X-ray computed tomography (CT) is a medical imaging framework. It takes measured projections of X-rays through two-dimensional cross-sections of an object from multiple angles and incorporates algorithms in building a sequence of two-dimensional reconstructions of the interior structure. This thesis comprises a review of the different types of algebraic algorithms used in X-ray CT. Using simulated test data, I evaluate the viability of algorithmic alternatives that could potentially reduce overexposure to radiation, as this is seen as a major health concern and the limiting factor in the advancement of CT [36, 34]. Most of the current evaluations in the literature [31, 39, 11] deal with low-resolution reconstructions and the results are impressive, however, modern CT applications demand very high-resolution imaging. Consequently, I selected ve of the fundamental algebraic reconstruction algorithms (ART, SART, Cimmino's Method, CAV, DROP) for extensive testing and the results are reported in this thesis. The quantitative numerical results obtained in this study, con rm the qualitative suggestion that algebraic techniques are not yet adequate for practical use. However, as algebraic techniques can actually produce an image from corrupt and/or missing data, I conclude that further re nement of algebraic techniques may ultimately lead to a breakthrough in CT. / UOIT

Iterative CT reconstruction algorithms are superior to the standard convolution backpropagation (CBP) methods when reconstructing from a small number of views (hence less radiation), but are computationally costly. To reduce the execution time, this work implements and tests a parallel approach to iterative algorithms using a cluster of workstations, which is a low cost system found in many offices and non-academic sites. A previous implementation showed little speedup because of the significant cost of inter-processor communication. In this thesis, several data partitioning methods are examined, including some image tiling methods that exploit the spatial locality demonstrated by local CT. Using these methods, computation can proceed locally, without the need for inter-processor communication during every iteration. A relative speedup of up to 17 times is obtained using 25 processors, demonstrating that good performance can be obtained running computationally intensive CT reconstruction algorithms on distributed memory hardware. / October 2007

Iterative CT reconstruction algorithms are superior to the standard convolution backpropagation (CBP) methods when reconstructing from a small number of views (hence less radiation), but are computationally costly. To reduce the execution time, this work implements and tests a parallel approach to iterative algorithms using a cluster of workstations, which is a low cost system found in many offices and non-academic sites. A previous implementation showed little speedup because of the significant cost of inter-processor communication. In this thesis, several data partitioning methods are examined, including some image tiling methods that exploit the spatial locality demonstrated by local CT. Using these methods, computation can proceed locally, without the need for inter-processor communication during every iteration. A relative speedup of up to 17 times is obtained using 25 processors, demonstrating that good performance can be obtained running computationally intensive CT reconstruction algorithms on distributed memory hardware.

The Kaczmarz method is an iterative algorithm that solves overdetermined systems of linear equalities. This paper studies a system of linear equalities and inequalities. We use the block version of the Kaczmarz method applied towards the equalities with the simple randomized Kaczmarz scheme for the inequalities. This primarily involves combining Needell and Tropp's work on the block Kaczmarz method with the application of a randomized Kaczmarz approach towards a system of equalities and inequalities performed by Leventhal and Lewis. We give an expected linear rate of convergence for this kind of system and find that using the block Kaczmarz scheme for the equalities can improve the rate compared to the simple Kaczmarz method.

Iterative CT reconstruction algorithms are superior to the standard convolution backpropagation (CBP) methods when reconstructing from a small number of views (hence less radiation), but are computationally costly. To reduce the execution time, this work implements and tests a parallel approach to iterative algorithms using a cluster of workstations, which is a low cost system found in many offices and non-academic sites. A previous implementation showed little speedup because of the significant cost of inter-processor communication. In this thesis, several data partitioning methods are examined, including some image tiling methods that exploit the spatial locality demonstrated by local CT. Using these methods, computation can proceed locally, without the need for inter-processor communication during every iteration. A relative speedup of up to 17 times is obtained using 25 processors, demonstrating that good performance can be obtained running computationally intensive CT reconstruction algorithms on distributed memory hardware.

We present a new theoretical model and reconstruction results for a new class of fast x-ray CT machine -- the Real Time Tomography (RTT) system, which uses switched sources and an offset detector array. We begin by reviewing elementary properties of the Radon and X-ray transforms, and limited angle tomography. Through the introduction of a new continuum model, that of sources covering the surface of a cylinder in R³, we show that the problem of three-dimensional reconstruction from RTT data reduces to inversion of the three-dimensional Radon transform with limited angle data. Using the Paley-Wiener theorem, we then prove the existence of a unique solution and give comments on stability and singularity detection. We show, first in the two-dimensional case, that the conjugate gradient least squares algorithm is suitable for CT reconstruction. By exploiting symmetries in the system, we then derive a method of applying CGLS to the three-dimensional inversion problem using stored matrix coefficients. The new concept of source firing order is introduced and formalised, and some novel visualisations are used to show how this affects aspects of the geometry of the system. We then perform a detailed numerical analysis using the condition number and SVD of the forward projection matrix $A$, to show that the choice of firing order affects the conditioning of the problem. Finally, we give reconstruction results from both simulated phantoms and real experimental data that support the numerical analysis.

This project implements a parallel algorithm for Computed Tomography based on the Algebraic Reconstruction Technique (ART) algorithm. This technique for reconstructing pictures from projections is useful for applications such as Computed Tomography (CT or CAT). The algorithm requires fewer views, and hence less radiation, to produce an image of comparable or better quality. However, the approach is not widely used because of its computationally intensive nature in comparison with rival technologies. A faster ART algorithm could reduce the amount of radiation needed for CT imaging by producing a better image with fewer projections. A reconstruction from projections version of the ART algorithm for two dimensions was implemented in parallel using the Message Passing Interface (MPI) and OpenMP extensions for C. The message passing implementation did not result in faster reconstructions due to prohibitively long and variant communication latency. The shared memory implementation produced positive results, showing a clear computational advantage for multiple processors and measured efficiency ranging from 60-95%. Consistent with the literature, image quality proved to be significantly better compared to the industry standard Filtered Backprojection algorithm especially when reconstructing from fewer projection angles. / October 2006

This project implements a parallel algorithm for Computed Tomography based on the Algebraic Reconstruction Technique (ART) algorithm. This technique for reconstructing pictures from projections is useful for applications such as Computed Tomography (CT or CAT). The algorithm requires fewer views, and hence less radiation, to produce an image of comparable or better quality. However, the approach is not widely used because of its computationally intensive nature in comparison with rival technologies. A faster ART algorithm could reduce the amount of radiation needed for CT imaging by producing a better image with fewer projections. A reconstruction from projections version of the ART algorithm for two dimensions was implemented in parallel using the Message Passing Interface (MPI) and OpenMP extensions for C. The message passing implementation did not result in faster reconstructions due to prohibitively long and variant communication latency. The shared memory implementation produced positive results, showing a clear computational advantage for multiple processors and measured efficiency ranging from 60-95%. Consistent with the literature, image quality proved to be significantly better compared to the industry standard Filtered Backprojection algorithm especially when reconstructing from fewer projection angles.

This project implements a parallel algorithm for Computed Tomography based on the Algebraic Reconstruction Technique (ART) algorithm. This technique for reconstructing pictures from projections is useful for applications such as Computed Tomography (CT or CAT). The algorithm requires fewer views, and hence less radiation, to produce an image of comparable or better quality. However, the approach is not widely used because of its computationally intensive nature in comparison with rival technologies. A faster ART algorithm could reduce the amount of radiation needed for CT imaging by producing a better image with fewer projections. A reconstruction from projections version of the ART algorithm for two dimensions was implemented in parallel using the Message Passing Interface (MPI) and OpenMP extensions for C. The message passing implementation did not result in faster reconstructions due to prohibitively long and variant communication latency. The shared memory implementation produced positive results, showing a clear computational advantage for multiple processors and measured efficiency ranging from 60-95%. Consistent with the literature, image quality proved to be significantly better compared to the industry standard Filtered Backprojection algorithm especially when reconstructing from fewer projection angles.

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