Electrical impedance tomography for piecewise constant domains using boundary element shape-based inverse solutions.

Shape-based solutions have recently received attention for certain ill-posed inverse problems. Their advantages include implicit imposition of relevant constraints and reduction in the number of unknowns, especially important for nonlinear ill-posed problems. We apply the shape-based approach to current-injection electrical impedance tomography (EIT) reconstructions. We employ a boundary element method (BEM) based solution for EIT. We introduce two shape models, one based on modified B-splines, and the other based on spherical harmonics, for BEM modeling of shapes. These methods allow us to parameterize the geometry of conductivity inhomogeneities in the interior of the volume. We assume the general shape of piecewise constant inhomogeneities is known but their conductivities and their exact location and shape is not. We also assume the internal conductivity profile is piecewise constant, meaning that each region has a constant conductivity. We propose and test three different regularization techniques to be used with either of the shape models. The performance of our methods is illustrated via both simulations in a digital torso model and phantom experiments when there is a single internal object. We observe that in the noisy environment, either simulated noise or real sources of noise in the experimental study, we get reasonable reconstructions. Since the signal-to-noise ratio (SNR) expected in modern EIT instruments is higher than that used in this study, these reconstruction methods may prove useful in practice.