AIR Tools II

AIR Tools II, Version 1.0

A MATLAB Package of Algebraic Iterative Reconstruction Methods, Improved Implementation (for Matlab Version 9.0 or later)

 To download this software package go to: where we maintain the software. It is distributed under the 3-Clause BSD License (a license file iw provided as part of the package). The package and the algorithms are described in the paper: P. C. Hansen and J. S. Jørgensen, AIR Tools II: Algebraic Iterative Reconstruction Methods, Improved Implementation, Numerical Algorithms, 79 (2018), pp. 107-137. doi:10.1007/s11075-017-0430-x.
This MATLAB package provides efficient, robust and flexible implementations of algebraic iterative reconstruction (AIR) methods for computing regularized solutions to discretizations of inverse problems, such as computed tomography. All our methods are equipped with stopping rules as well as heuristics for computing a good relaxation parameter, and we also provide several test problems from tomography. The package is intended for users who want to experiment with algebraic iterative methods and their convergence properties.

The software is a much expanded and improved version of the package AIR Tools from 2012, based on a new modular design. In addition to improved performance and memory use, we provide more flexible iterative methods, a column-action method, new test problems, new demo functions and - perhaps most important - the ability to use function handles The package includes some of the most common Algebraic Iterative Reconstruction (AIR) methods, divided into three classes.

• ART: algebraic reconstruction technique. Methods that sequentially involve one row at a time. The original algorithm is due to Kaczmarz; the name ART originates from the seminal paper by Gordon, Bender and Herman.
• CART: column-action reconstruction technique. Methods that sequentially involve one column at a time; these methods are sometimes referred to as column-relaxation methods.
• SIRT: simultaneous iterative reconstruction technique. Methods that simultaneously involve all the rows at a time, and therefore are based on matrix multiplications.
For all methods we provide several strategies for choosing the relaxation parameter as well as several stopping rules. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide a "training" algorithm that finds the optimal parameter for a given problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the NCP criterion; for the first two methods "training" can be used to finde the optimal discrepancy parameter. The package also includes seven test problems from tomography.

The function phantomgallery can generate a number of random phantomes:

A partial translation of AIR Tools II to Fortran is available from GitHub: FAIRtools.

The original work was carried out as part of the project CSI: Computational Science in Imaging, funded by the Danish Research Council for Technology and Production Sciences. The present version was developed during the project High-Definition Tomography, funded by an ERC Advanced Research Grant.

A course that uses this package (as well as Regularization Tools) can be found at this page:

The Original Software

The old version of the software, called AIR Tools, will not be maintained. It is still available here:

Software Dedicated to Computed Tomography

AIR Tools II contains generel-purpose software for linear inverse problems. Dedicated software packages for computed tomography are, e.g.:
• The ASTRA Toolbox, a MATLAB and Python toolbox of high-performance GPU primitives for 2D and 3D tomography.
• SNARK14, a programming system for the reconstruction of 2D images from 1D projections.
• TIGRE, a tomographic iterative GPU-based reconstruction toolbox.
For a more complete overview, see the software overview by Topomedia.