛，开销可忽略不计，并且在使用多个子集时可以减少OS伪影。
