Monotonicity-Based Shape Reconstruction in Electrical Impedance Tomography

Current-voltage measurements in electrical impedance tomography (EIT) can be partially ordered with respect to definiteness of the associated self-adjoint Neumann-to-Dirichlet operators. With this ordering, a pointwise larger conductivity leads to smaller current-voltage measurements, and smaller conductivities lead to larger measurements. We present a converse of this simple monotonicity relation and use it to solve the shape reconstruction (a.k.a. inclusion detection) problem in EIT. The outer shape of a region where the conductivity differs from a known background conductivity can be found by simply comparing the measurements to that of smaller or larger test regions.