United Kingdom.

An integral equation method for the inverse conductivity problem. Abstract: 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. 1. Introduction It is well-known that the inverse conductivity problem is an extremely ill-posed, non-linear inverse problem, and there has been much interest in determining the class of conductivity distributions that can be recovered from the boundary data, as well as in the development of related reconstruction algorithms. The interest in this problem has been generated by both difficult theoretical challenges and by the important medical, geophysical and industrial applications of this problem. Much theoretical work has been related to the approach of Calderón concerning the bijection between the Neumann-to-Dirichlet operator, which relates the distribution of the injected currents to the boundary values of the induced electrical potential and the conductivity inside the region [1–4]. The reconstruction procedures that have been proposed include a wide range of iterative methods based on formulating the inverse problem as a nonlinear optimisation problem. These techniques are quite demanding computationally particularly when addressing the three dimensional problem. This concern has encouraged the search for reconstruction algorithms which reduce the computational demands either by using some a priori information e.g. [5–7] or by developing non iterative procedures. Some of these methods [6, 7] use a factorization approach while others are based on reformulating the inverse problems in terms of integral equations [8–10]. In [9] we presented one such approach which we applied to a two dimensional bounded domain while in [10] we extended this work to an unbounded three dimensional region. A feature of this method is the use of a priori information to find the regularized conductivity. This represents an additional requirement of our method but in some cases information on the internal structure of the object is already …