A 3-D reconstruction algorithm for the linearized inverse boundary value problem for Maxwell's equations

The goal of impedance imaging is to recover the electric and magnetic properties within a region from electromagnetic measurements on the boundary of that region. The mathematical model we consider for this problem is the time-harmonic formulation of Maxwell's equations. We find analytical solutions to the forward boundary value problem for some simple cases and investigate the implications for making images of human tissue. These solutions to the forward problem enable us to discuss the circumstances for which two low-frequency approximations called the static and quasi-static models provide sufficiently accurate results in impedance imaging. We design and implement a three-dimensional reconstruction algorithm for the time-harmonic form of Maxwell's equations. This method recovers perturbations in the conductivity and permittivity within a box-shaped region from applying magnetic fields on the boundary and measuring the corresponding electric fields. The algorithm is derived from a linearizing approximation and thus yields useful images when the perturbations in the unknowns are small. The choice of fields to apply to maximize the information obtained from the data is based on the spectral properties of the boundary map. The discretization of the unknown is determined by choosing a solution which has minimal {dollar}Lsb2{dollar}-norm. We analyze the success of the algorithm by considering various sample reconstructions from synthetic data.