Graduate Thesis Or Dissertation

π-line reconstruction formulas in computed tomography

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  • X-ray computed tomography is a noninvasive imaging modality capable of reconstructing exact density values of 3D objects. Computed tomography machines are deployed across the world to provide doctors with an image that reveals more detail than a standard x-ray image. We investigate algorithms based on exact computed tomography reconstruction formulas where the backprojection depends on the π-line of the point to be reconstructed. This includes the helical inversion formulas of Katsevich and Pan et al. Our work provides numerical analysis, insight into the algorithms and practical applications of π-line reconstruction formulas. A certain derivative with respect to source position appears in many π-line reconstruction formulas. Its accurate implementation is critical for the performance of the algorithms and there are several numerical methods available to calculate the derivative. New error estimates are derived for the numerical methods to calculate the derivative in the fan-beam setting with the curved detector geometry. Theoretical justification for previously proposed methods is provided. Numerical results from simulated data are presented to confirm the theoretical results. The helical inversion formulas of Katsevich and Pan et al have been extended to a larger class of source trajectories. The generalized formulas depend on π-lines and a method is presented to calculate π-lines for a variable pitch and radius helix. We introduce the set called the region of backprojection that identifies all points that are reconstructed from the measured data of the current source position. The region of backprojection is described for the helix and the circle. A characteristic artifact is found in reconstructions from formulas that depend on π-lines. The boundary of the region of backprojection is hypothesized as the cause of the artifact. A mathematical framework is presented to identify the location of the artifact in the reconstruction. This "comet tail artifact" is also present in reconstructions based on π-lines with improperly aligned measured data. An error term for the misaligned data is developed and an algorithm is presented to correctly align the data. Implementation details for the reconstruction methods discussed are provided.
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