GKB Methods for X-Ray Computed Tomography with an Unmatched Back Projector

Abstract: In large scale X ray Computed Tomography (CT) inverse problems, the forward and back projectors are often generated using different discretizations. This discrepancy leads to unmatched pairs of projections, resulting in inconsistent normal equations. Consequently, employing the Conjugate Gradient method does not produce a useful solution. For matched operator pairs, the Golub Kahan bidiagonalization (GKB) method provides an efficient solution strategy. It works by projecting the original large-scale problem onto a lower-dimensional subspace, enabling the solution to be computed via a singular value decomposition of a sparse lower bidiagonal matrix. To address unmatched-pair problems in CT, we propose the AB and BA GKB algorithms as preconditioned forms of the GKB. These methods are straightforward to implement and allow for parameter tuning. We provide a discussion on the theoretical computational costs of our proposed algorithms in terms of floating point operations and compare with existing methods. While many Krylov methods tend to amplify noise in solutions, leading to semiconvergence, our proposed algorithms demonstrate greater resilience against this effect. We validate the effectiveness of our approach through numerical examples across various CT problems, showcasing its ability to deliver more stable solutions.

• The paper introduces AB-GKB and BA-GKB methods that replace Aᵀ with B to effectively address unmatched back projectors, stabilizing CT reconstructions.
• It demonstrates through numerical experiments on CT phantoms that GKB variants mitigate semiconvergence and achieve improved reconstruction accuracy compared to GMRES-based methods.
• The study highlights practical scalability via a matrix-free approach and efficient SVD computations on sparse, lower bidiagonal projected systems.

GKB-Based Algorithms for X-Ray CT Reconstruction with Unmatched Back Projectors

Problem Context and Motivation

The paper addresses large-scale inverse problems arising in X-ray Computed Tomography (CT), where the discretization of physical operators leads to forward and back projectors ($A$ and $B$) that are often constructed via distinct numerical schemes. This results in unmatched projector pairs, i.e., $B \ne A^\top$, which invalidates classical matched normal-equation formulations and renders standard iterative solvers like CGLS or LSQR ineffective. Existing Krylov subspace solvers for unmatched pairs—such as AB-GMRES and BA-GMRES—are prone to noise amplification due to semiconvergence, and their projected systems are computationally expensive due to dense upper Hessenberg matrices.

Proposed Methods

The paper introduces the AB-GKB and BA-GKB algorithms—preconditioned variants of the Golub–Kahan Bidiagonalization (GKB) technique—for reconstructing images in settings with unmatched back projectors. These methods operate by applying GKB to the composite operators $AB$ (AB-GKB) or $BA$ (BA-GKB), projecting the inverse problem onto a lower-dimensional Krylov subspace, and solving the reduced problem via the SVD of a sparse lower bidiagonal matrix. Unlike classical GKB (which is mathematically equivalent to LSQR for matched pairs), these variants replace $A^\top$ with $B$ in the algorithmic framework, thus allowing for numerical stability and efficiency with unmatched operators.

While AB-GKB and BA-GKB incorporate unmatched pairs, their subspace projection facilitates efficient parameter tuning, and the algorithms remain matrix-free, requiring only matrix-vector products with $A$ and $B$.

Theoretical and Computational Analysis

The authors rigorously analyze computational complexity—counting floating-point operations (FLOPS) for each algorithmic step and contrasting GKB-based methods with GMRES-based approaches. The key findings include:

• GKB methods produce projected systems with lower bidiagonal matrices ($C_k$), whose SVD is much cheaper ($B$0 FLOPS) than the SVD of upper Hessenberg matrices ($B$1) from GMRES ($B$2 FLOPS), given equivalent subspace sizes.
• AB-GKB and BA-GKB incur higher overall computational costs than their GMRES counterparts due to more frequent applications of $B$3, $B$4, and their transpose—even though GKB benefits from enhanced sparsity in the projected system.
• For overdetermined cases ($B$5), AB-variants are more expensive; for underdetermined cases, BA-variants exhibit higher complexity.

Numerical Results

The paper reports extensive experiments on three CT phantom problems, testing both overdetermined and underdetermined scenarios, with SNR levels ranging from 20 to 28 dB. Results are assessed via relative reconstruction error (RRE) for each method, with iterations terminated by either the Discrepancy Principle (DP)—which leverages noise estimates—or the Residual Norm Stagnation (RNS) criterion.

Main Numerical Findings:

• Semiconvergence Mitigation: AB-GKB and BA-GKB demonstrate less severe semiconvergence and yield more stable reconstructions than AB-GMRES and BA-GMRES, especially at moderate and high noise levels.
• Accuracy: GKB variants achieve comparable (often slightly superior) RRE compared to GMRES methods, except in the very earliest iterations.
• Computational Efficiency: GMRES methods are computationally less demanding and terminate in fewer iterations, but GKB approaches project onto sparser subspaces, with theoretical advantages in noise resilience.
• Stopping Criteria: The DP emerges as the superior stopping rule, consistently terminating algorithms near optimal RRE, while RNS is somewhat less reliable (especially for GKB methods).

Practical and Theoretical Implications

The application of GKB variants for unmatched projectors in CT presents significant practical advantages. By leveraging Krylov subspace projection and efficient SVD computation, these methods stabilize reconstructions and mitigate noise amplification, both of which are critical in medical imaging and industrial tomography where unmatched operator pairs are unavoidable. Matrix-free implementation with only forward/backward operator application facilitates scalability to large-scale problems.

From a theoretical perspective, the analysis provided clarifies the impact of projector mismatch on subspace-based regularization methods, extending the landscape beyond classical matched scenarios and offering new insight into the spectral properties and convergence behavior of Krylov solvers.

Future developments may focus on incorporating advanced regularization strategies within the GKB framework, hybrid approaches, and optimal parameter selection schemes (such as GCV or weighted predictive risk), aimed at further improving robustness, reducing computational overhead, and enabling real-time tomographic reconstruction for high-dimensional datasets.

Conclusion

AB-GKB and BA-GKB algorithms constitute robust Krylov-subspace-based solvers for X-ray CT inverse problems with unmatched projectors. They combine sparsity and spectral regularization via GKB with operational flexibility, thereby addressing both computational and semiconvergence limitations found in classical GMRES-based approaches. Although GKB methods incur higher computational costs per iteration, they provide improved noise resilience and reconstruction accuracy, particularly when paired with the Discrepancy Principle as the stopping criterion. This methodology has immediate applicability in large-scale tomographic imaging and motivates further research into Krylov-based regularization for ill-posed, non-normal systems in computational imaging.