Mathematical aspects of impedance imaging.

Abstract

The mathematical problem of reconstructing the unknown variable conductivity of an isotropic medium from a knowledge of boundary currents and voltages is an active area of mathematical research. In terms of impedance imaging the analytical problem is essentially the question 'is there only one conductivity distribution which could have produced this set of measurements?' In mathematical parlance this is an 'identification problem' or 'inverse problem' for an unknown coefficient in an elliptic partial differential equation. Recent results have come close to settling the analytical problem. Kohn and Vogelius have shown that the piece-wise analytic conductivity distributions can be identified by boundary measurements and Sylvester and Uhlmann have shown that a smooth conductivity can be identified in the three-dimensional case and, provided the conductivity is close enough to uniformity, in the two-dimensional case also. The practical numerical problem of designing a numerical algorithm is far from completely understood. Mathematically the problem is one of solving a non-linear functional equation. A common numerical technique for tackling this type of problem is to employ the Newton-Raphson method. This approach is considered in this paper and compared with some of the algorithms appearing in the bioengineering literature. It is observed that, to varying degrees, these methods approximate the Newton-Raphson method.