Numerical methods of electrical impedance tomography

Abstract

The second reconstruction algorithm this thesis considers is the D-bar method. The D-bar method solves the full non-linear inverse problem. This method relies on solving for the complex geometrical optics (CGO) solutions to Schrödinger equation. To compute these CGO solutions, the D-bar equation must be solved numerically. This thesis introduces and analyzes a simple, easy-to-use finite difference solver for the D-bar equation. This solver is proven to be convergent with second-order accuracy. The D-bar method via this finite difference solver is then compared to Calderón's method.

Lastly, using both of these methods, a first attempt at recovering a ventilation-perfusion index from human data is explored.

The first direct method considered is a linearized reconstruction algorithm called Calderón's method. The original theory does not specify any particular domain, but previous numerical implementations of this algorithm rely on a circular domain. I developed a numerical implementation that works for non-circular domains, since the domains of interest in medical imaging applications are non-circular. When the boundary is modeled with a radius that changes with the angle, the data function is approximated by a similar series to the circular case, except now the inner products, which themselves need to be expanded into series, represent the entries of a transformation of the discrete Dirichlet-to-Neumann map. Qualitative and quantitative effects of correct domain modeling are demonstrated.

In electrical impedance tomography (EIT), the internal conductivity of a body is determined from measurements made on the boundary of the body. The goal of EIT is to reconstruct the conductivity inside the body from a dataset of boundary currents and voltages. This thesis is focused on numerical implementations of direct methods for reconstructing the conductivity.