The inverse conductivity problem is the mathematical problem that must be solved in order for electrical impedance tomography systems to be able to make images. Here we show how this inverse conductivity problem is related to a number of other inverse problems. We then explain the workings of an algorithm that we have used to make images from electrical impedance data measured on the boundary of a circle in two dimensions. This algorithm is based on the method of least squares. It takes one step of a Newton's method, using a constant conductivity as an initial guess. Most of the calculations can therefore be done analytically. The resulting code is named NOSER, for Newton's One-Step Error Reconstructor. It provides a reconstruction with 496 degrees of freedom. The code does not reproduce the conductivity accurately (unless it differs very little from a constant), but it yields useful images. This is illustrated by images reconstructed from numerical and experimental data, including data from a human chest.
THE PROBLEM AND ITS CONNECTION WITH OTHER INVERSE PROBLEMS
Electrical impedance imaging systems apply currents to the surface S of a body B , measure the induced voltages at the surface, and from this information reconstruct an approximation to the conductivity in the interior [l-31. The reader the amount entering, which implies Js Ap')dS, = 0 .
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Electrical impedance imaging systems not only apply current, but also measure voltages on the boundary u ( @ ) = V ( @ ) for p' on S .
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