|
Applied Math Seminar
Electric impedance tomography with resistor networks
|
|
Electric impedance tomography consists in finding the conductivity inside a body from electrical measurements taken at its surface. This is a severely ill-posed problem: any numerical inversion scheme requires some form of regularization. We present inversion schemes that address the instability of the problem by seeking a sparse parametrization of the unknown conductivity. Specifically, we consider finite volume grids of size determined by the measurement precision, but where the node locations are to be determined adaptively. A finite volume discretization can be thought of as a resistor network, where the resistors are essentially averages of the conductivity over grid cells. We show that the model reduction problem of finding the smallest resistor network (of fixed topology) that can predict meaningful measurements of the Dirichlet-to-Neumann map is uniquely solvable for a broad class of measurements. We propose a simple inversion method that is based on an interpretation of the resistors as conductivity averages over grid cells, and an iterative method that improves such reconstructions by using sensitivity information on the changes in the resistors due to small changes in the conductivity. A priori information can also be incorporated to the latter method. |