Many results in the EIT literature focus on the continuum problem of determining a possibly anisotropic conductivity distribution from boundary measurements. In practice however, we must discretise the domain and use some numerical technique to solve the problem. Due to its flexibility, the finite element method is often used to solve the forward problem. Using this method the conductivity we attempt to reconstruct only affects the data through the system matrix. For piecewise constant conductivity, the entries of the system matrix are equivalent to the conductances of resistors in an electrical network that has the same topology as the finite element mesh. Since the conductivity only appears in the system matrix as these edge conductances and given the edge conductances we would like to answer the questions: is it possible to determine a conductivity distribution for this mesh that fits the edge conductances and if so is the conductivity distribution unique.
In this talk, we will first consider the problem of embedding a 3D simplicial complex in 3D Euclidean space which is an extension of the classical cartography problem of Tycho Brahe. We then extend this to the problem of finding an embedding given the edge conductances computed from various conductivity distributions.