On the uniqueness of the Calderón problem and its application in electrical impedance tomography

Abstract

This thesis addresses several questions about the uniqueness and reconstruction of the conductivity γ from knowledge of the boundary information encapsulated in the Dirichlet-to-Neumann map Λγ . This problem is well-known in the literature as Calder ́on problem. In two dimensions, we extend the uniqueness of Calder ́on problem in two dimensions for complex conductivities with curves of discontinuity based on the stationary phase method and the introduction of new exponentially growing solutions. In three dimensions, we extend the result established by Nachman for real conductivities with two derivatives, by noting that most of the proof holds with the need of extending some of the results to encapsulate the complex case. Moreover, we establish a methodology to recover the complex conductivity from small complex frequencies, but some open questions are left about this reconstruction process. Furthermore, we reduce the differentiability condition for uniqueness to hold. We have shown that the Dirichlet-to-Neumann map uniquely determines complex conductivities with one derivative. Our approach is completely novel and introduces a quaternionic analysis approach to deal with the problem in three dimensions. With the quaternionic framework we also introduce a possible path to show uniqueness for real conductivities in L∞. This is a step in the direction of a complete answer to Calder ́on’s question in three dimensions. This problem is also relevant for practical applications, in particular medical imaging where it is used in Electrical Impedance Tomography (EIT). For practical implementations, a reconstruction algorithm is required to transform the boundary measurements into a conductivity profile. We use iterative methods to obtain a reconstruction method and our goal is to provide a simple and effective way to compute the required Jacobian matrix This approach is based in automatic differentiation (AD) tools . We show that AD can be used to efficiently and effectively compute the Jacobian matrix of a numerical method that simulates the voltages measurements. Further, we show that this computation is as effective as analytical closed-forms applied in general iterative method in order to reconstruct the conductivity profile.