ADVANCES IN ELECTRICAL CAPACITANCE
TOMOGRAPHY
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Qussai Marashdeh, B.S., M.S.
*****
The Ohio State University
2006
Dissertation Committee:
Approved by
Fernando Teixeira, Adviser
Stanley Ahalt
Liang-Shih Fan
Adviser
Graduate Program in
Electrical Engineering
c Copyright by
Qussai Marashdeh
2006
ABSTRACT
Electrical tomography techniques for process imaging are very prominent for industrial applications due to their low cost, safety, high capture speed, and suitability
for different vessel sizes. Among electrical tomography techniques, electrical capacitance tomography has been the subject of extensive recent research due to its noninvasive nature and capability of differentiating between different phases based on
permittivity distribution. Research in electrical capacitance tomography is inherently interdisciplinary, and areas of research in it can be categorized as: (1) sensor
design, (2) hardware electronics, (3) and image reconstruction. Work presented in
this dissertation includes developments in image reconstruction and sensor design.
Work on image reconstruction presented in this dissertation include developments
of both forward and inverse solutions. A feed forward neural network based forward
solver has been developed for fast and relatively accurate forward solutions. The
forward solver has been integrated into a Hopfield optimization reconstruction technique to provide a fully non-linear image reconstruction process. In addition, a 3D
volume image reconstruction has been developed by extending the 2D neural network
multi objective image reconstruction technique (NN-MOIRT) to 3D applications, and
inclusion of new objective functions tailored for 3D imaging.
Developments on sensor related topics provided in this dissertation are 3D capacitance sensor designs for 3D imaging and non-invasive capacitance sensors for
ii
simultaneous permittivity/conductivity imaging. In the former case, a 3D sensor
with axial variation in field distribution has been used for volume imaging based on
the developed Hopfield 3D optimization image reconstruction. In the latter case,
an extension of the conventional capacitance sensor based on capacitance and power
measurements has been provided for simultaneous imaging of permittivity and conductivity distributions.
iii
This is dedicated to my Mother and Father, Brothers and Sisters
iv
ACKNOWLEDGMENTS
I thank Dr. Warsito for the intellectual support, guidance, encouragement, and
motivation through out my research which made this dissertation possible.
I thank Prof. L.-S. Fan for his advising, teaching, encouragement, and support
through out my graduate research. It has been an honor being a research associate
among his distinguished group.
I also thank Prof. F. L. Teixeira and express my appreciation to him. This work
would not have been possible without his efforts. It has been a great honor for me to
be a graduate student at OSU.
And I thank Mohammad Hadi, Allisa Park, and Bing Du for their contribution.
v
VITA
October 23, 1978 . . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Alabama, USA
2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. Electrical Engineering,
University of Jordan
2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. Electrical Engineering,
The Ohio State University
2001-present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research Associate,
The Ohio State University.
PUBLICATIONS
Research Publications
Q. Marashdeh, and L.-S. Fan, “Electrostatic Tomography for Process Measurement And Control Through Boundary Treatment”. United States Patent Application,January 2006.
Q. Marashdeh, W. Warsito, and L.-S. Fan, “Non-invasive electrical resistive-capacitive
tomography (ERCT) for simultaneous conductivity/permittivity imaging of multiphase flow”. United States Patent Application,January 2006.
Q. Marashdeh, and F.L. Teixeira, “Sensitivity Matrix Calculation for Fast Electrical
Capacitance Tomography (ECT) of Flow Systems”. IEEE Transactions on Magnetics, 40(2):1204–1207, March 2004.
Q. Marashdeh, W. Warsito, L.-S. Fan, and F.L. Teixeira, “Nonlinear Forward Problem Solution for Electrical Capacitance Tomography Using Feed-Forward Neural Network”. IEEE Sensors Journal, 6(2):441–449, April 2006.
W. Warsito, Q. Marashdeh, and L.-S. Fan, “Electrical Capacitance Volume-Tomography
(ECVT)”. Submitted to IEEE Sensors Journal.
vi
Q. Marashdeh, W. Warsito, L.-S. Fan, and F.L. Teixeira “Multimodal Tomography
System Based on ECT Sensors”. Submitted to IEEE Sensors Journal.
Q. Marashdeh, W. Warsito, L.-S. Fan, and F.L. Teixeira “Non-linear Image Reconstruction Technique for ECT using a Combined Neural Network Approach”. Submitted to Meas. Sci. & Tech..
Q. Marashdeh, and F.L. Teixeira “Perturbative Approach to Compute Sensitivity
Matrix Elements in Electrical Capacitance Tomography (ECT) of Flow Systems”.
Proceedings of the 2003 IEEE Antennas and Propagation International Symposium,
vol. 2, pp. 185-188, Columbus, OH, June 22-27, 2003. .
Q. Marashdeh, and F.L. Teixeira “Sensitivity Matrix Calculation for Fast Electrical
Capacitance Tomography (ECT) of Flow Systems”. Proceedings of the COMPUMAG
- 14th Conference on the Computation of Electromagnetic Fields, vol.3, pp. 104-105,
Saratoga Springs, NY, July 13-17, 2003.
W. Warsito, Q. Marashdeh, and L.-S. Fan, “3D-ECT: sensor design and image reconstruction”. Proceedings of 4th World Congress on Industrial Process Tomography,
vol. 1, pp. 82-87, Aizu, Japan, September 5-9, 2005.
Q. Marashdeh, W. Warsito, and L.-S. Fan, “Feed Forward Neural Network Solution
for Non-Linear Forward Problem in ECT”. Proceedings of 4th World Congress on
Industrial Process Tomography, vol. 2, pp. 746-751, Aizu, Japan, September 5-9,
2005.
Q. Marashdeh, W. Warsito, L.-S. Fan, and F.L. Teixeira, “Solution of Non-Linear
Forward Problems in Electrical Capacitance Tomography Using Neural Networks ”.
Proceedings of the 2005 IEEE Antennas and Propagation International Symposium,
vol. 1A, pp. 181-184, Washington DC, July 3-8, 2005.
Q. Marashdeh, W. Warsito, L.-S. Fan, and F.L. Teixeira, “Electrical Capacitance
Tomography Sensor Design for 3-D Applications”. 2005 URSI North American Radio
Science Meeting Digest, p. 301, Washington, DC, July 3-8, 2005.
Q. Marashdeh, M. Hadi, W. Warsito, and L.-S. Fan, “On the ECT Sensor Based
Dual Imaging Modality System for Electrical Permittivity and Conductivity Measurements”. Proceedings of 5th World Congress on Particle Technology, Orlando,
Florida USA, April 23-27, 2006.
vii
L.-S. Fan, W. Warsito, Q. Marashdeh, A.-H. A. Park, and B. Du, “Electrostatic
Tomography for Multiphase Process Imaging”. Proceedings of 5th World Congress
on Particle Technology, Orlando, Florida USA, April 23-27, 2006.
B. Du, Q. Marashdeh, W. Warsito, A.-H. A. Park, and L.-S. Fan, “Development of
Electrical Capacitance Volume Tomography (ECVT) and Electrostatic Tomography
(EST) for 3D Density and Charge Imaging of Fluidized Bed System”. Submitted to
Fluidization Conf. 2007.
FIELDS OF STUDY
Major Field: Electrical Engineering
Studies in:
Electromagnetics
Prof. Fernando Teixeira
Communication and Signal Processing Prof. Stanley Ahalt
viii
TABLE OF CONTENTS
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
Chapters:
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2
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Review of reconstruction techniques . . . . . . . . . . . . . . . . . . . . .
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2.1
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1.3
1.4
2.
2.2
2.3
Technology overview . . . . . . . .
Electrical Capacitance Tomography
1.2.1 ECT reconstruction . . . .
1.2.2 Multi modal tomography .
Dissertation objectives . . . . . . .
Dissertation outline . . . . . . . .
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(ECT) .
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Direct techniques . . . . . . . . . . .
2.1.1 Linear back projection . . . .
2.1.2 Singular value decomposition
2.1.3 Tikhonov regularization . . .
Iterative reconstruction techniques .
NN-MOIRT . . . . . . . . . . . . . .
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3.
Nonlinear Forward Problem Solution for Electrical Capacitance Tomography Using Feed Forward Neural Network . . . . . . . . . . . . . . . . . .
3.1
3.2
3.3
3.4
3.5
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Non-linear Image Reconstruction Technique for ECT using Combined Neural
Network Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 ECT forward problem basic equations . . . . . . . . . . . .
4.1.2 Iterative image reconstruction techniques . . . . . . . . . .
Combined feed-forward and analog neural network technique for
non-linear image reconstruction . . . . . . . . . . . . . . . . . . . .
4.2.1 Forward problems using feed forward neural networks . . . .
4.2.2 Inverse problem solution via Hopfield neural network . . . .
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Stability and convergence performance . . . . . . . . . . .
4.4.2 Image reconstruction results . . . . . . . . . . . . . . . . . .
Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . .
53
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Electrical Capacitance Volume Tomography . . . . . . . . . . . . . . . .
69
4.2
4.3
4.4
4.5
5.
Problem Statement . . . . . . . . . . . . . . . . . . . . .
3.1.1 ECT system . . . . . . . . . . . . . . . . . . . .
3.1.2 Forward problem in image reconstruction process
NN techniques for the forward ECT problem . . . . . .
3.2.1 Architecture . . . . . . . . . . . . . . . . . . . .
3.2.2 Training . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Generalization . . . . . . . . . . . . . . . . . . .
Experiments and sensor data pre-processing . . . . . . .
3.3.1 Data collection . . . . . . . . . . . . . . . . . . .
3.3.2 Capacitance data rearranging and filtering . . . .
Results and discussion . . . . . . . . . . . . . . . . . . .
Summary and Conclusion . . . . . . . . . . . . . . . . .
23
5.1
5.2
5.3
49
49
50
Principle of ECT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.1 Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.2 Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . 73
Multicriterion Optimization Image Reconstruction Technique (MOIRT) 76
5.2.1 Multicriterion Optimization Image Reconstruction Problem
76
5.2.2 Solution With Hopfield Neural Network . . . . . . . . . . . 77
Sensor Design and Sensitivity Map . . . . . . . . . . . . . . . . . . 81
x
5.4
5.5
5.6
6.
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reconstruction Results . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
87
92
Multimodal Tomography System Based on ECT Sensors . . . . . . . . .
96
6.1
6.2
6.3
6.4
6.5
7.
ECT Sensor Data . . . . . . . . . . . . . .
6.1.1 ECT sensor . . . . . . . . . . . . .
6.1.2 Equivalent Lumped-Circuit Models
Sensitivity matrix . . . . . . . . . . . . .
6.2.1 Capacitance matrix . . . . . . . .
6.2.2 Power matrix . . . . . . . . . . . .
Reconstruction . . . . . . . . . . . . . . .
Recontruction Results and Discussion . .
Conclusions . . . . . . . . . . . . . . . . .
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98
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Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1
7.2
7.3
7.4
7.5
Reconstruction techniques . . . . . . . . .
Forward problem . . . . . . . . . . . . . .
3D volume tomography . . . . . . . . . .
Multi-modal electrical tomography . . . .
Future work . . . . . . . . . . . . . . . . .
7.5.1 3D neural network forward solver .
7.5.2 3D sensor design . . . . . . . . . .
7.5.3 Multi-modal electrical tomography
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119
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125
Appendices:
A.
Finite Element Method for Solving The ECT Forward Problem . . . . . 128
B.
3D Reconstruction Related Issues . . . . . . . . . . . . . . . . . . . . . . 131
B.1 3D sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Neural networks forward solver for 3D reconstruction . . . . . . . .
B.2.1 Data preprocessing . . . . . . . . . . . . . . . . . . . . . . .
B.2.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Extended objective functions for 3D reconstruction . . . . . . . . .
B.3.1 The use of correlation in process tomography: . . . . . . . .
B.3.2 The use of correlation function and a prior information in 3D
ECT image reconstruction: . . . . . . . . . . . . . . . . . .
xi
131
133
133
137
138
138
140
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
xii
LIST OF FIGURES
Figure
Page
1.1
ECT system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2
ECT sensor with seven electrodes . . . . . . . . . . . . . . . . . . . .
5
2.1
Reconstruction results of different shapes LBP, ILBP, SIRT and NNMOIRT reconstruction techniques. . . . . . . . . . . . . . . . . . . .
22
3.1
ECT sensor with seven electrodes . . . . . . . . . . . . . . . . . . . .
26
3.2
Neural network with multiple layers in matrix form . . . . . . . . . .
29
3.3
Mean square error for different permittivity distributions obtained from
different rod sizes, where r is the rod size . . . . . . . . . . . . . . .
40
Linear mapping of different predicted capacitance vectors with respect
to measured capacitance for different permittivity distributions . . .
41
Predicted capacitance vector using both NN and LFP compared to
measured capacitance of the same permittivity distribution . . . . .
42
Comparison with Landweber reconstruction of Gas-Solid flow in terms
of convergence rate with both LFP and NN integrated as forward problem solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Convergence rate of NN and LFP integrated in Landweber iterative
reconstruction technique for the centered 1 inch tube in figure 3.6 . .
44
Reconstruction results using Landweber iterative technique for different permittivity distributions with both LFP and NN integrated as
forward problem solvers . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.4
3.5
3.6
3.7
3.8
xiii
4.1
Neural Network with one intermediate layer. . . . . . . . . . . . . . .
54
4.2
Mean square error with respect to reconstructed image vector for: (a)
Case 1: non-linear update based on ILBP, (b) Case 2: semi-linear
update (LFP for forward solution and NN-MOIRT for update), (c)
Case 3: non-linear update (NN for forward solution and NN-MOIRT
for update). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Reconstruction results using linear, semi linear, and nonlinear update
reconstruction for three different regime flows. . . . . . . . . . . . . .
65
Reconstruction results for the flow regimes in Figure 4.3 with a thresholding filter applied. . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Reconstruction results using linear, semi-linear, and non-linear reconstruction for four 1-inch diameter rods in a 4-inches diameter vessel
based on simulated results. . . . . . . . . . . . . . . . . . . . . . . . .
67
Reconstruction results using semi-linear and non-linear reconstruction
for 1 & .75-inch diameter rods in a 4-inches diameter vessel based on
experimental measurements. . . . . . . . . . . . . . . . . . . . . . . .
68
Sensor designs and volume image digitization: (a) Triangular sensor,
(b) Rectangular sensor, (c) Image digitization. . . . . . . . . . . . . .
83
Three-dimensional sensitivity maps: (a) Triangular sensor, (b) Rectangular sensor (The electrode pair number is in Figure 5.1) . . . . .
85
Axial sensitivity distribution for all 66 capacitance readings: (a) Triangular sensor, (b) Rectangular sensor; the dead zones are the areas
indicated by the dashed line . . . . . . . . . . . . . . . . . . . . . . .
86
4 Reconstruction results of a sphere in the center and the edge of
sensing domains using LBP technique: (a) (b) Triangular sensor, (c)
(d) Rectangular sensor . . . . . . . . . . . . . . . . . . . . . . . . . .
88
Reconstruction results of a sphere in the center and the edge of sensing
domains using Landweber technique: (a) (b) Triangular sensor, (c) (d)
Rectangular sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.3
4.4
4.5
4.6
5.1
5.2
5.3
5.4
5.5
xiv
5.6
Reconstruction results of a sphere in the center and the edge of sensing
domains using NN-MOIRT: (a) (b) Triangular sensor, (c) (d) Rectangular sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
3D image of actually falling sphere reconstructed using Landweber
technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
5.8
3D image of actually falling sphere reconstructed using NN-MOIRT .
94
6.1
Cross section of ECT sensor consisting of six electrodes surrounding a
cylindrical vessel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
5.7
6.2
Normalized electric field |E/E0 | as a function of conductivity σ . . . . 104
6.3
Normalized electric field |E/E0 | as a function of relative permittivity r 105
6.4
Power dissipation inside the pixel for the normalized electric field in
Figure 6.2 as a function of conductivity σ . . . . . . . . . . . . . . . . 106
6.5
Power vectors of forward solutions for the flow distribution depicted
at the bottom right corner. The electrical properties for (ring) zone B
are = 5 and σ = 0, whereas for (central) zone A, = 1 and σ varies
as indicated by the legend. . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6
Reconstruction results for the original distribution values depicted on
the left (diffusion dominated case). . . . . . . . . . . . . . . . . . . . 113
6.7
Reconstruction results for the original distribution values depicted on
the left (convection dominated case). . . . . . . . . . . . . . . . . . . 114
6.8
Reconstruction results for the original distribution values depicted on
the left (convection dominated case). . . . . . . . . . . . . . . . . . . 115
6.9
Reconstruction results for the original distribution depicted on the left.
Because of the reduced value of the skin depth at ring zone B, the
reconstruction fails to reproduce the original distribution. . . . . . . . 116
6.10 Reconstruction results for a two-sphere case, with the original distribution values depicted on the left. . . . . . . . . . . . . . . . . . . . . 117
xv
6.11 Reconstruction of simulated data for a 2 sphere case with the original
distribution depicted on the left of the Figure . . . . . . . . . . . . . 117
7.1
Adaptive 3D sensor composed of small plate elements. Different shapes
of plates and planes can be formed by connecting the small plates
together. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.1 ECT volume tomography verses conventional 3D ECT. . . . . . . . . 132
B.2 Sensitivity matrix of a trapezoidal sensor.
. . . . . . . . . . . . . . . 134
B.3 Sensitivity matrix of a square double plane sensor. . . . . . . . . . . . 135
B.4 Sensitivity matrix of a square triple plane sensor. . . . . . . . . . . . 136
xvi
CHAPTER 1
INTRODUCTION
Tomography by definition refers to the process of exploring the internal characteristics of a specified region through integral measurements related to the internal
characteristics of the specified domain [5] [73] [33] [26]. Traditionally, the term tomography has been used for the process of obtaining 2D cross sections called tomograms.
However, recent developments in different tomography discipline have expanded the
tomography concept to include 3D and 4D (with time domain included) imaging [14]
[51] [92] [63].
Tomography can be classified into two types: direct and indirect. In the former,
a method of visual recording not visible to the human eye is used e.g. X-ray or
infrared imaging [60] [22]. Whereas in the later, boundary measurements related to
the internal characteristics of the object of interest are used for image reconstruction
[18] . In indirect tomography, many physical quantities implemented through different
tomography systems can be used as the boundary measurement quantity. However,
from engineering point of view, an acceptable tomography technique is one which is
[98]:
1. Non-invasive: it requires no direct contact between the sensor and the object
or domain of interest .
1
2. Non-intrusive: it does not change change or disturb the nature of the object
being explored.
In process application, the choice of using a particular tomography modality depends on the nature of the flow under investigation, the required information about
the process, the size of the process vessel, and the environment surrounding the
process operation [70]. Process tomography techniques for industrial process imaging
are assessed based on their complexity, cost, safety, and ability to capture real time
dynamic flows [107]. The wide range of available tomography techniques, and the
problems associated with each tomography modality makes the tomography field inherently interdisciplinary . Tomography researchers are often required to have knowledge in physics, electronics, mechanics, mathematics, and programing.
Industrial processes tend to be complex in nature, and measurement and control
operations are used to improve products, simplify process, and increase efficiency.
Multi dimensional measurements through topographic techniques have provided a
leap toward estimation of the process state for better process control [70]. Implementation of process tomography can be achieved through a variety of techniques.
However, the most conspicuous techniques are those based on measurement of electrical properties, through utilization of capacitive, conductive, and inductive nature
of materials under investigation . Variation in electrical properties of different flow
components provided process measurement and imaging capabilities using electrical
tomography systems. The increased interest in electrical tomography techniques for
process applications has been motivated by their low construction cost, high speed,
safety, and suitability for various sizes of vessels [9] [19]. Nevertheless, the relatively
low resolution of reconstructed images, nonlinearity, and the ill posedness of system
2
equations pose a major challenge when dealing with electrical tomography systems
[67]. In this dissertation, work has been mainly focused on electrical capacitance
tomography (ECT) for process applications.
1.1
Technology overview
Research in tomography systems can be classified into three categories: (1) sensor
[21] [39], (2) data acquisition and hardware [103], and (3) reconstruction techniques
[105]. Sensor design, performance, and associated problems depend on the tomography modality being used. In electrical tomography sensors, encountered problems
are usually the soft field nature of the sensor, and ill posed response of the sensor
to different location in the imaging domain. Problems in the data acquisition of
electrical tomography sensors are mainly the low level power of sensed signal and
low signal to noise ratio (SNR). The low level of acquired signal is usually reflected
on the sensor dimensions. Increased sensor area provides higher SNR. However, a
trade off of lower spatial resolution of reconstructed images is usually associated with
increasing the sensor dimensions. Reconstruction techniques involve the process of
solving the inverse problem for finding the electrical property distribution from the
measured capacitance data. The reconstruction process highly depends on the sensor
under consideration.
1.2
Electrical Capacitance Tomography (ECT)
ECT was first developed in early 1980s for process imaging, and it has since been
applied to gas/solid, and gas/liquid flows [96]. More recently, three-phase imaging
of gas-liquid-solid has been realized through recent developments in reconstruction
3
Figure 1.1: ECT system
techniques [88]. As mentioned before, has the advantage of being non-invasive and
non-intrusive. Moreover, it provides information about the electric properties of different flow components. However, implementation of ECT is challenged by its low
spatial resolution. In ECT, the highest recorded resolution typically does not exceed
3% of the imaging doamin [89]. Reconstruction resolution in ECT is restricted by
the soft field nature of the sensor, the ill posedness of the system equations, and the
number of plates used in the sensor. However, increasing the number of sensor plates
is expected to increase the reconstruction resolution under the assumption that SNR
remains fixed. In practice, in a typical ECT sensor the capacitance level could be of
the order of Femto Farads.
The sensor hardware in ECT is typically composed of a number of n electrodes
surrounding the wall of the process vessel as illustrated in Figure 1.2, the number of
independent capacitance measurement available in a such a configuration is
4
n(n−1)
.
2
Figure 1.2: ECT sensor with seven electrodes
The permittivity distribution is related to capacitance measurement according to
Poisson equation:
∇ · (ε(x, y)∇φ(x, y)) = −ρ(x, y),
(1.1)
where (x, y) is the permittivity distribution, φ(x, y) is electrical potential distribution
and ρ(x, y) is charge distribution. The Poisson equation is a linear partial differential
equation in terms of φ(x, y). The nonlinearity of ECT results from the nonlinear dependence between the field distribution φ(x, y) (and hence the measured capacitance
data) and the unknown solution (x, y).
The mutual capacitance between two pairs of electrodes is given by the ratio
between stored charge and potential difference as in equation (1.2).
Cij =
Qj
,
4Vij
5
(1.2)
Where Cij is the mutual capacitance and 4Vij the potential difference between
electrodes i and j respectively, and Qj is the charge on the receiving electrode which
is found by applying Gauss law:
Qj =
I
Γj
(x, y)∇φ(x, y) · n̂dl,
(1.3)
where Γj is a closed path enclosing the detecting electrode and n̂ is a unit vector
normal to Γj . Using equations (1.2) and (1.3), the mutual capacitance is calculated
as:
Cij =
1 I
(x, y)∇φ(x, y) · n̂dl,
4Vij Γj
(1.4)
Equation (1.4) is known as the forward problem equation of ECT, in which the capacitance is calculated for a given permittivity distribution and boundary conditions.
On the other hand, the inverse problem is the process of estimating the permittivity
distribution from the measured capacitance data.
1.2.1
ECT reconstruction
One of the most critical challenges for the wide use of ECT in industrial process
applications is the relatively low accuracy of reconstructed images with the commonly
used reconstruction techniques. Reconstruction in ECT is considered a challenging
task due to the soft field nature of the technique, in which the field distribution in the
region of interest is dependent on the property distribution in a non-linear fashion
[45].
6
Inspired with the reconstruction techniques used in hard field tomography, in
which the field distribution is independent from the property distribution, linearization approximate methods are used. One of the most common approaches is the
sensitivity model. The sensitivity model is based on subdividing the domain into a
number of pixels, and obtaining the response of the sensor as a linear sum of responses
when the permittivity of one pixel is perturbed. The constructed sensitivity map is
then used to map back the integral measurement response (capacitance measurement)
to the permittivity map through projection methods [102]. However, this method is
only applicable when the dielectric contrast of the material being imaged is very low.
Moreover, the obtained images are exposed to a smoothing effect resulting in blurred
images.
Improvement in reconstruction results is obtained by implementing iterative reconstruction techniques, which is used to solve the non-linear mapping between capacitance measurement and permittivity distribution [72]. In iterative reconstruction
techniques the reconstructed image is modified by calculating the capacitance value
of the image vector (solving the forward problem) and updating the image according
to the difference between measured and calculated capacitance values (solving the
inverse problem). Another form of iterative reconstruction techniques are optimization techniques [89] [55] [56]. In optimization techniques, a set of objective functions
that include the difference between measured and calculated capacitance values; are
minimized to obtain the most likely image. Optimization techniques possess the advantage of including the characteristics of the desired image in the reconstruction
process.
7
Most literature on ECT has been focused on solving the inverse problem. In recent
developments, Warsito and Fan introduced a new reconstruction technique based on
minimizing a set of objective functions using hopfield neural networks [89]. This
technique is capable of imaging multi phase flows in 3-D directly from capacitance
measurements for the first time [93]. The case of 3D image reconstruction depends
mainly on the sensitivity variation along all three directions provided by the sensor, as
well as the capability of the reconstruction technique of dealing with a more severely
ill posed 3D inverse problem (compared to the 2D case).
Although the forward problem solution has a major effect on the over all performance of iterative image reconstruction techniques, it has been received relatively
less attention compared to the inverse problem. The forward problem has been approached using one of the following methods: 1) linearization methods [35], 2) bruteforce numerical methods such as finite element method [24] [4] [44], boundary value
method [23] [54] and finite difference method [20], and 3) analytical methods [2] [47]
[58]. Analytical methods are restricted to highly symmetric structures, whereas numerical methods suffer the problem of excess computational time and resources to
obtain an acceptable solution. Linearization methods are the ones more commonly
used.
1.2.2
Multi modal tomography
Multi-modal tomography is defined as using one or more sensing methods to obtain
different characteristics of constituents in the imaging domain [8] . The use of multimodal tomography systems in process engineering has been motivated by the need
for monitoring and measuring complex process involving multiple components. An
8
example of using a multi-modal tomography system is in oil industry; in which it is
required to obtain information about gas and water components in a pipeline of oil
extracted from a well. A dual ECT and γ-ray tomography system has been used in
this case [28]. The ECT sensor was used for imaging water distribution with higher
permittivity relative to oil, whereas the γ-ray sensor was used for gas imaging based
on the density difference between oil and gas.
Generally, there are three different approaches for implementing multi-modal tomography in multi-phase flow systems:
1. Using two or more different sensing techniques: in this case, two different sensors acquiring different signals are used [34]. This method suffers from high
cost, complexity, long time required for sensing and reconstruction, interference
between sensors electronics, and the fact that signal being acquired from both
sensors needs to be coordinated for real-time applications. Examples of this
approach include the oil pipe line case mentioned above.
2. Using a reconstruction technique capable of differentiating between different
phases: an example of this approach is the neural network multi criterion reconstruction technique (NN-MOIRT) [89]. In this case, images of three phase
flow are obtained by using two sigmoid functions in the reconstruction process.
However, using this approach limits the phase differentiation process from the
signal acquired from the sensor, which depends on one physical property only.
3. Using an inherently multi-modal system: examples of this approach are electric
impedance tomography (EIT) for simultaneous conductivity and permittivity
imaging [69].
9
Electrical tomography systems in general belong to the inherently multi-modal category [98]. Regarding ECT, the system is treated as being a single modal system of
permittivity imaging based on capacitance measurements. However, this approach is
based on static analysis of the ECT sensor. ECT can be extended to a multi modal
systems through consideration of the time varying excitation signal (quasi-static).
1.3
Dissertation objectives
The tomography field is inherently interdisciplinary, and ECT in no exception.
Work in this dissertation span different areas of ECT research. However, the main
objectives are:
1. Develop a forward problem solver based on neural networks to over come the inaccuracy of linearization techniques, while maintaining a high speed comparable
to linearization techniques.
2. Integrate the forward solution into non-linear reconstruction techniques for better reconstruction results.
3. Implement 3D image reconstruction for ECT based on the developed NNMOIRT reconstruction technique
4. Redesign the ECT sensor for compatibility with 3D image reconstruction techniques
5. Extending ECT sensor applicability for simultaneous permittivity and conductivity imaging based on quasi-static analysis.
10
1.4
Dissertation outline
In Chapter 2, a review of reconstruction techniques for electrical capacitance tomography is provided. Reconstruction techniques are classified in this chapter into
single step and iterative. Emphasis on recent developments of optimization techniques
is provided through the neural network multi criterion iterative reconstruction technique (NN-MOIRT). A comparison of different techniques in terms of reconstruction
performance is also provided.
Forward problem solution is a critical step in an iterative reconstruction process.
In Chapter 3, a new forward solution based on feed forward neural networks is presented. The new technique combines the advantages of fast prediction and relatively
accurate performance. The new forward solver is compared to the commonly used
linear forward projection and results are provided. Moreover, the performance of the
forward solver in iterative reconstruction is discussed.
In Chapter 4, the feed forward neural network forward solver is integrated into
the NN-MOIRT reconstruction technique. Such a combination of forward and inverse solvers is referred to as nonlinear image reconstruction. The new combined
reconstruction technique is compared to a full linear and a semi linear reconstruction.
In traditional 3D electrical tomography imaging, a 3D image is obtained through
a combination of 2D cross sectional images. In Chapter 5, a new capacitance volume imaging technique for real time 3D imaging is developed. The new technique
is capable of obtaining a whole 3D image from integral measurements without 2D
to 3D interpolation. The development of the 3D reconstruction technique in this
chapter is accompanied by a 3D sensor design compatible with 3D reconstruction.
Comparison of 3D reconstruction using different 3D sensors is provided based on 3D
11
linear back projection, 3D iterative linear back projection, and volume tomography
reconstruction technique.
Electrical tomography is inherently multi-modal in general. Imaging of multiple
electrical properties has been carried out through intrusive techniques such as electrical impedance tomography, based on current injection. In Chapter 6, a new nonintrusive method for permittivity and conductivity imaging based on ECT sensors
is proposed. The new method has the advantage of using already developed reconstruction techniques. Reconstruction results of both permittivity and conductivity
are provided through iterative linear back projection.
Conclusion of this dissertation is provided in Chapter 7, along with Suggestions
for future work.
12
CHAPTER 2
REVIEW OF RECONSTRUCTION TECHNIQUES
Electrical tomography generally belongs to the soft field category of sensors. In
soft field tomography, the interrogating field is dependent on the electrical property
distribution in the imaging domain [37]. The interrogating field and the electrical
property are represented in this through a partial differential equation with nonlinear coefficients. The effect of soft field sensors on image reconstruction appears in
the ill-posed inverse problem. A problem is defined to be ill-posed if [6]:
1. A solution exists for the problem.
2. The solution is unique.
3. The dependence of the solution on the electrical property is continuous.
The uniqueness of electrical tomography inverse problems in the isotropic case has
been presented under different assumptions [78] [65] [36]. Never the less, it is the third
condition of a well-posed problem that is usually violated when dealing with electrical
tomography inverse problems. The condition of continuity refers to the robustness
and stability of the solution. The violation of this condition in an electrical sensor;
e.g. ECT sensor; is attributed to two main factors: (1) the dramatical difference of
13
sensor response to different locations of electrical property distributions; the soft field
problem; and (2) the low level of sensed signal and the consequently low signal to
noise ration [104]. The combination of relatively high noise level and ill-posed inverse
problem contributes to the complication of reconstruction process.
Since no general method for solving the reconstruction problem exist, different
techniques are developed by researchers which can be classified mainly into direct
(single step ) techniques; in which the image is obtained from measured capacitance in
one mathematical step; and iterative techniques in which a set of objective functions
are maximized/minimized iteratively.
In most reconstruction techniques; whether single step or iterative; the sensitivity
matrix is used for forward or backward projection between the boundary measurements and the reconstructed image. The sensitivity matrix is a linearization of the
non-linear field to physical property distribution [35] [99].
2.1
Direct techniques
Direct techniques are based mainly on the sensitivity matrix model [35]. Sensitivity matrix is built by measuring the capacitance response for permittivity perturbations, pixels, over the spatial domain in the sensing area. The basic form of
sensitivity model assumes the sensitivity does not change as a function of permittivity distribution. Each element of the matrix is obtained from the response of a pair
of electrodes to a perturbation of high electrical permittivity in the imaging domain.
The elements are then normalized based on the response of the sensor filled with high
and low permittivity materials according to the following equation:
Sij =
Cij − Cijl
Cijh − Cijl
14
(2.1)
where Sij is the sensitivity matrix element of the jth capacitance pair and ith pixel,
Cij is the measured capacitance, Cijl and Cijh is the capacitance when the sensor is
filled with low and high permittivity material respectively.
2.1.1
Linear back projection
conditions (landweber paper) Linear back projection (LBP) was the first to be
used for image reconstruction in ECT. In LBP, the capacitance is assumed to be
formed from a superposition collection of different high permittivity pixels, and it
can be written as a function of sensitivity matrix as in equation (2.2), which is a
linear mapping from permittivity distribution to capacitance measurement through
sensitivity matrix.
C = SG
(2.2)
where C is the capacitance vector, S is the sensitivity matrix and G is the image
vector.
In order to obtain the permittivity distribution from the measured capacitance
vector, equation (2.2) should be solved. However, the sensitivity matrix does not is
generaly a rectangular matrix, and it does not have a direct inverse. Moreover, it is
ill-posed, ill-conditioned, and its elements are not completely independent [101]. In
LBP, the non-linear interaction between pixels, which is a function of permittivity
distribution as well as permittivity value is ignored. As a result, LBP tends to perform
better with lower permittivity difference. And the image vector is obtained through a
linear mapping from the capacitance vector using the transpose of sensitivity matrix
as in equation (2.3). Results from LBP are generally blurred as the image is formed
15
from overlaping projection proviging a bias in the image background.
G = ST C
2.1.2
(2.3)
Singular value decomposition
Due to the ill-possed nature of inverse problem in ECT, the direct inverse of
the (generally non-square) sensitivity matrix can not be calculated, and a natural
substitute is used, known as pseudoinverse. The pseudoinverse method provides the
least norm solution. Considering equation (2.2), an image can be obtained through
a pseudo inverse operation performed as follows:
S = U ΣV T
(2.4)
where columns of U are eigenvectors of SS T , and columns of V are eigenvectors
of S T S, and Σ is diagonal matrix of the same size as S. The square roots of the
nonzero eigenvalues of both SS T and S T S form the diagonal of matrix Σ. Thus the
pseudoinverse of matrix S becomes:
S + = V Σ−1 U T
(2.5)
And reconstruction equation becomes:
G = S +C
2.1.3
(2.6)
Tikhonov regularization
Regularization methods have been developed to solve ill-posed problems. Tikhonov
regularization is used to solve the ill-posed inverse problem in ECT [80]. The inverse
is produced by adding a regularization parameter. The mathematical details are summarized in Equations (2.7-2.9) . The quality of reconstructed images depend strongly
16
on the value of regularization parameter. Equation (2.3) written in its exact form
becomes:
S T SG = S T C
(2.7)
G = (S T S)−1 S T C
(2.8)
In the equation above, S T S is not an invertible matrix, a regularization parameter
is introduced for matrix to be inverted.
G = (S T S + µI)−1 S T C
(2.9)
Where µ is a regularization parameter, and I is the identity matrix. The quality of
the solution depends highly on the value of the regularization parameter.
2.2
Iterative reconstruction techniques
The non-linearity of ECT reconstruction imposes a restriction on the accuracy of
reconstructed images using single step techniques. An improvement is obtained by
solving the inverse problem iteratively. It is almost impossible to obtain satisfactory
results from non-iterative reconstruction algorithms. As a result, iterative techniques
are prevalently used in ECT [105] [20]. Iterative algorithms are based on solving for
the capacitance values from the current permittivity distribution, and producing an
updated image based on the difference between calculated and measured capacitance.
Iterative reconstruction techniques may be classified into two groups [89]: (1)
Algebraic reconstruction techniques (ART) [66] , and (2) optimization techniques
[38]. Generally, ART techniques are based on solving equation (2.10) iteratively to
estimate the image vector.
Gk+1 = Gk + β k S T (C m − y(Gk ))
17
(2.10)
Where k is iteration number, C m is the measured capacitance, β k is a relaxation
factor of iteration k, and y(Gk ) is the forward problem solution of image vector Gk .
ART reconstruction techniques differ in applying the relaxation factor β k , the update
method of the image vector, and in solving the forward problem. A commonly used
ART technique is the iterative linear back projection (ILBP). ILBP is an iterative
generalization of the well known LBP reconstruction technique. The forward problem
in ILBP is solved using forward projection as in equation (2.2). In case of all capacitance data are used to update the image vector at once, the algorithm is referred to
as simultaneous image reconstruction technique (SIRT) [75].
On the other hand, optimization techniques are based on minimizing or maximizing a set of objective functions. Optimization techniques tend to perform better than
ART techniques for two main reason: (1) ECT reconstruction is an ill-posed problem, there is no unique solution for the reconstructed image. Optimization techniques
provides the most likelihood image with respect to the objective functions used. (2)
ART techniques minimize MSE, which does not have information about the nature
of reconstructed image.
Different objective functions have been used in literature. For example, neural
network multi criterion optimization technique (NN-MOIRT) minimizes three objective functions using hopfield neural networks [89]. NN-MOIRT uses an energy
function composed of the objective functions: (1) mean square error (2) entropy
function (3) smoothness and peakedness function. The technique is capable of imaging multi phase flows. A mathematical description of the NN-MOIRT is provided in
the following section.
18
2.3
NN-MOIRT
The NN-MOIRT is an optimization reconstruction techniques based on Hopfield
neural networks [30] [31]. The optimization techniques minimizes three objective
function:
(1) Weighted square error function:
1
f1 (G) = ω1 ky(G) − C m k2
2
(2.11)
where y(G) is the forward problem solution of image G, and C m is the measured
capacitance vector.
(2) Entropy function
f2 (G) = ω2
N
X
Gj ln(Gj )
(2.12)
j=1
(3) Smoothness and peakedness function:
1
f3 (G) = ω3 (GT XG + GT G)
2
(2.13)
where 0 ≤ ω1 ≤ 1, 0 ≤ ω2 ≤ 1, 0 ≤ ω3 ≤ 1 are weighting constants, and X is a high
pass filter matrix. The energy function is a combination of all objective functions:
E(G) =
3
X
fj (G)
(2.14)
j=1
The optimization problem is solved using hopfield neural network summarized as
follows:
(1) The output variable is obtained through a linear mapping of the internal state of
the neuron.
Gj = vj = fΣ (uj )
19
(2.15)
where uj is the internal state of the neuron, and fΣ is the activation function given
by:


 0
if
if
if
fΣ (uj ) =  βuj + ξ

1
uj ≤ −ξ/β
− ξ/β < uj < 1 − ξ/β
1 − ξ/β ≤ uj
(2.16)
where β and ξ are constants determining the slope of the linear function.
(2) The energy function in hopfield network is formulated as:
E(G) =
3
X
wi fi (G) +
i=1
N
X
ψ(zj ) +
j=1
M Z
X
Gj
0
l=1
fΣ−1 dG
(2.17)
where N is the number of independent capacitance measurement, M is the number of
pixels in the image vector, ψ is an increasing function for zi > 0 and zero otherwise.
The first term is the objective functions defined in equation (2.14), the second term
is a constraint that forces z(t) ≤ 0 where zi is defined in equation (2.18), and the last
term motivates the output to be in the range 0 ≤ Gj ≤ 1.
z(t) =
M
X
j=1
(yj (G(t)) − Cjm )
(2.18)
(3) The objective function in equation (2.17) is integrated into the partial differential
equation of hopfield network, the update equation is found to be:
u0 (t) = −
1
∇G = −[ω1 (1 + lnG(t)) + ω2 y 0 (G(t))z(t) +
C0
u(t)
ω3 (XG(t) + G(t)) + y 0 (G(t))δ(z(t))] −
τ
(2.19)
where:
∂ψ
= δ(z(t)) =
∂(z(t))
(
0
αz(t)
if
if
z(t) < 0
z(t) > 0
(2.20)
Using equation (2.16) for the transfer function, the update equation becomes:
0
Gj (t + ∆t) = Gj (t) + βuj ∆t
20
(2.21)
A comparison of different reconstruction algorithms is depicted in Figure 2.1. LBP,
ILBP, SIRT, and NN-MOIRT reconstruction techniques are used to solve the inverse
problem, the measured capacitance data are obtained using single plan 12 electrode
sensor. A white gaussian noise was added to the capacitance data to test the reliability
of each reconstruction technique. It is clear from the results in Figure 2.1 that NNMOIRT reconstruction technique is superior in terms of quality of reconstruction
images as well as immunity to noisy measurements.
NN-MOIRT reconstruction technique is believed to be the first to introduce 3-D
image reconstruction directly from the measured capacitance data. Further details
on the 3D image reconstruction are provided in Chapter 5.
21
Model
Model distributions
distributions
Figure 2.1: Reconstruction results of different shapes LBP, ILBP, SIRT and NNMOIRT reconstruction techniques.
22
CHAPTER 3
NONLINEAR FORWARD PROBLEM SOLUTION FOR
ELECTRICAL CAPACITANCE TOMOGRAPHY USING
FEED FORWARD NEURAL NETWORK
In general, tomography can be classified into hard field and soft field tomography.
In hard field tomography (X-ray CT), the interrogating field is distributed independently from property distribution in direct path from the transmitting to the receiving
sensor. In soft field tomography, the interrogating field is a highly non-linear function of the (physical) constitutive property (e.g. electric permittivity) distribution
of interest. Both electrical impedance tomography (EIT) and electrical capacitance
tomography (ECT) belong to the soft field category. Although systems based on hard
field tomography are easier to deal with in terms of image reconstruction, ECT is gaining increased acceptance as a robust tool for industry and laboratory applications due
to its fast data acquisition speed, low construction cost, safety, and applicability for
a wide range of vessel sizes [108].
Generally, two types of reconstruction techniques can be used for ECT image
reconstruction: non-iterative and iterative algorithms. Because of the non-linear
relationship between the measured capacitance and the permittivity distribution in
ECT, non-iterative reconstruction algorithms usually do not give satisfactory results.
23
As a result, iterative techniques are prevalently used in ECT [20]. Iterative algorithms
for ECT are based on obtaining an estimate of the unknown permittivity distribution
from the capacitance data (inverse problem), and calculating the capacitance based
on the estimated permittivity distribution to update the image in the next scheme
(forward problem). This process is repeated iteratively until the capacitance error is
decreased to a satisfactory value [20].
Although the forward problem solution plays a crucial role in the quality of the
reconstructed image as well as on the speed of reconstruction process, most work on
ECT has been focused on improving the inverse problem solution, while relatively
little attention has been given to the forward problem. The forward problem is dealt
with generally three approaches: (1) linearization techniques [35]; (2) brute-force
numerical methods such as finite element method [4] [82], boundary value method
[54] and finite difference method [20] and; (3) (pseudo) analytical methods [58], [47],
[2]. Despite the fact that analytical methods can provide accurate and relatively fast
solutions, they are limited to very simple geometries with symmetric permittivity
distributions, and are not applicable to industrial tomography systems with complex
dynamic flows. On the other hand, numerical methods can provide fairly accurate
solutions for arbitrary property distributions. However, this accuracy occurs at the
expense of excessive computational time and resources. In terms of industrial applications, speed, accuracy, and simplicity are key factors in defining the overall quality
of the method used. In this regard, linearization methods provide relatively fast and
simple solution. However, they suffer in terms of accuracy due to the non-linear
nature of the electrical tomography.
24
In this work, we introduce a new approach for solving the non-linear forward
problem in soft tomography systems based on feed forward neural networks (NN) with
regularization. The measured capacitance data for the network training are organized
and filtered in such a way to better reflect the geometry of the sensor and combat
the ill-conditioning problem of ECT. The NN forward problem solution method is
then implemented in a image reconstruction technique based on iterative linear back
projection (ILBP). A comparison with regular ILBP technique is performed. The use
of ILBP reconstruction technique is chosen due to its widespread use in ECT.
3.1
Problem Statement
3.1.1
ECT system
An ECT system is generally composed of three different units: 1) the capacitance
sensor 2) the data acquisition and processing hardware and 3) the computer system
for image reconstruction process, control, and display. The sensor, as depicted in
Figure 3.1, consists of a number n of electrodes placed around the region of interest
providing
n(n−1)
2
number of independent capacitance measurement used for image
reconstruction. The electric field distribution inside the region of interest is a function
of permittivity distribution according to Poisson equation:
∇ · (ε(x, y)∇φ(x, y)) = −ρ(x, y),
(3.1)
where (x, y) is permittivity distribution, φ(x, y) is electrical potential distribution
and ρ(x, y) is charge distribution. The Poisson equation is a linear partial differential
equation in terms of φ(x, y). The nonlinearity of ECT results from the nonlinear dependence between the field distribution φ(x, y) (and hence the measured capacitance
data) and the (unknown) permittivity distribution (solution) (x, y). The mutual
25
Figure 3.1: ECT sensor with seven electrodes
capacitance between two pair of electrodes, source and detector, is obtained through
equation:
Cij =
Qj
,
4Vij
(3.2)
where Cij represents the mutual capacitance between electrodes i and j, 4Vij the
potential difference, and Qj is the charge on the sensing electrode which is found by
applying Gauss law.
Qj = −
I
Γj
(x, y)∇φ(x, y) · n̂dl,
(3.3)
where Γj is a closed path enclosing the detecting electrode and n̂ is a unit vector
normal to Γj .
3.1.2
Forward problem in image reconstruction process
The forward problem is the process of determining the output response of an ECT
system given the permittivity distribution in the region of interest. The importance of
fast forward solutions is manifested when iterative algorithms for image reconstruction
are used. In iterative algorithms, the image obtained from reconstruction is updated
by minimizing the error between the measured capacitance data and the forward
26
solution for a predicted permittivity distribution. This process is repeated iteratively
until a pre-defined criteria is met, so multiple forward solutions become necessary.
Obtaining explicit forward solutions from equations (3.1)-(3.3) via brute-force numerical techniques is a time-consuming task, and hence alternative techniques must
be explored. The most common method used to solve the forward problem in image reconstruction process is linear forward projection (LFP) [35]. LFP method is
based on the sensitivity model. The sensitivity model is an implementation of the
superposition theorem, in which the forward solution is obtained as a linear sum of
capacitance measurements from perturbations in permittivity distribution. Based on
this model, the mutual capacitance as a function of permittivity distribution can be
written as:
C = SG
(3.4)
where, C is the capacitance vector, S is the sensitivity matrix, and G is an image
vector representing the permittivity distribution. This technique suffers in a smoothing effect and lack of accuracy due to the linearization of an inherently non-linear
problem of electrical tomography.
The overall performance of the proposed forward problem solution is better appreciated when fully integrated in image reconstruction algorithms. In this regard,
both NN and LFP were integrated in Landweber iterative reconstruction technique,
which is a form of iterative linear back projection (ILBP). ILBP is an iterative generalization of the commonly used LBP reconstruction technique, in which the image
vector is updated iteratively to minimize the error between measured and calculated
capacitance data according to:
Gk+1 = S T C k + β(S T (Cm − y(Gk )))
27
(3.5)
where the calculated capacitance is obtained from the reconstructed image using a
forward problem solver. In the above, G is the image vector, k is the iteration number,
S is the sensitivity matrix, β is a factor controlling reconstruction convergence, and
y(Gk ) is the forward problem solution of image vector Gk . A constraint is applied to
equation (3.5) to the benefits of the so-called projected Landweber iteration [101].
3.2
3.2.1
NN techniques for the forward ECT problem
Architecture
Artificial NN are composed of simple processing elements, neurons, organized
in different layers and communicating with each other [77]. Each interconnection
between two neurons is associated with a weight that specifies the strength of the
connection. NN play an important role in various applications and posses the property of being a universal approximator, i.e, for any function of arbitrary degree, there
is a feed forward neural network able to approximate it [32]. NN are considered
an attractive choice for modelling nonlinear and complex problems because of their
robustness, ability to withstand noise, their universal approximation property, and
ability to predict and extrapolate information hidden in the training data, in a process
known as neural network learning [12]. An important aspect in the image reconstruction process is the speed in the prediction once trained without need for linearization
assumptions.
A multi-layer feed-forward (MLFF) NN consists of a number of neurons organized
in multiple layers as depicted in Figure 3.2. Each neuron is connected with a weight to
all neurons in the adjacent layers. The value of each weight represents the relevance
of the particular connection in the network structure. Each neuron output is mapped
28
Figure 3.2: Neural network with multiple layers in matrix form
to a transfer function. A sigmoid function is usually employed to map unbounded
data to the bounded range of the transfer function [64]. Different sigmoid functions
can be used with different ranges. For example, a very popular function is the logistic
sigmoid function with a range of [0,1] given by:
f (x) =
1
1 + exp(−αx)
(3.6)
where f is the logistic sigmoidal, x is its input, and α is a slope parameter. It can be
shown that a neural network with continuous transfer function in the output layer is
a universal approximator [11]. The input to each neuron in a certain layer is obtained
according to:
ξil = bli +
nl
X
Wijl Ojl−1
(3.7)
j=1
where i, j are the neuron numbers and l is the layer number under consideration, bli
is a bias term added to the input, Wijl is the weight connecting output of neuron j
of previous layer to neuron i, Ojl−1 is the output of neuron j in layer l − 1, and nl
is the number of neurons in layer l. The output of neuron i is obtained by applying
the transfer function in equation (3.6) to the neuron input in equation (3.7) which
29
results in:
Oil = f (ξil ) =
1+
1
P l
+ nj=1
Wijl Ojl−1 ))
exp(−α(bli
(3.8)
In a MLFF network, the output of a neuron is a function of all neurons in preceding
layers. An input-output mapping takes the form of nested nonlinear functions as:
Oi1 = fΣ1 = f (bli +
Oi2 = fΣ2 = f (b2i +
n1
X
j=1
n
2
X
Wijl xj )
Wij2 Oj1 )
j=1
Oil
= fΣl =
f (bli
+
nl
X
..
.
Wijl Ojl−1 )
j=1
OiL = fΣL = f (bLi +
nL
X
..
.
WijL OjL−1 )
(3.9)
j=1
where xj is the input to neuron j in the input layer, fΣl is the transfer function of
layer l, and L is the total number of layers in the network.
3.2.2
Training
Training of neural networks in general is based on error-correction methods, which
compare the output of the network to the desired response for error estimation, and
updates the weight vector until the error is minimized. For error-correction methods
to be applied in MLFF networks, the desired output of each layer has to be predetermined. The only data available to train a MLFF are the input and desired output of
the network as a whole. There is no explicit method available to determine the error
in the hidden neurons layers.
One of commonly used techniques to train a MLFF NN is the back propagation
learning algorithm [77]. In the back propagation technique, the error of the output
30
layer is propagated backwards to the hidden layers, and their weights are updated
accordingly. The training process starts by defining an objective function for the
weight update. The mean square error objective function J commonly used has the
form:
J=
nL
1 X
(di − OiL )2
2nL i=1
(3.10)
where nL is the number of neurons in the output layer and di is the desired output of
neuron i. Each weight is updated using the gradient descent method, which uses the
gradient of the objective function in equation (3.10) to determine the weight update
according to the following equation:
Wijl (n + 1) = Wijl (n) − η
∂J
∂Wijl (n)
(3.11)
where n is the iteration number and η is the learning rate. In MLFF, the partial
derivative of the objective function with respect to weights in hidden layers, and with
a transfer function given by equation (3.6), is given by [64]:
∂J
∂Wijl
nL
L
∂J ∂OiL
1 X
L ∂Oj
=
=−
(dj − Oj )
∂OiL ∂Wijl
nL j=1
∂Wijl
(3.12)
where,
∂yj
∂Wijl
0
0
0
= fΣL
· fΣL−1
. . . · fΣ1
· xj
0
fΣL
(u) =
∂fΣL (u)
= αfΣL (1 − fΣL ).
∂u
The gradient technique for updating the weights suffers the so-called saturation
problem. Saturation occurs when a neuron input is very large in magnitude with
respect to 1/α, where α is defined in equation (3.6). In such a case, the input is
mapped by the sigmoid function to the flat range of that function. The convergence
of the weight update process is then influenced by the near zero derivative of the
31
sigmoid function, and the error will not affect the update efficiently. The saturation
problem becomes severe when dealing with ill-conditioned problems as the variation
of weights is generally very large.
In this work, a resilient propagation RPROP updating algorithm is used for
weight update [68]. RPROP performs a direct adaption of the weight step based
on the sign of the gradient rather than its value. This method has the advantage of
avoiding the saturation problem described above. In RPROP the weights are updated
according to:
∆ij (n) =

 λ(+) ∆ij (n − 1)
 λ(−) ∆ij (n − 1)
if
if
∂J
∂J
l (n−1) ∂W l (n)
∂Wij
ij
∂J
∂J
l (n−1) ∂W l (n)
∂Wij
ij
≥0
<0
0 < λ(−) < 1, 1 < λ(+)
∆Wijl (n)

 −∆ij (n)
=
+∆ij (n)
if
if
∂J
l (n)
∂Wij
∂J
l (n)
∂Wij
(3.13)
(3.14)
≥0
<0
Wijl (n + 1) = Wijl (n) + ∆Wijl (n)
(3.15)
(3.16)
where ∆ij is the update value, ∆Wijl (n) is the weight step update between neurons
i and j in layers l and l − 1 respectively, n is the iteration number.
If the gradient changes sign, the previous update is canceled and a new update is
used with smaller step. The weight update step is increased as long as the derivative
maintains the same sign. Note that the weight update step does not depend on the
magnitude of the derivative, rather on its sign.
3.2.3
Generalization
In Section 3.2.2 the training of MLFF was discussed in terms of minimizing the
mean square error between the output of the data and the desired response of the
32
network. The performance of the network and training process is judged by the prediction ability of the network over arbitrary data. Networks have good generalization
if they give good prediction over general data.
The training data selection and the network architecture play an important role in
MLFF generalization. A representative training data set with input-output mappings
of the relationship to be approximated is a basic condition for good generalization.
On the other hand, there is no straightforward procedure to determine the MLFF NN
structure based on the input and output data. Most NN topologies are constructed
experimentally. The nature of the problem is also important in this regard. For
example, ill-conditioned problems are usually more difficult to generalize than wellconditioned problems. The ill-conditioning of the ECT forward problem results from
large variance in the electric field magnitude inside the sensing domain and also from
the soft field nature of ECT [10]. Solving the ill-conditioned forward problem in ECT
using NN requires a modification of the training algorithm as well as the objective
function since small changes on input parameters may cause significant changes in
the output. This is particularly important when noise is present in the measured
capacitances used during training, which is the case in practice.
To improve generalization, a modification is introduced to the objective function
in equation (3.10) [7] by adding a regularization term according to :
J =γ
nl
nL
L nX
l−1 X
X
1
1 X
(Wijl )2
(di − OiL )2 + (1 − γ) PL
2nL i=1
n
i=1 i l=1 j=1 i=1
(3.17)
where γ is a regularization parameter. The regularization term added in equation
(3.17) suppresses weights with high values and enforces smaller weights and biases in
the network, causing the response of the network to be smoother. The regularization
parameter γ needs to be optimized and set carefully. A large value of γ makes
33
the network more vulnerable to over-fitting, whereas small values of γ prevent the
network from approximating well the function represented by the training data. In
this work, γ is chosen by trial and error. The overall sensitivity of the network
regularization parameter depends on the particular network structure and the degree
of ill-conditioning of the problem.
3.3
3.3.1
Experiments and sensor data pre-processing
Data collection
An ECT sensor of 12 electrodes in a cylindrical arrangement as illustrated Figure
3.1 is used to collect the capacitance data set used for training. The data is collected
off-line based on experimental capacitance measurements of dielectric square rods of
different dimensions (.25, .5, .75, 1 and 2 inches) placed in different representative
locations (500 for each rod) within the vessel. The acquisition hardware is from
Process Tomography LTD. and operates at a rate of 100 frames per second. The rod
in each case was placed at arbitrary locations within the sensing domain to produce
different permittivity distributions. A total of 66 normalized mutual capacitances
were stored for each rod in one location. Each capacitance vector was then filtered
and processed as described in 3.3.2. The filtered capacitance vector is then used for
network training. A feed forward NN comprising of an input layer with 400 neurons
(for a 20 × 20 image resolution), 2 hidden layers each with 40 neurons, and one
output layer of 66 neurons (network output) was constructed. Training the NN is
based on prior experimental data. The training time required for the network used in
this work was about 5 hours on a personal computer. Since there is no general and
systematic rule for constructing an optimal NN to fit a given application, the network
34
architecture here was chosen based on trial and error. For consistency, the trained
NN should be applied in conjunction with the same sensor hardware used to collect
the training data. A set of data not used in training is used to test the ability of the
NN in solving the forward problem. The transfer function used is a logistic sigmoid
function for all hidden layers, and a linear function for the output layer. The network
was trained using RPROP algorithm to minimize the objective function composed by
mean square error and regularization term as in equation (3.17).
3.3.2
Capacitance data rearranging and filtering
Both forward and inverse problems in ECT deal with a mapping between the
normalized capacitance and the image vector. The variance of the capacitance vector
largely depends on the location of high permittivity pixels in the image vector, whereas
the absolute value of the capacitance is more closely related to the permittivity value
of the pixels in the image vector. The response of a single capacitance measurement
to a perturbation in the image vector depends on both the location of the sensor
and detector electrodes, as well as the separation between them with respect to the
location of the perturbed pixel. For a cylindrical sensor as shown in Figure 3.1,
the response of measured capacitance for a single pair of electrodes to pixels in the
center of the domain is directly proportional to the separation between the sending
and receiving electrodes in the pair. The absolute measured capacitance for a pair
of plates changes as a function of distance between the plates. As a result, the
sensitivity of the measured capacitance to noise increases as the distance between the
plates increases.
35
Training the NN based on minimizing MSE does not guarantee convergence to
the right solution, specially when the measured capacitance is sensitive to noise. For
this reason, optimization techniques (OT) used in image reconstruction are more successful in reconstruction than algebraic reconstruction techniques (ART) [89]. ART
techniques are mainly based on minimizing an error function, whereas OT are based
on minimizing various sets of objective functions, which includes some form of error
function in most cases.
In this work, the measured capacitance vector is reorganized according to the distance between the pair of electrodes used to acquire each measurement. For example,
capacitance measurements obtained from plates separated by 2 plates from either
sides are grouped in a sub-vector of the total capacitance vector. Each sub-vector is
then processed using an averaging filter according to the following equation.
g =
C(k)
N
1 X
Cb m (i − k)F (i)
N i=1
(3.18)
g is the filtered capacitance sub-vector, N is the length of the averaging
where C(k)
filter, Cb m is the normalized measured capacitance sub-vector, and F is an averaging
filter. The method is summarized as follows:
1. The measured capacitance data is first normalized according to [44]:
Cb m =
Cm − Ce
Cf − Ce
(3.19)
where C e , C f are the capacitance vectors when the sensor is entirely filled with
low and high permittivity respectively, and C m is the measured capacitance
data.
36
2. The measured capacitance data is reordered in sub-vectors according to the
distance between the sensing and receiving electrode used to acquire each capacitance measurement.
3. Sub-vectors with largest distance between sending and receiving electrodes
(group 1), as well as sub-vectors nearest to the ones with the largest distance
sub-vector (group 2), are pre-processed independently using the averaging filter
in equation (3.18).
4. The resulting filtered sub-vectors from step (3), together with all the remaining
capacitance sub-vectors, are used in training the NN, where the permittivity
distributions and the capacitance vectors are the inputs and outputs of the
network respectively.
The response of each capacitance subvector to an image is dependet on the location of the image. Objects located in the center of the domain are mainly detected
by group 1 and group 2 subvectors. The capacitances in subvectors 1 and 2 have two
main properties:
(a) Lower frequency components than other subvectors. Thus, the use of an averaging
(low-pass) filter is not expected to distort the information in the subvector signal.
(b) Lower signal levels than other subvectors, which makes them more vulnerable for
noise.
Since the noise in the raw measurement data is close to a (uncorrelated) white
Gaussian noise, the low-pass filtering is able to reduce the noise component.
37
3.4
Results and discussion
In Figure 3.3, different permittivity distributions with different rod sizes are used
to compare results from both NN and linear forward projection (LFP) in terms of
mean square error (MSE) between measured C m and predicted C p capacitance data
defined as:
M SE =
N
1 X
[C m (i) − C p (i)]2
N i=1
(3.20)
The rod in each case was placed at arbitrary locations within the sensing domain to
produce 100 different permittivity distributions. The NN solution performs better
than LFP in terms of MSE. However, these results do not necessarily reflect similarities between measured and predicted permittivity distribution data, rather they
deal with the error as the absolute difference squared between measured and predicted capacitance data. The same set of permittivity distributions used to produce
the results in Figure 3.3 were used for comparison through mapping with respect to
measured data, and prediction comparison is depicted in Figure 3.4. In Figure 3.4,
the capacitance predictions from NN and LFP are plotted against the capacitance
data obtained from measurements. The circles in the Figure represent capacitance
prediction from NN solver, whereas the dots are represent the results using a LFP
solver. The straight line correspond to a perfect capacitance prediction (i.e., predicted
values equal measured ones). It is clear from Figure 3.4 that the results using NN
technique are better correlated with the measured capacitance, as compared to the
LFP technique.
The nature of the particular permittivity distribution is an important variable for
defining the performance of the forward problem solver. Images in Figure 3.4, for
38
which NN and LFP yield similar performance, are related to a permittivity distribution where the high permittivity values (rod) is in the region of more ”linear” in the
field distribution, i.e. in the center of the domain (where the field is weaker). This
result is consistent with the nature of LFP technique, whose performance is degraded
when the degree of non-linearity of the problem is increased. Both NN and LFP predictions for a 2 inch annular flow are plotted in Figure 3.5 and compared to measured
capacitance of the same permittivity distribution used in prediction. Data predicted
using NN is more correlated to the measured capacitance than LFP, however, the
absolute value of predicted capacitance is scaled down when compared to measured
capacitance. This effect is a result of the limited range of data used in training.
The convergence rate for both methods was compared and tested on a 2 inch
annular flow. In this case, the experiment was done by fixing a 2 inch diameter
hollow cylinder in the center of the sensor. The region between the inner sensor
wall and the outer cylinder boundary was filled with solid particles (hence, a sharp
transition between gas and solid phases is present). The measured capacitance data
is obtained from a 12 electrode sensor surrounding a 4 inch diameter domain. The
reconstructed image size has 20 × 20 pixels, and the flow is composed of high (solid)
and low (gas) relative permittivities of 3.8 and 1 respectively. The results in Figure
3.6 show (qualitatively) that NN forward solution provides a better convergence than
LFP when integrated in the Landweber reconstruction algorithm. The convergence
rate was compared quantitatively for the flow in Figure 3.6. The MSE depicted in
Figure 3.7 is calculated for each iteration using Equation (3.20).
In Figure 3.8, both NN and LFP were integrated in Landweber iterative image
reconstruction and tested with different permittivity distributions for different flow
39
Figure 3.3: Mean square error for different permittivity distributions obtained from
different rod sizes, where r is the rod size
40
Linear mapping comparison of NN and LFP
1.2
NN Prediction
LFP Prediction
Exact Solution
Normalized predicted Capacitance
1
0.8
0.6
0.4
0.2
0
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized measured capacitance
Figure 3.4: Linear mapping of different predicted capacitance vectors with respect to
measured capacitance for different permittivity distributions
41
NN & LFP prediction comparision
1.2
Exact
NN
LFP
Normalized capacitance value
1
0.8
0.6
0.4
0.2
0
−0.2
0
10
20
30
40
50
60
70
Capacitance pair number
Figure 3.5: Predicted capacitance vector using both NN and LFP compared to measured capacitance of the same permittivity distribution
42
Permittivity Distribution Iteration Landweber
NN−Landweber
1
2
3
4
Figure 3.6: Comparison with Landweber reconstruction of Gas-Solid flow in terms of
convergence rate with both LFP and NN integrated as forward problem solvers
43
Convergence rate
−1
10
MSE
NN
LFP
−2
10
−3
10
0
2
4
6
8
10
12
14
16
Number of iterations
Figure 3.7: Convergence rate of NN and LFP integrated in Landweber iterative reconstruction technique for the centered 1 inch tube in figure 3.6
44
Permittivity Distribution
Landweber
NN−Landweber
Figure 3.8: Reconstruction results using Landweber iterative technique for different
permittivity distributions with both LFP and NN integrated as forward problem
solvers
45
regimes (1 inch tube, two 1 inch tubes, half flow, and annular flow, respectively),
under the same conditions described above. Again, NN performs better in terms of
quality and accuracy of the reconstructed image than LFP. In these examples, the NN
solution has typically required only 10 milli-seconds for reconstruction per iteration,
making it suitable for real-time image reconstruction.
3.5
Summary and Conclusion
In this work, a new technique for solving the non-linear forward problem in soft
field tomography has been introduced. The technique is based on multi-layer perceptron feed-forward neural networks with regularization for reordered and filtered
normalized capacitance data. Comparison with commonly used LFP forward problem
solution showed superiority of the proposed technique in terms of accuracy, quality of
reconstruction results, as well as convergence rate when integrated in Landweber reconstruction technique. In addition, the developed technique overcomes the problem
of excessive time and computer resources necessary when using brute-force numerical techniques for the forward problem. The described technique is fast and easy
to implement in any iterative reconstruction algorithm. The main limitations of the
technique are the in terms of training time and prior information required. Sufficient
training data has to be collected, and data has to be representative of the problem
for successful prediction. In addition, the training process has to be redone if the
hardware sensor design is changed.
46
CHAPTER 4
NON-LINEAR IMAGE RECONSTRUCTION
TECHNIQUE FOR ECT USING COMBINED NEURAL
NETWORK APPROACH
Electrical capacitance tomography (ECT) is gaining increased attention as a powerful imaging tool in industrial applications due to its non-invasive and non-intrusive
nature. Tomography systems involve the reconstruction of a physical property distribution inside a domain of interest from a set of integral measurements acquired by
installing a number of sensors around the domain of interest and measuring signals
which depend on physical properties in the imaging region. ECT has several advantages as fast data acquisition speed, low construction cost, safety, and applicability
for a wide range of vessel sizes [89]. However, the use of ECT is challenged by its
’soft field’ nature, in which the sensing field is a highly non-linear function of the
(physical) constitutive property (the electric permittivity) of interest [50]. The nonlinearity problem due to the ’soft field’ effect causes non-trivial solution to the image
reconstruction problem, in which no analytical solution is available.
Image reconstruction is commonly implemented using either non-iterative or iterative algorithms. Because of the non-linear relation between the sensing field and
the permittivity distribution, iterative techniques are more prevalent in ECT [61].
47
Iterative techniques are based upon minimizing the error between measured and calculated capacitance data sets, and by updating the reconstructed image accordingly.
In most ECT iterative reconstruction techniques, the image is updated based on linearization of the relation between permittivity distribution and capacitance values.
This linearization process is usually referred to as sensitivity model. In the sensitivity
model, the domain of interest is divided into small pixels, and the capacitance data
is obtained as a linear sum of different perturbations composing the overall permittivity distribution. Based on the sensitivity model, forward and inverse solutions can
be obtained through a linear forward projection (LFP) and linear backward projection (LBP), respectively, of the image vector and the capacitance measurement onto
the sensitivity matrix [100]. The appeal of the sensitivity model for solving both
forward and inverse problems stems from its convenient implementation and fast solution speed. However, a sensitivity model provides poor accuracy for the non-linear
problem, with the error increasing with the degree of permittivity contrast between
the phases being imaged. To overcome such limitations, a forward solver based on
a multilayer feed-forward neural network (MLFF-NN) has been recently introduced
for ECT [49]. This new forward solution technique is suitable to forward problems
in highly non-linear ECT problems by providing better accuracy than LFP, while
maintaining fast speed.
In this work, we introduce a new iterative approach for solving the combined
non-linear forward/inverse problem in soft-field ECT. The new method uses a nonlinear update of the image vector for the reconstruction process by means of a analog Hopfield network based on a neural-network multi-criteria optimization image
reconstruction technique (HN-MOIRT), while employing MLFF-NN for solving the
48
forward problem. The new updating technique eliminates the instability problem
usually encountered in iterative linear modeling. In addition, an improvement in
reconstruction results is obtained using non-linear update when compared to semilinear update techniques (e.g., by combining LFP with HN-MOIRT). Advantages of
combining distinct neural-network techniques to solve both the forward and inverse
problems include a better approximation to the forward and inverse non-linear ECT
problems, computational speed, and inherent parallelism (of the network).
4.1
4.1.1
Theory
ECT forward problem basic equations
An ECT system is composed of a n electrodes surrounding a pipe or vessel. The
number of independent capacitance measurements is equal to the n(n − 1)/2 independent electrode pairs. The measured capacitances are used to reconstruct the
permittivity distribution inside the vessel. The electric potential depends on the
permittivity distribution according to the Poisson equation
∇ · (ε(x, y)∇φ(x, y)) = −ρ(x, y),
(4.1)
where ε(x, y) is the permittivity distribution, φ(x, y) is the electric potential, and
ρ(x, y) is the charge distribution. Note that Poisson equation is a linear partial
differential equation in terms of φ(x, y). The non-linearity in ECT refers to the nonlinear dependency of φ(x, y) on (x, y). The mutual capacitance between two pairs of
electrodes, source and detector, is obtained through
Cij =
Qj
,
∆Vij
49
(4.2)
where Cij represents the mutual capacitance between electrodes i and j , ∆Vij the
potential difference, and Qj is the charge on the sensing electrode, found by applying
Gauss law below
Qj =
I
Γj
ε(x, y)∇φ(x, y) · n̂dl,
(4.3)
where Γj is a closed path enclosing the sensing electrode and n̂ is the unit vector
normal to Γj .
4.1.2
Iterative image reconstruction techniques
The inverse problem involves finding the permittivity distribution from a set of
measured capacitance values. The forward problem is the process of determining the
output response of an ECT system when the permittivity distribution is known. Due
to the non-linear dependence between measured capacitance and permittivity distribution, iterative reconstruction techniques are widely used in ECT to obtain better
reconstruction results, in comparison to single step reconstruction techniques. Fast
forward solution methods are particularly important when iterative algorithms for
image reconstruction are used. This is because, in iterative algorithms, the image obtained from reconstruction is updated by minimizing the error between the measured
capacitance data and the forward solution for a given permittivity distribution. This
process is repeated iteratively until a pre-defined criteria is met, so multiple forward
solutions are necessary.
Iterative image reconstruction techniques in ECT can be classified into two basic
categories: (1) algebraic reconstruction techniques (ART), in which the image is
updated to minimize the error between measured capacitance and the forward solution
for a reconstructed image (permittivity distribution), and (2) optimization techniques
50
in which a set of objective functions are optimized to meet certain image constraints,
including an error term similar to the one present in ART techniques. In both cases,
the minimization of the error based on the measured capacitance depends on the
gradient of the forward solution, and the update equation takes the form
Gk+1 = Gk − αJ(Gk ),
(4.4)
where Gk+1 is the image vector at iteration k +1, α is a relaxation factor, and J(Gk ) is
the gradient of the error between forward solution for the image vector at iteration k
and the measured capacitance vector. In ECT, the relationship between permittivity
distribution and measured capacitance data is given by eqs. 4.1-4.3.
Commonly used methods for solving the forward problem are based on (1) numerical techniques such as finite elements [24], boundary elements [23],and finite
differences [20], and (2) linearization techniques [35]. Although more accurate, bruteforce numerical techniques are time-consuming, and hence linearization techniques
are considered a more favorable choice for real-time applications. The most common
linearization method to solve the forward problem is linear forward projection (LFP),
based on the sensitivity model. The sensitivity model is based on considering the superposition of each individual pixel, in which the forward solution is obtained as a
linear sum of capacitance values obtained from small perturbations in permittivity
distribution. Based on this model, the update equattion 4.4 is approximated as
Gk+1 = Gk + αS T (C − SGk ),
(4.5)
where C is the measured capacitance vector, and S is the sensitivity matrix. As
mentioned, linearization techniques suffer in terms of accuracy due to the non-linear
nature of electrical tomography. In eq. 4.5, linearization is used for solving both
51
forward and inverse problems. An improvement in the iterative reconstruction process
can be achieved by integrating a non-linear forward solver such that
Gk+1 = Gk + αS T (C − y(Gk )),
(4.6)
where y(Gk ) is the non-linear forward solution of the image vector Gk . The reconstruction technique in this case is referred to as iterative semi-linear back projection
reconstruction [49]. Another form of semi-linear reconstruction can be implemented
through using a linear forward solver (LFP) in a nonlinear iterative reconstruction
technique (Hopfield Neural Network Multi Criterion Optimization Technique HNMOIRT), the combine forward and inverse solvers are referred to as LFP-HNNMOIRT as in the following equation.
Gk+1 = Gk + αJ(C − SGk , Gk ),
(4.7)
In an attempt to solve the forward problem while attaining the merits of both linearization (speed) and numerical techniques (accuracy), a NN based solution is developed. The new NN technique posses the advantage of differentiability, in which a
nonlinear image reconstruction update can be implemented for equation 4.4, without
the need for further approximations as in equations 4.5 and 4.6.
We should also point out that optimization reconstruction techniques have proved
superior over algebraic reconstruction techniques due to the ill-posedness nature of
the inverse problem in ECT. This constitutes an additional challenge to the reconstruction problem when computational and experimental noise are present. In this
case, finding the solution based on minimization of the forward error function alone
(algebraic reconstruction) does not guarantee an optimum solution. In this work,
the FNN forward solver is integrated into the HN-MOIRT which is a multi-criteria
52
optimization technique. The combined forward and inverse solver is referred to as
FNN-HN-MOIRT.
4.2
4.2.1
Combined feed-forward and analog neural network technique for non-linear image reconstruction
Forward problems using feed forward neural networks
Artificial neural networks (NN) are composed of simple processing elements, neurons, organized in different layers and communicating with each other [77]. The
feed-forward NN (FNN) posses the property of being universal approximators, i.e,
for any function of arbitrary degree, there is a feed forward neural network able to
approximate it [57]. In addition, NN are robust to noise and able to predict and extrapolate information hidden in the training data, an ability characterized as neural
network learning [97]. For these reasons, NN are considered an attractive choice to
model non-linear problems.
A multi-layer NN (ML-NN) consists of a number of neurons organized in multiple
layers, as depicted in Fig. 4.1. Each neuron is connected with a weight to all neurons
in adjacent layers. The value of each weight represents the relevance of the particular
connection in the network structure. Each neuron output is mapped to a transfer
function. A sigmoid function is usually employed to map unbounded data to the
bounded range of the transfer function [64]. The following sigmoid function is used
for this case
fΣ (x) =
1
1 + exp(−αx)
(4.8)
A neural network with continuous transfer function in the output layer is a universal approximator [11]. For a NN using a back-propagation algorithm for weight
53
Weights to layer 1
Weights to output layer
X1
W11
W13
f11
W1,1
f12
Output 1
W2,1
b11
W12
b12
W3,1
W2,1
W2,2
W2,3
f21
b21
X2
W1,2
f22
f31
W3,2
W3,2
W3,3
b31
X3
b22
W2,2
W3,1
Figure 4.1: Neural Network with one intermediate layer.
54
Output 2
update, the transfer function is used to calculate the output of a NN layer, whereas
the derivative of the transfer function is used to adjust the weights of different connections in the NN. In a multi-layer feed-forward (MLFF-) NN, the output of a neuron
is a function of the neurons in preceding layers according to
Oil = fΣ (bli +
n
X
Wijl Ojl−1 ) =
j=1
1+
exp(−α(bli
1
P
+ nj=1 Wijl Ojl−1 ))
(4.9)
where i and l are the neuron and layer number under consideration respectively, bl is
a bias term added to each input, Wijl is the weight connecting the output of neuron
j of previous layer to neuron i, and Ojl−1 is the output of neuron j in layer l − 1. The
input-output mapping takes the form of nested non-linear functions as
Oi1 = fΣ (bli +
n1
X
Wijl rj )
j=1
Oi2 = fΣ (b2i +
n2
X
Wij2 Oj1 )
j=1
Oil = fΣ (bli +
nl
X
..
.
Wijl Ojl−1 )
(4.10)
j=1
where rj are the elements of the input vector, and L is the total number of layers in
the network.
4.2.2
Inverse problem solution via Hopfield neural network
In this work, a multi-criteria optimization image reconstruction technique with
backward analog Hopfield neural-network (HN) solution, the HN-MOIRT, developed
by Warsito and Fan [89] is used to solved the image reconstruction inverse problem
iteratively.
The multi-criteria optimization based image reconstruction technique seeks an
image through a combination of the following three basic criteria: (i) largest entropy,
55
(ii) least weighted square error between the measured data set and the estimated value
calculated from the reconstructed image, and (iii) is local and smooth, with relatively
small peakedness. The three associated objective functions to be minimized are,
respectively, given by
h1 (G) = γ1
N
X
Gj ln Gj
(4.11)
j=1
1
h2 (G) = γ2 kfN N (G) − Ck
2
1 h3 (G) = γ3 GT XG + GT G
2
(4.12)
(4.13)
where γ1 , γ2 and γ3 are normalization constants, and X is an N × N non-uniformity
(spatial smoothing) matrix. The function fN N represents the forward solution using
MLFF-NN. The minimization problem seeks an image vector G for which the value
of multi-objective function H(G) = [h1 (G), h2 (G), h3 (G)]T is minimized simultaneously. This is stated mathematically as

minimize


 G∈Π



so that
E(G) =
P3
i=1
wi hi (G),
,
(4.14)
fN N (G) − C ≤ 0
where wi are relative weights operating on the objective function hi (G), such that
P
i
wi = 1. The feasible Π set is defined by the following linear constraints
n
Π = G ∈ RN |SG ≤ C, G ≥ 0, C ∈ RM
o
(4.15)
To solve the optimization problem using the Hopfield neural network technique, the
normalized permittivity values encoded in Gj are mapped into the output variable νj
for neuron j bounded by 0 and 1. The output variable mapped again by means of
a continuous and monotonic increasing sigmoid function of the internal state of the
neuron. For the Hopfield network, the following sigmoid function is employed
Gj = FΣ (uj ) =
exp(uj ) − exp(−uj )
exp(uj ) + exp(−uj )
56
(4.16)
where uj is the internal state variable of the neuron j. The Hopfield neural network
energy for the multi-criteria image reconstruction technique is written as:
E(G) = w1 γ1
N
X
1
Gj ln Gj + w2 γ2 kfN N (G) − Ck
2
j=1
M
N Z Gj
X
1 X
1 T
T
FΣ−1 (G)dG (4.17)
Ψ(zi ) +
γ3 G XG + G G +
2
R0 j=1 0
i=1
where R0 is the resistor of the neuron in the Hopfield network, and FΣ−1 (G) is the
inverse sigmoid function in the Hopfield network. Furthermore, Ψ(zi ) is a penalty
function defined as
∂Ψ
= θ(zi ) =
∂zi
(
0 if, zi ≤ 0
αzi if, zi > 0
(4.18)
where
zi = fN N,i (G) − Ci
(4.19)
Taking C0 as the specific capacitor of neuron and redefining R0 C0 , γ1 /C0 , γ2 /C0 ,
and γ3 /C0 as τ , γ1 , γ2 , and γ3 respectively, the time evolution of the internal state
variable u(t) of neurons in the Hopfield network becomes
1
∇E(G)
C0
u(t)
= −
− w1 γ1 (1 + ln G(t)) − w2 γ2 (fN0 N (G) − C)
τ
u0 (t) = −
−w3 γ3 (XG(t) − G(t)) − fN0 N (G)θ(fN N (G) − C),
(4.20)
where
fN0 N (G) =
∇E(G) =
L
Y
∂fN N (G)
∂fN N (G) ∂OL−1
∂O1
=
·
·
·
=
(1 − fN2 N,l ), (4.21)
∂Gj
∂OL−1 ∂OL−2
∂Gk l=1
"
∂E(G)
∂E(G) ∂E(G)
,
,···
∂G1
∂G2
∂GN
#T
,
u(t) = [u1 (t), u2 (t), · · · , uN (t)]T ,
57
G(t) = [G1 (t), G2 (t), · · · , GN (t)]T ,
z(t) = [z1 (t), z2 (t), · · · , zN (t)]T .
The image vector (permittivity values) Gj becomes the output of jth neuron and is
calculated from the sigmoid function
Gj (t) = FΣ (uj (t)),
j = 1, 2, · · · , N
(4.22)
We stress here that the inverse solver used does not involve training based on prior
data. The Hopfield network is based on multi-objective function minimization that
does not requires knowledge of any prior data. Thus, the error generated from the
MLFF-NN used in solving the forward problem is the only training error present.
4.3
Experimental setup
The experimental data was collected using a 10 cm internal diameter column with
a 12 electrode sensor around it. A set of circular dielectric rods with different sizes
was used for data collection. The data acquisition system used is from PTL (Process
Tomography Ltd., UK), and a set of 66 capacitance measurements is generated for
each frame. Different permittivity distributions were generated for MLFF-NN training by moving the rods inside the imaging domain. The training set consisted of
2,000 permittivity distributions and their associated measured capacitance data. A
MLFF-NN having one input layer with 400 neurons, two hidden layers with 10 and
20 neurons in each respectively, and an output layer with 66 neurons is employed.
A tangent sigmoid transfer function was used in the first two layers whereas a linear
transfer function was used for the third layer. The network was trained to predict the capacitance vector from a given permittivity distribution using a resilient
58
-propagation (RPROP) updating algorithm [68]. The training process was performed
using a PC computer with Pentium 2 GHz processor, and 1 GB RAM. The training
time required varies depending on the size of the training set and the size of the FNN
utilized. In our case, the training time was about 1 hour. For training a network that
is capable of predicting data outside the training set, the training data is required to
be representative of the complexity of the problem. To avoid over fitting, the training
set in this work was collected using rods with dielectric constant equal to 3.7, having
different sizes, and placed in different locations. The MLFF-NN based forward solution has a one-time cost for collecting data and training. Once the appropriate NN
representation is found, the NN forward solution is very fast. A typical time required
for obtaining a forward solution using the present NN is 10 ms.
Testing data were collected from permittivity distributions representing different
flow regimes (half flow, central flow, two column flow, and single rod distribution) to
validate the prediction capabilities of the trained MLFF-NN, as well as its integration
into the HN-MOIRT image reconstruction algorithm. The non-linear update image
reconstruction is implemented according to eq. 4.4, and the gradient is calculated
according to eqs. 4.22 and 4.20. Our MLFF-NN + HN-MOIRT based non-linear
reconstruction method is compared with both a Landweber technique as in eq. 4.5,
and a semi-linear reconstruction technique as in eq. 4.7. As mentioned before, in the
semi-linear method, the forward solution is obtained using LFP, while the inverse
solution is obtained using HN-MOIRT.
59
4.4
4.4.1
Results and discussion
Stability and convergence performance
In Fig. 4.2, the stability of the non-linear update reconstruction (Case 3: MLFFNN + HN-MOIRT) is compared to both the fully linear (Case 1: Landweber) and
semi-linear reconstruction (Case 2: LFP + HN-MOIRT). The relaxation factor, which
controls the image update in the full linear case, is an important element in stability
analysis. A low relaxation factor results in a more stable algorithm, but in slower
convergence. On the other hand, a high relaxation factor leads to faster convergence
at the expense of stability. The optimum relaxation factor is a function of the image
itself, and different techniques for determining an optimal value have been proposed
in literature. A study on the optimal choice of the relaxation factor is beyond the
scope of this work, and for simplicity we set it to unity here.
As seen in Fig. 4.2, the errors in Cases 2 and 3 remain bounded as a function of
the iteration number. The error in Case 1 represents an unstable system in which
the error first decreases and then increases exponentially. The stability of the cases
where MLFF-NN is used to solve the forward problem is a direct effect of the use
of a sigmoid function in the solution. The sigmoid function maps any input to a
finite, bounded domain (for the tangent sigmoid case, the input is mapped to [1,1]). As a result, the output is stabilized by restricting its range. Moreover, the use
of constraints for the HN-MOIRT iterative reconstruction provide stability further
assurance during the reconstruction process.
In general, the degree of stability depends on the NN prediction capability. Two
further observations should be made at this point. First, the error in Case 1 can
converge to a higher error level than Case 3 when the latter is stable. Second the error
60
for Case 1 in Fig. 4.2 diverges for large relaxation factor. As the NN represents the
relation between capacitance and permittivity distribution, both these observations
are consequences of the particular training and architecture used for the MLFF-NN.
Different results are expected by using different MLFF-NN training and architectures.
4.4.2
Image reconstruction results
Fig. 4.3 shows reconstruction results using linear, semi-linear, and non-linear updates. The non-linear update technique performs better than the other two techniques in terms of reconstruction quality. However, all three reconstruction results
are contaminated by noise. This is expected since the measured capacitance used
in reconstruction is corrupted by noise. The problem of noisy data can be minimized by proper network architecture and training. Using training data which are
non-contradictory and representative of the problem is also important to avoid over
fitting and to improve generalization. Generalization is the prediction ability of the
network. Fig. 4.4 was obtained by applying a threshold filter to the results in Fig. 4.3.
The threshold filter is a binary filter applied with a thresold level of 0.5 in our case.
The filter sets each pixel in the reconstructed image of a value greater than or equal
to 0.5 to unity, and less than 0.5 to zero. The measured capacitance data was used for
reconstruction, as discussed above. The non-linear update reconstruction again shows
an improvement over the other two techniques in terms of quality of reconstructed
images.
The performance of the MLFF-NN + HN-MOIRT reconstruction technique is
tested on simulated capacitance data of multiple phantoms and compared to LFP
+ HN-MOIRT and Landweber techniques as depicted in Fig. 4.5. It is clear that
61
the MLFF-NN + HN-MOIRT reconstruction provides better results. Applying the
thresholding filter described earlier shows that the non-linear reconstruction algorithm
provides the best correlation between the final image and the original distribution.
Further comparisons are provided in Fig. 4.6, in which the non-linear and semi-linear
reconstructions are tested on experiment data. Again, the MLFF-NN + HN-MOIRT
provides better reconstruction. In the case of experimental data mesurements, the
reconstruction is performed off-line.
4.5
Summary and Conclusion
In this work, a new non-linear technique is implemented for image reconstruction
in ECT systems. The technique is based on multi-layer perceptron feed-forward
neural network (MLFF-NN) with regularization for solving the forward problem, and
and a Hopfield network with multi-criteria optimization (HN-MOIRT) for the image
reconstruction update. The image update is performed by minimizing the error of
the predicted capacitance based on gradient calculation of the MLFF-NN. The new
image update technique overcomes instability problems usually faced in implementing
sensitivity models for image reconstruction. An improvement in reconstructed images
is also verified using the non-linear update when compared to linear and semi-linear
techniques.
The described technique is fast and can be easily integrated in any iterative reconstruction algorithm. Typically, it takes about 10 ms to obtain a forward solution and
about 20 ms to calculate the gradient of the network at each iteration, in a conventional 2 GHz processor. However, sufficient and representative training data has to be
62
collected for successful prediction. In addition, the training process is hardware dependent. Further improvements can expected by optimizing the network architecture
and the training algorithms. This will be the topic of a future work.
63
Image Error
0
10
MSE
NN−HN−MOIRT
LFP−HN−MOIRT
Landweber
−1
10
−2
10
0
100
200
300
400
500
600
700
800
900
1000
Iteration
Figure 4.2: Mean square error with respect to reconstructed image vector for: (a)
Case 1: non-linear update based on ILBP, (b) Case 2: semi-linear update (LFP for
forward solution and NN-MOIRT for update), (c) Case 3: non-linear update (NN for
forward solution and NN-MOIRT for update).
64
Original image
Landweber
LFP−HN−MOIRT
FNN−HN−MOIRT
Figure 4.3: Reconstruction results using linear, semi linear, and nonlinear update
reconstruction for three different regime flows.
65
Original image
Landweber
LFP−HN−MOIRT
FNN−HN−MOIRT
Figure 4.4: Reconstruction results for the flow regimes in Figure 4.3 with a thresholding filter applied.
66
Original Distribution
Landweber
LFP−HN−MOIRT
FNN−HN−MOIRT
Figure 4.5: Reconstruction results using linear, semi-linear, and non-linear reconstruction for four 1-inch diameter rods in a 4-inches diameter vessel based on simulated
results.
67
LFP−HN−MOIRT
Time
FNN−HN−MOIRT
0s
1s
2s
3s
Figure 4.6: Reconstruction results using semi-linear and non-linear reconstruction
for 1 & .75-inch diameter rods in a 4-inches diameter vessel based on experimental
measurements.
68
CHAPTER 5
ELECTRICAL CAPACITANCE VOLUME
TOMOGRAPHY
The recent progress in development of process tomography has provided more
insights into the complex multiphase flow phenomena in many industrial processes,
including pneumatic conveying, oil pipe lines, fluidized beds, bubble columns and
other chemical and biochemical processes [98]. Tomography in process applications
is capable of monitoring, both continuously and simultaneously, the local and global
dynamic behavior of the gas bubbles and the solid particles in a non-invasive manner. Among available tomography techniques, electrical tomography, including both
resistance and capacitance modalities, is considered the most promising for dynamic
flow imaging. The technique has a relatively high temporal resolution, up to few
milliseconds, with sufficient spatial resolution, up to 1 to 3 % of column diameter.
The high speed capability of electrical tomography systems is demonstrated in the
recent development for up to 1000 frames per second capture rate [87]. In our earlier
work on the development of a real time ECT, we have demonstrated the accuracy of
an image reconstruction technique based on the Hopfield neural network optimization
(neural network multi-criteria image reconstruction technique, NN-MOIRT) [89]-[91].
69
A 3-dimensional tomography image is usually generated by stacking up tomograms
(2D images) [74] [15] [52] [84] [53] [3]. However, the image of a whole volume can not
be represented by that obtained from the averaged capacitance measurement along
the sensor axial direction using the 2D sensor. Because the 3D image in this case
could only be generated from a static or slow moving object, it is termed as ’static’
3D imaging or ’quasi’ 3D imaging. The 3D imaging cannot be applied to situations
with a fast moving object or highly fluctuating multiphase flow media. To date, a 3D
imaging of multiphase flow using this technique is only possible in pseudo-3D mode.
For conventional ECT, 2D ECT in particular, the tomogram is reconstructed
from a capacitance sensor, which is in fact geometrically three-dimensional. Unlike
electromagnetic transmission tomography, a slice imaging is not possible for ECT
due to the extended length of the electrode. The 2D image obtained is thus a result
from projection of the object on a cross-section by assuming no variation in the axial
direction. Therefore, the 2D ECT is actually unreal in the sense that the threedimensional object needs to be assumed to have an infinite length. This is one of
major drawbacks of conventional ECT, and becomes problematic when the variation
in the permittivity along the axial direction is significant. Fortunately, electrical
tomography, either resistance or capacitance, has a potential for volumetric imaging,
as electrical current or wave, spreads to three-dimensional space. The ’soft field’
effect of the electrical field is once considered as one disadvantage of the technique
for imaging application, but it may be advantageous in terms of volume imaging.
In this study, we develop a technique to reconstruct simultaneously a volume image
of a region inside the vessel from capacitance measurement data using capacitive
sensor electrodes attached to the wall of the vessel. Due to the ’soft field’ nature of
70
the electrical field, the capacitance measurement can be made using arbitrary shapes
of electrodes and vessels. The term ’volume tomography’ instead of 3D tomography
stems from the fact that the technique generates simultaneously information of the
volumetric properties within the sensing region of the vessel with an arbitrary shape.
The terminology is also chosen to differentiate the technique from a ’static’ 3D or
quasi-3D tomography technique. The development of the technique primarily includes
the evaluation of the capacitance tomography sensor design and the volume image
reconstruction algorithm. The tests on capacitance data sets obtained from actual
measurements are also presented to demonstrate the validity of the technique for real
time, volume imaging of a moving object.
5.1
5.1.1
Principle of ECT
Forward Problem
The ECT involves tasks of collecting capacitance data from electrodes placed
around the wall outside the vessel (forward problem) and image reconstruction from
the measured capacitance data (inverse problem). The capacitance is measured based
on the Poisson equation which can be written in three-dimensional space as:
∇ · (ε(x, y, z)∇φ(x, y, z)) = −ρ(x, y, z),
(5.1)
where ε(x, y, z) is the permittivity distribution; φ(x, y, z) is the electrical field distributions; ρ(x, y, z) is the charge density. The measured capacitance Ci of the i-th pair
between the source and the detector electrodes is obtained by integrating Eq. 5.1:
Ci =
1 I
ε(x, y, z)∇φ(x, y, z)dA,
∆vij Ai
71
(5.2)
where ∆vij is the voltage difference between the electrode pair; Ai is the surface
area enclosing the detector electrode. Equation 5.2 relates the dielectric constant
(permittivity) distribution, ε(x, y, z), to the measured capacitance Ci .
The forward problem is dealt with generally in three approaches: linearization
techniques [47][46][35]; brute-force numerical methods such as finite element method
[4] and; (pseudo) analytical methods [2]. Despite the fact that analytical methods
can provide accurate and relatively fast solutions, they are limited to very simple
geometries with symmetric permittivity distributions, and are not applicable to industrial tomography systems with complex dynamic structures. On the other hand,
numerical methods can provide fairly accurate solutions for arbitrary property distributions. They, however, consume excessive computational time which is impractical
for tomography application with iterative image reconstruction. In this regard, linearization methods provide relatively fast and simple solutions, though they show a
smoothing effect on a sharp boundary of the reconstructed image. The smoothing
effect is reduced as th number of iterations increases in the reconstruction process.
Linearization techniques using the so-called sensitivity model [35], [100] are based
on the electrical network superposition theorem in which the domain (the cross section
of the sensor) is subdivided into a number of pixels. The response of the sensor; in
this case; becomes a sum (linear model) of interactions when the permittivity of one
pixel in the domain is changed by a known amount. This is similar to the first order
series expansion approach for ’hard field’ tomography [27]. Based on the sensitivity
model, eq. 5.3 can be written as:
Ci = −
X
j
1 I
∇φ(x, y, z)dA,
εj
∆vij Ai
72
(5.3)
The integration part divided by the voltage difference is called as sensitivity, which
can be derived as [11]:
Sij (xk , yk , zk ) =
Z
V0
Ei (x, y, z)Ej (x, y, z)
dxdydz∇φ(x, y, z)dA,
Vi Vj
(5.4)
where Ei (= −∇φ(x, y, z)) is the electrical field distribution vector when i-th electrode
is activated with voltage Vi while the rest of electrodes are grounded, and Ej is the
electrical field distribution vector when j-th electrode is activated with voltage Vj and
the rest of electrodes are grounded. V0 is the volume of k-th voxel. Eq. 5.2 then can
be written in matrix expression as:
C = SG,
(5.5)
where C is the M-dimension capacitance data vector; G is N-dimension image vector;
N is the number of voxels in the three-dimensional image; and M is the number of
electrode-pair combinations. Specifically, N is equal to n × n × nL , where n is the
number of voxels in one side of an image frame (layer); nL is the number of layers.
The sensitivity matrix S has a dimension of M × N .
5.1.2
Inverse Problem
The image reconstruction process is an inverse problem involving the estimation
of the permittivity distribution from the measured capacitance data. In eq. 5.6, if the
inverse of S exists, then the image can be easily calculated. However, in most cases,
especially in electrical tomography, the problem is ill-posed, i.e. there are fewer
independent measurements than unknown pixel values, thus the inverse of matrix
S does not exist. The simplest way to estimate the image vector is using a back
73
projection technique [35], [100], i.e.
G = ST C,
(5.6)
In eq. 5.6 all measurement data are simply back-projected (added up) to estimate
the image. This technique is called as linear back projection (LBP). Though the
reconstructed image is heavily blurred due to a smoothing effect, a rough estimation
of the original shape of the image can be obtained
To obtain sharper reconstructed images, usually an iterative process is employed.
The iterative image reconstruction process involves finding methods for estimating the
image vector G from the measurement vector C, and to minimize the error between
the estimated and the measured capacitance, under certain conditions (criteria), such
that:
SG ≤ C,
(5.7)
The most widely used iterative method to solve the problem in 2D ECT is Landweber
technique, also called as iterative linear back projection technique (ILBP), which is
a variance of a steepest gradient descent technique commonly used in optimization
theory [105]. The technique aims at finding image vector G which minimizes the
following least square error function.
f (G) =
1
1
kSG − Ck2 = (SG − C)T (SG − C)
2
2
(5.8)
The iteration procedure based on the steepest gradient descent technique becomes:
Gk+1 = Gk − αk ∇f (Gk ) = Gk − αk ST (SGk − C)
(5.9)
where αk is a penalty factor of iteration k-th, which is usually chosen as a constant.
The problem with the Landweber technique is that the reconstructed image is dependent on the number of iteration, and the convergence is not always guaranteed.
74
As seen in Eq. 5.9, the image vector is corrected iteratively by capacitance difference
∆C(= SGk − C) multiplied by the sensitivity ST and the penalty factor. When the
number of capacitance data is limited, the capacitance difference ∆C becomes insignificant, and the image iteratively is corrected by the sensitivity ST , producing the
so-called ”sensitivity-caused artifacts” that generates an image focused in the most
sensitive parts of the imaging domain. This is why the reconstructed image based on
Landweber technique has a better resolution near the wall (higher sensitivity) than
the center region (lower sensitivity).
Other techniques based on Tikhonov regularization [43] and simultaneous algebraic reconstruction technique (ART) and simultaneous iterative reconstruction technique (SIRT) [105] are also used widely. The majority of techniques are based on using
a single criterion, i.e. least square error function. However, the least square error does
not necessarily give rise to an accurate image, since it does not contain any information concerning the nature of a ’desirable’ solution [27]. More than one objective
function is required to be considered simultaneously in order to choose the ’best compromise solution’ or the best probability of the answer among possible alternatives.
This is especially the case for 3D reconstruction, as the number of unknown, voxel
values, is considerably increased with the same number of measurement data as in 2D
reconstruction. The probability problem even worsens in case of noise contaminated
data. Increasing the number of electrodes will definitely increase the probability of
obtaining a desirable solution, but usually a maximum number of electrodes defined
due to the limitation of the minimum size of the electrode to overcome the noise.
Thus, Multi-criterion optimization using more than one objective function is needed
to reduce the possibility of alternative solutions, and hence reduce the non-uniqueness
75
of the problem in obtaining a more definitive solution. The implementation of more
than one objective function thus yields a higher probability of obtaining an accurate
solution (estimation) in the image reconstruction. The multi-criterion optimization
image reconstruction technique for volume image reconstruction is described below.
5.2
5.2.1
Multicriterion Optimization Image Reconstruction Technique (MOIRT)
Multicriterion Optimization Image Reconstruction Problem
In this work, a multi-criterion optimization based image reconstruction technique
developed earlier by Warsito and Fan [89] for solving the inverse problem for 2D ECT
is extended to solve the inverse problem for the 3D ECT. The optimization problem
finds the image vector that minimizes simultaneously the four objective functions:
negative entropy function, least square errors, smoothness and small peakedness function, and 3-to-2D matching function. It is important to note that in addition to the
least square error objective function, all the other functions involved in the reconstruction process collectively define the nature of the desired image based on the
analysis of the reconstructed image. Thus, the error, which is generated from the linearized forward solver and propagated to the reconstructed image through the least
square objective function, is minimized with the other objective functions applied.
The negative entropy function used here is defined as:
h1 (G) = γ1 δ1 G ln G,
δ1 =
76
(
1 if, Gj > 0
0 if, Gj = 0
(5.10)
Here, γ1 is a normalized constant between 0 and 1. The least weighted square error
of the capacitance measurements is:
1
h2 (G) = γ2 kSG − Ck2
2
(5.11)
where S is the 3D sensitivity matrix with a dimension of M by N, and M is the
corresponding number of the measured capacitance data. γ2 is a normalized constant
between 0 and 1. The smoothness and small peakedness function is defined as:
1 h3 (G) = γ3 GT XG + GT G
2
(5.12)
Here X is a N by N non-uniformity matrix. γ3 is a constant between 0 and 1. An
additional objective function for the 3D image reconstruction is required to match
the 3D reconstructed image to the 2D, namely 3-to-2D matching function, which is
defined as:
1
h4 (G) = γ4 kH2D G − G2D k2
2
(5.13)
Here, H2D is a projection matrix from 3D into 2D, having dimensions of N × N2D ,
N2D is the number of voxels in one layer of the 3D volume image vector G, and
γ4 is a constant between 0 and 1. The 2D image vector is the 2D solution of the
inverse problem in the image reconstruction. Finally, the multi-criteria optimization
for the reconstruction problem is to choose an image vector for which the value of the
multi-objective functions is minimized simultaneously.
5.2.2
Solution With Hopfield Neural Network
Hopfield and Tank [31] proposed a technique based on a neural network model to
solve optimization problem. In particular, they presented a mapping of the traveling
salesman problem onto neural networks. Since then, Hopfield model neural networks
77
(or simply Hopfield nets) have been used to successfully address many difficult optimization problems, including image restoration [59][76][109] and image reconstruction
for ’hard field’ tomography [85], [86] and ’soft field’ tomography [89], [91]. Their advantages over more traditional optimization technique lie in their potential for rapid
computational power when implemented in electrical hardware, and the inherent parallelism of the network [71].
To solve the image reconstruction problem, the image voxel value Gj to be reconstructed is mapped into the neuron output variable vj in the Hopfield nets. The
output variable is a continuous and monotonic increasing function of the internal
state of the neuron uj :
Gj = υj = fΣ (uj )
(5.14)
where fΣ is called activation function with typical choice of the form:
fΣ (uj ) = [1 + exp(−βuj )]−1
(5.15)
Here β is a steepness gain factor that determines the vertical slope and the horizontal
spread of the sigmoid-shape function. By using such a non-linear sigmoid-shape
activation function, the neuron output is forced to converge between 0 and 1.
The behavior of a neuron in the network is characterized by the time evolution of
the neuron state uj governed by the following differential equation [29]:
C0j
∂E(G)
duj
=−
dt
∂Gj
(5.16)
where C0j is an associated capacitance in the j-th neuron,E(G) is the total energy of
the Hopfield nets. The time constant of the evolution is defined by:
τ = R0j C0j
78
(5.17)
where R0j is the associated resistance. The overall energy function of the network
includes a sum of the constraint functions (objective functions), and penalty functions
over violation of the constraints. Using the same approach as Warsito and Fan [89],
the overall networks energy function corresponding to the optimization problem above
becomes:
E(G) =
4
X
wi hi (G) +
i
2
X
k
Ψ(z ) +
k=1
N
X
1 Z
j=1 Rj
Gj
0
fΣ−1 (G)dG
(5.18)
The first term in Eq. (18) is the interactive energy among neurons based on the
objective functions described above. The second term is related to the violation
constraints (penalty functions) to the three weighted square error functions which
must also be minimized. The third term encourages the network to operate in the
interior of the N-dimensional unit cube (0 ≤ Gj ≤) that forms the state space of the
system, and N is the number of neurons in the Hopfield nets, which is equal to the
number of voxels in the digitized volume image. In the second term of Eq. 5.18, where
z1j = SG − C, z2j = H2D G − G2D . The constraint function Ψ(αk zk ) = Ψ(αk zki )
which is defined as:
∂Ψ
= δ(αk zki ) =
∂zki
(
0 if, zki ≤ 0
αk zki if, zki > 0
(k = 1, 2, 3)
(5.19)
Substituting all the objective functions in Eqs. 5.10 to 5.13 into Eq. 5.19, the overall
network energy function becomes:
1
1 1
E(G) = γ1 δ1 G ln G + γ2 kz1 k2 + γ3 GT XG + GT G + γ4 kz2 k2
2
2
2
N
X
1 Z Gj −1
+Ψ(α1 z1 ) + Ψ(α2 z2 ) +
fΣ (G)dG
j=1 Rj 0
(5.20)
Equation 5.20 can be solved, for example, using Euler’s method to obtain the time
evolution of the network energy. The form of penalty parameter αk is chosen as [3]:
αk (t) = α0k + ς k exp(−η k t)
79
(5.21)
Here α0k , ς k and η k are positive constants. The penalty parameter provides a mechanism for escaping local minima by varying the direction of motion of neurons in such
a way that the ascent step is taken largely by the penalty function in initial steps.
The value of the penalty factor reduces as the algorithm proceeds.
For simplicity, choosing R0j = R0 and C0j = C0 , and redefining R0 C0 , γ1 /C0 to
γ1 /C0 as, τ , γ1 to γ4 , respectively, the time evolution of the internal state variable of
neurons in the networks becomes:
u0 (t) = −
u(t)
− γ1 W1 ⊗ (1 + ln G(t)) + γ2 W2 ⊗ ST z1
τ
T
T
+γ3 (W3 ⊗ XG(t) − G(t)) − γ4 W4 ⊗ HT
2D z2 − S δ(α1 z1 ) − H2D δ(α1 z1 ) (5.22)
where
u0j (t) =
4
X
l=1
duj (t)
,
dt
j = 1, 2, 3, · · · , N,
Wl = [wl,1 , wl,2 , · · · , wl,N ]T ,
wl,j = 1,
j = 1, 2, 3, · · · , N,
u(t) = [u1 (t), u2 (t), · · · , uN (t)]T ,
G(t) = [G1 (t), G2 (t), · · · , GN (t)]T ,
G2D (t∞ ) = [G2D,1 (t∞ ), G2D,2 (t∞ ), · · · , G2D,N 2D (t∞ )]T ,
⊗ denotes an array multiplication (element-by-element product), and t∞ indicates
the asymptotic solution of 2D image reconstruction using the Hopfield network. The
neuron state is updated as uj (t + ∆t) = uj (t) + uj (t)∆t . The neuron output which
corresponds to the voxel value is updated as:
υj (t + ∆t) = Gj (t + ∆t) = fΣ (uj (t + ∆t)),
= Gj (t) + fΣ0 (uj (t))u0j (t)∆t,
80
(5.23)
(5.24)
Here fΣ0 (u) =
dfΣ (uj )
.
du
The weights are updated every iteration step as:
(t)
where,
(t)
∆W1 /∆Wl
Wlt+∆t = P4
(t)
(t)
l=1 ∆W1 /∆Wl
Wlt = fl (G(t + ∆t) − fl (G(t)))
(5.25)
(5.26)
The stopping rule is used when changes in the firing rates become insignificant, i.e.
for all voxels |∆G(t) << 1| .
5.3
Sensor Design and Sensitivity Map
In two-dimensional ECT, the sensitivity matrix only has variation in radial (x- and
y-axes) directions, assuming infinite length of the electrode in the z-direction. Imaging a three-dimensional object requires a sensitivity matrix with three-dimensional
variation, especially in the axial (z-axis) direction to differentiate the depth along
the sensor length. Therefore, the fundamental concept of the electrical capacitance
sensor design for the 3D volume imaging is to distribute equally the electric field
intensity (sensitivity) all-over the three-dimensional space (control volume). This
concept relates to the sensitivity variance (the difference between the maxima and
minima) and the sensitivity strength (the absolute magnitude). Two sensor designs
are considered here, and their performances for 3D volume imaging is evaluated, i.e.
a 12-electrode triangular sensor arranged in one plane and a 12-electrode rectangular
sensor arranged in triple planes as shown in Figure 5.1(a) and (b). The choice of the
electrode number is based on the data acquisition system available for experiment,
which is, but not limited to, 12 channels. For the rectangular sensor, the electrodes
81
are arranged in one plane shifted to another to distribute the electrical field intensity
more uniformly in the axial direction and to increase the radial resolution up to twice
the radial resolution of a 4-electrode sensor. The radial resolution of the rectangular
sensor with this electrode arrangement, thus, equals to 8-electrode sensor per plane.
The sensitivity maps for the two capacitance sensors are presented in Figure 5.2.
The sensitivity maps show distributions of sensitivity variation in three-dimensional
space. For the triangular sensor, the sensitivity maps of capacitance readings between
any electrode pair have a three-dimensional variation. On the other hand, it is only
the sensitivity maps of capacitance readings between inter-plane electrode pairs that
provide a three-dimensional variation in the rectangular electrode case. The maps
show relatively comparable axial and radial sensitivity variation for the rectangular sensor, but less equally for the triangular sensor. Equal sensitivity variation all
over the sensing domain is essential to avoid an artifact or image distortion in the
reconstruction result due to inequality in the sensitivity strength distribution. For
the rectangular sensor, the largest magnitude in the sensitivity is found in the sameplane electrode pair capacitance reading, while the lowest is in the electrode pair
between first and third layers. The magnitude of the sensitivity strength does not
affect significantly the image reconstruction process, but it relates largely to the SNR
in the capacitance measurement. As seen in Figure 5.2, the sensitivity strength in
the first and third layers of electrode pairs is one order less in magnitude than that of
the same-plane electrode pair. Therefore, the capacitance measurement between first
and third planes is very sensitive to noise. A very careful manufacture of the sensor
is then required. The capacitance measurement between inter-plane electrode pairs
is related mostly to the horizontal length of the rectangular electrode, and almost
82
8
7
1
(a)
1
3
11
(b)
(c)
Figure 5.1: Sensor designs and volume image digitization: (a) Triangular sensor, (b)
Rectangular sensor, (c) Image digitization.
83
independent of the axial length of the electrode. Therefore, a consideration of the
horizontal length of the electrode must be given in manufacturing the rectangular
sensor. The sensor design and arrangement selected provides almost the same radial
resolution all over the planes, never the less, the axial resolution slightly differs in
every plane. Figure 5.3 shows the axial sensitivity distribution for all 66 electrode
pairs for both sensors. No much variation is observed for the triangular sensor in the
middle region of the sensing zone. This region gives no differentiation in the image
reconstruction process, and becomes a dead-zone in which a convergence is difficult
to achieve. For the rectangular sensor, the dead zones are found in the bottom (layer
numbers 1 to 3) and top (layer numbers 18 to 20) portion of the sensor domain. The
dead zones for the rectangular sensor can be removed by considering only the effective volume of the sensing domain, i.e. layers 4 to 17. All reconstructed images for
the rectangular sensor presented in this paper, unless otherwise stated, belong to the
effective sensor domain.
5.4
Experiments
The experiment is conducted using a 12-channel data acquisition system (DAM200TP-G, PTL Company, UK). The ECT system is comprised of the capacitance sensor,
sensing electronics for data acquisition, and a computer system for image reconstruction. The sensors include two types of 12-electrode systems as shown in Figure 5.1.
The length of sensing domain is 10 cm and column diameter is 10 cm. The data
acquisition system is capable of capturing image data up to 80 frames per second.
There are 66 combinations of independent capacitance measurements between electrode pairs from 12-electrode sensor systems. The test object is a dielectric sphere
84
Layer 16
0 .0 1
Layer 16
0 .0 2
0 .0 1
0
0
-0 . 0 1
-0 . 0 1
Layer 10
0 .0 1
Layer 10
0 .0 2
0 .0 1
0
0
-0 . 0 1
-0 . 0 1
Layer 4
0 .0 1
Layer 4
0 .0 2
0 .0 1
0
0
-0 . 0 1
-0 . 0 1
Pair #1- #7
Pair #1- #8
(a)
Layer 16
x 10
0.0 2
1
0
0
-0 . 0 2
-1
Layer 10
x 10
0.0 2
1
0
0
-0 . 0 2
-1
Layer 4
x 10
0.0 2
-3
-3
-3
Layer 16
Layer 10
Layer 4
1
0
0
-0 . 0 2
-1
Pair #1- #3
Pair #1- #11
(b)
Figure 5.2: Three-dimensional sensitivity maps: (a) Triangular sensor, (b) Rectangular sensor (The electrode pair number is in Figure 5.1)
85
0.35
0.3
Normalized sensitivity [-]
0.25
0.2
0.15
0.1
0.05
0
-0.05
0
2
4
6
8
10
12
Layer number [-]
14
16
18
20
14
16
18
20
(a)
0.3
0.25
Normalized sensitivity [-]
0.2
0.15
0.1
0.05
0
-0.05
-0.1
0
2
4
6
8
10
12
Layer numbe [-]
(b)
Figure 5.3: Axial sensitivity distribution for all 66 capacitance readings: (a) Triangular sensor, (b) Rectangular sensor; the dead zones are the areas indicated by the
dashed line
86
(I.D. = 1/4 column I.D., relative permittivity = 3.8). The image is reconstructed on
20x20x20 resolution based on the algorithm described above. The volume image digitization is shown in Figure 5.1 c. The reconstruction process and data post-processing
are run on a Pentium 4 machine, 3 GHz and a memory of 2 GB.
5.5
Reconstruction Results
Figures 5.4 to 5.6 show the comparisons of reconstruction performance using the
two electrode designs and the three reconstruction algorithms: Linear Back Projection (LBP), Landweber Technique (ILBP, Iterative Linear Back Projection) and
NN-MOIRT. The iteration number is set as 100 for all cases. The reconstructions are
based on actual capacitance measurements of dielectric objects: located in the center
of the sensing domain and another sphere located half inside the sensing domain.
Each row in every Figure contains two slice images of X-Z and Y-Z cuts in the first
two columns and one 3D image in the 3rd column. The 3D image is an isosurface
display with an isovalue of 0.5 of the maximum permittivity.
Figure 5.4 shows the reconstruction results based on the LBP technique. Elongation in axial direction of the reconstructed images occurs to both objects in the
single-plane triangular sensor. The axial elongation effect is expected as the sensitivity variation in the axial direction for the triangular electrode is insignificant
as compared to that in the radial direction (See Figure 5.3 a). For the rectangular sensor (Figures 5.4 c and d), the technique gives relatively accurate shapes of
the objects though a smoothing effect appears in the sharp boundary of the reconstructed images. The contrasts between low and high permittivity regions in the
reconstructed images are relatively uniform in both radial and axial directions. The
87
(a)
(b)
(c)
(d)
Figure 5.4: 4 Reconstruction results of a sphere in the center and the edge of sensing
domains using LBP technique: (a) (b) Triangular sensor, (c) (d) Rectangular sensor
88
conserved shape and the uniform contrast in the reconstructed image relate largely to
the sensitivity variation, and thus spatial resolution, corresponding to the electrode
design. This indicates that the triple-plane rectangular electrode gives relatively more
uniform sensitivity variation in both the radial and axial directions as compared to
the single-plane triangular sensor. Figure 5.5 shows the reconstruction results using
Landweber Technique (iterative LBP). The reconstructed images are severely distorted in all cases for both sensor designs. An elongation effect is also observed for
the triangular sensor. The reconstructed images appear to be directed toward the
sensing sites with relatively stronger sensitivities, which correspond to the junctions
between electrodes, causing a distortion and elongation due to a ”sensitivity-caused
artifact” as described in Section 5.1.2. The distortion may also arise from noises contained in the capacitance data. The reconstructed volume images using NN-MOIRT
algorithm are shown in Figure 5.6. For the triangular sensor; although elongation
effect is still observed; the results are much better compared to those using LBP
and Landweber techniques. The effect of noise to the reconstructed image is also
minimal as compared on that using Landweber technique. For rectangular sensor,
the reconstructed images are almost perfect except the contrast which is less clear as
compared to those of the triangular sensor. By using a rectangular sensor arranged
in three planes, thus increasing sensitivity variation in the axial direction, it can resolve the elongation problem which is caused by a non-uniform sensitivity strength
between the axial and radial directions. However, with the same number of electrodes as in the triangular sensor, the spatial resolution for the rectangular electrode
is decreased, resulting in less contrast in the reconstructed image. Increasing the
number of electrodes per plane for the rectangular sensor will increase the contrast
89
(a)
(b)
(c)
(d)
Figure 5.5: Reconstruction results of a sphere in the center and the edge of sensing
domains using Landweber technique: (a) (b) Triangular sensor, (c) (d) Rectangular
sensor
90
Figure 5.6: Reconstruction results of a sphere in the center and the edge of sensing
domains using NN-MOIRT: (a) (b) Triangular sensor, (c) (d) Rectangular sensor
91
between low and high permittivities in the reconstructed image. Figures 5.7 and 5.8
show a series of instantaneous volume-images of the same dielectric sphere as used
in figures 5.4 to 5.6 when falling through the inside of the sensor based on image
reconstruction results using the Landweber technique and NN-MOIRT. A distortion
in the shape of reconstructed images from level to level is observed in the results using Landweber technique. On the other hand, the shape of the reconstructed images
using NN-MOIRT is relatively conserved at every level, verifying the capability of the
algorithm to resolve, to some extent, the effect of ”sensitivity-caused artifact”. This
result also indicates that the technique requires less number of measurement data to
generate the same image quality as produced by Landweber technique. The capability to minimize the effect of ”sensitivity-caused artifact” is essential, in particular
for volume imaging, as there will always be non-uniformity in the sensitivity strength
due to the ’soft field’ effect. The use of entropy function and the distribution of the
weight coefficients to the different objective functions are considered to be effective
in minimizing the effect of ”sensitivity-caused artifact”. Both factors are unique to
the NN-MOIRT algorithm. Distribution of weight coefficients is made in such a way
to provide a uniform speed of convergence in each voxel.
5.6
Conclusion
The dynamic volume imaging technique based on the principle of ECT has been
developed. The technique enables a real time 3D imaging of a moving object; or a
real time volume imaging (4D); and allows a total interrogation of a whole volume
within the sensor domain. The work has successfully reconstructed experimental data
of actually moving object for the first time. The technique is feasible for real time
92
Figure 5.7: 3D image of actually falling sphere reconstructed using Landweber technique
93
Figure 5.8: 3D image of actually falling sphere reconstructed using NN-MOIRT
94
volume imaging of multiphase flow systems. Volume imaging of multiphase systems in
conduits such as pipe bends, T-junctions, conical vessels or other complex geometrical
systems, where no other technique is available in the past, is also possible in the near
future. The technique also opens possibility for real time 3D medical imaging of
human body.
95
CHAPTER 6
MULTIMODAL TOMOGRAPHY SYSTEM BASED ON
ECT SENSORS
Industrial flow processes tend to be complex in nature, and often involve a variety of components in a combination gas, liquid, and solid phases. Process (flow)
tomography has provided a leap forward in real-time flow estimation for improved industrial process monitoring and control [70]. Implementation of process tomography
have been achieved through a variety of techniques. The most conspicuous techniques
are those based on measurement of electrical properties, through the utilization of
capacitive, conductive, or inductive nature of the flow components under investigation. In most cases, electrical tomography is implemented based on measurements of
a single constitutive property, viz. permittivity for electrical capacitance tomography
(ECT) or conductivity for impedance (resistivity) tomography. However, the need
for real-time imaging of complex processes involving multiphase components have
motivated in recent years the development of imaging systems exploiting multiple
electrical properties [5], i.e., multimodal tomography.
Multimodal tomography is, in general, implemented through three different strategies [89]: (i.) integration of two or more sensor tomography hardware components
into one imaging system (e.g., gamma-ray and ECT tomography [8]), (ii.) use of
96
reconstruction techniques capable of differentiating between different components
and phases based on a single sensing signal (e.g., multicriteria reconstruction techniques [89]), and (iii.) use of single sensor hardware to acquire different signals corresponding to different electrical properties (e.g., impedance tomography sensors for
imaging permittivity and conductivity [108]). Although the first strategy is fast, it
has a major disadvantage in terms of its high cost and instrumentation complexity
(added hardware). In addition, the data acquisition needs to be carefully coordinated for real-time applications to yield consistent data at different time frames.
The second strategy is the least costly to implement. However, it yields relatively
longer reconstruction times due to more involved reconstruction algorithms. The third
strategy is inherently multi-modal since it provides all required information (on different electrical properties) using the same sensor hardware and same reconstruction
technique. Moreover, integration of such systems with multi-modal reconstruction
techniques can provide independent data for different phases in the imaging domain.
For example, obtaining both capacitive and conductive (impedance) flow information
simultaneously is beneficial in many applications [108], particularly when the flow
under consideration is a mixture of components with widely different conductivity
and permittivity constants such as oil flow along a pipeline.
Electrical impedance tomography (EIT) has been extensively used for both medical and industrial applications [13]. Although EIT commonly refers to (unimodal) resistivity tomography systems, it can also be used for permittivity/conductivity imaging by considering amplitude and phase measurements of the interrogating signal.
However, such applications depend on current injection [17], which requires direct
97
contact between the sensor and imaging domain. This is not viable when an insulating element separates the flow of interest from the sensor system itself, as in the case
of many industrial processes. In this case, one common requirement for the tomographic system is to be both non-invasive (i.e., not in direct contact with the domain
of interest) and non-intrusive (i.e., not to affect the process under examination) [98].
In this work [48], a new non-invasive multimodal tomography system is proposed
based on the use of ECT sensor technology. Unlike usual ECT sensor operation
which assumes a static interrogating field, the interrogating field of the proposed system operates under quasi-static conditions. The ECT sensor system is employed to
simultaneously measure variations in both capacitance and power corresponding to
permittivity and conductivity distributions, respectively, within the sensing domain
(vessel). A dual capacitance/power sensitivity matrix is obtained and used in approximate image reconstruction algorithm based on iterative linear back projection
(ILBP). The system performance is tested on different permittivity and conductivity
flow distributions, showing very good results.
6.1
6.1.1
ECT Sensor Data
ECT sensor
ECT sensors as the one depicted in Fig. 6.1 are usually non-invasive and nonintrusive. An ECT sensor generally consists of n electrodes placed around the region of
interest, providing n(n − 1)/2 independent mutual capacitance measurements. Unlike
usual EIT sensors that use direct current injection as excitation signal, ECT sensors
rely on a time varying driving signal [108]. In a typical ECT system, the frequency
of the excitation signal is about 1 − 10 MHz, and the sensor size is less than few
98
Figure 6.1: Cross section of ECT sensor consisting of six electrodes surrounding a
cylindrical vessel.
meters in either dimension. As a result, the wavelength is much larger than the size
of the sensor and a static or quasi-static approximation can be employed to describe
the field behavior. ECT analysis in the literature is carried out by assuming a static
approximation for the electric field distribution (modulated by the time variation).
In the static approximation, the coupling between the electric and magnetic field
coupling is ignored, with the effect of the later is ignored for ECT purposes.
Applying a quasi-static approximation in Maxwell’s equations, the electric field
distribution obeys the following equation:
∇ · (σ + jω)∇φ = 0,
(6.1)
~ = −∇φ is the electric field intensity, ω is the
where φ is the electric potential, E
angular frequency, σ is the conductivity, and is the permittivity.
99
The mutual capacitance Cij between any two pair of electrodes i and j (source
and detector) is given by
1 I
∇φij · n̂dl,
Cij = −
4Vij Γj
(6.2)
where 4Vij is the potential difference, Γj is a closed surface (or path in 2-D) enclosing
the detecting electrode, and n̂ is the unit normal vector to Γj . Moreover, the r.m.s.
power dissipated by a conductive object in the domain of interest, given the potential
distribution φij due to the source electrode i and grounded detector electrode j, is
given by
1 ZZ
σ|∇φij |2 dS,
Pij =
2 Ω
(6.3)
Equations (6.2) and (6.3) relate the permittivity and conductivity distributions to
(global) measurements of capacitance and power, respectively. The solution for Cij
and Pij given (x, y) and σ(x, y) constitutes the forward problem. The process of
obtaining (x, y) and σ(x, y) from capacitance and power measurements constitutes
the inverse problem.
6.1.2
Equivalent Lumped-Circuit Models
It is useful to express field relations in terms of equivalent lumped-circuit models
whenever possible [1] [81] [41]. In the present scenario, equivalent circuit models for
each parallel plate pair of the ECT sensor can be constructed as a parallel association
of a lumped resistor Rij and capacitor Cij , characterizing ohmic losses and mutual
capacitance, respectively, between each parallel plate pair ij of the ECT sensor. These
lumped elements have different values whether an empty vessel case is considered or
an object is present in the imaging domain. In particular, the equivalent resistor is
zero if the background medium associated to an “empty” vessel scenario is lossless.
100
The equivalent parameters can be extracted both from measurements and from field
calculations, and then used for the reconstruction process, as detailed below.
In terms of the field distribution and the constitutive parameters, the resistance
Rij and the change on Cij can be approximated as
−1
∆Vij2 Z Z
f 2
,
σ|∇φij | dS
Rij =
2
Ω
1 I e ∇φeij − f ∇φfij · n̂dl,
∆Cij =
∆Vij Γj
(6.4)
(6.5)
where the superscript e refers to the empty vessel case, and f to the filled (loaded)
vessel case.
6.2
Sensitivity matrix
Obtaining inverse solutions for σ and f given Rij and ∆Cij data can be time
consuming [105]. The main difficulty for ECT sensors (and for electrical tomography
sensors in general) is the soft-field nature of the problem [37]. In soft-field tomography,
the interrogating field (potential) φ depends on the electric property distributions σ
and f in a nonlinear, complicated fashion. Moreover, there is no analytic solution for
the forward problem in general, and accurate solutions can only be obtained in general via brute-force computational techniques [54]. The large computational resources
and long computational time required make these techniques costly and impractical
for fast (real-time) reconstruction and monitoring. A popular approximation strategy used in practice consists on a linearization of the problem using sensitivity matrix
models [35]. The sensitivity matrix is built by considering small perturbations (pixels) filled with a material of higher permittivity or conductivity. The response of the
sensor for each perturbation is organized in a (sensitivity) matrix according to their location in the sensing domain to form a basis set for (linear) reconstruction. Moreover,
101
the forward solution in this case is obtained as a superposition of sensor responses to
different perturbations, commonly referred to as linear forward projection (LFP). The
sensitivity matrix acts as a forward projection matrix between electric property distribution and sensor boundary measurement. The transpose of the sensitivity matrix
can be used for back projection. Back projection techniques are used to solve the inverse problem and include variants such as linear back projection (LBP) [35], iterative
linear back projection (ILBP) [37], pseudo-inverse projection [105], and regularized
projection [62].
In this work, a dual sensitivity matrix that includes both capacitance and power
measurement data is constructed and used for solving both forward and inverse problems in an approximate fashion. The dual matrix elements are approximated based
on the electric field distribution in the empty sensor scenario.
6.2.1
Capacitance matrix
As discussed above, the sensitivity matrix is constructed by calculating the system
response to different electric property perturbations in the sensing domain. In case
of capacitance tomography, the perturbed property is electric permittivity.
An approximate (first-order perturbation) method for sensitivity matrix calculation has been introduced in [47] based on approximating the capacitance difference
introduced by a perturbation using the electric field from an empty sensor. The difference in capacitance is related directly to the difference in total stored energy caused
by the permittivity pixel. This energy difference is composed of two components [47]:
internal to the pixel ∆Wint = a(i α2 −e )|E~0 |2 /2 ≡ βint |E~0 |2 , and external to the pixel
∆Wext = aγ 2 e |E~0 |2 ≡ βext |E~0 |2 , were a is the effective pixel volume, i is the pixel
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permittivity, α is the pixel polarizability (a geometric factor), e is the background
permittivity around the pixel, E~0 is the (unperturbed) electric field amplitude at the
pixel location, γ is a constant related to the relative permittivity values. The constants βint and βext are introduced to simplify the final equations. Combining these
two equations, we have ∆C = 2(βext + βint )|E~0 |2 /(∆V )2 . In order to solve for the
sensitivity matrix, the sensor model has to be solved only once in the empty case.
The sensitivity matrix is normalized according to the capacitances measured for
the empty and fully dielectric filled vessels as
∆Cijm =
∆Cij
,
− Cije
Cijf
(6.6)
where ∆Cijm represents normalized sensitivity matrix elements, and Cijf and Cije are
the elements for filled and empty vessels, respectively. From the previous considerations (linear approximation), this would produce positive normalized capacitances in
all cases. In practice, due to nonlinearities in the actual problem, negative values for
the sensitivity matrix elements can occur. In general, the efficacy of the above linearization approach is established based on its integration into iterative approximate
reconstruction algorithms, as detailed in Section V.
6.2.2
Power matrix
Similarly to the capacitance matrix, each element in the power matrix linearizes
the relation between the conductive (heating) loss and a small conductive pixel perturbation in an insulating background given by Eq.(6.3), integrated over a (small)
pixel volume with conductivity σ.
The electric field is found by solving eq. (6.1) and using eq.6.3 to determine
the dissipated power. In order to simplify this calculation, we assumed that the
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Normalized E field
0
10
ε =1
ε =10
ω =10 MhZ
−1
|E/E0|
10
−2
10
−3
10
−4
10
−6
10
−5
10
−4
10
−3
10
−2
σ
10
−1
10
0
10
1
10
Figure 6.2: Normalized electric field |E/E0 | as a function of conductivity σ
104
Figure 6.3: Normalized electric field |E/E0 | as a function of relative permittivity r
105
Resistive power loss for normalized E
−3
10
ε =1
ε =10
−4
ω=10 MhZ
10
−5
Pn
10
−6
10
−7
10
−8
10
−6
10
−5
10
−4
10
−3
10
−2
σ
10
−1
10
0
10
1
10
Figure 6.4: Power dissipation inside the pixel for the normalized electric field in Figure
6.2 as a function of conductivity σ
106
perturbation is a small spherical pixel with uniform conductivity and an (effective)
uniform electric field in its interior that is a function of the unperturbed (empty
vessel) electric field E~0 and a correction factor f (ω, σ, ) to be determined. In this
way, eq. (6.3) is written
1
∆P = aσ|E~0 f (ω, σ, )|2 .
2
(6.7)
In this approximation, the forward problem needs to be solved once for |E~0 | to determine the power matrix. However, f (ω, σ, )) is dependent upon the solution of
Poisson equation since, in practice, σ can vary by many orders of magnitude. The
factor f (ω, σ, ) is approximated differently at three regimes, so that ∆P is approximated as follows
1. Diffusion regime (σ ω) :
10−3
1
∆P ≈ aσ |E~0 |
2
σ
!2
(6.8)
2
(6.9)
2. Convection regime (σ ω):
1
1.88
∆P ≈ aσ |E~0 |
2
r
3. Mixed regime (σ ≈ ω) :
1
1
∆P ≈ aσ |E~0 |
2
|σ + jω|
!2
(6.10)
The approximation for f is illustrated by FEM simulation results shown in Figs.
6.2, 6.3, and 6.4. The FEM results considers a small spherical pixel with 10 mm
radius placed in a 10 MHz (actual ECT sensor operation frequency) uniform electric
field, and show the results for the normalized electric field and associated power loss.
Note that maximum power dissipation occurs when σ ≈ ω. The power matrix is
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normalized analogously to the capacitance matrix. However, since the the power
dissipated in the empty vessel case is zero, the power sensitivity matrix elements
remains positive, regardless of nonlinearities being neglected.
It is important to point out that the power matrix calculation provided above is
based on the assumption that the skin depth δs is much greater than the physical
size of the sensing domain. Otherwise, the power measurement technique will not
provide a correct estimation of the actual conductivity distributions. Reduced skin
depths distort capacitance measurement as the electric field will have a fast decay
and may not pervade the entire sensing domain. This puts a limit on the maximum
conductivity feasible to be imaged. A possible remedy is to decrease the frequency of
operation so as to increase the skin depth for a given conductivity. However, lower
frequencies produced lower (power) signals, characterizing a fundamental trade-off.
6.3
Reconstruction
The soft field nature and ill-posedness of the inverse problem are the main problems encountered in the reconstruction step. Although more accurate nonlinear inverse problem solutions have been the subject of intensive research, the most commonly applied reconstruction techniques are still based on (linearized) sensitivity
models. In this work, iterative linear back projection (ILBP) is used for approximate
image reconstruction. In ILBP, both forward and inverse problems are solved iteratively to minimize the residual image error. In contrast to traditional ILBP based on
a single modality sensitivity matrix, a dual modality sensitivity matrix is employed
here. The first component of the matrix represents the capacitance perturbation,
108
whereas the second component refers to the conductivity perturbation, as detailed in
the previous Section.
In ILBP, the image vector is updated iteratively to minimize the error between
measured and calculated data according to:
h
Gk+1 = Gk + τ S T M − SGk
i
(6.11)
where the calculated data is obtained from the (last) reconstructed image using linear
forward projection. In the above, G is the image vector, k is the iteration number,
S is the sensitivity matrix, τ is a factor controlling the reconstruction convergence,
and M is the boundary measurement. In the dual modality reconstruction performed
here, G and M are both complex vectors, with real part associated with the permittivity distribution and measured capacitance, respectively, and imaginary part with
conductivity distribution and measured dissipated power, respectively.
Further improvements on the image quality of the approximate ILBP reconstruction are generally implemented through applying (postmode) image constraints, regularization parameters, and/or image processing filters. For example, a limiting constraint of maximum and minimum reconstructed pixel values can be applied to equation (6.11) for benefit of the so-called projected Landweber iteration![101]. In this
work, a thresholding filter is applied to the reconstructed images in their postmode.
This filter is aimed at removing smooth transitions between regions of different electrical property due to blurring caused by ILBP.
Implementing iterative reconstruction techniques requires solving the forward and
inverse problems multiple times. The use of sensitivity matrix models in its present
form also reiles on the assumptions that the field distributions that are not strongly affected by the permittivity and conductivity pixel distributions for the complementary
109
measurements, viz., power and capacitance measurements, respectively. Although
this assumption does not remain valid in general, it is valid in convection dominated
and diffusion dominated regimes. In both cases, one electrical property is dominant
over the other in determining the measurement perturbation, and reconstruction can
be performed independently for the permitttivity and conductivity distributions.
6.4
Recontruction Results and Discussion
A 10 MHz ECT sensor layout composed of 12 electrodes as depicted in Fig. 6.1 is
used to assess the multimodal tomography system performance proposed here. Simulations for sensitivity calculations and boundary measurements are carried out using
FEM, where a dual sensitivity matrix for capacitance and power perturbations is constructed based on the electric field distribution in the empty sensor state. Boundary
measurements of capacitance and power, as well as sensitivity elements, are normalized for image reconstruction according to
cm =
M
Mm − Me
Mf − Me
(6.12)
where M e , M f are the boundary values (power and capacitance) when the vessel
domain is entirely filled with low and high values of the corresponding electric property
(conductivity for power and permittivity for capacitance), and M m is the measured
data. The reconstruction process and data forward simulations were carried out on a
Pentium IV computer, with 2 GHz processor and a 1 GB RAM memory. The sensing
domain is a cylindrical vessel with 1 m radius. Inside the sensing domain, various fluid
flow distributions are simulated as indicated by the original distributions in Figs. 6.6,
6.7, 6.8, and 6.9. In this case, a cylindrical flow zone with 0.5 m radius (zone A) and
a cylindrical ring flow zone with 0.667 m inner radius and 0.95 m outer radius (zone
110
B), with each zone having a different pair σ and , co-exist. The background medium
is set as = 1, σ = 0. Permittivity values refers here to relative permittivity. The
corresponding reconstruction results using ILBP are shown in the same figures.
In Fig. 6.5, power measurements for each capacitor plate pair are provided for the
flow distribution when relative permittivity of both zone A and zone B is 5, and the
conductivity values are indicated in the Figure. Although the measured dissipated
power changes with the conductivity, it does not uniquely predicts the conductivity
value. This is because the dissipated power may depend on the permittivity constant
as well, as shown in Fig. 6.4.
As discussed in Section 6.3, the relative value of conductivity with respect to permittivity is an important variable for defining the performance of reconstruction. In
Fig. 6.6, the high value of conductivity constant in the zone A (diffusion dominated)
enables the solution to converge to two distinct regions of (pure) permittivity and conductivity maps. In Fig. 6.7, the electric field distribution is mainly controlled by the
permittivity constant due to relatively small values of the conductivity. As a result,
the permittivity reconstruction captures both zone A and zone B distributions. The
conductivity reconstruction, on the other hand, is again able to reconstruct the (less)
conductive zone A (convection dominated) satisfactorily. Thus in the case of either
convection- or diffusion-dominated cases, an independent reconstruction of permittivity and conductivity can be implemented. Fig. 6.8 shows a further example, where
small conductivity is present in both zone A and zone B, and both permittivity and
conductivity reconstructions perform satisfactorily. The skin depth in all these cases
is much larger than the size of the imaging domain, and the permittivity distribution
is reconstructed without significant influence from the conductivity distribution.
111
Figure 6.5: Power vectors of forward solutions for the flow distribution depicted at
the bottom right corner. The electrical properties for (ring) zone B are = 5 and
σ = 0, whereas for (central) zone A, = 1 and σ varies as indicated by the legend.
112
A: Original Distribution
σ =0
ε =5
0
B: Conductivity Distribution C: Permittivity distribution
σ =1
ε =5
0.5
1
0
0.5
1
0
0.5
1
Figure 6.6: Reconstruction results for the original distribution values depicted on the
left (diffusion dominated case).
However, as menioned above, recontruction for a diffusion dominated regime may
fail for conductivity values producing skin depths below vessel size. This is illustrated
in Fig. 6.9, where the effect of a skin depth of δs ≈ 0.18m in a 1 m radius vessel is
depicted. It is clear from the figure that boundary measurements are unrepresentative
of the electrical property distribution.
Further examples are illustrated in Fig. 6.10 and Fig. 6.11 where two separate
with different electrical properties bubbles coexist. In the example of Fig. 6.10, both
permittivity and conductivity distribution are captured, but the permittivity distribution of the high conductivity region is affected by the reduced skin depth there (note
113
A: Original Distribution
σ =0
ε =5
0
B: Conductivity Distribution C: Permittivity distribution
σ =10−5
ε =5
0.5
1
0
0.5
1
0
0.2
0.4
0.6
0.8
Figure 6.7: Reconstruction results for the original distribution values depicted on the
left (convection dominated case).
the assymetry in the permittivity reconstruction). In the example of Fig. 6.11, permittivity and conductivity distribution are captured without significant skin effect.
Note that all reconstruction images provided are normalized to unity.
6.5
Conclusions
In this work, a new non-invasive multimodal EIT/ECT system based on the ECT
sensor hardware has been discussed. The new tomography system is based on quasistatic analysis of ECT sensor with both capacitance and power measurements used for
permittivity and conductivity imaging. A dual sensitivity matrix based on linearizing
assumptions is obtained and used in an approximate ILBP reconstruction algorithm.
114
A: Original Distribution
−5
σ =10
ε =5
0
B: Conductivity Distribution C: Permittivity distribution
σ =10−5
ε =5
0.5
1
0
0.5
1
0
0.2
0.4
0.6
0.8
Figure 6.8: Reconstruction results for the original distribution values depicted on the
left (convection dominated case).
The developed tomography system overcomes the need for sensor contact and invasion when impedance imaging is performed based on current injection. Reconstruction results based on representative data have showed the capability of the system in
providing approximate conductivity and permittivity maps via ILBP.
Direct use of traditional reconstruction techniques such as ILBP assumes independence between power and capacitance signals. This is valid when one electrical
property dominates over the other in determining the measurement data. On the
other hand, any correlation can be explored towards developing new reconstruction
techniques tailored for this problem in the future.
115
A: Original Distribution
σ =10
ε =5
0
B: Conductivity Distribution C: Permittivity distribution
σ =10−5
ε =5
0.5
1
0.1 0.2 0.3 0.4 0.5
0
1
2
3
Figure 6.9: Reconstruction results for the original distribution depicted on the left.
Because of the reduced value of the skin depth at ring zone B, the reconstruction
fails to reproduce the original distribution.
116
A: Original Distribution
σ =1
ε =5
B: Conductivity Distribution C: Permittivity distribution
σ =10−5
ε=5
0
0.5
1
0
0.5
1
0
0.5
1
Figure 6.10: Reconstruction results for a two-sphere case, with the original distribution values depicted on the left.
A: Original Distribution
σ =0
ε =5
0
B: Conductivity Distribution C: Permittivity distribution
−5
σ =10
ε =1
0.5
1
0
0.5
1
0
0.5
1
Figure 6.11: Reconstruction of simulated data for a 2 sphere case with the original
distribution depicted on the left of the Figure
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CHAPTER 7
CONCLUSION AND FUTURE WORK
The problem of ECT is inherently inter-disciplinary, and investigation of ECT systems has been focused on reconstruction techniques and hardware design. The hardware and reconstruction process have been deeply investigated over the last decade
[106] [83] [91]. Available commercial ECT systems are based on linear back projection
reconstruction technique applied on data typically obtained from 12 to 16 rectangular
capacitance plates. A number of studies have demonstrated acceptable reconstruction results based on iterative reconstruction techniques. However, application of
such techniques to on-line commercial ECT systems is limited by the excessive time
required for reconstruction. On the other hand, low signal to noise ratio of measured
capacitance from the available hardware restricts the number of plates mounted on the
sensor [83]. An increase on the number of plates has considerable effect on reducing
the ill-posedness of the ECT problem and increasing the resolution of reconstructed
images.
The main goal of this work was to introduce a commercial ECT system capable
of providing on-line volume imaging with higher image resolution. Such development
requires the application of iterative reconstruction with a speed high enough to capture changes of multi phase flows in real time. The inverse ECT problem has been
118
solved based on the 3D NN-MOIRT reconstruction technique. This reconstruction
algorithm overcomes the problem of real time imaging through the capability of being
implementable on a hardware chip. The forward problem has been solved based on
feed forward neural networks for improved image reconstruction. Again, the developed forward solver has the advantage of being realized on hardware chips making
real time iterative reconstruction possible. One important conclusion of this work is
that real time volume imaging using ECT is possible and expected to appear as a
commercial product in the near future.
7.1
Reconstruction techniques
Recent trends of reconstruction techniques have been directed toward regularizationbased reconstruction and multi-criterion optimization reconstruction. In the former
case, a form of regularization is applied during reconstruction to reduce noise effects
in the reconstructed image, and to improve transitions between regions with different electrical properties. ECT constitutes an ill-posed sensor problem in which the
capacitance response for different locations of permittivity perturbations can change
dramatically. As a result, the signal to noise ration of capacitance data obtained in
response of a permittivty change in the center of the domain is much smaller than that
for a near wall location. Occasionally, the noise level for a centrally located purturbation is even greater than the capacitance difference signal. Thus, using the sensitivity
matrix for image reconstruction requires the inversion of an ill-conditioned system
matrix. Regularization is applied through different methods, i.e., by modifying the
eigenvalues for improving the condition number of the system matrix [95] or by applying a form of image filtering during reconstruction [49]. The error difference between
119
the measured and calculated capacitance is minimized iteratively. This method of
reconstruction does not account for the quality of the resulting image during reconstruction. The reconstruction criteria is dependent only on the measured capacitance
vector (mean square error criteria), which may be not completely representative of
the original permittivity distribution due to noise contamination. In the ideal (noise
free) case, the mean square error criteria is enough for image reconstruction and can
lead to a very good approximation to the exact distribution.
On the other hand, in multi criterion optimization based reconstruction techniques, a set of objective functions based on the measured capacitance and quality of
reconstructed images are optimized during the reconstruction process. Using a noisy
capacitance vector for image reconstruction does not guarantee convergence to the
correct image. Objective functions depending on the quality of reconstructed image
are used to direct the reconstruction process toward the most acceptable image. The
use a Hopfield neural network for optimization is an important step toward real time
imaging as it can be realized on electronic hardware leading to a very fast iterative
reconstruction process.
7.2
Forward problem
The forward problem in iterative reconstruction is an important step for convergence to the desired image. In this work, a feed forward neural network has been
applied for this purpose. Using neural networks for solving this problem has been
motivated by their high speed, accuracy in case of proper training, and ability to
adapt to the complexity of a severely ill-posed ECT problem. Moreover, the ability
120
of realizing a trained network on an electronic chip provides a route to developing
real time 3D commercial ECT systems.
Forward solutions produced by the proposed method have demonstrated the feasibility of the neural network technique for image reconstruction.
7.3
3D volume tomography
Applications of 3D ECT have been also carried out through quasi-3D imaging.
In quasi-3D imaging, the 3D image is obtained through stacking 2D images at different cross sections. Image processing has been applied in some cases to reduce the
mismatch between different 2D stacked frames. The superiority of optimization techniques, and more specific the 3D NN-MOIRT, has been demonstrated through the
direct 3D imaging (volume imaging). The 3D ECT problem is more ill-posed than
the 2D problem, and the ration of unknowns to measurement data is far greater than
in the 3D case. The results demonstrated in this work have shown the capability of
3D NN-MOIRT for obtaining volume images directly from measurement data. The
3D reconstruction has been modified to include objective functions suitable for 3D
applications. The 3D NN-MOIRT reconstruction technique has been validated based
on experimental measurements.
It is important to mention that the success of any volume reconstruction technique
depends on the 3D sensor being used. Generally, a 3D sensor is one capable of
providing sensitivity variation along the axial direction which can be utilized for 3D
image reconstruction. Different parameters of the sensor can be optimized to obtain
such variation. However, a usable 3D sensor has been introduced for the first time
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in this work. It is expected that tomography research will focus on optimizing 3D
sensors in the near future.
The impact of this recent development on industrial processes is expected to be
significant. Until recently, theories of chemical reaction and multi-phase process has
been based on assumption and observations proved in some cases to be wrong. The
development of real time 3D imaging is an important step toward a better understanding of chemical reaction evolution and process dynamics.
However, full exploitation of these recent developments requires the integration
of the developed reconstruction technique along with the 3D sensor and appropriate
data acquisition hardware into a commercial product. Work is currently in progress
toward this goal.
7.4
Multi-modal electrical tomography
Imaging of different electrical properties has been a mean of studying complex
flows due to the diversity they provide for the interrogating signal. Traditionally, the
diversity of interrogating signal has been explored through invasive methods such as
current injection in electrical impedance tomography (EIT). In this work, a multimodal tomography system based on electrical capacitance sensor has been developed
through joint use of capacitance and power measurements for permittivity and conductivity imaging respectively. The new system combines the merits of multi-modal
systems in terms of increased imaging capability, and electrical tomography systems
in terms of speed and safety. In addition, the developed system is non-invasive since
it uses the ECT sensor for capacitance and power measurement.
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The proposed multi-modal system uses the same reconstruction techniques developed for electrical tomography systems in general. However, reconstruction can be
improved if the correlation between permittivity and conductivity maps is accounted
for during reconstruction.
7.5
Future work
As this work was aimed at improving the overall performance of ECT systems
for commercial use, several topics have been investigated such as forward and inverse
solvers, 3D reconstruction and multi-modal tomography. However, the improvements
provided in this dissertation do not provide a complete remedy for different challenges
in ECT technology, and research is on going for finding better alternatives.
Suggestion for future work may include all aspects of electrical tomography technology. Yet, the most prominent suggestions based on this dissertation are highlighted
in the following.
7.5.1
3D neural network forward solver
The electrical tomography problem is generally ill-posed and non-linear in all its
disciplines. These problems have commonly been tackled through iterative reconstruction techniques. In the forward problem, different methods have been used to
obtain a satisfactory solution.
Research so far has focused on the 2D scenario regarding both inverse and forward
solutions. In a development provided in this work, a 3D reconstruction technique
capable of obtaining whole volume images directly from boundary measurements has
been presented. This new development is based on using the sensitivity matrix for
solving the forward problem in the iterative scheme. The sensitivity matrix has
123
been; to a certain extent; used successfully in iterative reconstruction. Nevertheless,
its use reveals several limitations due to the ill-posed and non-linear nature of the
problem. In the 3D case, these problems are exacerbated as the 3D problem is more
ill-posed compared to 2D, and stronger in non-linearity. The severity of ill-posedness
of the problem comes from the increased number of unknowns for the same number
of boundary measurement data compared to the 2D case. The stronger non-linearity
is mainly caused by the variation of electric field along the axial direction, which is
assumed to be uniform in the 2D case.
Based on the characteristics of the 3D problem, the need for a fast non-linear 3D
forward problem solver is even more urgent. Thus, feed forward neural networks can
be investigated for solving this problem due to their characteristics discussed before.
It should be noted that the neural network is more involved in the 3D case, and an
investigation on optimal regularization parameters is necessary. Moreover, the neural
network topology has to be chosen accordingly. On this aspect, and radial basis
functions might provide better results for 3D.
7.5.2
3D sensor design
In 3D reconstruction, the sensor hardware plays a major role in defining the
quality of reconstructed images. In this regard, two types of sensor hardwares have
been exploited in Chapter 5. However, defining the appropriate sensor for a certain
application is not a trivial matter, and further different types of sensors need to be
investigated for a better understanding of the problem. The easiest way of introducing
a variation of the field along the axial direction is by changing the number of plates
and the plate shape. In Appendix B, different sensitivity maps for different sensors are
124
provided. Yet, choosing the appropriate shape usually requires a change in the sensor
design, which is a costly operation both in time and money. One possible solution is
the design of adaptive 3D sensor that can change the sensor plate shape and number of
planes electronically. The new sensor would take a shape similar to the one depicted
in Figure 7.1 where different synthetic sensors can be formed by combining the signals
from the small plate elements. It is important to note that combining different plate
elements in different ways would result in different sensitivity matrices. The problem
can be solved based on combining sensitivity matrices from different elements to form
the new sensitivity matrix.
7.5.3
Multi-modal electrical tomography
ECT sensors are used for permittivity imaging through capacitance measurement.
In this work, we have illustrated the use of ECT sensors for simultaneous permittivity
and conductivity imaging through capacitance and power measurements, in an inherently multi-modal electrical tomography. For this technology to be implemented in
industrial applications, it is suggested to design a data acquisition system capable
of dual capacitance and power measurements. Research in tomography hardware is
well established and the realization of this technology into a commercial product is
expected to enhance imaging capabilities of multi-phase systems.
Reconstruction for the combined permittivity/conductivity maps can be implemented based on already developed reconstruction techniques for different electrical
tomography modalities. However, the power and capacitance signals are correlated,
and a new reconstruction technique capable of including the nonlinear interaction
125
Figure 7.1: Adaptive 3D sensor composed of small plate elements. Different shapes
of plates and planes can be formed by connecting the small plates together.
126
between permittivity and conductivity distributions is another possible direction of
future work.
127
APPENDIX A
FINITE ELEMENT METHOD FOR SOLVING THE ECT
FORWARD PROBLEM
Although the sensitivity matrix model for approximating the forward solution of
an ECT problem is the most popular, other techniques have also been investigated in
literature. The Finite Element Method (FEM) is the most prominent among bruteforce computational techniques for electrical tomography systems forward problem
solving. The FEM is based on dividing the problem domain into a finite number of
elements. The elements can be triangles or quadrilaterals in 2D applications, or more
generally, tetrahedrons or hexahedra in 3D applications. The process of combining the
cells into a one complete structure that discretizes the spatial domain is referred to as
mesh generation. Mesh generation is a challenging process in which the appropriate
sizes of elements had to be found to adequately discretize the problem [79].
In ECT, the main equation to be solved is Poisson equation
∇ · (ε(x, y, z)∇V (x, y, z)) = −ρ(x, y, z),
(A.1)
where (x, y, z) is permittivity distribution, V (x, y, z) is electrical potential distribution, and ρ(x, y, z) is charge distribution. The solution, V can be approximated in
128
terms of a set of basis functions as:
Ṽ (x, y, z)) =
N
X
αj φj (x, y, z),
(A.2)
j=1
where N is the number of basis functions used, αi is a scaling factor, and φi (x, y, z)
is the ith basis function. Since a finite number of basis function is being used, the
residual error in this case is found as:
∇ · (ε(x, y, z)∇Ṽ (x, y, z)) + ρ(x, y, z) = r(x, y, z),
(A.3)
where r(x, y, z) is the residual. One approach of finding the approximate solution
using finite elements is to enforce a zero residual error on certain points of the domain
(point matching). In this case the enforcing equation takes the shape:


N
X
j=1

αj ∇ · (ε(x, y, z)∇φj (x, y, z))
+ ρ(xi , yi , zi ) = 0,
(A.4)
xi ,yi ,zi
where i = 1, · · · , N . Transforming the problem to a matrix equation leads to
[Kij ] [αj ] = [−ρ(xi , yi , zi )] ,
(A.5)
where Kij is the ith point solution of the j th basis function. Another form of minimizing the residual is through applying a testing function procedure; i.e.;
Z
0
1
Z
0
1
Z
0
1
h
i
∇ · (ε(x, y, z)∇Ṽ (x, y, z)) + ρ(x, y, z) ui (x, y, z)dxdydz = 0,
(A.6)
After integration by parts, we arrive at what is commonly referred to as the weak
form of the original problem. The solution in this case depends highly on the selection
of ui (x, y, z). Following Galerkin’s approach, the same set of basis function can be
used as testing functions. In a matrix form the above becomes:
[Kij ] [αj ] = [fi ] ,
129
(A.7)
where [Kij ] is the stiffness matrix and its elements are the weak form of the original problem with zero charge distribution for the it h basis function and j t h testing
function, and [fi ] is the forcing vector of the form:
Z
0
1
Z
0
1
Z
0
1
ρ(x, y, z)φi (x, y, z)dxdydz = 0,
(A.8)
where φi (x, y, z) is the it h testing function.
Applications of FEM to electrical tomography problems are commonly based on
setting a fixed mesh independent of the property distribution. This technique is used
to overcome the complexity and excessive time required for generating a new mesh
for each imaging frame or iteration during the reconstruction process.
130
APPENDIX B
3D RECONSTRUCTION RELATED ISSUES
B.1
3D sensor
Conventional 3D imaging (or 3D static imaging) is based on combining different
cross sections of 2D images to form a pseudo 3D image. This technique has been
applied to different modalities of tomography reconstruction and has demonstrated
successful results in some cases, especially in hard field tomography. In ECT, application of static 3D imaging has been limited by the relatively large size of sensor plate
and the severely ill-posed reconstruction problem. The sensor in ECT is bounded
by a minimum size of 5 cm in the axial direction due to the low SNR of sensed signal. Thus, 3D static imaging suffers major drawbacks in terms of image quality, as
illustrated in Figure B.1.
As mentioned earlier, capture of distribution variance along the axial direction
is based on a change in the interrogating signal along the same direction. In ECT,
this variation is achieved through different sensor designs. These designs are based
on exploiting the soft field nature of ECT for 3D imaging. Plate shapes and number
of planes are the main elements used for field variation. In Figures B.2-B.4, different
sensor designs with the associated field variation are provided. In Figure B.2, the 3D
131
Conventional Tomography
Static object
2D Image
Reconstruction
Static 3D
Reconstruction
Volume-Tomography
Static/Dynamic
3D Reconstruction
Static/Dynamic
3D object
Volume (3D)
Image Reconstruction
Figure B.1: ECT volume tomography verses conventional 3D ECT.
132
variation is established through modifying the plate shape. In Figures B.3 and B.4,
the use of different planes in distributing the sensor plates provides the required axial
field variation. Note that a trade off between axial and spatial resolution exists as
the number of planes is increased. Other forms of 3D sensor can be obtained through
using non-symmetrical plate shapes in a multi plane sensor.
B.2
Neural networks forward solver for 3D reconstruction
Solving the forward problem is an important part of any iterative image reconstruction available for tomography applications. Providing an accurate forward solution is part of the ongoing research toward better reconstruction results. In case of 3D
ECT volume reconstruction, the need for fast and accurate 3D forward solvers is even
more urgent as the problem is larger in size and more severly ill-posed when compared
to the 2D case. Following the successful implementation of 2D forward solvers based
on feed forward neural networks, some issues on the application of neural network
solutions for the 3D case are discussed next.
B.2.1
Data preprocessing
Although reconstruction in case of ECT is a nonlinear problem due to its soft
field nature, applying iterative techniques based on minimizing the mean square error is sufficient for obtaining an optimal inverse problem solution in the noise-free
case. However, in the practical applications capacitance measurement data is always
contaminated with noise. The noise has different effects depending on the level of the
measured capacitance value. The SNR is minimum for capacitance data corresponding to the center region of the imaging domain.
133
Plate 7
Plate 4
Plate 2
Plate 11
Plate 1
Figure B.2: Sensitivity matrix of a trapezoidal sensor.
134
Plate 9
Plate 11
Plate 5
Plate 2
Plate 1
Figure B.3: Sensitivity matrix of a square double plane sensor.
135
Plate 11
Plate 9
Plate 5
Plate 3
Plate 1
Figure B.4: Sensitivity matrix of a square triple plane sensor.
136
Taking the noise effect into account, the reconstruction problem is not unique, and
results obtained based on minimizing the forward problem error are not necessarily
optimal. The multi-criterion optimization role comes in defining optimal images based
on optimizing additional objective functions that define the nature of the desired
image. Moreover, since the overall problem of non-uniqueness is rooted on noise
contamination, the reconstruction resolution and quality is expected to improve if
the noise level is mitigated through data preprocessing.
In 3D reconstruction, the level of capacitance sensed signal is commonly lower
compared to the 2D case. As a result, noise affects the reconstruction result more
strongly. Moreover, the magnitude of the capacitance signal changes dramatically as
a function of pixel location in 3D. As a result, different pairs of capacitance plates
are affected differently depending on the distance between them.
Preprocessing the capacitance data has to take into account the shape of the
sensor and the distance between plates. It is recommended to reorganize the measured
capacitance vector into a number of sub-vectors depending on the distance between
plates, and preprocessing each sub-vector separately for noise mitigation.
B.2.2
Regularization
Regularization is commonly used to transfer ill-posed problems into well-posed
ones. Regularization is equivalent to applying smoothing constraints to guarantee
a stable solution. In ECT reconstruction, regularization has been applied directly
through regularization parameters, or indirectly through different image objective
functions in multi-criterion optimization algorithms.
137
In neural network forward problem training, regularization is applied trough different techniques to obtain a solution with acceptable generalization. In case of 3D
problem, applying regularization is even more urgent. It is suggested here to use radial basis function network (RBFN) for 3D forward problem prediction. Radial basis
functions possess the property of fine tuning the network to correct for a local region
in the mapping process. Whereas in FFNN, the training has to be repeated all over
again in case of unsatisfactory results. RBFN in its basic form consist of three layers
of input layer, radial basis functions layer, and output layer.
B.3
Extended objective functions for 3D reconstruction
In the extended 3D NN-MOIRT reconstruction, an additional objective function
for 2D-3D image matching has been included. This extra objective function term
assures that a 2D projection of the 3D image matches the 2D reconstructed image.
In this work, a new objective function based on the correlation function is being
investigated for incorporation in to 3D reconstruction.
B.3.1
The use of correlation in process tomography:
Transient analysis of multi-phase flow processes has been a major area of research
in recent years [90]. Comprehensive understanding of process characteristics depends
mainly on the ability of measuring dynamics of flow elements [40]. An important
factor that has received increased attention is the velocity measurement of flow components inside the process vessel (velocimetry).
Recently, successful implementation of ECT sensor for phase velocity measurement
has been presented in various scientific journals [42] [16] [25]. ECT velocimetery is
mainly implemented using one of two techniques:
138
• Auto correlation between different image frames using a single sensor plane
• Cross correlation between images obtained from two sensor planes along the
vessel
In both methods above, the correlation between two images acquired at a single location (auto-correlation), or at different locations (cross-correlation) is obtained
along a time interval. The time instant at which maximum correlation value is obtained is assumed to refer to the time required by the phase of interest to travel the
distance between the ECT dual plane sensor in the cross-correlation case, and to the
exposure time in the auto-correlation case [90].
Successful implementation of the presented correlation techniques is based on the
nature of the flow under consideration. The correlation difference is expected to be
larger in flows having a more random nature. As a result, correlation velocimetery
does not perform well for stratified and annular flows [16].
It is important to mention some challenges associated with implementing ECT
velocimetry techniques:
• The correlation between images from different sensor planes decreases with increasing the distance between planes. This is a direct result of the random
nature of the flow under consideration, where different phases vary in temporal
and spatial domains.
• The sensor size is relatively large with respect to the phase under consideration
in many cases.
• The low spatial resolution from ECT reconstruction makes it more difficult to
obtain accurate correlation measures.
139
The introduction of 3D ECT reconstruction directly from measured capacitance
provided a substitute for the correlation method used for velocity measure [93]. In
3D ECT images, the velocity is calculated by tracking the movement of the center
of the bubble or phase under consideration obtained from 3D reconstructed image.
Never the less, the use of correlation function still provides a potential for image
reconstruction enhancement, as discussed next.
B.3.2
The use of correlation function and a prior information
in 3D ECT image reconstruction:
The first known 3D reconstruction technique capable of obtaining satisfactory
images directly from measured capacitance is the 3D neural network multi criterion
optimization reconstruction technique (NN-MOIRT). The NN-MOIRT is based on
optimization of a set of objective functions related to minimizing the error between
measured and calculated capacitance, and to the nature of desired image [89]. Generally, optimization techniques enable the integration of various objective functions
suitable for different applications of tomography, and the NN-MOIRT is no exception
[94]. Thus, the present discussion of correlation and prior information for implemented in reconstruction is based on the framework of optimization techniques.
3D reconstruction is based on utilizing variation of electric field in axial and transverse directions for obtaining a 3D image. The objective functions used in 3D NNMOIRT are based on obtaining the most likely image from the measured capacitance.
In ECT, multiple candidates for a possible reconstructed image can occur as a result of
the ill-posedness of the ECT sensor. However, the objective functions applied are not
based on a prior knowledge of flow characteristics, rather on a general characteristics
of an accepted image.
140
In many cases of process tomography, a prior information of the flow under consideration is available. Utilization of such information have been a topic of research
in 2D image reconstruction. However, its implementation has not been investigated
in detail. In 3D reconstruction, the use of prior information is expected to improve
and speed up the reconstruction process even more strongly than in 2D. This is because the role of objective functions in defining the most likely image becomes more
important as the ill-posedness of the problem increases.
In a smart 3D ECT reconstruction technique, preliminary reconstruction results
would be obtained without integration of any prior information. The reconstruction
technique would then be able to extract some elementary characteristics of the imaged flow, and implement it in reconstruction of later frames. The reconstruction
technique would be able to obtain a flow characteristics by calculating the correlation
of successive images and trying to assess the general trend of the flow under consideration. In addition, the velocity of the flow would be measured for better estimation
of the speed of evolution of the targeted phase.
For two 3D images, the correlation function over a volume V0 can be defined as:
1 Z Z Z
A(x, y, z, t)B(x+∆x, y+∆y, z+∆z, t+∆t)dxdydz
RAB (∆x, ∆y, ∆z, ∆t) =
V0
x,y,z∈Vo
where A is the first image function, B is the second image function, and Vo is the
domain of integration. Assuming that the correlation function is applied for images
from two successive measurements, ∆t will be the time between the measurements,
which is a constant dependent on the acquisition hardware reading rate. As a result,
it can be omitted from the equation, and the study can be focused on the spatial
variation. However, it is important to note that knowing ∆t is important for velocity
measurements.
141
In image reconstruction, the image is displayed in a digitized form by dividing
the image into pixels. The discrete form the correlation function for two successive
images writes in the form:
ny nz
nx X
X
1 X
A(i, j, k)B(i + l, j + m, k + n)
RAB (l, m, n) =
V0 i=1 j=1 k=1
where, l, m and n, are the unit pixel displacements in the x, y and z directions respectively. And nx , ny and nz are the number of pixels in each direction.
For highly correlated images, the reconstruction would be simplified by using
a prior information from previous reconstructed image for the reconstruction being
processed at hand. On the other hand, for images with lower correlation, the influence
of previous images would decrease.
The use of correlation data in an objective function can be implemented in different
ways. The first method is based on minimizing the difference between the image being
reconstructed and the image reconstructed in the previous frame. The objective
function term to be minimized in this case is:
f (Gt+∆t ) = (1 − α)(Gt+∆t − Gt )
where Gt is the image vector at time t, and α is a relaxation factor to control the
correlation between images under consideration. The value of the α controls the effect
of correlation information in the reconstruction process. The α parameter could be
set based on the correlation information according to:
α=
q
(l∆x)2 + (m∆y)2 + (n∆z)2
√
3
V
where l∆x, m∆y and n∆z are the spatial distances for maximum correlation in the
x, y and z directions respectively, and V is the volume of correlation. It is noted that
142
obtaining the value for α requires prior information of the flow correlation characteristics being imaged. For the highest correlation value occurring at a larger spatial
difference, the correlation objective function would given less weight in reconstruction,
and vice versa. As the distance traveled by the phase under consideration is related
to its speed of evolution, less correlation is expected for larger spatial differences.
A more detailed objective function term could be implemented using the correlation function itself. The objective function to be maximized in this case would
be:
f (Gt+∆t ) = (1 − α)(
ny nz
nx X
X
1 X
Gt+∆t (i, j, k)Gt (i + l, j + m, k + n))
V i=1 j=1 k=1
where α is obtained the same way as above.
A prior knowledge of the correlation function can also be used for predicting the
smoothness of the reconstructed image. A large derivative of the correlation function
corresponds to sharp transition in the image, and a low value corresponds to smoother
images. A possible use of this information can be in the determination of the weight
of the smoothness objective function used in [89].
It is important to mention that calculating the correlation function is a time
consuming task. However, in a smart ECT system, this needs to be calculated only
at the start of the imagining process. The ECT reconstruction algorithm is expected
to find the characteristics of the flow under consideration based on calculation of the
correlation function from previous reconstructed images.
143
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