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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:686708 |
| Date | January 2011 |
| Creators | Thompson, William |
| Contributors | Lionheart, William |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.research.manchester.ac.uk/portal/en/theses/source-firing-patterns-and-reconstruction-algorithms-for-a-switched-source-offset-detector-ct-machine(97dc0705-45e2-4b7a-9ef3-1c8a58d5411a).html |
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