Deterministic Versus Randomized Kaczmarz Iterative Projection

Kaczmarz's alternating projection method has been widely used for solving a consistent (mostly over-determined) linear system of equations Ax=b. Because of its simple iterative nature with light computation, this method was successfully applied in computerized tomography. Since tomography generates a matrix A with highly coherent rows, randomized Kaczmarz algorithm is expected to provide faster convergence as it picks a row for each iteration at random, based on a certain probability distribution. It was recently shown that picking a row at random, proportional with its norm, makes the iteration converge exponentially in expectation with a decay constant that depends on the scaled condition number of A and not the number of equations. Since Kaczmarz's method is a subspace projection method, the convergence rate for simple Kaczmarz algorithm was developed in terms of subspace angles. This paper provides analyses of simple and randomized Kaczmarz algorithms and explain the link between them. It also propose new versions of randomization that may speed up convergence

Extended and Constrained Cimmino-type Algorithms with Applications in Tomographic Image Reconstruction

In the first part of this paper we propose an extension of Cimmino's reflections algorithm to inconsistent least squares problems. For proving convergence of the extended algorithm we first make a convergence analysis of the classical method that reveals other interesting aspects which don't appear in the convergence analysis made in the original paper by G. Cimmino. In the second part of the paper we introduce in both original and extended Cimmino algorithms constraining procedures and prove convergence of mixed algorithms in both cases to constrained classical or least squares solutions, respectively. In the third part of the paper we present numerical experiments on some algebraic image reconstruction models, with particular attention to tomographic particle image reconstruction, which illustrate the performance of both extended and constrained algorithms

Compressed Sensing Based Reconstruction Algorithm for X-ray Dose Reduction in Synchrotron Source Micro Computed Tomography

Synchrotron computed tomography requires a large number of angular projections to reconstruct tomographic images with high resolution for detailed and accurate diagnosis. However, this exposes the specimen to a large amount of x-ray radiation. Furthermore, this increases scan time and, consequently, the likelihood of involuntary specimen movements. One approach for decreasing the total scan time and radiation dose is to reduce the number of projection views needed to reconstruct the images. However, the aliasing artifacts appearing in the image due to the reduced number of projection data, visibly degrade the image quality. According to the compressed sensing theory, a signal can be accurately reconstructed from highly undersampled data by solving an optimization problem, provided that the signal can be sparsely represented in a predefined transform domain. Therefore, this thesis is mainly concerned with designing compressed sensing-based reconstruction algorithms to suppress aliasing artifacts while preserving spatial resolution in the resulting reconstructed image. First, the reduced-view synchrotron computed tomography reconstruction is formulated as a total variation regularized compressed sensing problem. The Douglas-Rachford Splitting and the randomized Kaczmarz methods are utilized to solve the optimization problem of the compressed sensing formulation. In contrast with the first part, where consistent simulated projection data are generated for image reconstruction, the reduced-view inconsistent real ex-vivo synchrotron absorption contrast micro computed tomography bone data are used in the second part. A gradient regularized compressed sensing problem is formulated, and the Douglas-Rachford Splitting and the preconditioned conjugate gradient methods are utilized to solve the optimization problem of the compressed sensing formulation. The wavelet image denoising algorithm is used as the post-processing algorithm to attenuate the unwanted staircase artifact generated by the reconstruction algorithm. Finally, a noisy and highly reduced-view inconsistent real in-vivo synchrotron phase-contrast computed tomography bone data are used for image reconstruction. A combination of prior image constrained compressed sensing framework, and the wavelet regularization is formulated, and the Douglas-Rachford Splitting and the preconditioned conjugate gradient methods are utilized to solve the optimization problem of the compressed sensing formulation. The prior image constrained compressed sensing framework takes advantage of the prior image to promote the sparsity of the target image. It may lead to an unwanted staircase artifact when applied to noisy and texture images, so the wavelet regularization is used to attenuate the unwanted staircase artifact generated by the prior image constrained compressed sensing reconstruction algorithm. The visual and quantitative performance assessments with the reduced-view simulated and real computed tomography data from canine prostate tissue, rat forelimb, and femoral cortical bone samples, show that the proposed algorithms have fewer artifacts and reconstruction errors than other conventional reconstruction algorithms at the same x-ray dose

Imaging with Diffraction Tomography

The problem of cross sectional (tomographic) imaging bf objects with diffracting sources is addressed. Specifically the area of investigation is the effect of multiple scattering and attenuation phenomena in diffraction imaging. This work reviews the theory and limits of first order diffraction tomography and studies iterative techniques that can be used to improve the quality of tomographic imaging with diffracting sources. Conventional (straight-ray) tomographic algorithms are not valid when used with acoustic or microwave energy. Thus more sophisticated algorithms are needed; First order diffraction tomography uses a linearized version of the wave equation and gives an especially simple reconstruction algorithm. This work reviews first order approximations to the scattered field and studies the quality of the reconstructions when the assumptions behind these approximations are violated. It will be shown that the Born approximation is valid when the phase change across the object is less than it and the Rytov approximation is valid when the refractive index changes by less than two or three percent. Better reconstructions will be based on higher order approximations to the scattered field. This work describes two fixed point algorithms (the Born and the Rytov approximations) and an algebraic approach to more accurately calculate the scattered fields. The limits of each of these approaches is discussed and simulated results are shown. Finally a review of higher order inversion techniques is presented. Each of these techniques is reviewed and some of their limitations are discussed

An Asynchronous Parallel Randomized Kaczmarz Algorithm

We describe an asynchronous parallel variant of the randomized Kaczmarz (RK) algorithm for solving the linear system $Ax=b$. The analysis shows linear convergence and indicates that nearly linear speedup can be expected if the number of processors is bounded by a multiple of the number of rows in $A$

Superiorization and Perturbation Resilience of Algorithms: A Continuously Updated Bibliography

This document presents a, (mostly) chronologically ordered, bibliography of scientific publications on the superiorization methodology and perturbation resilience of algorithms which is compiled and continuously updated by us at: http://math.haifa.ac.il/yair/bib-superiorization-censor.html. Since the beginings of this topic we try to trace the work that has been published about it since its inception. To the best of our knowledge this bibliography represents all available publications on this topic to date, and while the URL is continuously updated we will revise this document and bring it up to date on arXiv approximately once a year. Abstracts of the cited works, and some links and downloadable files of preprints or reprints are available on the above mentioned Internet page. If you know of a related scientific work in any form that should be included here kindly write to me on: [email protected] with full bibliographic details, a DOI if available, and a PDF copy of the work if possible. The Internet page was initiated on March 7, 2015, and has been last updated on March 12, 2020.Comment: Original report: June 13, 2015 contained 41 items. First revision: March 9, 2017 contained 64 items. Second revision: March 8, 2018 contained 76 items. Third revision: March 11, 2019 contains 90 items. Fourth revision: March 16, 2020 contains 112 item