Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector
This work addresses a convergence issue in large-scale computed tomography, providing a practical fix for practitioners using unmatched projector/backprojector pairs.
Algebraic iterative reconstruction methods with unmatched projector/backprojector pairs often fail to converge due to nonsymmetric iteration matrices. The authors propose a modified algorithm with a small shift parameter that guarantees convergence to a fixed point of a perturbed problem, and demonstrate its effectiveness on computed tomography test problems.
We consider algebraic iterative reconstruction methods with applications in image reconstruction. In particular, we are concerned with methods based on an unmatched projector/backprojector pair; i.e., the backprojector is not the exact adjoint or transpose of the forward projector. Such situations are common in large-scale computed tomography, and we consider the common situation where the method does not converge due to the nonsymmetry of the iteration matrix. We propose a modified algorithm that incorporates a small shift parameter, and we give the conditions that guarantee convergence of this method to a fixed point of a slightly perturbed problem. We also give perturbation bounds for this fixed point. Moreover, we discuss how to use Krylov subspace methods to efficiently estimate the leftmost eigenvalue of a certain matrix to select a proper shift parameter. The modified algorithm is illustrated with test problems from computed tomography.