Title:

The inverse conductivity problem : anisotropy, finite elements and resistor networks

EIT is a method of imaging that exists for a century, initially in geophysics and in recent years in medical imaging. Even though the practical applications of EIT go back to the early 20th century the systematic study of the inverse conductivity problem started in the late 1970s, hence many aspects of the problem remain unexplored. In the study of the inverse conductivity problem usually Finite Element Models are used since they can be easily adapted for bodies of irregular shapes. In this work though we use an equivalent approximation, the electrical resistor network, for which many uniqueness results as well as reconstruction algorithms exist. Furthermore resistor networks are important for EIT since they are used to provide convenient stable test loads or phantoms for EIT systems. In this thesis we study the transfer resistance matrix of a resistor network that is derived from nport theory and review necessary and sufficient conditions for a matrix to be the transfer resistance of a planar network. The so called “paramountcy” condition may be useful for validation purposes since it provides the means to locate problematic electrodes. In the study of resistor networks in relation to inverse problems it is of a great importance to know which resistor networks correspond to some Finite Element Model. To give a partial answer to this we use the dual graph of a resistor network and we represent the voltage by the logarithm of the circle radius. This representation in combination with Duffin’s nonlinear resistor network theory provides the means to show that a nonlinear resistor network can be embedded uniquely in a Euclidean space under certain conditions. This is where the novelty of this work lies.
