Browsing by Author "Syren, Joonas Emil"

In this thesis we consider the Dirichlet-to-Neumann map in Electrical Impedance Tomography (EIT). EIT is a tomography method which uses electrical currents and voltages to determine the conductivity distribution inside the measured object. The Dirichlet-to-Neumann map (DN map) takes the voltage on the boundary and gives the resulting current density on the boundary. This map can be approximated by a matrix known as the Dirichlet-to-Neumann matrix. In this thesis we analyse this matrix using Principal Component Analysis (PCA). In chapter 1 we give a short introduction to EIT with a brief history of the study and some applications of the method. The Dirichlet-to-Neumann map is derived in chapter 2. Constructing the DN map requires solving the Dirichlet problem which is derived from Maxwell?s equations. The Dirichlet problem and its solvability is studied in this chapter as well. Some of the concepts needed in this study can be found from the appendices. The method used for approximating the DN map is introduced in chapter 3. The approximated matrices are then analysed using PCA, which is described in the same chapter. PCA can be used to find the components where the variation is the largest and to reduce the dimension of the data using these components. We use a method known as Singular Value Decomposition (SVD) to reduce the dimension of the data and to compute the principal values and components. We computed the DN matrices with simulated data using different conductivity distributions. We chose the unit circle as our domain with a constant conductivity on the background and four anomalies with changing conductivity. The 4th chapter introduces the computations and the obtained results. The resulting principal components and values are shown in this chapter. An approximation of the data was made using the dimensionality reduction method described in chapter 3. The relative errors for different reconstructions are also shown in chapter 4. In the final chapter we discuss the results. Our goal was to find out how the four variables in the conductivity distributions affect the dimension of the hyperplane that the DN matrices form. It seems that even with four degrees of freedom the DN matrices vary most on a 2-dimensional plane. We also found out that in this case most of the principal components have almost no effect on the data.