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.

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.

Electromagnetic (EM) imaging schemes are inherently non-linear and ill-posed. Albeit there exist remedies to these fundamental problems, more efficient solutions are still being sought. To this end, in this thesis, the non-linearity is tackled in- corporating a multitude of techniques (ranging from Born approximation (linear), inexact Newton (linearized) to complete nonlinear iterative Landweber schemes) that can account for weak to strong scattering problems. The ill-posedness of the EM inverse scattering problem is circumvented by formulating the above methods into a minimization problem with a sparsity constraint. More specifically, four novel in- verse scattering schemes are formulated and implemented. (i) A greedy algorithm is used together with a simple artificial neural network (ANN) for efficient and accu- rate EM imaging of weak scatterers. The ANN is used to predict the sparsity level of the investigation domain which is then used as the L0 - constraint parameter for the greedy algorithm. (ii) An inexact Newton scheme that enforces the sparsity con- straint on the derivative of the unknown material properties (not necessarily sparse) is proposed. The inverse scattering problem is formulated as a nonlinear function of the derivative of the material properties. This approach results in significant spar- sification where any sparsity regularization method could be efficiently applied. (iii) A sparsity regularized nonlinear contrast source (CS) framework is developed to di- rectly solve the nonlinear minimization problem using Landweber iterations where the convergence is accelerated using a self-adaptive projected accelerated steepest descent algorithm. (iv) A 2.5D finite difference frequency domain (FDFD) based in- verse scattering scheme is developed for imaging scatterers embedded in lossy and inhomogeneous media. The FDFD based inversion algorithm does not require the Green’s function of the background medium and appears a promising technique for biomedical and subsurface imaging with a reasonable computational time. Numerical experiments, which are carried out using synthetically generated mea- surements, show that the images recovered by these sparsity-regularized methods are sharper and more accurate than those produced by existing methods. The methods developed in this work have potential application areas ranging from oil/gas reservoir engineering to biological imaging where sparse domains naturally exist.