Electrical impedance tomography with piecewise polynomial conductivities

Abstract

Given only the static boundary flux and potential, electrical impedance tomography solves the inverse problem for the conductivity distribution. A Gauss-Newton solution is presented to solve this nonlinear problem when the conductivity distribution is represented by a piecewise polynomial basis function. An efficient method is presented to solve for the Jacobian matrix. This efficiency is made possible because of the local support of the basis functions used to approximate the conductivity distribution and data collection using the four-electrode technique. A method is presented for the local support case to solve for the Jacobian constants, which are needed to assemble the Jacobian matrix. It is shown that when higher than piecewise constant conductivities are desired it is more efficient to model conductivity than resistivity. Results are presented showing simulated reconstructions using a piecewise constant conductivity representation, and a (bi)linear conductivity representation with C° continuity across conductivity elements. These results show that although the Gauss-Newton method performs very well, further work needs to be done in designing meshes that increase the conditioning of the approximate Hessian matrix.