Image Reconstruction in EIT

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Image reconstruction in electrical impedance

tomography

William Robert Breckon

May 1990

MIMS EPrint: 2008.48

Manchester Institute for Mathematical Sciences

School of Mathematics

The University of Manchester

Reports available from: http://www.manchester.ac.uk/mims/eprints

And by contacting: The MIMS Secretary



Manchester, M13 9PL, UK

ISSN 1749-9097


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Image Reconstruction in Electrical Impedance

Tomography

Ph.D. Thesis

William Robert Breckon

Oxford PolytechnicMay 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 recip-ients of these paper copies were Andrea Borsic in Italy, and Manuch Soleimani

in Iran. Whether despite or b ecause 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 Manch-

ester capable of scanning hundreds of sheets of paper and emailing 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 Iwill 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 http://www.maths.manchester.ac.uk/~bl,

and this thesis can be found on the MIMS e-print server by searchinghttp://eprints.ma.man.ac.uk/

Professor William Robert Breckon Lionheart



April 15th 2008

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Abstract

Image Reconstruction Electrical Impedance Tomography

W.R. Breckon

This thesis is concerned with Electrical Impedance Tomogaphy (EIT), a medi-cal imaging technique in which pictures of the electrical conductivity distribu-tion of the body are formed from current and voltage data taken on the bodysurface. The focus of the thesis is on the mathematical aspects of reconstruct-ing the conductivity image from the measured data (the reconstruction prob-lem). The reconstruction problem is particularly difficult and in this thesis itis investigated analytically and numerically. The aim of this investigation is to

understand why the problem is difficult and to find numerical solution meth-ods which respect the difficulties encountered. The analytical investigationof this non-linear inverse problem for an elliptic partial differential equationshows that while the forward mapping is analytic the inverse mapping is dis-continuous. A rigorous treatment of the linearisation of the problem is given,including proofs of forms of linearisation assumed by previous authors. It isshown that the derivative of the forward problem is compact. Numerical calcu-lations of the singular value decomposition (SVD) are given including plots ofsingular values and images of the singular functions. The SVD is used to set-tle a controversy concerning current drive patterns. Reconstruction algorithmsare investigated and use of Regularised Newton methods is suggested. A for-

mula for the second derivative of the forward mapping is derived which provestoo computationally expensive to calculate. Use of Tychonov regularisationas 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 au-thors for EIT reconstuction based on X-Ray CT. Details of the authors ownimplementation of a regularised Newton method are given. Finally the idea ofadaptive current patterns is investigated. An algorithm is given for the exper-imental determination of optimal current patterns and the integration of thistechnique with regularised Newton methods is explored. Promising numerical

results from this technique are given. The thesis concludes with a discussionof some outstanding problems in EIT and points to possible routes for theirsolution. An appendix gives brief details of the design and development of theOxford Polytechnic Adaptive Current Tomograph.


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Image Reconstruction in EIT

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