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Electrical impedance tomography is an imaging method with applications to geophysics and medical imaging. A new approximation is presented based on Nachman's 2-dimensional construction for closed domains. It improves upon existing approximations by extending the range of application from resolving 2 times the surface conductivity to imaging perfect conductors and insulators. With perfect knowledge of boundary data, this approximation exactly resolves a single conductive disc embedded in a homogenous domain. The problem, however, is ill-posed, and imaging performance degrades quickly as the distance from the boundary increases. The key to the approximation lies in (a) approximating Fadeev's Green's function (b) pre-processing measured voltages based on a boundary-integral equation (c) solving a linearized inverse problem (d) solving a d-bar equation, and (e) scaling the resulting image based on analytical results for a disc. In the development of the approximation, a new formula for Fadeev's Green's function is presented in terms of the Exponential Integral function. Also, new comparisons are made between reconstructions with and without solving the d-bar equation, showing that the added computational expense of solving the d-bar equation is not justified for radial problems. There is no discernible improvement in image quality. As a result, the approximation converts the inverse conductivity problem into a novel one-step linear problem with pre-conditioning of boundary data and scaling of the resulting image. Several extensions to this work are possible. The approximation is implemented for a circular domain with unit conductivity near the boundary, and extensions to other domains, bounded and unbounded should be possible, with non-constant conductivity near the boundary requiring further approximation. Ultimately, further research is required to ascertain whether it is possible to extend these techniques to imaging problems in three dimensions.

This thesis outlines two advancements in the field of limited data absorption tomography. First, a novel reconstruction algorithm integrating the use of level set methods is presented that incorporates the additional a priori knowledge of a distinct interface between the species of interest and co-flow. The added a priori further reduces the ill-posedness of the system to produce a final concentration distribution that explains the laser absorption measurements and is qualitatively consistent with advection/diffusion transport. The algorithm is demonstrated by solving a simulated laser tomography experiment on a turbulent methane plume, and is compared with the current state-of-the-art reconstruction algorithm. Given the limited number of attenuated measurements, accurate reconstructions are also highly dependent on the locations sampled by the measurement array. This thesis displays how the mathematical properties of the coefficient matrix, A, formed by the locations of the lasers, are related to the information content of the attenuation data using a Tikhonov reconstruction framework. This formulation, in turn, becomes a basis for a beam arrangement design algorithm that minimizes the reliance on additional assumed information about the concentration distribution. Using genetic algorithms, beam arrangements can be optimized for a given application by incorporating physical constraints of the beam locations. The algorithm is demonstrated by optimizing unconstrained and constrained arrangements of light sources and detectors. Simulated experiments are performed to validate the optimality of the arrangements.

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This thesis reports on research undertaken in the area of microwave tomography (MWT) where the goal is to find the dielectric profile of an object of interest using microwave measurements collected outside the object. The main focus of this research is on the development of inversion algorithms which solve the electromagnetic inverse scattering problem associated with MWT. Various regularization techniques for the Gauss-Newton inversion algorithm are studied and classified. It is shown that these regularization techniques can be viewed from within a single consistent framework after applying some modifications. Within the framework of the two-dimensional MWT problem, the inversion of transverse magnetic and transverse electric data sets are considered and compared in terms of computational complexity, image quality and convergence rate. A new solution to the contrast source inversion formulation of the microwave tomography problem for the case where the MWT chamber consists of a circular conductive enclosure is introduced. This solution is based on expressing the unknowns of the problem as truncated eigenfunction expansions corresponding to the Helmholtz operator for a homogeneous background medium with appropriate boundary conditions imposed at the chamber walls. The MWT problem is also formulated for MWT chambers made of conducting cylinders of arbitrary shapes. It is shown that collecting microwave scattered-field data inside MWT setups with different boundary conditions can provide a robust set of useful information for the reconstruction of the dielectric profile. This leads to a novel MWT setup wherein a rotatable conductive triangular enclosure is used to generate scattered-field data. Antenna arrays, with as few as only four elements, that are fixed with respect to the object of interest can provide sufficient data to give good reconstructions, if the triangular enclosure is rotated a sufficient number of times. Preliminary results of using the algorithms presented herein on data collected using two different MWT prototypes currently under development at the University of Manitoba are reported. Using the open-region MWT prototype, a resolution study using the Gauss-Newton inversion method was performed using various cylindrical targets.

In 3-dimensional positron emission tomography (PET), the proportion of the detected photon coincidences arising from scatter can be very high, up to 60%. A standard approach to scatter correction in 3D PET is to estimate the scattered coincidences and subtract them from the list of recorded events. More recently, novel techniques have been proposed in which the information carried by the scattered photons was extracted and directly used in the image reconstruction step. In this work, the algorithms were extended for use in non-TOF 3D PET systems and were qualitatively and quantitatively evaluated. Despite some promising initial outcomes, standard performance metrics of images reconstructed with each technique, representing the quality of images, was shown to be deteriorated under specified conditions. Further work is required to investigate potential benefits of the proposed algorithms for both ideal and clinical conditions.

Stem cell therapy is emerging as a possible method for treating many diseases and disorders, such as cardiovascular disease. In particular, stem cells may be able to revive the dead tissue caused by acute myocardial infarction (heart attack). Adipose-derived stem cells were labelled with 18F-fluorodeoxyglucose (FDG) and superparamagnetic iron oxide (SPIO) particles, for imaging with positron emission tomography (PET) and magnetic resonance imaging (MRI), respectively, and injected into several rats following induced myocardial infarction. Stem cell retention in the heart was investigated following three injection sites; two within the heart (intramyocardial and left intraventricular), and one easily accessible vein (tail vein). The PET and MR images were registered and the initial distributions analyzed using region of interest (ROI) analysis, to determine which injection method would result in the highest stem cell retention in the infarcted heart. The ROI results determined that the intramyocardial injection had the highest % injected dose (%ID) in the heart with 14 +/- 4%, followed by left intraventricular and tail vein with %IDs of 3.6 +/- 0.8% and 1.2 +/- 0.6%, respectively. The results indicate that stem cell delivery via intramyocardial injection should be utilized for optimal retention in the heart.

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