Analytical and Iterative Regularization Methods for Nonlinear Ill-posed Inverse Problems: Applications to Diffuse Optical and Electrical Impedance Tomography

Electrical impedance tomography (EIT) and Diffuse Optical Tomography (DOT) are imaging methods that have been gaining more popularity due to their ease of use and non-ivasiveness. EIT and DOT can potentially be used as alternatives to traditional imaging techniques, such as computed tomography (CT) scans, to reduce the damaging effects of radiation on tissue. The process of imaging using either EIT or DOT involves measuring the ability for tissue to impede electrical flow or absorb light, respectively. For EIT, the inner distribution of resistivity, which corresponds to different resistivity properties of different tissues, is estimated from the voltage potentials measured on the boundary of the object being imaged. In DOT, the optical properties of the tissue, mainly scattering and absorption, are estimated by measuring the light on the boundary of the tissue illuminated by a near-infrared source at the tissue\u27s surface. In this dissertation, we investigate a direct method for solving the EIT inverse problem using mollifier regularization, which is then modified and extended to solve the inverse problem in DOT. First, the mollifier method is formulated and then its efficacy is verified by developing an appropriate algorithm. For EIT and DOT, a comprehensive numerical and computational comparison, using several types of regularization techniques ranging from analytical to iterative to statistical method, is performed. Based on the comparative results using the aforementioned regularization methods, a novel hybrid method combining the deterministic (mollifier and iterative) and statistical (iterative and statistical) is proposed. The efficacy of the proposed method is then further investigated via simulations and using experimental data for damage detection in concrete