Image reconstruction in electrical impedance tomography

This thesis is concerned with Electrical Impedance Tomogaphy (EIT), a medical imaging technique in which pictures of the electrical conductivity distribution of the body are formed from current and voltage data taken on the body surface. The focus of the thesis is on the mathematical aspects of reconstructing the conductivity image from the measured data (the reconstruction problem). The reconstruction problem is particularly difficult and in this thesis it is investigated analytically and numerically. The aim of this investigation is to understand why the problem is difficult and to find numerical solution methods which respect the difficulties encountered. The analytical investigation of this non-linear inverse problem for an elliptic partial differential equation shows that while the forward mapping is analytic the inverse mapping is discontinuous. A rigorous treatment of the linearisation of the problem is given, including proofs of forms of linearisation assumed by previous authors. It is shown that the derivative of the forward problem is compact. Numerical calculations of the singular value decomposition (SVD) are given including plots of singular values and images of the singular functions. The SVD is used to settle a controversy concerning current drive patterns. Reconstruction algorithms are investigated and use of Regularised Newton methods is suggested. A formula for the second derivative of the forward mapping is derived which proves too computationally expensive to calculate. Use of Tychonov regularisation as well as filtered SVD and iterative methods are discussed. The similarities, and differences, between EIT and X-Ray Computed Tomography (X-Ray CT) are illuminated. This leads to an explanation of methods used by other authors for EIT reconstuction based on X-Ray CT. Details of the author's own implementation of a regularised Newton method are given. Finally the idea of adaptive current patterns is investigated. An algorithm is given for the experimental determination of optimal current patterns and the integration of this technique with regularised Newton methods is explored. Promising numerical results from this technique are given. The thesis concludes with a discussion of some outstanding problems in EIT and points to possible routes for their solution. An appendix gives brief details of the design and development of the Oxford Polytechnic Adaptive Current Tomograph