Towards quantitative computed tomography

Abstract

Computed tomography is introduced along with an overview of its diverse applications in many scientific endeavours. A unified approach for the treatment of scattering from linear scalar wave motion is introduced. The assumptions under which wave motion within a medium can be characterised by concourses of rays are presented along with comment on the validity of these assumptions. Early and conventional theory applied for modelling the behaviour of rays, within media for which ray assumptions are valid, are reviewed. A new computerised method is described for reconstruction of a refractive index distribution from time-of-flight measurements of radiation/waves passing through the distribution and taken on a known boundary surrounding it. The reconstruction method, aimed at solving the bent-ray computed tomography (CT) problem, is based on a novel ray description which doesn't require the ray paths to be known. This allows the refractive index to be found by iterative solution of a set of linear equations, rather than through the computationally intensive procedure of ray tracing, which normally accompanies iterative solutions to problems of this type. The preliminary results show that this method is capable of handling appreciable spatial refractive index variations in large bodies. A review containing theory and techniques for image reconstruction from projections is presented, along with their historical development. The mathematical derivation of a recently developed reconstruction technique, the method of linograms is considered. An idea, termed the plethora of views idea, which aims to improve quantitative CT image reconstruction, is introduced. The theoretical foundation for this is the idea that when presented with a plethora of projections, by which is meant a number greater than that required to reconstruct the known region of support of an image, so that the permissible reconstruction region can be extended, then the intensity of the reconstructed distribution should be negligible throughout the extended region. Any reconstruction within the extended region, that departs from what would be termed negligible, is deduced to have been caused by imperfections of the projections. The implicit expectation of novel schemes which are presented for improving CT image reconstruction, is that contributions within the extended region can be utilised to ameliorate the effects of the imperfections on the reconstruction where the distribution is known to be contained. Preliminary experimental results are reported for an iterative algorithm proposed to correct a plethora of X-ray CT projection data containing imperfections. An extended definition is presented for the consistency of projections, termed spatial consistency, that incorporates the region with which the projection data is consistent. Using this definition and an associated definition, spatial inconsistency, an original technique is proposed and reported on for the recovery of inconsistencies that are contained in the projection data over a narrow range of angles.