Reconstructions in impedance and optical tomography with singular interfaces

Singular layers modelled by a tangential diffusion process supported on an embedded closed surface (of co-dimension 1) have found applications in tomography problems. In optical tomography they may model the propagation of photons in thin clear layers, which are known to hamper the use of classical diffusion approximations. In impedance tomography they may be used to model thin regions of very high conductivity profile. In this paper we show that such surfaces can be reconstructed from boundary measurements (more precisely, from a local Neumann-to-Dirichlet operator) provided that the material properties between the measurement surface and the embedded surface are known. The method is based on the factorization technique introduced by Kirsch. Once the location of the surface is reconstructed, we show under appropriate assumptions that the full tangential diffusion process and the material properties in the region enclosed by the surface can also uniquely be determined.