Spectral element method for forward and inverse problems in electrical impedance tomography (EIT) and thermoacoustic tomography (TAT)

In this thesis, two imaging modalities, Electrical Impedance Tomography (EIT) and Thermoacoustic Tomography (TAT) have been studied. The forward problems for both modalities are solved by Spectral Element Method (SEM). The reconstruction algorithm for EIT is Distorted Born Iterative Method (DBIM). Experimental data has been reconstructed successfully for EIT. In Thermoacoustic Tomography, a direct reconstruction algorithm based on Back-propagation is proposed to solve the problems caused by acoustic inhomogeneity. A 3-D numerical breast phantom has been created to validate and test the inhomogeneous back-propagation method. Numerical results show that the distortions caused by the acoustic inhomogeneity have been greatly improved by the inhomogeneous back-propagation.; Electrical Impedance Tomography (EIT) is an imaging technique for the internal electrical conductivity distribution of a medium by measuring the low-frequency electrical potential on its boundary. The forward solver used in EIT is normally based on the conventional Finite Element Method (FEM). In this thesis, it is proposed that the Spectral Element Method (SEM) be used in the EIT forward problem. The SEM is capable of achieving better accuracy than the FEM with the same degree of freedom. The reconstruction algorithm used is the Distorted Born Iterative Method (DBIM), a version of regularized Gauss-Newton method.; 2-D and 3-D EIT systems have been constructed in our laboratory. The 2-D EIT imaging chamber is cylindrical in shape with 32 electrodes whereas the 3-D EIT system has a cone shape applicator with 128 electrodes. Both systems are controlled by switching circuits, which are responsible for measurement control. The results of the SEM forward solver have been validated with the measured data for both the 2-D and 3-D systems. Excellent images have been reconstructed from the measured data. The smallest object that the 3-D EIT system can image has a size of 20 mm.; Thermoacoustic Tomography (TAT) is another imaging modality in which a short electromagnetic (EM) wave is irradiated on a tissue and acoustic waves are generated due to the resulting thermoelastic expansion of the tissue. Images are reconstructed from the measured acoustic waves. To account for the inhomogeneity in TAT forward problem, a full wave forward solver is needed. The forward problem of TAT is based on Helmholtz's equation, rather than the Laplace's equation in EIT. However, the SEM developed for EIT can be readily modified to solve the TAT forward problem. With the introduction of a perfectly matched layer (PML), an SEM forward solver for TAT has been developed. Numerical results show that higher order SEM is able to achieve greater accuracy than the conventional FEM with the same degrees of freedom.; Most of the current reconstruction algorithms for TAT make an assumption that the tissue is acoustically homogeneous. This assumption works well under most circumstances, but it can cause significant degradation of images in heterogeneous media. In this thesis, a direct algorithm is proposed to solve the inhomogeneous reconstruction problem. The reconstruction algorithm is based on the back-propagation method and the inhomogeneous Green's function is evaluated by the Born approximation. The inhomogeneous medium itself is usually a source; the shape and size of the inhomogeneous medium can therefore be estimated. Together with prior information of acoustic speed of the medium, a 3-D inhomogeneity map can be formed. Numerical results show that the idea works very well for small contrast problems. The inhomogeneous back-propagation results show drastic improvements over the homogeneous back-propagation.; A 3-D realistic numerical breast phantom has been formed to study the back-propagation algorithms. The numerical breast phantom is derived from the female data set of Visible Human Project (VHP). Synthetic data is simulated by the SEM forward solver. An inhomogeneity map has been successfully...