Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector

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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.
Original languageEnglish
JournalSIAM Journal on Scientific Computing
Issue number3
Pages (from-to)A1822-A1839
Publication statusPublished - 2019


  • Unmatched transpose
  • Algebraic iterative reconstruction
  • Perturbation theory
  • Left-most eigenvalue estimation
  • Computed tomography

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