Kyriakopoulou, Dimitra;
(2024)
Analytical and Numerical Aspects of Tomography.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
The Boundary Control (BC) Method was originally formulated with respect to hyperbolic inverse problems. We extend this powerful method to elliptic and parabolic problems. In particular, by considering Boundary Value Problems as edge problems, we extend the functional analytic version of BC from commutative C* algebras to noncommutative ones, through the use of a suitably chosen pseudodifferential calculus for singular manifolds. We then apply this extended BC to the open Calderon´ conjecture, presenting a multitude of cases where the uniqueness argument holds. Via the pseudodifferential calculus extension to higher singularities by an iteration process, the extended BC is applied to the reconstruction of a Riemannian polyhedron. We also showed that for the standard (hyperbolic) BC the interface and corner detection of a Riemannian polyhedron can be achieved by introducing appropriate notions from physics (e.g. waveguides and 4-wave-mixing). For Electrical Impedance Tomography, i.e., for the 2D case of the Calderon´ problem, a novel algorithm is constructed that is simpler and faster compared to the well-known D-bar method, which is based on Complex Geometric Optics solutions. The following algorithms are implemented in the tomographic , extensively used library, Software for Tomographic Image Reconstruction (STIR): two 2D Positron Emission Tomography (PET), two 2D Single Photon Emission Computed Tomography (SPECT), and one 3D PET. These algorithms serve as templates for how to include in the library analytic reconstruction algorithms, without the need to extensively study the library. The PET algorithms constitute strong alternatives to STIR’s existing filtered back-projection PET algorithms in terms of image quality and/or speed, while the SPECT algorithms are the first of their kind within the library.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Analytical and Numerical Aspects of Tomography |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2024. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) Licence (https://creativecommons.org/licenses/by-nc-nd/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10202525 |
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