Flow tomography technology is a new technology which has developed rapidly
in recent years and the technology in solving the problem of multiphase flow
detection has great developmental potential and wide industrial application
prospect. Electrical capacitance tomography technique has the advantages of
low cost, wide application range, simple structure, non-invasive, good safety
performance and so on (Loser et al., 2001). The
electrical capacitance tomography system has inherent characteristic of nonlinear.
The number of capacitance values (projection data) which is independently measured
is limited and it is far less than the number of pixels of reconstructed image.
There is no analytical solution of inverse problems. At the same time, the stability
of solution for the ECT system is poor because of the nonlinear and "soft field
effect" and its ill-posedness is very serious, therefore, image reconstruction
is difficult (Liu et al., 2009).
Whether the algorithm can reconstruct images which are very close to the original
images or its speed of reconstructing images (Ghanbari,
2008) is fast, it decides the quality of the ECT images. LBP, Landweber
iterative algorithm, projection Landweber iterative algorithm (Gosselin
et al., 2009) and CG are common algorithms which are used for ECT
LBP is simple and it can fast reconstruct image. But the quality of image is
relatively poor and it is only a qualitative method (Zhiyao
et al., 2009). The projection Landweber iterative algorithm can obviously
improve the stability of iteration (Daoye et al.,
2009) and it can also effectively control the noise. But when it processes
the complicated flow pattern, it usually requires a large number of iterations
in order to achieve satisfactory results. So this defect restricts its applications
(Zhiyao et al., 2009). CG is suitable for the
coefficient matrix which is symmetric and positive definite. The image reconstruction
time of this method is short and it also has fast convergence for simple flow.
But the effect of the image is not ideal for the complicated flow pattern. In
order to solve above mentioned problems, this study proposes image reconstruction
algorithm which is based on adaptive Pulse Coupled Neural Network (PCNN).
MATERIALS AND METHODS
Basic principle of electrical capacitance tomography system: The working
principle of electrical capacitance tomography is as follows: The capacitive
sensor array is installed around the measured filed.
|| Composition of electrical capacitance tomography system with
When the density or distribution of the dielectric changes, the sensors will
detect the changes of capacitance values. Then it will transmit these values
to the computer to process and the computer can reconstruct the distribution
of the dielectric section. Finally the computer will calculate the parameters
of multiphase flow. ECT system is mainly composed of capacitor sensors, data
acquisition system and the computer which is responsible for reconstructing
images (Yang, 2010). The physical structure is shown in
At present, most of the ECT image reconstruction algorithms are based on the
linear model (Li and Yang, 2009). In the model, the
dielectric constant is mapped to the capacitor. After discretization, linearization
and normalization (Grudzien et al., 2010), the
model can be expressed as:
In the equation, C∈Rm represents the capacitance measurement
value, C∈RmHn represents the coefficient matrix (sensitivity
matrix), ∈∈Rn represents the medium distribution image
vector. The task of ECT image reconstruction is to solve the distribution of
dielectric constant ε with the given capacitance value C. The inverse problem
of ECT system is to reconstruct medium dielectric constant distribution map
in the detection zone by observing and measuring the capacitance measurement
values (Ge and Song, 2010). That is to solve the gray
values of each pixel in the imaging area.
Image reconstruction algorithm based on adaptive pulse coupled neural network:
Pulse coupled neural network is also known as the third generation artificial
neural network. It is a feedback network which is composed of a number of interconnecting
neurons. This structure is inspired by biological visual cortex model (Cheng
et al., 2008).
||Single neuron model of simplified pulse coupled neural network
Generally, each neuron of PCNN is composed of three parts: Input section,
connecting part (modulation section) and pulse generating part. Single neuron
model is shown in Fig. 2.
Image reconstruction algorithm: The capacitance values are measured between electrodes and they are used as original data of image reconstruction in ECT system. But these values are very different. They can vary several times to hundreds of times. So it is easy to make a big error in the result and this goes against the improvement of calculation accuracy. In the prior to image reconstruction, the measured capacitance and dielectric constant are normalized. The normalized form of capacitance C and dielectric constant variation ε are:
In the equation: CL and CH, respectively represent the capacitance between electrodes when measuring area is filled with air and the capacitance of electrodes when measuring area is filled with solid medium. εL and εH, respectively represent air and solid medium dielectric constant. represents gas/solid two-phase hybrid dielectric constant in the measuring zone.
After normalized, Eq. 1 of ECT system (the sensitivity of each pixel in the sensor measuring the area is not equal) can be transformed into:
||Structure diagram of training system of adaptive pulse coupled
In the equation: λ represents normalized capacitance matrix, K represents
normalized effective dielectric constant, S represents sensitivity coefficient
matrix. Therefore, the inverse problem can be expressed as K+ = S+λ,
in the equation, K+ is the least square solution of K, that is estimated
gray value of reconstructing image, S+ is the generalized inverse
matrix of S. The Eq. 3 is applied to the discrete mathematics
equations of PCNN (Cheng, 2009) and the following equation
can be obtained:
The subscript ij represents neuronal labeling. λij, Lij,
Uij and Eij, respectively represent feedback input, connecting
input, internal activities and dynamic threshold of the Nij capacitance
value in the matrix (Ma and Qi, 2006). M and W represent
connection weight matrixes (generally W = M). VF, VL and
VE, respectively represent amplification factors. aF,
aL and aE, respectively represent time decaying constant
and n represents the iteration number. Kij represents output with
two values, that is the image pixel gray value.
Principle of automatically setting parameters in adaptive pulse coupled neural network: In order to get optimal output, the algorithm uses BP algorithm ideas. It also uses the EBP criterion (LMS criterion and the gradient descent method) as PCNN learning standards. The adaptive parameter adjustment is proceed with the ignition cycle so as to achieve an adaptive pulse coupled neural network. The structure of the system is shown as (Fig. 3).
Ignition cycle analyzes the ignition statistical characteristics of each pixel and it can specifically and explicitly quantize the physical relationship between input and output of PCNN model:
The Eq. 6 is substituted into Eq. 9, then the following equation can be obtained:
There are four main parameters: W, β, αE and VE.
The setting of connecting matrix is relatively simple and its value is the inverse
of square of the distance between pixels. The setting values of the other three
major parameters can be obtained by using adaptive algorithm according to the
desired output. It is assumed that each neuron in PCNN is restarted only once
before ignition. If focusing on neurons of PCNN, the criterion of mean square
error is as follows:
The corresponding partial differential item is as follows:
Replacing the second Tactual in Eq. 12 with T,
then adaptive rules of each variable can be obtained as follows:
There are three key parameters (connecting strength β, time attenuation
constant of dynamic threshold aE and dynamic threshold amplitude
constant VE) in PCNN. Equations which are used to automatically set
these parameters can be deduced according to the above-mentioned rules.
Connecting strength β is one of important parameters of all parameters in PCNN, it directly influences the roughness of texture of the result image when it processes the image by using PCNN. In order to get the adaptive adjustment rule of connecting strength β, firstly partial differential equations of connecting strength β which is corresponding to firing cycle T should be solved:
Adaptive rule of parameter β can be obtained according to Eq. 13:
The ηβ is the adaptive learning rate of parameter β. The equation can be the adaptive rule in order to reduce the output error by adaptively adjusting the connecting strength β.
The time attenuation constant of dynamic threshold controls igniting time interval and it also influences the roughness of the processing image to some extent. Then the adaptive rule of time attenuation constant of dynamic threshold can be obtained as follows:
And the adaptive criterion of dynamic threshold magnitude constant can also be obtained based on the above principles:
The above deducted adaptive rules of key parameters can be applied directly
to each neuron. Then optimal parameter settings (for desired output of a specific
application) can be obtained. These parameters can make the PCNN more effective
when it does image processing tasks, while the generalization ability of the
network is improved. Each neuron uses different parameters when the firing cycle
of each neuron is adjusted.
RESULTS AND DISCUSSION
In order to verify the effectiveness of the algorithm, simulation test is done
with the 12-electrode system. Chen et al. (2008)
have designed and implemented the data acquisition system which is used to obtain
data of 12-electrode electrical capacitance tomography. The study also discusses
the pipeline grid division method. In this study, pipeline section is divided
into 1024 pixels with 32x32 grid when it is reconstructing images. There are
856 imaging units in the effective area of the pipeline section. Wang
et al. (2010) proposed a novel algorithm which is based on trust
region. The algorithm is used to reconstruct image for electrical capacitance
tomography system. The study selects typical flow patterns in the simulation
experiment. In this study, laminar flow, core flow and bubble flow are selected
to test the algorithm in the pre-set experiment. It makes statistics of filter
threshold when it is reconstructing image. The system reconstructs images by
using pulse coupled neural network and the quality of images will be compared
with the other images reconstructed by LBP, Landweber and CG. The simulating
calculation is done on a computer with dual-core Pentium 2 CPU and 2G memory
by using MATLAB.
The speed of reconstructing image is represented by using iterative number
N. The greater is N, the longer is the reconstruction time and this illustrates
the speed is slower. The algorithm sets N = 0 because the LBP method belongs
to the single step procession. Iterative step number N is determined by numerical
experiments. Chen et al. (2008) proposed a method
which measures parameters of two-phase flow and it can also reconstruct image
for electrical capacitance tomography. The equation of iterative error is discussed
in the study. The usual practice is that the iterative error should satisfy
the following equation:
Then the iteration is stopped. The space image error is chosen as evaluation index of image quality when the quality of the reconstructed image is analyzed. It is defined as follows:
|| Comparison of images which are reconstructed by using different
algorithms for 4 different flow patterns
||Errors of images which are reconstructed by using different
algorithms for 4 different flow patterns (%)
In the equation: gi(img) represents the reconstructed image vector. gi(imit) represents the distribution of media prototype image vector. I represents the index of imaging region dividing unit. n represents the total number of units in imaging area. The experimental results is shown in Table 1 (The black area represents the water and the white area represents transformer oil).
The initial parameters are setted as follows: parameters according to the expected
output are setted as: β = 0.20, aE = 0.1 and VE =
20. The parameters which are needed to adjust before adaptive network operates
should be setted as: β = 0.60, aE = 0.80 and VE =
25 and the other parameter is setted as: VL = 1. The experimental
result shows that the convergence rate of PCNN is accelerated obviously after
it is trained.
||The No. of iterations (times) of different algorithms which
are used to process 4 different flow patterns
The iterative number of the algorithm (adjustment of weights) is more than
100000 times in order to make the MSE network error reduce to 0.026. The training
error drops sharply in the first 10000 training cycles but the trend of the
decline of the error obviously slows down later.
For core flow and laminar flow, the images which are reconstructed by using
this algorithm are closer to the original pattern. The quality of the images
is better than that of the images which are reconstructed by other three algorithms.
This can be seen obviously from Table 1 and 2.
For complex bubbly flow, there is a certain gap between the imaging effect of
this algorithm and the original image pattern but it has improved greatly compared
with the other three algorithms. Experimental result shows that the iteration
steps of CG are the least which can be seen from Table 3,
while the iteration steps of this algorithm are more. After above analysis,
it is known that for the simple flow pattern and complex pattern, the quality
of the images which are reconstructed by using the algorithm in this study is
better. In addition, the images are more close to original images compared with
these images reconstructed by LBP, CG and Landweber algorithm. But the number
of iterations of this algorithm is larger.
This study presents a novel image reconstruction algorithm. It is based on pulse coupled neural network (PCNN) and it is used to process different flows in electrical capacitance tomography (ECT). The calculation steps of the algorithm are given after analyzing the basic principles of electrical capacitance tomography and PCNN. This algorithm is used in order to solve nonlinear and ill-posed problems. The values of main parameters of PCNN can be adjusted by applying gradient descent method. And it adjusts the adaptive rate in order to avoid falling into the local minimum problem. The algorithm has the advantages of small imaging error and high imaging precision. It is easy to meet the convergent requirement. Simulation results show that the quality of the images reconstructed by this algorithm is far better than LBP algorithm. In addition, they are also better than the images reconstructed by Landweber and CG and its reconstructed images are closer to the original pattern. So it provides a new and effective method for ECT image reconstruction.
This study is supported by National Natural Science Foundation of China (60572153, 60972127), Higher Education Doctoral Fund (200802140001), Natural Science Foundation of Heilongjiang Province (QC2012C059), Chunhui Plan of Ministry of Education (Z2007-1-15013) and Science and Technology Project of Department of Education in Heilongjiang Province (11541040, 12511097, 12531094).