A Novel Computational Imaging Algorithm for Electrical Capacitance Tomography

Abstract

High-precision images enable electrical capacitance tomography (ECT) to obtain more reliable measurement results, meaning that the reconstruction algorithm is particularly important. Some excellent numerical algorithms have successfully solved the inverse problem for ECT imaging, but their imaging quality is relatively low. To solve this problem, this paper proposes a new reconstruction algorithm based on regularized extreme learning machines (RELMs). The implementation of the algorithm is mainly divided into two steps: (1) according to a large number of training samples, the RELM model can be obtained by the iterative split Bregman (ISB) algorithm, which can describe the mapping relationship between the capacitance correlation coefficient and the imaging target well, and (2) the capacitance correlation coefficient is calculated, which is then used as input to the RELM model to predict the final imaging. Both simulation and experimental results show that the RELM algorithm achieves greater improvement in imaging quality and robustness, and provides new development ideas for the ECT.

1. Introduction

Electrical capacitance tomography is a promising process imaging technique. It enables visualization of multiphase flow by reconstructing the distribution of multiphase media in the pipe by measuring the capacitance vector between arrays of electrodes outside the pipe. ECT is a growing research topic, with advantages such as its wide range of applications, non-invasiveness, fast response time, high security performance, low cost, etc. It is widely used in the multiphase flow process detection of insulating media [1,2,3,4], such as monitoring flames and combustion conditions in engine combustion chambers, defect detection of thermal protection materials, measuring solid concentration in cyclone separator, pneumatic measurement of high pressure pulverized coal, etc.

Image reconstruction is an important process for visualizing the detection results of ECT systems. The imaging quality of the ECT determines whether the ECT technique can be practically implemented in engineering applications. However, ECT image reconstruction process is severely ill-conditioned, so it is necessary to study image reconstruction algorithms to weaken this property. At present, some excellent numerical algorithms have successfully solved the inverse problem for ECT imaging. Two major categories of ECT reconstruction methods exist—non-iterative and iterative image reconstruction algorithms [5]. The advantage of the non-iterative imaging algorithm is that as a “one-step” imaging algorithm, it has faster imaging speed and can complete simple imaging tasks. When faced with complex imaging targets, non-iterative imaging algorithms cannot reconstruct the measured objects well and the image quality is poor. Therefore, non-iterative imaging algorithms are mostly used as online imaging algorithms for qualitative analysis or to provide initial values for iterative imaging algorithms. Common iterative algorithms include the gene algorithm (GA) [6], the Landweber algorithm [7], the Newton–Raphson iterative algorithm (NRIA) [8], the Kalman filtering algorithm (KFA) [9], and the extreme learning machine (ELM) algorithm [10]. Compared to non-iterative imaging algorithms, iterative imaging algorithms improve the image reconstruction quality, but this improvement is limited by the high time cost due to many iterations and the difficulty in choosing the parameters of the algorithm. In the past, the aforementioned algorithms have been validated in a number of simple practical applications, which has motivated the development of the ECT techniques. However, as the demand for image reconstruction quality continues to increase, the above algorithms cannot meet the needs of the practical applications of the ECT. To achieve higher image reconstruction quality, this paper proposes a more efficient image reconstruction algorithm.

The stability and robustness of the conventional ELM algorithm itself are not very good, and the image quality is low when used directly for image reconstruction. To address this issue, combining the advantages of the ELM algorithm and the Tikhonov regularization (TR) method, a new image reconstruction algorithm aiming to enhance reconstruction accuracy is proposed. In this paper, the RELM method provides several appealing attributes enumerated in the following:

(1)

The implementation of the algorithm mainly consists of two stages: at the learning stage, according to the training sample set, the RELM model between capacitance correlation coefficient and imaging target is extracted; at the prediction stage, by taking the calculated capacitance correlation coefficient as the input of the RELM model, the corresponding reconstructed image can be predicted.

(2)

The solution method of the capacitance correlation coefficient is analyzed in detail. Based on the TR method, the objective function of the RELM algorithm is designed, in which the L₁-norm as data fidelity effectively reduces the influence of outliers in the input data on the reconstruction results, the second-order total variation (STV) norm and the L₁-norm as the regularizer effectively improves the stability of the numerical solution. The objective function is solved by the ISB method, which greatly improves the solution efficiency.

(3)

Numerical simulation and experimental results prove that the RELM algorithm improves the stability of the numerical solution, and the problem of poor imaging quality caused by inaccurate capacitance data is effectively alleviated.

The rest of this paper is organized as follows: in Section 2, we propose an imaging methodology based on the RELM. In Section 3, we detail the method for solving the capacitance correlation coefficient. In Section 4, the RELM model between the capacitance correlation coefficient and the imaging target is extracted from a large number of training samples. The numerical simulations and implementation details are described in Section 5 and Section 6. Finally, the conclusions are drawn in Section 7.

2. Imaging Methodology Based on Regularized Extreme Learning Machines

Imaging quality is crucial for ECT applications, so we must find image reconstruction algorithms with strong imaging capabilities. The idea of the RELM algorithm mainly comes from the actual ECT experimental procedure. Through many experiments, it is found that there is a certain relationship between the capacitance correlation coefficient and the imaging target. Therefore, it is particularly important to find the mapping relationship between the two and use this mapping relation to solve the problem of low reconstruction accuracy for the ECT. In this paper, based on the existing numerical methods, we propose a RELM-based computational image reconstruction algorithm that fully exploits the mapping relationship between capacitance correlation coefficients and imaging targets, and achieves high reconstruction accuracy. The specific implementation process can be carried out as follows:

(1)

The training sample set between the capacitance correlation coefficient and the imaging target is obtained through numerical simulation or experiment.

(2)

The mapping model between the capacitance correlation coefficient and the imaging target is obtained by the RELM method.

(3)

The capacitance correlation coefficient corresponding to the measured capacitance value is calculated. The calculation method is the same as that of the training sample set.

(4)

The calculated capacitance correlation coefficient is used as the input of the mapping model obtained in the second step, and the final reconstruction image is predicted.

The proposed RELM-based image reconstruction algorithm is different from the regular ELM algorithm, and their imaging process block diagrams are presented in Figure 1 and Figure 2, respectively. The RELM algorithm has the following advantages: (1) the algorithm effectively uses the prior information obtained from numerical simulations or experimental results, which greatly improves the quality of imaging results; (2) when the mapping model is obtained from the training sample set, the algorithm does not directly establish the imaging model between the measured capacitance and the imaging target, but rather establishes the mapping relationship model between the capacitance correlation coefficient and the true images of targets; and (3) based on the TR technology, the conventional ELM method is improved, which effectively enhances the stability of the numerical solution, and the solution efficiency is greatly improved by the ISB algorithm.

The RELM model has some limitations, such as resolution and number of sensors. When the number of sensors is different and the measured area is divided into different numbers of microelements, the capacitance correlation coefficients obtained are different, which means that the input of the model is different. Therefore, it is necessary to retrain the mapping model.

3. Solving for Capacitance Correlation Coefficients

For two random variables $X=\left\{x_1,x_2,\cdots x_m\right\}$ and $Y=\left\{y_1,y_2,\cdots y_m\right\}$, the correlation coefficient between them is as follows [11]:

$$\gamma \left(X,Y\right)=\frac{\sum _i\left(x_i-\overline{x_i}\right)\left(y_i-\overline{y_i}\right)}{\sqrt{\sum _i{\left(x_i-\overline{x_i}\right)}^2}\sqrt{\sum _i{\left(y_i-\overline{y_i}\right)}^2}}$$

(1)

where $\overline{x_i}$ and $\overline{y_i}$ are the mean values of $X$ and $Y$, respectively. $\gamma \left(X,Y\right)$ reflects the correlation degree of the two vectors ($X$ and $Y$), and its value is between −1 and 1.

According to the research conclusions of relevant scholars for decades, it has been a consensus in the industry that the image reconstruction model of the ECT can be expressed using the following mathematical formula when considering capacitive noise [12]:

$$C+\alpha =SG$$

(2)

where $C$ is the capacitance vector, $S$ is sensitivity matrix, $G$ is the permittivity vector, and $\alpha$ is noise capacitance vector. The image reconstruction process of the ECT is essentially a process of solving Equation (2). However, Equation (2) is only a basic model, and it is very difficult to solve it directly. Therefore, we must find effective numerical methods to solve it.

When the number of electrodes is n and the measured area is divided into t units, the high permittivity medium (${\epsilon }_2$) is placed in the first micro unit and the low permittivity medium (${\epsilon }_1$) is placed in the other units. All capacitances between electrodes are measured and normalized to obtain capacitance vector $C_{\mathrm{s}1}$. Similarly, the normalized capacitance matrix can be obtained as follows:

$$C_s={(C}_{\mathrm{s}1},C_{\mathrm{s}2},\cdots ,C_{si},\cdots ,C_{st})$$

(3)

where $C_{si}={(c_1,c_2,\cdots ,c_m)}^T$, $C_{si}\in R^{m\times 1}$, $C_s\in R^{N\times t}$, $m=\frac{n(n-1)}{2}$. $C_s$ is essentially a sensitivity field matrix.

When there is a medium to be measured in the measured area, the measured capacitance value is $C_d$($C_d\in R^{m\times 1}$). According to Equation (1), the capacitance correlation coefficient ${\gamma }_i=\gamma \left(C_{si},C_d\right)$ can be obtained, and the following capacitance correlation coefficient vector can be obtained in the same way.

$$\gamma =\left({\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_t\right)$$

(4)

where $\gamma \in R^{1\times t}$ is essentially a dielectric constant matrix and can be directly used for imaging. Reference [13] has confirmed the advantages of this method. In this paper, we take this further and use it as the input of the RELM model to enhance the imaging quality.

4. Regularized Extreme Learning Machine

In the process of numerical simulation and experiment, we found that there is a certain correspondence between the capacitance correlation coefficient and the imaging target. How to extract and utilize this relationship to improve the stability and reliability of reconstruction results has become the biggest challenge. In view of the excellent numerical performance of the ELM algorithm [14], it is used to extract nonlinear mapping relationships between the capacitance correlation coefficient and the imaging target.

4.1. Extreme Learning Machine

Machine learning algorithms represent nonlinear models of data relationships in a data-driven manner, enabling the effective prediction of new data. As a representative method of machine learning, the ELM algorithms are suitable for image reconstruction due to their advantages of theoretical analytical solutions and fast convergence [15]. The extreme learning machine belongs to the single hidden layer feedforward neural network (SLFN), and it can automatically adjust the structure based on the prediction results of current data segments without manually adjusting parameters and retraining the learning model [16].

For a group of training samples, $\left(x_i,y_i\right),i=\mathrm{1,2},\cdots ,N$, $x_i$ ($x_i={\left[x_{i1},x_{i2},\cdots x_{in}\right]}^T\in R^n$) is the i-th sample, $y_i$($y_i={\left[y_{i1},y_{i2},\cdots y_{im}\right]}^T\in R^m$) is its corresponding label, where n and m are the number of features and categories, respectively [16]. The basic structure of the ELM algorithm is shown in Figure 3. The output of the hidden layer and output layer can be expressed by the following equations [15]:

$$h=g(a,b,x)$$

(5)

$$h\left(x_i\right)V=y_i,\hspace{1em}i=\mathrm{1,2},\cdots ,N$$

(6)

where $g\left(x\right)$ is the activation function.

Equation (6) can be written in matrix form as follows:

$$HV=Y$$

(7)

where

$$H={\left[\begin{array}{cccc}g(a_1,b_1,x_1)& g(a_1,b_1,x_2)& \cdots & g(a_1,b_1,x_N)\\ g(a_2,b_2,x_1)& g(a_2,b_2,x_2)& \cdots & g(a_2,b_2,x_N)\\ ⋮& ⋮& \ddots & ⋮\\ g(a_L,b_L,x_1)& g(a_L,b_L,x_2)& \cdots & g(a_L,b_L,x_N)\end{array}\right]}^T$$

(8)

$$V=\left[\begin{array}{c}\begin{array}{c}{v}_1^T\\ v_2^T\end{array}\\ \begin{array}{c}⋮\\ v_L^T\end{array}\end{array}\right],\hspace{1em}Y=\left[\begin{array}{c}\begin{array}{c}{y}_1^T\\ y_2^T\end{array}\\ \begin{array}{c}⋮\\ y_N^T\end{array}\end{array}\right]$$

(9)

where $H$ is the output matrix of the hidden layer, $V$ is the output weight matrix connecting the hidden layer and the output layer, $Y$ is the output matrix of the output layer, a is the input weight, and b is the offset of the hidden layer [17].

In the ELM algorithm, the input weight, a, and the offset, b, are random values without manual setting. When they are determined, the unique output weight matrix, V, can be obtained according to Equation (7). The calculation flow of output weight matrix is shown in Figure 4.

4.2. Regularized Extreme Learning Machine

As a powerful machine learning algorithm, the ELM algorithm has been successfully applied to the ECT imaging, but the stability and robustness of the ELM algorithm itself are not very good. When the conventional ELM algorithm is directly used for image reconstruction, it leads to the following two problems: (1) when directly calculating the least square solution of the cost function, the user cannot change the parameters of the ELM algorithm according to the characteristics of the training sample set, which makes the controllability of the conventional ELM algorithm worse; and (2) when the data set contains some disturbances the conventional ELM algorithm will be seriously affected, making the imaging quality worse. To solve these intractable problems, this paper proposes the RELM algorithm based on the TR technique. The TR theory usually transforms ill-posed problems into optimization problems to solve. Its objective function consists of two parts: the data fidelity term and the regularizer [18]. The generalized expression is as follows:

$$\underset{V}{\min }\left\{P\left(V\right)+{\sum }_{i=1}^n{\lambda }_i{Q}_i\left(V\right)\right\}$$

(10)

where $P\left(V\right)$ is the data fidelity term, which measures the difference between $HV$ and $Y$, $Q_i\left(V\right)$ is the regularizer, and ${\lambda }_i$ is the regularization parameter, which adjusts the proportion of the data fidelity term and the regularizer, and ${\lambda }_i$ > 0 [19].

When designing the objective function based on the TR technology, introducing the prior information and the characteristics of the imaging targets into the inverse problem-solving process is helpful to improve the accuracy and stability of the numerical solution. Theoretically, a variety of estimation functions can be used as the data fidelity term and the regularizer, and different choices determine the different stability and accuracy of the numerical solution, so their selection is particularly important. Because reconstructed images in ECT generally have sparse features, the L₁ norm has the advantage of generating sparse solutions compared to other norms. Therefore, we deploy the L₁-norm as the data fidelity term to enhance the accuracy of numerical solution [20], whose mathematical expression is as follows:

$$P\left(V\right)={‖HV-Y‖}_1$$

(11)

In addition to the data fidelity term of the solution, the design of the regularizer is also very crucial. The first-order total variation (FTV) regularization has the ability to preserve the discontinuity characteristics of the measured region [21], but this method commonly leads to the ladder effect, so we deploy the STV regularization as the regularizer to get a more stable numerical solution. The STV norm has incomparable advantages in enhancing the sparse characteristics and improving imaging ability for the edge of the imaging target compared to the FTV norm [22]. However, when the STV regularization is employed alone, the imaging results may still have the ladder effect. Therefore, a new penalty function must be added to eliminate the imaging artifacts caused by the STV regularization. Because most imaging targets have sparse characteristics, we add additional sparse constraints through the L₁ regularization. The mathematical expression of the regularization term is as follows:

$$\mathrm{Q}\left(V\right)={\lambda }_1{‖V‖}_1+{\lambda }_2\left(‖D_{xx}V‖+‖D_{xy}V‖+‖D_{yx}V‖+‖D_{yy}V‖\right)$$

(12)

where $D_{xx}$, $D_{xy}$, $D_{yx},$ and $D_{yy}$ are different difference operators, and ${\lambda }_1$ and ${\lambda }_2$ are regularization parameters.

As a result of submitting Equations (11) and (12) into Expression (10), a new objective function can be obtained:

$$\underset{V}{\min }\left\{{‖HV-Y‖}_1+{\lambda }_1{‖V‖}_1+{\lambda }_2\left(\begin{array}{c}‖D_{xx}V‖\\ +‖D_{xy}V‖+‖D_{yx}V‖+‖D_{yy}V‖\end{array}\right)\right\}$$

(13)

Suppose that ${‖WV‖}_1=‖D_{xx}V‖+‖D_{xy}V‖+‖D_{yx}V‖+‖D_{yy}V‖$, then Expression (13) can be rewritten as:

$$\underset{V}{\min }\left\{{‖HV-Y‖}_1+{\lambda }_1{‖V‖}_1+{\lambda }_2{‖WV‖}_1\right\}$$

(14)

where $W={[D_{xx},D_{xy},D_{yx},D_{yy}]}^T$.

To solve Expression (14), the ISB algorithm is preferred. Expression (14) can be converted to the following form [23]:

$$\left\{\begin{array}{c}\underset{V}{\min }\left\{{‖d_1‖}_1+{\lambda }_1{‖V‖}_1+{\lambda }_2{‖d_2‖}_1\right\}\\ s. t. d_1=HV-Y,d_2=WV\end{array}\right.$$

(15)

According to the ISB algorithm theory, Expression (15) can be rewritten as follows:

$$\left(V_{k+1},d_{1,k+1},d_{2,k+1}\right)=\underset{V,d_1,d_2}{\min }\left\{\begin{array}{c}{‖d_1‖}_1+{\lambda }_1{‖V‖}_1+\frac{{\lambda }_3}{2}{‖d_1-HV+Y-b_{1,k}‖}_2^2\\ +{\lambda }_2{‖d_2‖}_1+\frac{{\lambda }_4}{2}{‖d_2-WV-b_{2,k}‖}_2^2\end{array}\right\}$$

(16)

$$b_{1,k+1}=b_{1,k}+H{V}_{k+1}-Y-d_{1,k+1}$$

(17)

$$b_{2,k+1}=b_{2,k}+W{V}_{k+1}-d_{2,k+1}$$

(18)

where ${\lambda }_1$ and ${\lambda }_2$ are regularization parameters.

Equation (16) can be decoupled into three simple sub equations for solution, as shown in Equations (19)–(21) [24]:

$$V_{k+1}=\underset{V}{\min }\left\{\begin{array}{c}{\lambda }_1{‖V‖}_1+\frac{{\lambda }_3}{2}{‖HV-Y+b_{1,k}-d_{1,k}‖}_2^2\\ +\frac{{\lambda }_4}{2}{‖WV+b_{2,k}-d_{2,k}‖}_2^2\end{array}\right\}$$

(19)

$$d_{1,k+1}=\underset{d_1}{\min }\left\{{‖d_1‖}_1+\frac{{\lambda }_3}{2}{‖d_1-H{V}_{k+1}+Y-b_{1,k}‖}_2^2\right\}$$

(20)

$$d_{2,k+1}=\underset{d_2}{\min }\left\{{\lambda }_2{‖d_2‖}_1+\frac{{\lambda }_4}{2}{‖d_2-W{V}_{k+1}-b_{2,k}‖}_2^2\right\}$$

(21)

Solving Equation (16) is equivalent to solving Equations (19)–(21). The solutions of the above three equations can be obtained by iteratively shrinking the threshold (IST) algorithm [25], and the iterative formats of their solutions are as follows:

$$V_{k+1}=soft(V_k-{\lambda }_3{H}^T\left(H{V}_k-Y+b_{1,k}-d_{1,k}\right)-{\lambda }_4{W}^T\left(W{V}_k+b_{2,k}-d_{2,k}\right),{\lambda }_1)$$

(22)

$$d_{1,k+1}=soft(H{V}_{k+1}-Y+b_{1,k},\frac{1}{{\lambda }_3})$$

(23)

$$d_{2,k+1}=soft(W{V}_{k+1}+b_{2,k},\frac{{\lambda }_2}{{\lambda }_3})$$

(24)

According to the above discussion, we have obtained a new image reconstruction algorithm, whose execution steps can be summarized as Algorithm 1.

Algorithm 1 The RELM algorithm

Input: parameter initialization

If the iterative condition is met, then

Update $V_{k+1}$: $V_{k+1}=soft(V_k-{\lambda }_3{H}^T\left(H{V}_k-Y+b_{1,k}-d_{1,k}\right)-{\lambda }_4{W}^T\left(W{V}_k+b_{2,k}-d_{2,k}\right),{\lambda }_1)$

Update $d_{1,k+1}$: $d_{1,k+1}=soft(H{V}_{k+1}-Y+b_{1,k},\frac{1}{{\lambda }_3})$

Update $d_{2,k+1}$: $d_{2,k+1}=soft(W{V}_{k+1}+b_{2,k},\frac{{\lambda }_2}{{\lambda }_3})$

Update $b_{1,k+1}$: $b_{1,k+1}=b_{1,k}+H{V}_{k+1}-Y-d_{1,k+1}$

Update $b_{2,k+1}$: $b_{2,k+1}=b_{2,k}+W{V}_{k+1}-d_{2,k+1}$

k++

Else

Output: $Y(H{V}_{k+1}=Y)$

end

The RELM algorithm is an improved and optimized version of the ELM technique, integrating the advantages of the ELM technique and TR technology, effectively overcoming the ill-conditioned nature of the ECT inverse problem and the adverse effects of capacitive inaccuracy. The RELM algorithm provides a significant improvement in reliability and reconstruction precision compared to the ELM technique. Its process is shown in Figure 5.

5. Numerical Simulation and Analysis

To validate the reconstruction effect of the regularized extreme learning machine (RELM) algorithm, we carried out a series of numerical simulation experiments. As a result of comparing the reconstruction results of the linear back projection (LBP) technique, the Tikhonov regularization (TR) technique, the truncated singular value decomposition (TSVD) technique, the conjugate gradient (CG) technique, the Landweber technique, the simultaneous iterative reconstruction technique (SIRT), the Gaussian process regression (GPR) method, the extreme learning machine (ELM) method, and the RELM technique, this section qualitatively and quantitatively analyzes the excellent numerical characteristics of the RELM algorithm. The square sensor with 12 electrodes, as shown in Figure 6, is adopted in this simulation study. The parameters of the sensor are listed in

Table 1. Structural parameters of the ECT sensor.

Parameter	Size
R1	80 mm
d1	12 mm
d2	9 mm
d3	60 mm
d4	63 mm

. Through numerical simulation and experimental measurements, we obtained 1500 sets of training samples in this study. The test sample set data is not included in the training sample set. In the sample generation process for the simulation experiments as well as the actual experiments, the inputs are the capacitance correlation coefficients and the outputs are the corresponding true images. In

Table 2. Training sample pairs.

	(a)	(b)	(c)	(d)	(e)	(f)	(g)
input	${\gamma }_a$	${\gamma }_b$	${\gamma }_{\mathrm{c}}$	${\gamma }_d$	${\gamma }_{\mathrm{e}}$	${\gamma }_f$	${\gamma }_g$
output

, we show several training sample pairs. We use relative error (RE) to evaluate the quality of reconstruction results, which is defined as follows:

$$RE=\frac{{‖{\widehat{\mathit{Y}}}_{\mathit{i}}-{\mathit{Y}}_{\mathit{i}}‖}_2}{{‖{\mathit{Y}}_i‖}_2}$$

(25)

where $Y$ is the gray value of the real image, and $\widehat{\mathit{Y}}$ is the gray value of the predicted image. The smaller the RE, the higher the reconstructed image quality.

5.1. Case 1

To test the performance of the RELM algorithm, seven groups of simulation experiments are carried out in this section, and the imaging quality of various algorithms is compared. The seven kinds of dielectric constant distributions shown in Figure 7 are taken as the phantom objects for numerical simulation, in which the high permittivity is the yellow area and the low permittivity is the blue area. In Figure 7a, the radius of the circle is 10 mm. For Figure 7b, the radius of the circle is 8 mm. The height of laminar flow in Figure 7c is 20 mm. In Figure 7d and Figure 7g, the sides of the square are 16 mm and 15 mm, respectively. For Figure 7e, there are two rectangles with a length of 60 mm and a width of 15 mm. In Figure 7f, the side length of the square is 10 mm, and the radius of the circle is 5 mm. The TR algorithm with the parameter value of 0.01 provides the initial value for the iterative algorithm mentioned in this paper. The parameters of the CG algorithm, the Landweber algorithm, and the SIRT are listed in

Table 3. Parameters of the CG algorithm.

Test Objects	(a)	(b)	(c)	(d)	(e)	(f)	(g)
Number of iterations	78	92	159	243	298	413	295
Relax factor	0.03	0.03	0.03	0.03	0.03	0.03	0.03

,

Table 4. Parameters of the Landweber algorithm.

Test Objects	(a)	(b)	(c)	(d)	(e)	(f)	(g)
Number of iterations	40	64	75	182	390	549	343
Relax factor	1	1	1	1	1	1	1

and

Table 5. Parameters of the SIRT.

Test Objects	(a)	(b)	(c)	(d)	(e)	(f)	(g)
Number of iterations	80	105	223	416	786	424	448
Relax factor	0.04	0.04	0.04	0.04	0.04	0.04	0.04

, respectively. The number of hidden layer nodes in both ELM and RELM is 1000. The reconstruction results of various algorithms are shown in Figure 8 and Figure 9. To quantitatively evaluate the reconstruction accuracy of the various algorithms, we calculate the REs, shown in

Table 6. Imaging errors (%).

	(a)	(b)	(c)	(d)	(e)	(f)	(g)
LBP	40.44	56.60	41.47	45.70	38.45	46.88	57.41
TR	32.21	51.99	71.78	92.09	59.92	42.67	57.19
TSVD	23.64	47.11	72.15	121.49	44.31	37.70	44.20
CG	24.67	25.50	29.13	30.69	32.15	25.39	20.61
Landweber	22.54	21.78	27.99	30.66	35.11	28.99	26.24
SIRT	20.05	21.08	27.51	29.85	32.31	24.54	28.07
GPR	9.06	8.79	9.64	8.69	10.31	7.62	7.01
ELM	9.81	7.63	8.45	7.61	8.78	7.93	7.19
RELM	2.13	2.06	1.09	1.68	1.49	1.84	1.31

, between the reconstructed and true permittivity.

The imaging results of the LBP technique, the TR technique, and the TSVD method are shown in Figure 8. Although they can roughly recognize the contour of the phantom objects, their imaging results are poor when faced with complex imaging targets. In Figure 9, we can see that the reconstruction results of iterative algorithm, i.e., the CG technique, the Landweber technique, and the SIRT, are significantly superior over non-iterative algorithms. However, the imaging results of all three algorithms still have distortion, which cannot reconstruct the phantom objects well.

In Figure 9(a4–g4) and Figure 9(a5–g5), the GPR algorithm and the ELM algorithm take full use of the advantages of the machine learning algorithm, which considerably improves the final imaging quality compared to the aforementioned traditional imaging algorithms in this paper. When the imaging target is complex or has a right angle, the imaging quality of the classical imaging algorithms is poor. Due to the learning of a large number of datasets, the GPR algorithm and the ELM algorithm can perform better on imaging tasks. However, from Figure 9(a4–g4) and Figure 9(a5–g5), we see that for seven kinds of imaging targets, the imaging results of the ELM algorithm and the GPR algorithm still have different degrees of artifacts. From Figure 9(a6–g6), we observe that the RELM algorithm has less artifacts compared to the ELM algorithm, and the imaging results are closest to the phantom objects. The reason why the RELM can improve the image reconstruction accuracy in engineering applications is that it not only has strong generalization ability and classification capabilities, but also improves the robustness of the numerical solution. From the above analysis, the RELM algorithm can be divided into two steps: (1) the RELM mapping model between the capacitance correlation coefficient and the imaging target is extracted based on the training samples, and (2) the capacitance correlation coefficient is calculated and then it is used as the input of the RELM model to predict the final imaging. The implementation process is relatively simple, and there is no need to manually set the weights and offsets of the hidden layers, which is in line with the practical needs of the project.

High-quality reconstructed images are an important guarantee for reliable measurement results. From the results listed in Table 6, we see that, compared with other algorithms mentioned, the RELM algorithm achieves the highest imaging accuracy for different simulation targets. For phantom objects in Figure 7, the REs of the RELM algorithm are 2.13%, 2.06%, 1.09%, 1.68%, 1.49%, 1.84%, and 1.31%, respectively. The quantitative error comparison results successfully verify the RELM algorithm has high reliability in solving the inverse problem of ECT and greatly improves the reconstruction accuracy.

5.2. Case 2

The robustness of image reconstruction algorithm is another important index to evaluate the algorithm. In practical application, the ECT system will be affected by noise when collecting capacitance data. Small input errors will cause the output result to deviate significantly from the true result. Consequently, to make the simulation results of the RELM algorithm more convincing, we add 5% and 10% random noise to the original capacitance to assess the RELM algorithm. The phantom objects to be reconstructed and the imaging results are presented in Figure 7 and Figure 10 and Figure 11. We executed 100 computations, and the image error mean is displayed in

Table 7. Image error mean (%).

	(a)	(b)	(c)	(d)	(e)	(f)	(g)
0% noise	2.13	2.06	1.09	1.68	1.49	1.84	1.31
5% noise	3.78	3.56	2.69	2.16	2.64	2.93	2.27
10% noise	4.51	4.16	4.79	3.61	4.27	3.79	3.43

. The noise level is defined by the following equation:

$$\eta =\frac{‖\widehat{C}-C‖}{‖C‖}\times 100\%$$

(26)

where $\eta$ is the capacitance noise level, $\widehat{C}$ is the noisy capacitance data, and $C$ is the raw capacitance data.

From the reconstruction results in Figure 10 and Figure 11, we find that under the noise levels of 5% and 10%, the imaging results of phantom objects are still satisfactory, which shows that the RELM algorithm has strong robustness and stability under different noise levels. As can be seen quantitatively from the image error mean values in Table 7, the image errors still keep a small value under 10% noise level. The image error means of phantom object (a)–phantom object (g) are 4.51%, 4.16%, 4.79%, 3.61%, 4.27%, 3.79%, and 3.43%, respectively. The conclusion of the qualitative analysis and quantitative analysis shows that the RELM algorithm has a strong anti-noise performance under different levels of noise for various complex conditions.

5.3. Case 3

When acquiring the RELM mapping model between the capacitance correlation coefficient and the imaging target, it is first necessary to confirm the number of hidden layer nodes (NHLNs). In the actual numerical simulation and experimental process, it is found that the NHLN has an impact on the quality of imaging results. To explore the mechanism of interaction between image accuracy and the NHLN, a series of simulation experiments are carried out in this section, focusing on the relationship between the NHLN and the image errors of the imaging results. The relationship between image errors and NHLN is intuitively illustrated by numerical methods. The reconstructed targets are shown in Figure 7, and the image error as a function of the NHLN is shown in Figure 12.

From Figure 12, we observe that the image errors of the seven kinds of phantom objects decrease rapidly with the increase of the NHLN. When the NHLN is greater than 700, the image errors change very little. However, it is worth noting that although increasing the NHLN can increase the imaging quality of the RELM algorithm, it also increases the computational burden. Especially when the training sample set is large, it increases the time cost significantly. Therefore, in the actual ECT application, the relationship between image errors and the NHLN should be balanced, and the appropriate NHLN should be selected according to the actual needs.

6. Experiment Results and Analysis

A series of experiments are carried out to validate the performance of the proposed method in practical applications. As shown in Figure 13, the ECT imaging system is built by our research team and mainly consists of the following parts: (1) the sensor device. This part is a 12-electrode square sensor with a copper foil wrapped cylindrical shield on the outside, whose cross section is shown in Figure 7. The size parameters of the sensor are presented in Table 1. (2) Data acquisition system. The data acquisition system can switch up to 12 channels, and its resolution is 0.5 Ff. (3) Imaging system. In this experiment, quartz sand is used as the measured material with high dielectric constant, and air is used as the background material with low dielectric constant. The measured objects are plastic polycarbonate tubes filled with quartz sand of different diameters, as shown in (4) in Figure 13. We implement three sets of experiments, and their two-dimensional imaging targets are shown as (a)–(c) in Figure 14. These targets include a concentric rectangular ring (the side length of the large square = 40 mm, and the thickness = 15 mm), big circle (diameter = 16 mm), and small circle (diameter = 6 mm). The empty field calibration and the full field calibration are performed by filling the square tube with air and quartz sand, respectively. For the imaging targets (a)–(c) in Figure 14, the NHLN in the ELM algorithm and the RELM algorithm is 1000, and the parameters of various iterative algorithms are listed in

Table 8. Parameters of the CG algorithm.

Test Objects	(a)	(b)	(c)
Number of iterations	274	86	108
Relax factor	0.03	0.03	0.03

,

Table 9. Parameters of the Landweber algorithm.

Test Objects	(a)	(b)	(c)
Number of iterations	216	79	127
Relax factor	1	1	1

and

Table 10. Parameters of the SIRT.

Test Objects	(a)	(b)	(c)
Number of iterations	527	82	149
Relax factor	0.04	0.04	0.04

.

Table 11. Experimental results of non-iterative algorithms.

	(a)	(b)	(c)
LBP
TR
TSVD

and

Table 12. Experimental results of iterative algorithms.

	CG	Landweber	SIRT	GPR	ELM	RELM
(a)
(b)
(c)

are the images reconstructed of various imaging algorithms, and we quantitatively compare the imaging quality using REs, as presented in Figure 15.

From the image results in Table 11 and Table 12, we can see that the imaging quality of the iterative imaging algorithm is higher than that of the non-iterative imaging algorithm, but the distortion and deformation are still large. From Table 12, we can observe that the GPR algorithm and the ELM algorithm greatly improve the imaging quality, but their imaging results have a certain degree of artifacts due to its poor stability. The results in Table 12 indicate that the RELM algorithm provides a significant improvement for the four kinds of imaging targets, and they are better than other conventional imaging algorithms. In Figure 15, we can clearly see that the image errors of the RELM algorithm are the smallest for the four kinds of experiments, which are 1.43%, 2.54%, and 2.39%, respectively. From experimental results, we conclude that the RELM algorithm effectively solves the image reconstruction problem of the ECT and obtains satisfactory imaging results.

7. Conclusions

To alleviate the ill-conditioned characteristics of the ECT inverse problem, an image reconstruction algorithm based on the RELM is proposed. The main conclusions of this paper are summarized as follows:

(1)

Combined with the prior information of the imaging targets, the RELM algorithm improves the imaging quality. The implementation of the algorithm mainly includes two stages: at the first stage, according to a large number of training samples, the RELM model solved by the ISB algorithm between the capacitance correlation coefficient and the imaging target is extracted; at the second stage, the correlation coefficient of the measured capacitance of ECT system is calculated, which is then used as input for the RELM model to predict the final imaging.

(2)

The solution method of capacitance correlation coefficient is introduced. The newly designed objective function based on L1-norm and STV norm can better model the inverse problem of the ECT, which alleviates the adverse effects caused by the inaccuracy of capacitance data and enhances the sparse characteristics of imaging targets. The ISB method greatly improves the solution efficiency for the objective function.

(3)

Numerical simulation results verify the effectiveness of the RELM algorithm. For the seven kinds of complex flow patterns shown in Figure 7, the RELM algorithm achieves the best imaging quality and the smallest image errors compared to the other above-mentioned algorithms. Under different noise levels, the RELM algorithm has strong robustness and noise resistance. When the number of hidden layer nodes of the RELM is greater than 700, the image error remains at a low level and the attenuation degree is small.

(4)

In this study, a set of the ECT system is built to realize on-line imaging. The experimental results proved that the RELM algorithm has feasibility and effectiveness in the actual ECT image reconstruction process.

This paper proposes a new ECT imaging algorithm. Its advantages are verified by numerical simulations and experiments. The high-precision imaging quality of the RELM algorithm provides a reliable basis for studying the internal mechanism of the imaging target, which should be further studied in the future.