Reconstruction of Domain Boundary and Conductivity in Electrical Impedance Tomography Using the Approximation Error Approach

Electrical impedance tomography (EIT) is a highly unstable problem with respect to measurement and modeling errors. With clinical measurements, knowledge about the body shape is usually uncertain. Since the use of an incorrect model domain in the measurement model is bound to lead to severe estimation errors, one possibility is to estimate both the conductivity and parametrization of the domain boundary. This could in principle be carried out using the Bayesian inversion paradigm and Markov chain Monte Carlo sampling, but such an approach would lead in clinical situation to an impractical solution because of the excessive computational complexity. In this paper, we adapt the so-called approximation error approach for approximate recovery of the domain boundary and the conductivity. In the approximation error approach, the modeling error caused by an inaccurately known boundary is treated as an auxiliary noise process in the measurement model and sample statistics for the noise process are estimated based on the prior models of the conductivity and boundary shape. Using the approximation error model, we reconstruct the conductivity and a low rank approximation for the realization of the modeling error, and then recover an approximation for the domain boundary using the joint distribution of the modeling error and the boundary parametrization. We also compute approximate spread estimates for the reconstructed boundary. We evaluate the approach with simulated examples of thorax imaging and also with experimental data from a laboratory setting. The reconstructed boundaries and posterior uncertainty are feasible; in particular, the actual domain boundaries are essentially within the posterior spread estimates.