Medical imaging modalities such as computerized tomography (CT) using X-ray and magnetic resonance imaging (MRI) have been well established providing three-dimensional high-resolution images of anatomical structures inside the human body and computer-based mathematical methods have played an essential role for their image reconstructions. However, since each imaging modality has its own limitations, there have been much research efforts to expand our ability to see through the human body in different ways. Lately, biomedical imaging research has been dealing with new imaging techniques to provide knowledge of physiologic functions and pathological conditions in addition to structural information. Electrical impedance tomography (EIT) is one of such attempts for functional imaging and monitoring of physiological events. EIT is based on numerous experimental findings that different biological tissues inside the human body have different electrical properties of conductivity and permittivity. Viewing the human body as a mixture of distributed resistors and capacitors, we can evaluate its internal electrical properties by injecting a sinusoidal current between a pair of surface electrodes and measuring voltage drops at different positions on the surface. EIT is based on this bioimpedance measurement technique using multiple surface electrodes as many as 8 to 256. See Figs. 1.1a and 1.2. In EIT, we inject linearly independent patterns of sinusoidal currents through all or chosen pairs of electrodes and measure induced boundary voltages on all or selected electrodes. The measured boundary current-voltage data set is used to reconstruct cross-sectional images of the internal conductivity and/or permittivity distribution. The basic idea of the impedance imaging was introduced by Henderson and Webster in 1978 , and the first clinical application of a medical EIT system was described by Barber and Brown . Since then, EIT has received considerable attention and several review papers described numerous aspects of the EIT technique [8, 10, 14, 36, 49, 62]. To support the theoretical basis of the EIT system, mathematical theories such as uniqueness and stability were developed [2, 6, 16, 19, 25, 29, 38, 39, 48, 52, 57-59, 61] since Caldeŕon's pioneering contribution in 1980 .