The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations Ax=b. The randomized version of the Kaczmarz method put forth by Strohmer and Vershynin iteratively projects onto a randomly chosen solution space given by a single row of the matrix A and converges exponentially in expectation to the solution of a consistent system. In this paper we analyze two block versions of the method each with a randomized projection, that converge in expectation to the least squares solution of inconsistent systems. Our approach utilizes a paving of the matrix A to guarantee exponential convergence, and suggests that paving yields a significant improvement in performance in certain regimes. The proposed method is an extension of the block Kaczmarz method analyzed by Needell and Tropp and the Randomized Extended Kaczmarz method of Zouzias and Freris. The contribution is thus two-fold; unlike the standard Kaczmarz method, our methods converge to the least-squares solution of inconsistent systems, and by using appropriate blocks of the matrix this convergence can be significantly accelerated. Numerical experiments suggest that the proposed algorithm can indeed lead to advantages in practice.
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@cite_7: This revised and updated text presents the computational and mathematical procedures underlying data collection, image reconstruction, and image display in computerized tomography. New topics: the fast calculation of a ray sum for a digitized picture, the task-oriented comparison of reconstruction algorithm performance, blob basis functions and the linogram method for image reconstruction. Features: Describes how projection data are obtained and the resulting reconstructions are used; Presents a comparative evaluation of reconstruction methods; Investigates reconstruction algorithms; Explores basis functions, functions to be optimized, norms, generalized inverses, least squares solutions, maximum entropy solutions, and most likely estimates; Discusses SNARK09, a large programming system for image reconstruction; Concludes each chapter with helpful Notes and References sections. An excellent guide for practitioners, it can also serve as a textbook for an introductory graduate course.
@cite_8: The POCS-method (projection onto convex subsets) has been proposed as an efficient way of recovering a band-limited signal from irregular sampling values. However, both the ordinary POCS-method (which uses one sampling point at a given time, i.e. consists of a succession of projections onto affine hyperplanes) and the one-step method (which uses all sampling values at the same time) become extremely slow if the number of sampling points gets large. Already for midsize 2D-problems (e.g. 128 X 128 images) one may easy run into memory problems. Based on the theory of pseudo-inverse matrices new efficient variants of the POCS- method (so to say intermediate versions) are described, which make use of a finite number of sampling points at each step. Depending on the computational environment appropriate strategies of designing those families of sampling points (either many families with few points, or few families with many points, overlapping families or disjoint ones...) have to be found. We also report on numerical results for these algorithms.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
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@cite_21: Abstract We give a new method for direct reconstruction of three-dimensional objects from a few electron micrographs taken at angles which need not exceed a range of 60 degrees. The method works for totally asymmetric objects, and requires little computer time or storage. It is also applicable to X-ray photography, and may greatly reduce the exposure compared to current methods of body-section radiography.
@cite_32: Applied Iterative Methods is a self-contained treatise suitable as both a reference and a graduate-level textbook in the area of iterative algorithms. It is the first book to combine subjects such as optimization, convex analysis, and approximation theory and organize them around a detailed and mathematically sound treatment of iterative algorithms. Such algorithms are used in solving problems in a diverse area of applications, most notably in medical imaging such as emission and transmission tomography and magnetic-resonance imaging, as well as in intensity-modulated radiation therapy. Other applications, which lie outside of medicine, are remote sensing and hyperspectral imaging. This book details a great number of different iterative algorithms that are universally applicable.
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@cite_3: Abstract Using the results of Smith, Solmon, and Wagner [K. Smith, D. Solomon, S. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977) 1227–1270] and Nelson and Neumann [S. Nelson, M. Neumann, Generalizations of the projection method with application to SOR theory for Hermitian positive semidefinite linear systems, Numer. Math. 51 (1987) 123–141] we derive new estimates for the speed of the alternating projection method and its relaxed version in R m . These estimates can be computed in at most O ( m 3 ) arithmetic operations unlike the estimates in papers mentioned above that require spectral information. The new and old estimates are equivalent in many practical cases. In cases when the new estimates are weaker, the numerical testing indicates that they approximate the original bounds in papers mentioned above quite well.
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@cite_27: Algebraic reconstruction techniques (ART) are iterative procedures for recovering objects from their projections. It is claimed that by a careful adjustment of the order in which the collected data are accessed during the reconstruction procedure and of the so-called relaxation parameters that are to be chosen in an algebraic reconstruction technique, ART can produce high-quality reconstructions with excellent computational efficiency. This is demonstrated by an example based on a particular (but realistic) medical imaging task, showing that ART can match the performance of the standard expectation-maximization approach for maximizing likelihood (from the point of view of that particular medical task), but at an order of magnitude less computational cost. >
@cite_5: Spring balance apparatus including a weighing lever supported on a knife edge balance point with one side of the lever acted on by a mass to be weighed against a weighing spring coupled with the other side of the weighing lever, and including an arrangement for the compensation of a measuring error due to temperature change by modifying the ratio of transmission of forces applied to the weighing lever by means of a bimet allic element in which the inventive improvement comprises a knife edge member transferring the forces of the weighing spring to the other side of the weighing lever which knife-edge member has connected rigidly to it an actuating lever coupled to the bimet allic element to alter responsive to temperature changes the orientation of the knife edge member relative to the other side of the weighing lever and thereby modify the ratio of transmission of forces to compensate for measuring error that would otherwise occur due to temperature change.
@cite_34: Abstract The purpose of the paper is threefold: (1) To develop a useful error bound for the method of alternating projections which is relatively easy to compute and remember; (2) To exhibit a counterexample to a conjecture of Kayalar and Weinert; (3) To show that (in the case of at least three subspaces) any error bound which only depends on the angles between the various subspaces involved canneverbe sharp.
The Kaczmarz method was first introduced in the 1937 work of Kaczmarz himself @cite_30 . Since then, the method has been revitalized by researchers in computer tomography, under the name (ART) @cite_21 @cite_32 @cite_48 @cite_7 . Deterministic convergence results for the method often depend on properties of the matrix that are difficult to compute or analyze @cite_24 @cite_34 @cite_52 @cite_3 . Moreover, it has been well observed that random choice of row selection often speeds up the convergence @cite_5 @cite_27 @cite_8 @cite_48 .