zbMATH — the first resource for mathematics

Fixing nonconvergence of algebraic iterative reconstruction with an unmatched backprojector. (English) Zbl 1420.65031

65F10 Iterative numerical methods for linear systems
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F22 Ill-posedness and regularization problems in numerical linear algebra
15A18 Eigenvalues, singular values, and eigenvectors
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
Full Text: DOI arXiv
[1] A. Biguri, M. Dosanjh, S. Hancock, and M. Soleimani, TIGRE: A MATLAB-GPU toolbox for CBCT image reconstruction, Biomed. Phys. Eng. Express, 2 (2016), 055010; the software available from https://github.com/CERN/TIGRE.
[2] T. Elfving and P. C. Hansen, Unmatched projector/backprojector pairs: Perturbation and convergence analysis, SIAM J. Sci. Comput., 40 (2018), pp. A573–A591. · Zbl 1383.65035
[3] T. Elfving, P. C. Hansen, and T. Nikazad, Semiconvergence and relaxation parameters for projected SIRT algorithms, SIAM J. Sci. Comput., 34 (2012), pp. A2000–A2017. · Zbl 1254.65044
[4] W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, Springer, New York, 2016. · Zbl 1347.65063
[5] P. C. Hansen, The discrete Picard condition for discrete ill-posed problems, BIT, 5 (1990), pp. 658–672. · Zbl 0723.65147
[6] P. C. Hansen, Test matrices for regularization methods, SIAM J. Sci. Comput., 16 (1995), pp. 506–512. · Zbl 0820.65020
[7] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998.
[8] P. C. Hansen, Regularization Tools Version 4.0 for MATLAB 7.3, Numer. Algorithms, 46 (2007), pp. 189–194. · Zbl 1128.65029
[9] P. C. Hansen, Oblique projections and standard-form transformations for discrete inverse problems, Numer. Linear Algebra Appl., 20 (2013), pp. 250–258.
[10] P. C. Hansen and J. S. Jørgensen, AIR Tools II: Algebraic iterative reconstruction methods, improved implementation, Numer. Algorithms, 79 (2018), pp. 107–137.
[11] R. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, UK, 2013.
[12] P. M. Joseph, An improved algorithm for reprojecting rays through pixel images, IEEE Trans. Medical Imag., 3 (1982), pp. 192–196.
[13] D. A. Lorenz, S. Rose, and F. Schöpfer, The randomized Kaczmarz method with mismatched adjoint, BIT, 58 (2018), pp. 1079–1098.
[14] K. Meerbergen and D. Roose, Matrix transformations for computing rightmost eigenvalues of large sparse non-symmetric eigenvalue problems, IMA J. Numer. Anal., 16 (1996), pp. 297–346. · Zbl 0856.65033
[15] K. Meerbergen and M. Sadkane, Using Krylov approximations to the matrix exponential operator in Davidson’s method, Appl. Numer. Math., 31 (1999), pp. 331–351. · Zbl 0942.65036
[16] F. Natterer, The Mathematics of Computerized Tomography, Classics in Appl. Math. 32, SIAM, Philadelphia, 2001.
[17] R. C. Nelson, S. Feuerlein, and D. T. Boll, New iterative reconstruction techniques for cardiovascular computed tomography: How do they work, and what are the advantages and disadvantages?, J. Cardiovascuar Computed Tomography, 5 (2011), pp. 286–292.
[18] G. L. G. Sleijpen and H. A. Van der Vorst, A Jacobi–Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 401–425. · Zbl 0860.65023
[19] D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357–385. · Zbl 0763.65025
[20] G. W. Stewart, A Krylov–Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 601–614.
[21] G. W. Stewart and J. G. Sun, Matrix Perturbation Theory, Academic Press, Boston, 1990.
[22] W. van Aarle, W. J. Palenstijn, J. Cant, E. Janssens, R. Bleichrodt, A. Dabravolski, J. De Beenhouwer, K. J. Batenburg, and J. Sijbers, Fast and flexible X-ray tomography using the ASTRA toolbox, Opt. Express, 24 (2016), pp. 25129–25147; the software available from http://www.astra-toolbox.com.
[23] E. van den Berg and M. P. Friedlander, Spot—A Linear-Operator Toolbox, http://www.cs.ubc.ca/labs/scl/spot/.
[24] S. van der Maar, K. J. Batenburg, and J. Sijbers, Experiences with Cell-BE and GPU for tomography; in Embedded Compter Systems: Architectures, Modeling, and Simulation, Proceedings 9th International Workshop SAMOS, K. Bertels, N. Dimopoulos, C. Silvano, and S. Wong, eds., Springer, Berlin, 2009, pp. 298–307.
[25] G. L. Zeng and G. T. Gullberg, Unmatched projection/backprojection pairs in iterative reconstruction algorithms, IEEE Trans. Med. Imaging, 19 (2000), pp. 548–555.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.