image reconstruction in eit
TRANSCRIPT
-
7/27/2019 Image Reconstruction in EIT
1/181
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
School of Mathematics
The University of Manchester
Manchester, M13 9PL, UK
ISSN 1749-9097
-
7/27/2019 Image Reconstruction in EIT
2/181
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
School of Mathematics
The University of Manchester
April 15th 2008
1
-
7/27/2019 Image Reconstruction in EIT
3/181
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.
-
7/27/2019 Image Reconstruction in EIT
4/181
-
7/27/2019 Image Reconstruction in EIT
5/181
-
7/27/2019 Image Reconstruction in EIT
6/181
-
7/27/2019 Image Reconstruction in EIT
7/181
-
7/27/2019 Image Reconstruction in EIT
8/181
-
7/27/2019 Image Reconstruction in EIT
9/181
-
7/27/2019 Image Reconstruction in EIT
10/181
-
7/27/2019 Image Reconstruction in EIT
11/181
-
7/27/2019 Image Reconstruction in EIT
12/181
-
7/27/2019 Image Reconstruction in EIT
13/181
-
7/27/2019 Image Reconstruction in EIT
14/181
-
7/27/2019 Image Reconstruction in EIT
15/181
-
7/27/2019 Image Reconstruction in EIT
16/181
-
7/27/2019 Image Reconstruction in EIT
17/181
-
7/27/2019 Image Reconstruction in EIT
18/181
-
7/27/2019 Image Reconstruction in EIT
19/181
-
7/27/2019 Image Reconstruction in EIT
20/181
-
7/27/2019 Image Reconstruction in EIT
21/181
-
7/27/2019 Image Reconstruction in EIT
22/181
-
7/27/2019 Image Reconstruction in EIT
23/181
-
7/27/2019 Image Reconstruction in EIT
24/181
-
7/27/2019 Image Reconstruction in EIT
25/181
-
7/27/2019 Image Reconstruction in EIT
26/181
-
7/27/2019 Image Reconstruction in EIT
27/181
-
7/27/2019 Image Reconstruction in EIT
28/181
-
7/27/2019 Image Reconstruction in EIT
29/181
-
7/27/2019 Image Reconstruction in EIT
30/181
-
7/27/2019 Image Reconstruction in EIT
31/181
-
7/27/2019 Image Reconstruction in EIT
32/181
-
7/27/2019 Image Reconstruction in EIT
33/181
-
7/27/2019 Image Reconstruction in EIT
34/181
-
7/27/2019 Image Reconstruction in EIT
35/181
-
7/27/2019 Image Reconstruction in EIT
36/181
-
7/27/2019 Image Reconstruction in EIT
37/181
-
7/27/2019 Image Reconstruction in EIT
38/181
-
7/27/2019 Image Reconstruction in EIT
39/181
-
7/27/2019 Image Reconstruction in EIT
40/181
-
7/27/2019 Image Reconstruction in EIT
41/181
-
7/27/2019 Image Reconstruction in EIT
42/181
-
7/27/2019 Image Reconstruction in EIT
43/181
-
7/27/2019 Image Reconstruction in EIT
44/181
-
7/27/2019 Image Reconstruction in EIT
45/181
-
7/27/2019 Image Reconstruction in EIT
46/181
-
7/27/2019 Image Reconstruction in EIT
47/181
-
7/27/2019 Image Reconstruction in EIT
48/181
-
7/27/2019 Image Reconstruction in EIT
49/181
-
7/27/2019 Image Reconstruction in EIT
50/181
-
7/27/2019 Image Reconstruction in EIT
51/181
-
7/27/2019 Image Reconstruction in EIT
52/181
-
7/27/2019 Image Reconstruction in EIT
53/181
-
7/27/2019 Image Reconstruction in EIT
54/181
-
7/27/2019 Image Reconstruction in EIT
55/181
-
7/27/2019 Image Reconstruction in EIT
56/181
-
7/27/2019 Image Reconstruction in EIT
57/181
-
7/27/2019 Image Reconstruction in EIT
58/181
-
7/27/2019 Image Reconstruction in EIT
59/181
-
7/27/2019 Image Reconstruction in EIT
60/181
-
7/27/2019 Image Reconstruction in EIT
61/181
-
7/27/2019 Image Reconstruction in EIT
62/181
-
7/27/2019 Image Reconstruction in EIT
63/181
-
7/27/2019 Image Reconstruction in EIT
64/181
-
7/27/2019 Image Reconstruction in EIT
65/181
-
7/27/2019 Image Reconstruction in EIT
66/181
-
7/27/2019 Image Reconstruction in EIT
67/181
-
7/27/2019 Image Reconstruction in EIT
68/181
-
7/27/2019 Image Reconstruction in EIT
69/181
-
7/27/2019 Image Reconstruction in EIT
70/181
-
7/27/2019 Image Reconstruction in EIT
71/181
-
7/27/2019 Image Reconstruction in EIT
72/181
-
7/27/2019 Image Reconstruction in EIT
73/181
-
7/27/2019 Image Reconstruction in EIT
74/181
-
7/27/2019 Image Reconstruction in EIT
75/181
-
7/27/2019 Image Reconstruction in EIT
76/181
-
7/27/2019 Image Reconstruction in EIT
77/181
-
7/27/2019 Image Reconstruction in EIT
78/181
-
7/27/2019 Image Reconstruction in EIT
79/181
-
7/27/2019 Image Reconstruction in EIT
80/181
-
7/27/2019 Image Reconstruction in EIT
81/181
-
7/27/2019 Image Reconstruction in EIT
82/181
-
7/27/2019 Image Reconstruction in EIT
83/181
-
7/27/2019 Image Reconstruction in EIT
84/181
-
7/27/2019 Image Reconstruction in EIT
85/181
-
7/27/2019 Image Reconstruction in EIT
86/181
-
7/27/2019 Image Reconstruction in EIT
87/181
-
7/27/2019 Image Reconstruction in EIT
88/181
-
7/27/2019 Image Reconstruction in EIT
89/181
-
7/27/2019 Image Reconstruction in EIT
90/181
-
7/27/2019 Image Reconstruction in EIT
91/181
-
7/27/2019 Image Reconstruction in EIT
92/181
-
7/27/2019 Image Reconstruction in EIT
93/181
-
7/27/2019 Image Reconstruction in EIT
94/181
-
7/27/2019 Image Reconstruction in EIT
95/181
-
7/27/2019 Image Reconstruction in EIT
96/181
-
7/27/2019 Image Reconstruction in EIT
97/181
-
7/27/2019 Image Reconstruction in EIT
98/181
-
7/27/2019 Image Reconstruction in EIT
99/181
-
7/27/2019 Image Reconstruction in EIT
100/181
-
7/27/2019 Image Reconstruction in EIT
101/181
-
7/27/2019 Image Reconstruction in EIT
102/181
-
7/27/2019 Image Reconstruction in EIT
103/181
-
7/27/2019 Image Reconstruction in EIT
104/181
-
7/27/2019 Image Reconstruction in EIT
105/181
-
7/27/2019 Image Reconstruction in EIT
106/181
-
7/27/2019 Image Reconstruction in EIT
107/181
-
7/27/2019 Image Reconstruction in EIT
108/181
-
7/27/2019 Image Reconstruction in EIT
109/181
-
7/27/2019 Image Reconstruction in EIT
110/181
-
7/27/2019 Image Reconstruction in EIT
111/181
-
7/27/2019 Image Reconstruction in EIT
112/181
-
7/27/2019 Image Reconstruction in EIT
113/181
-
7/27/2019 Image Reconstruction in EIT
114/181
-
7/27/2019 Image Reconstruction in EIT
115/181
-
7/27/2019 Image Reconstruction in EIT
116/181
-
7/27/2019 Image Reconstruction in EIT
117/181
-
7/27/2019 Image Reconstruction in EIT
118/181
-
7/27/2019 Image Reconstruction in EIT
119/181
-
7/27/2019 Image Reconstruction in EIT
120/181
-
7/27/2019 Image Reconstruction in EIT
121/181
-
7/27/2019 Image Reconstruction in EIT
122/181
-
7/27/2019 Image Reconstruction in EIT
123/181
-
7/27/2019 Image Reconstruction in EIT
124/181
-
7/27/2019 Image Reconstruction in EIT
125/181
-
7/27/2019 Image Reconstruction in EIT
126/181
-
7/27/2019 Image Reconstruction in EIT
127/181
-
7/27/2019 Image Reconstruction in EIT
128/181
-
7/27/2019 Image Reconstruction in EIT
129/181
-
7/27/2019 Image Reconstruction in EIT
130/181
-
7/27/2019 Image Reconstruction in EIT
131/181
-
7/27/2019 Image Reconstruction in EIT
132/181
-
7/27/2019 Image Reconstruction in EIT
133/181
-
7/27/2019 Image Reconstruction in EIT
134/181
-
7/27/2019 Image Reconstruction in EIT
135/181
-
7/27/2019 Image Reconstruction in EIT
136/181
-
7/27/2019 Image Reconstruction in EIT
137/181
-
7/27/2019 Image Reconstruction in EIT
138/181
-
7/27/2019 Image Reconstruction in EIT
139/181
-
7/27/2019 Image Reconstruction in EIT
140/181
-
7/27/2019 Image Reconstruction in EIT
141/181
-
7/27/2019 Image Reconstruction in EIT
142/181
-
7/27/2019 Image Reconstruction in EIT
143/181
-
7/27/2019 Image Reconstruction in EIT
144/181
-
7/27/2019 Image Reconstruction in EIT
145/181
-
7/27/2019 Image Reconstruction in EIT
146/181
-
7/27/2019 Image Reconstruction in EIT
147/181
-
7/27/2019 Image Reconstruction in EIT
148/181
-
7/27/2019 Image Reconstruction in EIT
149/181
-
7/27/2019 Image Reconstruction in EIT
150/181
-
7/27/2019 Image Reconstruction in EIT
151/181
-
7/27/2019 Image Reconstruction in EIT
152/181
-
7/27/2019 Image Reconstruction in EIT
153/181
-
7/27/2019 Image Reconstruction in EIT
154/181
-
7/27/2019 Image Reconstruction in EIT
155/181
-
7/27/2019 Image Reconstruction in EIT
156/181
-
7/27/2019 Image Reconstruction in EIT
157/181
-
7/27/2019 Image Reconstruction in EIT
158/181
-
7/27/2019 Image Reconstruction in EIT
159/181
-
7/27/2019 Image Reconstruction in EIT
160/181
-
7/27/2019 Image Reconstruction in EIT
161/181
-
7/27/2019 Image Reconstruction in EIT
162/181
-
7/27/2019 Image Reconstruction in EIT
163/181
-
7/27/2019 Image Reconstruction in EIT
164/181
-
7/27/2019 Image Reconstruction in EIT
165/181
-
7/27/2019 Image Reconstruction in EIT
166/181
-
7/27/2019 Image Reconstruction in EIT
167/181
-
7/27/2019 Image Reconstruction in EIT
168/181
-
7/27/2019 Image Reconstruction in EIT
169/181
-
7/27/2019 Image Reconstruction in EIT
170/181
-
7/27/2019 Image Reconstruction in EIT
171/181
-
7/27/2019 Image Reconstruction in EIT
172/181
-
7/27/2019 Image Reconstruction in EIT
173/181
-
7/27/2019 Image Reconstruction in EIT
174/181
-
7/27/2019 Image Reconstruction in EIT
175/181
-
7/27/2019 Image Reconstruction in EIT
176/181
-
7/27/2019 Image Reconstruction in EIT
177/181
-
7/27/2019 Image Reconstruction in EIT
178/181
-
7/27/2019 Image Reconstruction in EIT
179/181
-
7/27/2019 Image Reconstruction in EIT
180/181
-
7/27/2019 Image Reconstruction in EIT
181/181