Resistor networks and optimal grids for the numerical solution of electrical impedance tomography with partial boundary measurements

Abstract

The problem of Electrical Impedance Tomography (EIT) with partial boundary measurements is to determine the electric conductivity inside a body from the simultaneous measurements of direct currents and voltages on a subset of its boundary. Even in the case of full boundary measurements the non-linear inverse problem is known to be exponentially ill-conditioned. Thus, any numerical method of solving the EIT problem must employ some form of regularization. We propose to regularize the problem by using sparse representations of the unknown conductivity on adaptive finite volume grids known as the optimal grids, that are computed as part of the problem. Then the discretized partial data EIT problem can be reduced to solving the discrete inverse problems for resistor networks. Two distinct approaches implementing this strategy are presented. The first approach uses the results for the EIT problem with full boundary measurements, which rely on the use of resistor networks with circular graph topology. The optimal grids for such networks are essentially one dimensional objects, which can be computed explicitly using discrete Fourier transform and rational interpolation with continued fractions. We solve the partial data problem by reducing it to the full data case using the theory of extremal quasiconformal (Teichmuller) mappings. The second approach is based on resistor networks with the pyramidal graph topology. Such network topology is better suited for the partial data problem, since it allows for explicit treatment of the inaccessible part of the boundary. We present a method of computing the optimal grids for the networks with general topology (including pyramidal), which is based on the sensitivity analysis of both the continuum and the discrete EIT problems This is the first study of the optimal grids for the case, where reduction to one dimension is not possible. We present extensive numerical results for the two approaches. We demonstrate both the optimal grids and the reconstructions of smooth and discontinuous conductivities in a variety of domains. The numerical results show several advantages of our approaches compared to the traditional optimization-based methods. First, the inversion based on resistor networks is orders of magnitude faster than any iterative algorithm. Second, our approaches are able to correctly reconstruct both smooth and discontinuous conductivities including those of very high contrast, which usually present a challenge to the iterative or linearization-based inversion methods. Finally, our method does not require any form of artificial regularization via penalty terms. However, our method allows for such regularization to incorporate prior information in the solution.