Image Reconstruction in Electrical Impedance Tomography

W.R. Breckon

PhD Thesis Oxford Polytechnic 1990


Abstract

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.

Mathematically this consists of a non-linear inverse problem for an elliptic partial differential equation. A review of necessary mathematical techniques is given in Chapter~2, together with details of the formulation of the problem. In Chapter~3 a rigorous treatment of the linearisation of the problem is given, including proofs of forms of linearisation assumed by previous authors. Chapter~4 details the finite element technique used in the forward modelling of the EIT problem. The ill-posedness of the inverse problem is investigated in Chapter~5. This includes a proof of the discontinuity of the non-linear forward mapping, and the compactness of the derivative of the forward problem. 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 in Chapter~6. The 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 explaination of methods used by other authors for EIT reconstuction based on X-Ray CT. The chapter concludes with details of the authors own implementation of a regularised Newton method.

Chapter~7 developes the idea of adaptive current patterns. 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 concliudes 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.


A scanned pdf file of the thesis is now available on the MIMS e-print server as 2008.48: Image reconstruction in electrical impedance tomography


Contents

1 Introduction 1

2 Mathematical Preliminaries 12

3 Linearisation 30

4 Finite Element Modelling 43

5 Ill-posedness of the Inverse Problem 57

6 Reconstruction Algorithms 79

7 Adaptive Methods 104

8 Conclusions and Further Work 113

A Design of an Adaptive Current Tomograph 117

B Published Papers