Impedance tomography and layer stripping. (English) Zbl 0807.92013

Anger, Gottfried (ed.) et al., Inverse problems: principles and applications in geophysics, technology, and medicine. Proceedings of the international conference held in Potsdam, Germany, August 30 - September 3, 1993. Berlin: Akademie Verlag. Math. Res. 74, 307-321 (1993).
When an electric potential is applied to the surface of a conducting body, the resulting current flux across this surface depends on the internal resistance of the body. For this reason, voltage and current measurements made on the boundary of an object can be used to probe the internal structure of the body. We shall discuss the problem of determining as much information as possible about the internal resistivity from measurements of voltage potentials and corresponding current fluxes at the boundary. This problem, which is often referred to as Impedence Computed Tomography or Electrical Impedence Imaging, arose in geophysics from attempts to determine the composition of the earth. More recently it has been proposed as a potentially valuable diagnostic tool for the medical and biological sciences. The first general mathematical formulation of the problem is due to Calderón, who also described the solution to a linearized version of the problem he had posed. Since then there has been significant progress in both theoretical and applied aspects of the subject. Methods which had been developed to study the impedance tomography problem have found new applications to both scattering theory and inverse spectral problems.
This review is in two sections, the first section introduces the Dirichlet to Neumann map – which plays a key role in the mathematical formulation of the problem – and provides a brief description of the basic methods introduced in the mid to late eighties for proving uniqueness theorems. In the second section we discuss a more recent approach to the problem of reconstruction based on the method of layer stripping.
For the entire collection see [Zbl 0777.00024].


92C55 Biomedical imaging and signal processing
35Q92 PDEs in connection with biology, chemistry and other natural sciences
78A70 Biological applications of optics and electromagnetic theory
35J25 Boundary value problems for second-order elliptic equations