Image reconstruction in electrical impedance tomography. Breckon, William Robert. MIMS EPrint:

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Image reconstruction in electrical impedance tomography Breckon, William Robert 1990 MIMS EPrint: 2008.

1 Image reconstruction in electrical impedance tomography Breckon, William Robert 1990 MIMS EPrint: Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN

2 Image Reconstruction in Electrical Impedance Tomography Ph.D. Thesis William Robert Breckon Oxford Polytechnic May 1990 Over the years I have been asked to provide copies of this thesis. When I wrote the thesis my ability, and the technology available at the time, made it difficult to include graphics electronically and I do not have the electronic sources for them. Indeed they were produced either by Fortran libraries long since neglected or hand drawn. This limitation was a blessing in that requests for the thesis were directed to me personally rather than an anonymous server, and I was able to make personal contact with each person, often young graduate students new to the field, who requested a paper copy. I provided this in exchange for a brief account of why they were interested in EIT and what they intended to do. Some notable recipients of these paper copies were Andrea Borsic in Italy, and Manuch Soleimani in Iran. Whether despite or because of my encouragement and thesis they both went on to make significant contributions, both constructing EIT systems and implementing working reconstruction algorithms at Masters level in their own countries. At a time when these tasks were far from routine. I had the privilege to supervise them both for their PhDs and they continue to contribute to this and other areas of inverse problems. With the advent of a photocopier in the School of Mathematics at Manchester capable of scanning hundreds of sheets of paper and ing the portable document format file, my first thought was to scan this thesis. Looking over it it all seems very dated and we have moved on so much, but it is still requested by those new to the field. I hope the reader will gain some benefit and forgive any errors they find. Now that the thesis is available on the preprint server I will miss hearing from those starting out in EIT. So please feel free to contact me anyway and tell me what you are doing or planning to do in this area. At the time of writing my contact details are on the web site and this thesis can be found on the MIMS e-print server by searching Professor William Robert Breckon Lionheart School of Mathematics The University of Manchester April 15th

3 Abstract Image Reconstruction Electrical Impedance Tomography W.R. Breckon 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.

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Principal Moderator s Report

Principal Moderator s Report Centres are reminded that the deadline for coursework marks (and scripts if there are 10 or fewer from the centre) is December 10 for this specification. Moderators were pleased