Determining the boundary of inclusions with known conductivities using a Levenberg-Marquardt algorithm by electrical resistance tomography

Electrical resistance tomography (ERT) is a non-intrusive technique to image the electrical conductivity distribution of a closed vessel by injecting exciting current into the vessel and measuring the boundary voltages induced. ERT image reconstruction is characterized as a severely nonlinear and ill-posed inverse problem with many unknowns. In recent years, a growing number of papers have been published which aim to determine the locations and shapes of inclusions by assuming that their conductivities are piecewise constant and isotropic. In this work, the boundary of inclusions is reconstructed by ERT with a boundary element method. The Jacobian matrix of the forward problem is first calculated with a direct linearization method based on the boundary element, and validated through comparison with that determined by the finite element method and analytical method. A boundary reconstruction algorithm is later presented based on the Levenberg-Marquardt (L-M) method. Several numerical simulations and static experiments were conducted to study the reconstruction quality, where much importance was given to the smoothness of boundaries in the reconstruction; thus, a restriction of the curve radius is introduced to adjust the damping parameter for the L-M algorithm. Analytical results on the stability and precision of the boundary reconstruction demonstrate that stable reconstruction can be achieved when the conductivity of the objects differs much from that of the background medium, and convex boundaries can also be precisely reconstructed. Contrarily, the reconstructions for inclusions with similar conductivities to the background medium are not stable. The situation of an incorrect initial estimation of the inclusions' number is numerically studied and the results show that the boundary of inclusions could be correctly reconstructed with a splitting/merging function under the aforementioned proper operation condition of the present algorithm.