Hybrid ABBA-GMRES for Unmatched Backprojectors in Large Scale X-Ray Computerized Tomography

TL;DR

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.

Abstract

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.

Figures (13)

• Figure 1: Relative reconstruction error norms for different solvers using an unmatched backprojector, i.e. ${\bf B}\neq{\bf A}^{\top}$.
• Figure 2: Relative reconstruction error norms for different solvers using the same system matrix ${\bf A}$ and a matched backprojector, i.e. ${\bf B}={\bf A}^{\top}$.
• Figure 3: Left panel: True image of size $128\times 128$. Right panel: noisy sinogram from 50 view angles and 2.5% Gaussian noise.
• Figure 4: Relative reconstruction error norms and SSIM for AB-GMRES with different regularization parameter choices. The markers correspond to the iterations with minimum RRE for each method.
• Figure 5: Relative reconstruction error norms and SSIM for BA-GMRES with different regularization parameter choices. The markers correspond to the iterations with minimum RRE for each method.