Electrical impedance tomography with resistor networks

Abstract

We introduce a novel inversion algorithm for electrical impedance tomography in two dimensions, based on a model reduction approach. The reduced models are resistor networks that arise in five point stencil discretizations of the elliptic partial differential equation satisfied by the electric potential, on adaptive grids that are computed as part of the problem. We prove the unique solvability of the model reduction problem for a broad class of measurements of the Dirichletto-Neumann map. The size of the networks is limited by the precision of the measurements. The resulting grids are naturally refined near the boundary, where we measure and expect better resolution of the images. To determine the unknown conductivity, we use the resistor networks to define a nonlinear mapping of the data that behaves as an approximate inverse of the forward map. Then we formulate an efficient Newton-type iteration for finding the conductivity, using this map. We also show how to incorporate a priori information about the conductivity in the inversion scheme.

Overdetermined partial boundary value problems on finite networks

“…For instance, Sylvester and Uhlmann treated in [9,18] the uniqueness of solution; Curtis, Ingerman and Morrow have worked on critical circular planar networks conductivity reconstruction [12,13,14,16]; Borcea, Druskin, Guevara and Mamonov have gone into EIT problems in depth and their last works on the subject treat numerical conductivity reconstruction [6,7,8].…”

Dirichlet-to-Robin maps on finite networks

“…This point of view is important in applications since finite network models arise in finite volume discretizations of the elliptic partial differential equation that model the continuos inverse problem, see [5,6,11]. In this framework, the characterization of the networks that allow the recovery of the conductance is essential.…”

Circular resistor networks for electrical impedance tomography with partial boundary measurements

“…We follow the approach in [11,49] and parametrize the unknown conductivity on adaptive grids which we call optimal. The number of grid points is limited by the precision of the measurements and their location is determined as part of the inverse problem.…”

“…The optimal grids considered in [11,49] and in this paper are computed with an approach based on rational approximation techniques. They are called optimal because they give spectral accuracy of the DtN map with finite volumes on coarse grids.…”

“…We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11,49] for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on so-called optimal grids that are computed as part of the problem.…”