AIR tools

AIR tools -- a MATLAB package of algebraic iterative reconstruction methods. We present a MATLAB package with implementations of several algebraic iterative reconstruction (AIR) methods for discretizations of inverse problems. These so-called row action methods rely on semi-convergence for achieving the necessary regularization of the problem. Two classes of methods are implemented: Algebraic reconstruction techniques and simultaneous iterative reconstruction techniques. In addition we provide a few simplified test problems from medical and seismic tomography. For each iterative method, a number of strategies are available for choosing the relaxation parameter and the stopping rule. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide a new “training” algorithm that finds the optimal parameter for a given test problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the normalixed cumulative periodogram criterion; for the first two methods “training” can be used to find the optimal discrepancy parameter.


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  1. Jakob S. Jørgensen, Evelina Ametova, Genoveva Burca, Gemma Fardell, Evangelos Papoutsellis, Edoardo Pasca, Kris Thielemans, Martin Turner, Ryan Warr, William R. B. Lionheart, Philip J. Withers: Core Imaging Library - Part I: a versatile Python framework for tomographic imaging (2021) arXiv
  2. Liu, Yong; Gu, Chuan-Qing: On greedy randomized block Kaczmarz method for consistent linear systems (2021)
  3. Nikazad, T.; Karimpour, M.: Column-oriented algebraic iterative methods for nonnegative constrained least squares problems (2021)
  4. Behling, Roger; Bello-Cruz, J.-Yunier; Santos, Luiz-Rafael: The block-wise circumcentered-reflection method (2020)
  5. Benvenuto, Federico; Jin, Bangti: A parameter choice rule for Tikhonov regularization based on predictive risk (2020)
  6. Cueva, Evelyn; Courdurier, Matias; Osses, Axel; Castañeda, Victor; Palacios, Benjamin; Härtel, Steffen: Mathematical modeling for 2D light-sheet fluorescence microscopy image reconstruction (2020)
  7. Gazzola, Silvia; Kilmer, Misha E.; Nagy, James G.; Semerci, Oguz; Miller, Eric L.: An inner-outer iterative method for edge preservation in image restoration and reconstruction (2020)
  8. Jia, Zhongxiao: Regularization properties of Krylov iterative solvers CGME and LSMR for linear discrete ill-posed problems with an application to truncated randomized SVDs (2020)
  9. Lei, Yunwen; Zhou, Ding-Xuan: Convergence of online mirror descent (2020)
  10. Webber, James W.; Quinto, Eric Todd: Microlocal analysis of a Compton tomography problem (2020)
  11. Webber, James W.; Quinto, Eric Todd; Miller, Eric L.: A joint reconstruction and lambda tomography regularization technique for energy-resolved x-ray imaging (2020)
  12. Wu, Nianci; Xiang, Hua: Projected randomized Kaczmarz methods (2020)
  13. Zheng, Zhe; Ng, Michael; Wu, Chunlin: A globally convergent algorithm for a class of gradient compounded non-Lipschitz models applied to non-additive noise removal (2020)
  14. Censor, Yair; Heaton, Howard; Schulte, Reinhard: Derivative-free superiorization with component-wise perturbations (2019)
  15. Dong, Yiqiu; Hansen, Per Christian; Hochstenbach, Michiel E.; Brogaard Riis, Nicolai André: Fixing nonconvergence of algebraic iterative reconstruction with an unmatched backprojector (2019)
  16. Gazzola, Silvia; Hansen, Per Christian; Nagy, James G.: IR tools: a MATLAB package of iterative regularization methods and large-scale test problems (2019)
  17. Gibali, Aviv; Mai, Dang Thi; Vinh, Nguyen The: A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications (2019)
  18. Hansen, Per Christian; Dong, Yiqiu; Abe, Kuniyoshi: Hybrid enriched bidiagonalization for discrete ill-posed problems. (2019)
  19. Heaton, Howard; Censor, Yair: Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems (2019)
  20. Hu, Yunyi; Andersen, Martin S.; Nagy, James G.: Spectral computed tomography with linearization and preconditioning (2019)

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