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We present an image reconstruction algorithm for the Inverse Conductivity Problem based on reformulating the problem in terms of integral equations. We use as data the values of injected electric currents and of the corresponding induced boundary potentials, as well as the boundary values of the electrical conductivity.We have used a priori information to find a regularized conductivity distribution by first solving a Fredholm integral equation of the second kind for the Laplacian of the potential, and then by solving a first order partial differential equation for the regularized conductivity itself. Many of the calculations involved in the method can be achieved analytically using the eigenfunctions of an integral operator defined in the paper.

The importance of accurate mathematical modelling in the development of image reconstruction algorithms for electrical impedance tomography (EIT) has been discussed in a number of recent papers. It is particularly important in iterative reconstruction schemes where the forward problem of calculating the electric potential from Neumann boundary data is solved many times. One area which needs to be considered it the mathematical modelling of the electrodes used in the technique. In this paper we discuss one of the more sophisticated models which has been proposed and present the results of a number of numerical and analytic calculations which we have made as a contribution to the understanding of this question.

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We present an analytic treatment of the inverse problem of reconstructing the electrical conductivity of the lower half space from electrical measurements performed on its surface. As the domain under consideration is infinite, the inversion requires the knowledge of data up to infinite distances. One way of overcoming this problem is to approximate the half space by a large cylinder and to use an asymptotic estimate for data at large distances. We have transformed the governing differential equation into an integral equation and regularized it by the use of a priori information. In this way we obtain a stable Fredholm integral equation of the second kind for a regularized conductivity distribution. This equation can be solved either numerically or by using its eigenfunctions which we have computed explicitly.