Hybrid GKB Methods for X-Ray Tomography Problems with Unmatched Back Projector

X ray Computed Tomography (CT) is a widely used imaging modality in medical, industrial, and scientific applications.

In practical CT implementations, forward, A, and back projection, B, operators are often constructed using different discretization schemes to improve computational efficiency on available software and hardware platforms.

Consequently, the resulting projector pair is generally unmatched, meaning that the back projector is not the exact adjoint of the forward projector.

This mismatch alters the mathematical properties of the reconstruction problem and can affect the convergence behavior of iterative reconstruction methods.

Previous studies have proposed AB and BA Golub Kahan bidiagonalization (GKB) methods, as well as GMRES and hybrid GMRES methods, for solving CT reconstruction problems with unmatched projector pairs, demonstrating reduced semiconvergence effects compared with conventional approaches.

In this work, we develop hybrid variants of the AB- and BA-GKB algorithms by incorporating regularization within the projection process.

The singular value decomposition for the operator on the projected space is used to efficiently reconstruct the solution, and to automatically select the regularization parameter using either the L-curve or generalized cross-validation.

Numerical experiments on several CT reconstruction problems demonstrate the effectiveness of the proposed hybrid methods in improving robustness against semiconvergence.