Image Reconstruction Method Along Electrical Field Centre Lines Using

image reconstruction method along electrical field centre lines using a modified mixed normalization model for electrical capacitance tomogra

Image reconstruction method along electrical field centre lines using
a modified mixed normalization model for electrical capacitance
tomography
Lifeng Zhang1 and Wuliang Yin2
1.
Department of Automation, North China Electric Power University,
Baoding 071003, China
2.
School of Electrical and Electronic Engineering, The University of
Manchester, Manchester M13 9PL, UK
E-mail: [email protected]
Abstract: During the process of image reconstruction for electrical
capacitance tomography (ECT), normalization of measured capacitance
values is carried out with the low and high permittvity. The parallel
normalization model (PM) and series normalization model (SM) are most
commonly used. In recent years, using different combination methods
such as electrical field centre line (EFCL) for PM and SM, several
mixed normalization models (MM) obtained better description of the
permittivity distribution of the two-phase media in pipe. In this
paper, a new method of determining the weight factors of the PM and SM
sensitivity matrix to form the MM sensitivity matrix is proposed. This
weight factors are determined according to the minimum distance
between the element and EFCL. Simulation and experimental test were
carried out and the results show that both the accuracy and the shape
fidelity can be improved obviously.
Keywords: two-phase flow, electrical capacitance tomography, image
reconstruction, mixed normalization model, electrical field centre
line
1.
Introduction
Two-phase flow exists widely in industrial processes such as petroleum
refining, chemical engineering and electricity generation. Electrical
capacitance tomography (ECT) is one kind of process tomography (PT)
technique that was developed during the late 1980s, which can provide
visualization measurement results of two-phase flow in real time
through the cross-sectional reconstructed images of the permittivity
distribution in pipe [1, 2]. Using the accurately reconstructed
images, other parameter such as void fraction of two-phase flow can be
calculated [3-6]. Unfortunately, ECT image reconstruction is a typical
ill-posed problem and its solution is unstable [7]. As a result, the
accuracy of image reconstruction algorithms will limit the application
of ECT in industrial field. A variety of algorithms have been studied
to improve the visualization quality [8-15]. Image algorithms of ECT
can be divided into non-iterative and iterative algorithms. For any
kind of image reconstruction algorithms, sensitivity matrix and
capacitance measurement normalization model are necessary.
As can be seen in [16], capacitance measurement normalization model is
the description of two-phase media distribution. The parallel
normalization model (PM) is the firstly and commonly used model [17].
Yang and Byars presented the series normalization model (SM) and
obtained the better reconstructed images [18]. Dong and Guo used the
combined parallel and series normalization, in which the optimal
weight factor was obtained by minimizing the error between the
measured capacitance and the capacitance estimated from the image
using Tikhonov regularization algorithm [16]. For all the above
algorithms, the sensitivity matrix calculated with the parallel
normalization model is still used.
In 2001, Loser et al. considered the influence of permittivity
distribution on the electrical field lines in the imaging area and
calculated the new sensitivity matrix [19]. Followed that, Kim et al.
presented the sensitivity matrix generated using mixed normalization
model (MM) based on the electrical field centre line (EFCL) in 2007
[20]. Zhang and Wang presented a new combined normalization model (CM)
to calculate the new sensitivity matrix, and the optimal weight factor
for the mixed capacitance normalization measurements was determined
using the Landweber iterative algorithm with optimal step length in
2009 [21]. Following the work of [21], a new method to obtain the
sensitivity matrix based on the minimum distance between the element
and EFCL is presented. Simulation and experiments were carried out and
the results showed that the quality of reconstruction images can be
improved obviously compared with PM, SM and CM.
2.
Theory of capacitance normalization model for ECT
2.1. Parallel and series normalization models for ECT
The 12-electrode ECT sensor can be seen in figure 1(a). The electrode
pair i−j is treated as an ideal parallel-plate capacitance sensor.
and is the high and low permittivity of two-phase
media, respectively. Figures 1(b) and (c) show the parallel
capacitance model and series capacitance model of parallel-plate
capacitor, in which the proportion of media with high permittivity are
( ) and ( ), respectively. The
capacitance values are and when the permittivity of
media between two electrodes is and , respectively.

(a) (b) (c)
Figure 1. Sensor and capacitance model: (a) 12-electrode ECT sensor, (b)
parallel capacitance model and (c) series capacitance model.
For the parallel model shown in figure 1(b), the measured capacitance
can be expressed as:
(1)
So we can obtain as follows:
(2)
As to the series model shown in figure 1(c), the measured capacitance
can be expressed as:
(3)
And hence can be expressed as follows:
(4)
It can be seen from equations (2) and (4) that the proportion of high
permittivity media and can be calculated using
parallel and series capacitance normalization models according to the
distribution.
For ECT system, when the parallel normalization model (PM) or series
normalization model (SM) are used for all the electrode pairs, the PM
and SM for ECT system can be expressed as the following equations (5)
and (6) [21].
(5)
(6)
where and are the normalized capacitance vectors based
on PM and SM respectively. is the measured capacitance vector,
while and are the capacitance vectors when the pipe is
full of high and low permittivity material, respectively.
The corresponding sensitivity matrix of PM and SM can be obtained
according to equations (7) and (8) [21].
(7)
(8)
where and are the sensitivities of the kth element
when the capacitance of the electrode pair i−j is measured and is
normalized based on PM and SM, respectively. is the correction
factor with regard to the area of the kth in-pipe element.
PM and SM are commonly used for ECT image reconstruction to simplify
the calculation. However, it can be seen from figure 2 that the
two-phase flow regimes are very complicated and the process of the
transformation from one kind of flow regime to another one is speedy.
Furthermore, for the flow regimes shown in figure 2, neither PM nor SM
can be used for any electrode pair i−j to accurately calculate the
proportion of high permittivity media because the media distribution
between different electrode pair i−j cannot be simply described as
parallel or series model.

Figure 2. Four typical flow regimes.
Additionally, for the existence of ‘soft-field’ effect, the relation
between capacitance and media distribution is nonlinear, which can be
illustrated by figures 3 and 4 in detail.

Figure 3. Flow regimes: (a) one object close to electrode 7, (b) one
object close to electrode 1, and (c) synthesized flow regime by (a)
and (b).
There is one object close to electrode 7 in figure 3(a) and another
object close to electrode 1 in figure 3(b). When these two objects
exist simultaneously in pipe, we can obtain figure 3(c). When
electrode 1 is excited, the corresponding 11 capacitance values
between electrode 1 and the remaining electrodes 2-12 ( ,
, , ) for the three flow regimes in figures 3 were
calculated, which were shown in figure 4.

Figure 4. 11 Capacitance values between electrode 1 and electrodes
2-12.
For the reason of the nonlinear relation between measured capacitance
values and the permittivity distribution and the ‘soft-field’
characteristic. It can be seen from figure 4 that all the 11
capacitance values of flow regime in figure 3(c) are not equal to the
sum of the capacitance values of figure 3(a) and (b), which means the
PM is not accurate. Meanwhile, it can be inferred that the use of SM
will also be inaccurate. In order to describe the media distribution
more accurately, the new capacitance normalization model needs to be
further studied.
2.2. The modified mixed normalization model for ECT
The schematic of the 12-electrode ECT sensor can be seen in figure 5(a),
in which the EFCL between electrodes 1 and 9 is shown. It can be seen
from figure 5(a) that the imaging area in pipe is divided into two
areas by EFCL between electrodes 1 and 9. It is the same for other
electrode pair. The mesh grid using in the finite element method (FEM)
is shown in figure 5(b), in which is the minimum distance
between the kth element in area 1 or area 2 of the pipe and the EFCL
of the electrode pair i−j. The diameter of the pipe is 125 mm. The
number of grid in the imaging area of pipe is 768.

(a) (b)
Figure 5. Schematic graph of ECT sensor: (a) EFCL and (b) mesh grid.
The mixed normalization model (MM) for ECT can be expressed as
equation (9) and the corresponding sensitivity matrix can be obtained
using equation (10).
(9)
(10)
where and are the normalized capacitance based on MM
and the weight factor of PM and SM for the electrode pair i−j,
respectively. is the sensitivity of the kth element when the
capacitance of the electrode pair i−j is measured and is normalized
based on MM. is the weight factor of PM and SM for the
calculation of the sensitivity of the kth element in imaging area.
It can be seen from figure 5(b) that the dissection elements fall into
the area 1 or area 2 based on the EFCL of the electrode pair i−j. For
all the elements in area 1 or area 2, the maximum value of can
be calculated. Thus, can be calculated according to the ratio
of to the maximum distance in area 1, when the kth element
belongs to area 1. Otherwise, will be calculated using the
ratio of to the maximum distance in area 2. The calculation of
can be expressed in equation (11).
(11)
where and is the maximum value of the minimum
distances for the elements in area 1 and area 2, respectively.
Using equation (11), the calculated for the mesh grid of
different electrode pair showed in figure 5(b) can be seen in figure
6.
The calculated for electrode pairs 1-12, 1-9 and 1-7 is shown in
figure 6, respectively. There are two images in figure 6 to show the
results of for each electrode pair, in which the left image
with the plotted EFCL is the 2D description of and the grey of
each triangle element is the value of . During the image
reconstruction, the 32×32 square image reconstruction mesh grid is
used and the number of grid inside the imaging area is 812. Hence the
right image in figure 6 is the 3D description of corresponding
to the 812 pixels inside the pipe.
It can be seen form figure 6 that is close to zero when the kth
element is near to the EFCL in area 1 or area 2 which is divided by
the corresponding EFCL, and the 3D description of on the right
image is more clearly to show the value of . And then, it can
be seen from equation (10) that the corresponding capacitance
normalization model tends to be SM when is close to zero. As
the element is farer from the EFCL, the weight factor of PM will be
larger, and the corresponding capacitance normalization model tends to
be PM.

(a) electrode pair 1-12

(b) electrode pair 1-9

(c) electrode pair 1-7
Figure 6. of different electrode pair.
2.3. The new mixed normalization model for ECT
According to the method proposed in section 2.2, the new sensitivity
matrix can be obtained. Subsequently, image reconstruction
based on the MM and the Landweber iterative algorithm with optimal
step length was adopted as the following equation [23, 24].
(12)
where , is the step length of the kth iteration.
is the grey matrix of the kth reconstructed image. Sensitivity
matrix S used in the equation (12) is .
In the former research, the weight factor which is used in
equation (9) and the calculation formula of are presented in
[21].
3.
Results
3.1. Simulation results
Oil-gas two-phase flow is selected as the simulation object and 4
typical flow phantoms are studied. The relative permittivity of the
oil and gas is set to 3 and 1, respectively. Image reconstruction
based on PM, SM, CM and MM is carried out using Matlab software. The
size of each visualization image is 32 ×32, and the number of pixels
in the circular imaging area is 812.
The ECT sensor with 12 electrodes, which is illustrated in figure 7,
is used to conduct the simulation reconstructions. An earthed
shielding with 200 mm in the length and 170 mm in the diameter is used
to enclose the sensor to reduce the external electromagnetic
interferences. The diameter of the pipe is 125 mm and the thickness of
the pipe wall is 10 mm. The electrodes are made of Titanium metal. The
length, width and thickness of each electrode is 62.5 mm, 30 mm and 1
mm, respectively. Naturally, the imaging area is also a circle with
the diameter of 125 mm.

Figure 7. Layout of the 12-electrode ECT sensor for simulation.
Simulation experiment was carried for the four phantoms shown in
figure 8. After 200 iterations using the Landweber iterative algorithm
with optimal step length, reconstructed images based on PM, SM, MM and
the combined normalization model (CM) presented in [21] were shown in
figure 8.

Figure 8. Reconstructed images for considered phantoms based on PM,
SM, CM and MM.
To evaluate the quality of reconstructed images quantitatively, the
relative image error (IE) and correlation coefficient (CC) are
calculated as follows [7]
(13)
(14)
where and is the true permittivity distribution of the
test object and the reconstructed permittivity distribution,
respectively. N is the number of the pixels in the imaging area. In
our simulation, N equals 812. Reconstructed images with higher quality
will have smaller IE and larger CC.
The quantitative index IE and CC of the reconstructed images showed in
figure 8 are calculated, which can be seen in Tables 1 and 2,
respectively.
It can be seen from figure 8 that the quality of reconstructed image
for the object on the centre can be enhanced obviously using MM
compared with PM, SM and CM. For all the phantoms in figure 8, the
reconstructed images obtained based on PM have many artefacts.
Furthermore, there is serious distortion for phantom b and two objects
cannot be distinguished. Although object on the centre in phantoms c
and d can be seen from the reconstructed images, it cannot be
separated clearly with other objects.
Table 1. Calculated IE corresponding to four phantoms in figure 8.
Phantom
PM
SM
CM
MM
a
0.7502
0.4134
0.3015
0.2396
b
0.7877
0.5368
0.4248
0.3362
c
0.7297
0.7104
0.5989
0.4766
d
0.7740
0.7707
0.5082
0.4173
Table 2. Calculated CC corresponding to four phantoms in figure 8.
Phantom
PM
SM
CM
MM
a
0.6268
0.7134
0.7796
0.8396
b
0.5014
0.5868
0.6854
0.7548
c
0.5627
0.6204
0.6995
0.7189
d
0.5251
0.5907
0.6527
0.7402
When the SM is used, the quality of reconstructed images can be
improved, from which the object on the centre can be clearly
distinguished, but artefacts still exist. Additionally, the shape
fidelity of the objects in the reconstructed images is not high. After
using CM and MM, the quality of reconstructed images is further
improved. Not only the objects can be distinguished clearly from the
reconstructed images, but also the better shape fidelity of objects
can be obtained. Even though the quality of reconstructed images can
be enhanced obviously both by CM and MM compared with PM and SM, if
the images were analysed carefully, it can be noticed that the
reconstructed images with MM are better than those of CM. It also can
be seen from the calculated IE and CC listed in Tables 1 and 2, the
reconstructed images using MM have the smallest IE and largest CC.
3.2. Experimental test
Experimental test was carried out using the digital ECT system based
on FPGA developed by Tianjin University [24, 25], which can be seen in
figure 9. The ECT system contains 24 capacitance measurement channels,
which is designed for two plane of 12-electrode ECT sensor. It also
can be used according to custom-defined scheme. The online image
reconstruction speed is about 120 frame/s which is suitable for
industrial process measurement in real-time.

Figure 9. Digital ECT system.
In the two-phase flow experiment, a 12-electrode sensor pipe with the
inner diameter of 125 mm was used. In flow pattern a (bubble flow),
there are two Perspex rods with the diameter of 20 mm. Flow patterns b
and c are both the laminar flow plus one Perspex rod, and the Perspex
rod is more close to the stratified flow in flow pattern c compared
with flow pattern b. It is hard for flow patterns b and c to obtain
reconstructed images with high quality for the reason of the
soft-field effect. The image area is also divided into 812 pixels and
the image reconstruction results were showed in figure 10.

Figure 10. Reconstructed images for experimental test.
It can be seen from figure 10 that for the flow pattern a, the
reconstructed image based on PM is badly distorted, which can be
improved by using SM, but there still exist artefacts and the two rods
still cannot be separated clearly. When the CM and MM is used, the
quality of reconstructed images can be greatly enhanced. The two rods
can be separated clearly, but the rod close to the pipe wall can be
better reconstructed and the shape of the object in the reconstructed
images is very close to the real shape using MM than CM. The similar
effect can also be obtained for the flow patterns b and c, which means
reconstructed images with higher accuracy and shape fidelity can be
obtained based on MM.
4.
Conclusions
A new capacitance mixed normalization model (MM) for a 12-electrode
ECT sensor is presented in this paper. It is based on the combination
of parallel normalization model (PM) and series normalization model
(SM). The electrical field centre line (EFCL) is adopted for the
determination of the weight factor, which is calculated based on the
minimum distance between the element and EFCL, and the corresponding
sensitivity matrix of MM can be obtained. During the process of image
reconstruction, Landweber iterative algorithm with optimal step length
is used to calculate the weight factor for the normalized capacitance
measurements. Simulation and experimental test were carried out and
the results show that both the accuracy and the shape fidelity can be
improved obviously using MM. Due to the soft-field effect, the quality
of reconstructed images, in the case of the object in the centre or
the existence of many objects, is relatively low. By the use of MM,
object in the centre or objects arranged close to each other can be
better reconstructed and this suggests that the influence of
soft-field effect can be mitigated.
Acknowledgments
The authors thank the National Natural Science Foundation of China
(No. 51306058) and the Fundamental Research Funds for the Central
Universities (No. 2017MS131) for supporting this research. Dr. Lifeng
Zhang would also like to thank the China Scholarship Council for
supporting his visit to the University of Manchester.
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