混合ABBA-GMRES用于大尺度X射线计算机断层扫描中不匹配的后投影器
Hybrid ABBA-GMRES for Unmatched Backprojectors in Large Scale X-Ray Computerized Tomography
AI总结：
本文提出混合AB-和BA-GMRES方法，通过引入Tikhonov正则化提升大尺度CT中不匹配投影器的重建质量与稳定性。
摘要：
在大规模X射线计算机断层扫描（CT）中，由于显式形成系统矩阵的成本过高，矩阵自由迭代方法是必不可少的。在实践中，正投影器和反投影器常常使用不同的离散化或加速方法，导致投影器对不匹配。这种不匹配违反了经典最小二乘求解器背后的对偶性假设，因此由此产生的迭代不再对应于真正的最小二乘问题，可能会表现出非对称或不一致的行为。先前的工作已经探讨了Krylov子空间求解器，如AB-GMRES和BA-GMRES，用于处理不匹配的投影器对，这些方法表现出半收敛的正则化行为。在匹配条件下，AB-GMRES和BA-GMRES分别减少为LSQR和LSMR。然而，在存在不匹配的投影器时，AB-和BA-GMRES已被观察到比经典最小二乘求解器产生更高质量的重建质量。在本文中，我们开发了混合AB-和BA-GMRES方法，将Tikhonov正则化直接纳入Krylov子空间迭代中。我们还检查了所提出的方法与LSQR和LSMR的混合变体之间的关系，考虑了匹配和不匹配的反投影器。我们提出了自动选择正则化参数的策略，包括基于L-曲线和广义交叉验证（GCV）的方法，并分析了它们对收敛行为和图像质量的影响。使用GPU加速的投影器在二维CT问题上的数值实验表明，所提出的混合AB-和BA-GMRES方法减轻了半收敛性，产生了更高质量的重建，并表现出比非混合对应物更稳定的停止行为。
英文摘要：
In large-scale X-ray computed tomography (CT), matrix-free iterative methods are essential due to the prohibitive cost of explicitly forming the system matrix. In practice, forward projectors and backprojectors are often implemented with different discretizations or accelerations, leading to unmatched projector pairs. This mismatch violates the adjointness assumptions underlying classical least-squares solvers, so the resulting iterations no longer correspond to a true least-squares problem and can exhibit non-symmetric or inconsistent behavior. Prior work has explored Krylov subspace solvers such as AB-GMRES and BA-GMRES to handle unmatched projector pairs, where these methods exhibit semi-convergent regularizing behavior. Under matched conditions, AB-GMRES and BA-GMRES reduce to LSQR and LSMR, respectively. However, in the presence of unmatched projectors, AB- and BA-GMRES have been observed to yield improved reconstruction quality compared to classical least-squares solvers. In this paper, we develop hybrid AB- and BA-GMRES methods that incorporate Tikhonov regularization directly into the Krylov subspace iterations. We also examine the relationship between the proposed methods and hybrid variants of LSQR and LSMR, considering both matched and unmatched backprojectors. We propose automatic strategies for selecting regularization parameters, including approaches based on the L-curve and generalized cross validation (GCV), and analyze their effect on convergence behavior and image quality. Numerical experiments on two-dimensional CT problems using GPU-accelerated projectors demonstrate that the proposed hybrid AB- and BA-GMRES methods mitigate semi-convergence, produce higher-quality reconstructions, and exhibit more stable stopping behavior than their non-hybrid counterparts.
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