D‐Bar Method for Electrical Impedance Tomography with Discontinuous Conductivities

Abstract

The effects of truncating the (approximate) scattering transform in the D‐bar reconstruction method for two‐dimensional electrical impedance tomography are studied. The method is based on the uniqueness proof of Nachman [Ann. of Math. (2), 143 (1996), pp. 71–96] that applies to twice differentiable conductivities. However, the reconstruction algorithm has been successfully applied to experimental data, which can be characterized as piecewise smooth conductivities. The truncation is shown to stabilize the method against measurement noise and to have a smoothing effect on the reconstructed conductivity. Thus the truncation can be interpreted as regularization of the D‐bar method. Numerical reconstructions are presented demonstrating that features of discontinuous high contrast conductivities can be recovered using the D‐bar method. Further, a new connection between Calderón’s linearization method and the D‐bar method is established, and the two methods are compared numerically and analytically.