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

1.1 Electrical Impedance Tomography 1

1.2 Impedance Measurement In Medicine 1

1.3 Medical Imaging 2

1.4 Description of EIT 3

1.5 Brief History 5

1.6 EIT at Oxford Polytechnic 9

1.7 Structure of this Thesis 10

2.1 Introduction 12

2.2 Regions 13

2.3 Current and Voltage 13

2.4 Simple Examples 14

2.4.1 Uniform disk 14

2.4.2 Concentric Anomaly 15

2.5 Partial Differential Equations 16

2.5.1 Sobolev Spaces 16

2.5.2 Boundary Conditions 19

2.5.3 Weak formulation 21

2.5.4 Existence , Uniqueness and Continuous Dependence 22

2.5.5 Transfer Impedance 23

2.6 Compact Linear Operators 24

2.7 Calculus in Banach spaces 26

2.7.1 Derivatives and Order of Convergence 27

2.7.2 Taylor's Theorem 28

3.1 Introduction 30

3.2 Approach 31

3.3 Choice of Space for $\gamma $ 33

3.4 Direct Form 33

3.5 Interpretation as a source 37

3.6 Integral Form 37

3.7 Operator Form 39

3.8 Translation to Inverse scattering problem 39

3.9 Historical Note 42

4.1 Choice of Forward Modelling Technique 43

4.2 Theory of the Finite Element Method 44

4.2.1 Preliminaries 44

4.2.2 Approximation Space 45

4.2.3 The System Matrix 48

4.2.4 Application of Boundary Conditions 48

4.3 Specification of Conductivities 49

4.4 The NAG Finite Element Library 50

4.5 Implementation of the Forward Modelling Program 51

4.5.1 History of fwprob 51

4.5.2 Choice of Language 51

4.5.3 Program Details 51

4.5.4 Trouble Shooting 52

4.6 Mesh Generation, Numbering and Refinement 53

4.7 Numerical Calculation of the Derivative Matrix 55

5.1 Ill-posedness 57

5.2 Existence of an Inverse 58

5.3 Continuity of the Inverse 59

5.4 The Linearised Problem 61

5.5 Singular Value Decomposition 62

5.6 Calderòn Fields 65

5.7 Fourier Series for Calderòn Fields 67

5.8 Ill-posedness of Linearised Problem 68

5.9 Numerical Calculation of SVD 69

5.9.1 Implementation 69

5.9.2 Results and Interpretation 70

5.10 Polar or Adjacent? 75

6.1 Introduction 79

6.2 Non-linear Data Fitting 80

6.2.1 How many minima? 80

6.2.2 Newton's Method for a critical point 81

6.2.3 Calculation of the second derivative 83

6.3 Regularised Newton Methods 84

6.3.1 The Levenberg-Marquardt Method 84

6.3.2 Using the SVD. 86

6.3.3 Iterative Methods 87

6.4 X-Ray CT 89

6.5 Radon Transform Inversion 92

6.5.1 Not a Radon Transform 92

6.5.2 Consistent Updates 94

6.6 Application of ART to EIT 95

6.7 Implementation of Regularised Newton Methods 97

6.7.1 Reconstructing the Moat Object 97

6.7.2 The Effect of Data Errors 97

6.7.3 The Positivity Constraint 98

7.1 Which Measurements to Make? 105

7.2 Two-norm Optimal Currents 105

7.3 Algorithms for Eigen-functions 106

7.3.1 Power Method 106

7.3.2 Higher Eigen-functions 107

7.3.3 Critique of the Measurement Model 108

7.4 Reconstructing With Optimal Currents 109

7.5 Numerical Results 110

7.6 Point Optimal Currents 110

8.1 Summary of this Work 113

8.2 The Future of EIT 113

8.3 Some Leads on the Problems 114

8.3.1 Electrode Modelling 114

8.3.2 Electrode Placement and Boundary Shape Errors 116

8.4 Final Remarks 116

A.2 System Overview 118

A.3 Circuit Details 120

A.3.1 Motherboard 120

A.3.2 Electrode-Interface Board 125

A.4 Software Drivers 128

A.5 Problems with the Design 130

A.6 Calibration and Testing 131

A7 Future Work 132