Iterative computational imaging method for flow pattern reconstruction based on electrical capacitance tomography

Journal Pre-proofs Review Iterative computational imaging method for flow pattern reconstruction based on electrical capacitance tomography Hongbo Guo, Shi Liu, Hongyan Chen, Shanxun Sun, Jiankang Ding, Hongqi Guo PII: DOI: Reference:

S0009-2509(19)30922-4 https://doi.org/10.1016/j.ces.2019.115432 CES 115432

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Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

13 May 2019 20 September 2019 11 December 2019

Please cite this article as: H. Guo, S. Liu, H. Chen, S. Sun, J. Ding, H. Guo, Iterative computational imaging method for flow pattern reconstruction based on electrical capacitance tomography, Chemical Engineering Science (2019), doi: https://doi.org/10.1016/j.ces.2019.115432

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Iterative computational imaging method for flow pattern reconstruction based on electrical capacitance tomography Hongbo Guo 1,*, Shi Liu 2, Hongyan Cheng 1, Shanxun Sun 1, Jiankang Ding 3 and Hongqi Guo 4

1. School of Control and Computer Engineering, North China Electric Power University, Changping District, Beijing, 102206, China. 2. School of Energy, Power and Mechanical Engineering, North China Electric Power University, Changping District, Beijing 102206, China. 3. Engineering Technology Center, North China Power Engineering Co., Ltd. of China Power Engineering Consulting Group, Xicheng District, Beijing, 100120, China. 4. Electrical Department, Shanxi Yanbei Coal Industry School, Pingcheng District, Datong, 037005, China. * Correspondence: Hongbo Guo; E-mail address: [email protected]; Tel.: +86-010-61772482.

Abstract: The electrical capacitance tomography (ECT) is a promising measurement technique, which tries to reconstruct the permittivity distribution in a measurement domain by solving an inverse problem. Low quality images narrow the applicability of the technique. To address the challenge, a new cost function, which considers model deviation and measurement noises, is devised to model the ECT reconstruction problem. The soft thresholding method and the fast-iterative shrinkage thresholding technique (FIST) are embedded into the iterative split Bregman (ISB) method to solve the devised objective functional. The numerical and experimental results indicate that the proposed ECT imaging technique not only mitigates the ill-posed nature, but also improves the reconstruction quality.

1

Keywords: Flow pattern imaging; Image reconstruction; Split Bregman algorithm; Regularization method; Electrical capacitance tomography

1. Introduction As an effective industrial measurement technique, the electrical capacitance tomography (ECT) has many significant merits and has been deployed to visualize the permittivity distribution in process industries, such as the measurement of the gas-oil flow in a vertical riser section (Omar, et al., 2018), the visualization measurement of the surface shift in a wet soil freezing process (Liu et al., 2016), the imaging of the moisture flow process in the cementbased material (Voss et al., 2016), the real-time deposition monitoring of pipelines in a waxy crude oil flow process (Mei et al., 2015), the visualization measurement of the fluid flow (Qiu et al., 2014) and the moisture content distribution of granules (Rimpiläinen et al., 2012) in a fluidized bed, the flow pattern identification of the gas-liquid flow in a trickle bed (Atta et al., 2010; Wang et al., 2014), the imaging of the filtration of two suspension flows process in a monolith reactor (Hamidipour et al., 2010), etc. A suitable image reconstruction algorithm is critical in practical applications, but the inherent ill-posed nature makes the solution process intractable. In order to overcome the above disadvantage and obtain a high-quality solution, numerous imaging algorithms have been proposed. There are two kinds of imaging methods, the non-iterative methods and the iterative imaging techniques. The popular non-iterative methods involve the linear back-projection method

2

(LBPM) (Sun et al., 2015; Xie et al., 1989), the Tikhonov regularization technique (TRT) (Lei et al., 2011; Peng et al., 2007), the truncated singular value decomposition method (TSVDM) (Li et al., 2011), etc. The LBPM is simple, and achieves the online imaging. However, the method often yields low-quality reconstructions. The TRT is a prevalent technique for tackling ill-posed problems, but its applicability is narrowed by the over-smoothing effect. The TSVDM ensures the numerical stability by discarding the small singular values of the sensitivity matrix. In practices, it is difficult to determine a proper truncation thresholding value. The non-iterative algorithms can achieve fast reconstruction, but often lead to incorrect reconstructions. Common iterative methods include the iteration Landweber algorithm (ILA) (Yang et al., 1999), the algebraic reconstruction technique (ART) (Li et al., 2015; Wen et al., 2010), the simultaneous iterative reconstruction technique (SIRT) (Ren et al., 2015; Su et al., 2000), the conjugate gradient method (CGM) (Zhao et al., 2007), the sparsity representation based the imaging method (Zhang et al., 2015), etc. The ILA is a popular imaging method for the ECT reconstruction. But the algorithm is plagued by the semi-convergence, causing large reconstruction errors. The reconstruction quality of the ART and the SIRT is similar, but the algorithms require implementing multiple iterations before obtaining an acceptable solution (Lei et al., 2017; Peng et al., 2000). The CGM, utilizing the conjugate gradient direction of the objective function, has a low computing complexity. Using the sparsity regularization to execute the image reconstruction is a good idea, and a crucial task is to seek for a suited sparse representation method. Recently, some scholars have proposed many iterative methods. For example, Ye et al. (2017) reported the low-rank decomposition based imaging method. Moura et al. (2017) introduced

3

the redundant sensitivity matrix based algorithm. The Rudin-Osher-Fatemi model has been employed to solve the ECT inverse problem (Chen et al., 2017; Wu and Tai, 2010). Lei et al. (2009) extended the TRT to increase the quality of a solution, and the simulation results tested the effectiveness of the proposed method. Yan et al. (2014) proposed an effective iterative algorithm with the fuzzy thresholding, which reduced imaging errors and decreased the number of iterations. Hu et al. (2016) constructed a Landweber based method, which utilized interelectrode capacitance to select automatically the optimal sensitive matrix. The experiments verified that the algorithm can increase the reconstruction performance and reduce deformations. The total variation regularization method has been introduced by Yang et al. (2010) for the ECT imaging task, and its feasibility and effectiveness were numerically tested. The total variation method is a popular regularization method, which has found wide applications. But, it is difficult for the method to reconstruct high-quality images because of the staircasing effect (Lysaker and Tai, 2006). The above-mentioned algorithms emphasize the inaccuracy of capacitance data, but ignore model deviation that deteriorates imaging results. To address such difficulty, we propose an effective imaging model highlighting model deviation and measurement noises. By devising a cost function, in this study we transform the inverse problem to be an optimization problem. A crucial task in devising the cost function is to seek for a suited regularizer. Researchers have developed many effective regularizers for better reconstructions, such as the L0 norm, the L1 norm, the L2 norm, the Lp norm, the total variation norm, and more. The L0 norm can ensure the sparsity of a solution, but its solution is intractable. In order to address the above problem, the scholars present the L1 norm to overcome the drawbacks.

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(Zhang et al., 2011; Zhao et al., 2018). Therefore, in this work the L1 norm, as a regularizer, is inserted into the cost function. After a loss model is determined, another key topic is to search for a powerful numerical method. Common solvers, e.g., the alternating direction method of multipliers method (Bot and Csetnek, 2018; Jiao et al., 2016), the iterative split Bregman (ISB) method (Cai et al., 2010; Elvetun and Nielsen, 2016; Isaac et al., 2017), the homotopy theory (Ghannadi et al., 2018; Trandinh et al., 2018), etc., can solve the problem. In this study, an effectively hybrid algorithm that takes advantage of the ISB method, the fast-iterative shrinkage thresholding (FIST) method and the soft thresholding method is proposed to solve the developed objective function more effectively. We develop a powerful imaging model in conjoint with a new solver to reduce reconstruction artifacts in this paper, and the contributions of this work are summarized as below: (1) A powerful imaging model is proposed to emphasize the inaccuracy of the measurement data and model deviation, and a new cost function is proposed to model the ECT inverse problem, in which the L1 norm is used as the data fidelity term to alleviate the sensitivity of solution to noises and the L1 norm of the unknown variable is used as regularizer to strengthen the assumed sparsity of the imaging targets. (2) The soft thresholding method and the FIST method, as effective sub-problem solvers, are embedded into the ISB method to solve the devised cost function, reducing the computation complexity and cost. (3) Numerical results indicate that the devised imaging model not only reconstructs fine details but also improves the robustness compared with the other popular imaging techniques.

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(4) The experiment results implies that the proposed method is competent in practical reconstruction tasks, and leads to the improved imaging quality. The structure of this paper is organized as follows. Section 2 introduces the ECT imaging model. Section 3 introduces a new cost function, which is followed by our solver in Section 4. Section 5 provides the numerical and experimental results, and analyzes the comparison results, which is followed by the conclusions of this study in Section 6.

2. Problem formulation The ECT tries to reconstruct the distribution of permittivity by solving an inverse problem (Neumayer et al., 2012; Warsito and Fan, 2001). In this section, we introduce a common imaging model, which is followed by our extended imaging model. 2.1. Common imaging model In practical applications, the electrostatic field within a sensor can be mathematically described by (Haddadi and Maddahian, 2016):

  ( (h, g ) (h, g ))  0

(1)

where  (h, g ) and  (h, g ) represent the permittivity and the potential at point (h, g ) . Therefore a simplified mathematical model for calculating the capacitance is formulated as (Yang and Peng, 2003):

Cij  

Q 1     (h, g)(h, g)d V V 

(2)

where C ij and Q are the capacitance and the electric charge, respectively; V and  stand for the electric potential between the electrode i and j and the surface functions of electrodes i and j, respectively. 6

For convenient computation, the ECT image reconstruction is usually linearized by (Yang et al., 1999; Yang and Peng, 2003):

Ms  y

(3)

where y stands for the capacitance vector with the dimensionality of m  1 ; s stands for the permittivity distribution with the dimensionality of n  1 ; M is called the sensitivity field matrix, and its dimensionality is m  n . With the focus on the measurement noise, the ECT imaging model can be rewritten as (Ye et al., 2016):

Ms  y  r

(4)

where r represents a measurement noise vector with the dimensionality of m  1 . 2.2. Extended imaging model Without the consideration of the inaccuracy of the imaging model, Eq. (4) only focuses on the measurement noise. The model deviation mainly derives from the following two aspects: (1) Eq. (4) is an approximation model of the physical measurement process; (2) the sensitivity matrix is approximately calculated by the finite element method (Lei et al., 2011; Yang and Peng, 2013). To address the above difficulties and improve the imaging quality, an extended model is introduced, i.e.,

Ms  l  y  r

(5)

where l means model deviation. One of main difference between Eq. (5) and the common ECT imaging models depends on whether the inaccuracy of the measurement data and model deviation are simultaneously considered.

7

The major task in Eq. (5) is to estimate the variables s and l . Eq. (5) is an ill-posed problem, and a small fluctuation in the input data will lead to a large change in the solution, making the final solution meaningless. Therefore, Eq. (5) results in the difficulty in the numerical computation. Special techniques are required to stabilize the numerical solution, such as regularization method (Bertero et al., 1988; Lei et al., 2009; Li et al., 2011; Tikhonov and Arsenin, 1978). As a result, finding a numerical algorithm that can improve the quality of a solution while stabilize the numerical solution is crucial. 3. Objective function It is not easy to solve Eq. (5) directly. In general, the TRT can tackle the inverse problem effectively, and has been used and studied in many fields. The method can transform the original problem into a general optimization problem by constructing a proper cost function with the data fidelity term and the regularizers (Bertero et al., 1988; Brezinski et al., 2003; Tikhonov and Arsenin, 1978). In this study, the TRT is employed, and thus the solution of Eq. (5) is converted into a general optimization problem in the form of (Guan et al., 2012; Vauhkonen et al., 1998):





min P( M , l , y )   i 1 i Qi ( s ) s ,l

n

(6)

where P(M, l, y) represents the data fidelity term; Qi (s ) stands for the regularizer; i  0 is called the regularization parameter. In order to deploy Eq. (6) to a specific problem, the three key problems should be addressed: (1) How to devise a suitable data fidelity term. (2) How to devise a suited regularizer to integrate important image priors. (3) How to devise an effective numerical method to solve the devised loss function.

8

3.1. Data fidelity term In actual measurements, the capacitance data usually contains noises, deteriorating final reconstructions. To address the challenge, in this study the L1 norm is used as the data fidelity term, i.e.,

P ( M , l , y )  ||Ms  y  l ||1 where

(7)

|| Ms  y  l ||1 means the L1 norm of the vectors, and can be specified as N

|| Ms  y  l ||1   | Ms j  y  l | . j 1

3.2. Regularizer After the data fidelity is determined, devising a suited regularizer is another important topic. The choice of the regularizer is crucial, which impacts the quality of solution directly (Zwaan et al., 2017). Since the L1 norm is popular in imposing the sparsity constraint of the expected solution, in this work we also use the norm as a regularizer, i.e.,

Q1 ( s ) || s ||1

(8)

Additionally, in this study the L2 norm is employed as a regularizer for the unknown variable

l:

Q 2 ( s ) || l ||22

(9) 1/2

 N  where || l ||2 means the L2 norm of the vectors, and can be specified as || l ||2    | l j |2  .  j 1  Submitting Eqs. (7-9) to Eq. (6) yields the final loss function:

min ||Ms  y  l ||1  1 || s ||1  2 || l ||22  s ,l

where 1  0 and 2  0 stand for the regularization parameters. We characterize the proposed objective functional:

9

(10)

(1) Different from common reconstruction models, in this study, an extended imaging model with the focus on the inaccuracy of the capacitance data and model deviation is proposed. (2) With the main motivation of reducing the effect of outliers and enhancing the imaging reconstruction quality, the L1 norm is used to measure the data fidelity. (3) The TRT is used to integrate the important image priors and stabilize the numerical solution, which will better the reconstruction quality.

4. Numerical method After the cost function is determined, devising an effective numerical method to solve it is another interesting topic. Eq. (10) has two non-smooth terms, and many methods are available for tackling the task. In this section, the FIST algorithm and the soft thresholding method, as sub-problem solvers, are embedded into the ISB method (Elvetun and Nielsen, 2016) to form an effective hybrid algorithm for solving the proposed objective functional. 4.1. Iterative split Bregman method The ISB method belongs to a popular optimization method, in which the variable splitting technique is employed to simplify computation (Cai et al., 2010; Xu et al., 2013; Yin et al., 2010). In essence, the technique tries to solve the following optimization task, i.e.,

min{  G ( s ) 1  K ( s )} s

(11)

where G ( s ) and K ( s ) are known functions. By introducing the equality constraint g  G ( s ) , Eq. (11) can be reformulated into:

min  g 1  K ( s )   s.t. g  G ( s ) Deploying the ISB technique to Eq. (12) yields the following sub-problems: 10

(12)

( s k 1 , g k 1 )  min{ || g ||1  K ( s )  s,g

 2

|| g  G ( s )  b k ||22 }

b k 1  b k  G ( s k 1 )  g k 1

(13)

(14)

where k is the number of iterations;  stands for a penalty parameter, b is an iteration variable (Cai et al., 2010; Goldstein and Osher, 2009). 4.2. Solution method By introducing an equality constraint u  Ms  y +l , Eq. (10) can be rewritten as a constraint optimization problem:

min || u ||1 1 || s ||1 2 || l ||22   u , s ,l  s.t. u  Ms  y  l

(15)

Deploying the ISB method, Eq. (15) can be rewritten as the following sub-problems:

   min ||u ||1  1 || s ||1  2 || l ||22  || u  ( Ms  y + l )  b k ||22  u , s ,l 2  

(16)

b k 1  b k +Ms k 1  y  l k 1  u k 1

(17)

Eq. (16) is split into three sub-problems:

   l k 1  min 2 || l ||22  || u k  ( Ms k  y +l )  b k ||22  l 2  

(18)

   u k 1  min ||u ||1  || u  ( Ms k  y +l k 1 )  b k ||22  u 2  

(19)

   sk1  min 1 || s ||1  || uk1  (Ms  y+l k1)  bk ||22  s 2  

(20)

The solution of Eq. (18) can be specified by:

l k 1 （22 +）-1 (u k  Ms k  y  b k )

(21)

The solution of Eq. (19) can be solved fast by the soft threshold operator:

u k 1  soft(( Ms k  y  l k +1  b k ),1  )

11

(22)

where soft(,  )  sign()  max(|  |  , 0) stands for the soft threshold operator (Donoho, 1995; Salahuddin et al., 2002). The solution of Eq. (20) is intractable, and therefore devising a more effective numerical method to solve it is required. The FIST algorithm (Beck and Teboulle, 2009) involves two important skills: the linearization approximation of the quadratic term and an accelerating technology. The skills are employed to speed up the convergence of the algorithm, which have been applied to different scenarios with impressive results. Therefore, in this paper the devised cost function is solved by the method. According to the FIST algorithm, we use the Taylor expansion to approximate Eq. (20):

W ( s )  W ( s k )  W ( s k )( s  s k ) 

1 ( s  s k ) 2  [( s  s k ) 2 ] 2a

(23)

1  s  s k 22 2a

(24)

Eq. (23) is rewritten as following:

W ( s )  W ( s k )  g ( s k )( s  s k )  where g ( s k ) is defined as:

g ( s k )   M T  Ms k  y  l k 1  b k  u k 1 

(25)

where the superscript T stands for the matrix transpose. From Eq. (24) and Eq. (25) yields the following formula, i.e.,

W (s) 

ag 2 ( s k ) 1  s  s k  ag ( s k ) 22 W ( s k )  2a 2

(26)

Finally, the updating scheme of s is specified as:

 1 ag 2 ( s k )  s k 1  min 1 || s ||1   s  s k  ag ( s k ) 22 W ( s k )   s 2a 2   Omitting constant term yields the solution of s :

12

(27)

 1  s k 1  min 1  s 1   s  s k  ag ( s k ) 22  s 2a  

(28)

The soft threshold operator can be used to solve Eq. (28) (Donoho, 1995; Salahuddin et al., 2002):

s k 1  soft(( s k  ag ( s k )), 1a)

(29)

In order to accelerate the convergence, s k 1 should be updated as follows:

  k 1  s k 1  s k 1   k 1   s k 1  s k    



where  k 1  1  1  4( k ) 2

2

(30)

and  0  1 .

According to the above discussion, the fast iterative split Bregman (FISB) algorithm as a new method is devised, and the execution steps can be summarized in Algorithm 1: Algorithm 1: The FISB algorithm Initialize the parameters for running the algorithm is provided. Loop goes until || s k 1  s k ||2 less than  : 1. Update  k 1 ; 2. Update l k 1 by solving Eq. (21); 3. Update u k 1 by solving Eq. (22); 4. Update s k 1 by solving Eq. (30); 5. Update b k 1 by solving Eq. (17); 6. k  k  1 ; End loop Return: The final result. where  stands for tolerance error.

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The FISB algorithm inherits the excellent properties from both of the TRT and the ISB method, which can be summarized as follows: (1) The ISB method, splitting a complicated optimization problem into simple computation, can tackle the devised objective functional effectively. (2) The FIST algorithm and the soft thresholding method, as sub-problem solvers, are deployed to reduce the costly calculation.

5. Numerical and experimental results Numerical and experiment results are used to test the FISB algorithm. To achieve the above purpose, we compare several popular imaging algorithms, such as, the LBPM, the TRT, the TSVDM, the SIRT, the CGM, the ILA and the ART. Measurement noise level,  and the image error,  are used to assess the robustness of the tested algorithms, which are defined by:



|| yOri  yNoi || 100% || yOri ||

(31)

|| sOri  sRec || 100% || sRec ||

(32)



where yOri means the original capacitance and y Noi is the noise data; sOri and sRec are the practical permittivity distribution and reconstructed permittivity distribution, respectively. A 12-electrodes ECT sensor model is simulated, and Fig. 1 provides a detailed schematic diagram. The circle with the radius of 50 mm is the conductive shield. The material of the red parts is copper, the black parts stand for glass, and the blue area configured by glass is the shell of tube and its thickness is 2 mm. There are 12 protective guards between the tube and the

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conductive shield. The MATLAB is used to perform the imaging methods. Conductive shield

Electrode

Test object

Protective guards

Tube

Fig. 1. ECT sensor model. Before solving the inverse problem, the capacitance data between every electrode pair and the sensitivity matrix must be solved. According to the sensor structure, the capacitance between every electrode pair can be obtained by solving Eq. (2), and we use the following equation to compute the sensitive matrix: Ti , j ( g , h)   



 i ( g , h)  j ( g , h) Vi



Vj

dgdh

(33)

The sensor sensitivity will change with the permittivity distributions, and Figs. 2(a-d) show four kinds of the sensitivity distributions. Owing to the linearization approximation of the imaging model, the sensitivity matrix is a Jacobian matrix that its each term may be positive or negative.

15

1

0.4

0.5

0.2

0

0

-0.5

-0.2

-1 -0.4

-1.5

-0.6 80

-2 80 60

60

80

80

40

20

60

40

60

40

20

20 0

40 20 0

0

0

a

b

0.4

0.1

0.2

0

0

-0.1

-0.2

-0.2

-0.4 80

-0.3 80 60

60

80 60

40 20

40

20

20 0

80 60

40

40

20 0

0

c

0

d

Fig. 2. Four kinds of sensitivity distributions. (a. Sensitivity distribution between electrodes 1-2; b. Sensitivity distribution between1-3; c. Sensitivity distribution between 1-4; d. Sensitivity distribution between 1-6.) 5.1. Numerical results We use four testing targets shown in Figs. 3(a-d) to assess the performances of the methods. In the imaging targets, the yellow region stands for glass and the blue region stands for air. Table 1 lists the details of the imaging targets. The initial solutions for the compared iteration algorithms are computed by the TRT with the regularization parameter 0.001. The relax factors of the iterative algorithms are 1 and the number of iterations of the iterative algorithms are listed in Table 2. The capacitance values for the imaging targets in Figs. 3(a-d) are shown in

16

Figs. 4(a-d). The results recovered by the non-iterative algorithms and the iterative algorithms are shown in Figs. 5(a1-c4) and Figs. 6(a1-e4), respectively. The image errors (IEs) are shown in Fig. 7, and the values of the algorithmic parameters of the proposed FISB algorithm are shown in Table 3, which is selected empirically.

a

b

c

d

Fig. 3. Imaging targets. Table 1. The details of the imaging targets. Imaging targets

Details

Fig. 3(a)

The inner radius is 30 mm and the external radius is 48 mm.

Fig.3(b)

The radii of the two objects are both 13 mm.

Fig. 3(c)

The radii of the two objects are both 13 mm.

Fig. 3(d)

The radii of the two objects are both 15 mm.

Table 2. The number of iterations of the iterative methods. Methods

Fig. 3(a)

Fig. 3(b)

Fig. 3(c)

Fig. 3(d)

CGM

20

20

20

20

SIRT

224

212

199

136

17

ILA

187

231

233

175

ART

279

319

362

260

a

b

c

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d Fig. 4. Capacitance values of imaging targets. (a. Capacitance values for the image target in Fig. 3(a); b. Capacitance values for the image target in Fig. 3(b); c. Capacitance values for the image target in Fig. 3(c); d. Capacitance values for the image target in Fig. 3(d).)

a1

a2

a3

a4

b1

b2

b3

b4

c1

c2

c3

c4

Fig. 5. Reconstruction results of the non-iterative solvers. (a1-a4. Reconstruction results of the LBPM for the imaging targets in Figs. 3(a-d); b1-b4. Reconstruction results of the TRT for the imaging targets in Figs. 3(a-d); c1-c4. Reconstruction results of the TSVDM for the imaging targets in Figs. 3(a-d).)

19

a1

a2

a3

a4

b1

b2

b3

b4

c1

c2

c3

c4

d1

d2

d3

d4

e1

e2

e3

e4

Fig. 6. Reconstruction results of the iterative solvers. (a1-a4. Reconstruction results of the 20

CGM for the imaging targets in Figs. 3(a-d); b1-b4. Reconstruction results of the ILA for the imaging targets in Figs. 3(a-d); c1-c4. Reconstruction results of the SIRT for the imaging targets in Figs. 3(a-d); d1-d4. Reconstruction results of the ART for the imaging targets in Figs. 3(a-d); e1-e4. Reconstruction results of the FISB algorithm for the imaging targets in Figs. 3(ad).)

LBPM TRT TSVDM CGM ILA SIRT ART FISB

image errors(%)

80

60

40

20

0

a

b

c

d

image targets

Fig. 7. Image error comparison for the imaging targets in Figs. 3(a-d). Table 3. Algorithm setting for the FISB algorithm. Imaging targets

1

2

N

Fig. 3(a)

0.01

0.1

15

Fig. 3(b)

0.02

0.2

15

Fig. 3(c)

0.01

0.2

15

Fig. 3(d)

0.05

0.3

15

The LBPM, the TRT and the TSVDM are three typical non-iterative algorithms. The methods can achieve the on-line reconstruction, but they result in incorrect reconstructions. In Figs. 5(a1-c4), we can find that the non-iterative methods lead to poor reconstructions. Such results indicate that it is difficult for the methods to ensure a reliable tomographic image. The reconstruction quality of the CGM, the ILA, the SIRT and the ART achieves the better 21

results than the non-iterative methods, and the artificial images are significantly reduced. The reconstruction results shown in Figs. 6(a1-d4) imply such conclusion. In Figs. 6(e1-e4), the image quality of the FISB method is superior to others methods. Such encouraging results derive from the following four respects. First, the inaccuracy of the capacitance data and model deviation have been jointly considered by the proposed algorithm. Second, the L1 norm-based data fidelity term improves the robustness and highlights the sparsity of the expected solution. Third, the ISB technique with the soft thresholding method and the FIST technique is deployed to solve the devised cost function. Fourth, the TRT is employed to ameliorate the reconstruction quality. From Fig. 7, we observe that the FISB method has the smallest IEs (i.e., 8.7%, 10.1%, 8.0% and 8.8% for the imaging objects (a-b), respectively). Such encouraging results indicate that the proposed FISB method is an effectiveness and reliability technique, and it can reconstruct satisfactory images. From a quantitative point of view, the proposed algorithm is superior to the traditional ones. 5.2. Reconstruction time The reconstruction time is crucial for actual applications, and therefore, reconstruction time is assessed for the compared solvers in this section. The results are listed in Table 4. Table 4. Reconstruction time for the iterative reconstruction techniques. Methods

Fig. 3(a)

Fig. 3(b)

Fig. 3(c)

Fig. 3(d)

CGM

0.03

0.04

0.04

0.04

ILA

0.03

0.04

0.04

0.04

SIRT

0.03

0.04

0.04

0.04

22

ART

0.03

0.04

0.04

0.04

FISB

1.3

1.5

1.5

1.5

From Table 4, we observe that the compared iterative solvers require implementing multiple iterations before obtaining a satisfactory solution and the reconstruction time of the proposed FISB is longest among the competing solvers. In the future, the proposed imaging method should be further studied to improve the computing efficiency. 5.3. Reconstruction performances under different noise levels In practical applications, measurement signals always contain noises, bringing a negative impact on the imaging quality. Therefore, in this section, it is necessary to examine the noise terms. The different noise levels, which can be set to 4%, 8% and 12%, respectively, are used to assess the robustness of the proposed FISB algorithm. The algorithmic parameters are equivalent to Case 5.1. Figs. 8(a1-c4) list the reconstruction results produced by the proposed FISB algorithm, and the IEs are shown in Fig. 9.

a1

a2

a3

a4

b1

b2

b3

b4

23

c1

c2

c3

c4

Fig. 8. Results Reconstructed by the FISB algorithm under different noise levels. (a1-a4. Reconstruction results under 4% noise level; b1-b4. Reconstruction results under 8% noise level; c1-c4. Reconstruction results under 12% noise level.) 100

100 LBPM TRT TSVDM CGM ILA SIRT ART FISB

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c

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Fig. 9. Reconstruction performance comparison for the imaging techniques under different

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noise levels. (a-d correspond to the imaging errors for the imaging targets in Figs. 3(a-d).) The excellent robustness of the proposed method is confirmed by the images in Figs. 8 and 9. According to the characteristics summarized by the section 5.1, we observe that the proposed FISB method can still ensure a satisfactory reconstruction quality with the increasing noise levels. 5.4. Reconstruction performances under different permittivity distributes In order to test the proposed FISB method further, different permittivity distributes (2.6, 3.5 and 5.5) are used in this study. The algorithmic parameters are the same as those in 5.1, and the IEs are showed in Fig. 10.

70

30 20 10 0

50 40 30 20

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image errors(%)

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LBPM TRT TSVDM CGM ILA SIRT ART FISB

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10 a

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image targets

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image targets

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a

b

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image targets

c

Fig. 10. Image errors under different permittivity distributes for the imaging targets. (a-c correspond to the imaging errors under different permittivity distributes (2.6, 3.5, 5.5).) From Fig. 10, we observe that the proposed method still leads to the smallest IEs compared with other methods. The reconstruction results indicate that the proposed imaging method can obtain a satisfactory solution under different permittivity distributes. 5.5. Regularization parameters The regularization parameters  1 and 

2

control the intensity of the regularization, and

it is crucial to choose a suited regularization parameter. As a result, Figs. 11(a-b) show the 25

variation tendency of the image errors under different regularization parameters. Specifically speaking, a large regularization parameter can ensure the numerical stability, but the regularization problem will deviate from the original inverse problems excessively, which leads to the distortion of the reconstruction images. On the contrary, a small value can ensure the consistence with the original problem, but it is difficult to keep the solutions stable. Therefore a suitable regularization parameter is highly desired by practical applications. From Figs. 11(a-b), the proposed method keeps relatively stable under a wide variation range of the regularization parameter, and the comparison results indicate that the FISB method is not sensitive to the regularization parameters. 45

30 Fig. 3(a)

40 35

Fig. 3(b)

Fig. 3(c)

30

Fig. 3(c)

Fig. 3(d)

image error(%)

image error(%)

Fig. 3(a)

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25 20 15

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Fig. 3(d)

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a

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Fig. 11. The variation tendency of the image errors under different regularization parameters. (a. Image errors under different 1 values with the fixed 2 ; b. Image errors under different

2 values with the fixed 1 .) 5.6. Sensitivity of regularization parameter values In this section, we use a set of non-symmetric testing targets with different radii shown in Figs. 12(a-b) to assess the performances of the proposed FISB method. The radii of the large

26

circle and small circle in Fig. 12(a) are 13mm and 8mm, respectively, and the radii of the circles in Fig. 12(b) are 10mm. The results recovered by the algorithms are shown in Figs. 13(a-h) and Figs. 14(a-h), respectively. In Figs. 15(a-f), reconstruction results under different regularization parameter values are used to further test the proposed algorithm, and the variation tendency of the corresponding image errors is shown in Fig. 16.

a

b

Fig. 12. Imaging targets.

a

b

c

d

e

f

g

h

Fig. 13. Reconstruction results. (a. Reconstruction results of the LBPM; b. Reconstruction results of the TRT; c. Reconstruction results of the TSVDM; d. Reconstruction results of the 27

CGM; e. Reconstruction results of the ILA; f. Reconstruction results of the SIRT; g. Reconstruction results of the ART; h. Reconstruction results of the FISB algorithm, respectively.)

a

b

c

d

e

f

g

h

Fig. 14. Reconstruction results. (a. Reconstruction results of the LBPM; b. Reconstruction results of the TRT; c. Reconstruction results of the TSVDM; d. Reconstruction results of the CGM; e. Reconstruction results of the ILA; f. Reconstruction results of the SIRT; g. Reconstruction results of the ART; h. Reconstruction results of the FISB algorithm, respectively.)

a

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c

d

e

f

Fig. 15. Reconstruction results of the FISB algorithm. (a and d stand for reconstruction results when 1  0.01 and 2  0.1 ; b and e stand for reconstruction results when

1  0.04 and 2  0.3 ; c and f stand for reconstruction results when 1  0.08 and 2  0.4 .)

2 2

image error(%)

20

2

=0.1 =0.3 =0.4

15 10 5 0 Fig. 20(a) 0.08

imaging targets

0.04

Fig. 20(b) 0.01

1

Fig. 16. The variation tendency of the image errors under different regularization parameter values. From Figs. 13-15, we find that the FISB method show better reconstruction performances than the competing solvers. In Fig. 16, we can observe that the IEs are less sensitive to the small changes of regularization parameter values. Such encouraging results imply that the proposed 29

FISB method is competent in handling ECT image reconstruction problems. The proposed imaging method belongs to one of regularization methods, and the reconstruction quality is dependent on regularization parameter values. In above comparisons, we observe that the IEs change with regularization parameter values. Currently, selecting an optimal regularization parameter value is challenging so far, and in the field of the ECT image reconstruction, the regularization parameter value is often determined empirically. Future studies should put more emphasis on the adaptive selection of the regularization parameter. 5.7. Experimental results In order to test the performance of the presented method, two experiments were carried out on the 8-electrodes circular sensor. In this study, we use air (the permittivity is 1.0) and quartz sand (the permittivity is 3.0) as experimental materials. The ECT system is shown in Fig. 17.

Fig. 17. ECT system. The sensitivity distributions between different electrode pairs are shown in Figs. 18(a-d).

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-0.02 40 30

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Fig. 18. Sensitivity distributions between the electrode pairs. Two targets are employed in the experiment. The radius of the circle is 30 mm in Fig. 19(a) and 50 mm in Fig. 19(b), respectively. The capacitance values for the testing targets are shown in Figs. 20(a-b). The results reconstructed by all above algorithms, are shown in Figs. 21-23. The parameters are the same as those in 5.1. The pixels of each image are 32×32, and the IEs are listed in Fig. 24. The regularization parameter values of the imaging target in Fig. 19(a) are

1  0.02 , 2  0.4 , and the regularization parameter values of the imaging target in Fig. 19(b) are 1  0.03 , 2  0.6 .

31

a

b

Fig. 19. Experimental targets. 12

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pF

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pF

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b Fig. 20. Capacitance values of the two experimental targets.

a

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c

d

e

f

Fig. 21. Reconstruction results of the non-iterative solvers. (a-b. Reconstruction results of the LBPM for the imaging targets Figs. 19(a-b); c-d. Reconstruction results of the TRT for the imaging targets in Figs. 19(a-b); e-f. Reconstruction results of the TSVDM for the imaging targets in Figs. 19(a-b).)

a

b

c

d

e

f

g

h

Fig. 22. Reconstruction results of the iterative solvers. (a-b. Reconstruction results of the CGM for the imaging targets in Figs. 19(a-b); c-d. Reconstruction results of the ILA for the imaging targets in Figs. 19(a-b); e-f. Reconstruction results of the SIRT for the imaging targets in Figs. 19(a-b); g-h. Reconstruction results of the SIRT for the imaging targets in Figs. 19(a-b).)

33

a

b

Fig. 23. Reconstruction results of the FISB algorithm for the imaging targets in Figs. 19(a-b). 30

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Image error(%)

Image error(%)

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Target (a)

LBPM TRT TSVDM CGM

ILA

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Target (b)

a

b

Fig. 24. Image errors comparison under different imaging techniques for the imaging targets in Figs. 19(a-b). The reconstruction results shown in Figs. 21-23 indicate that the proposed algorithm produces better reconstruction results than the other compared algorithms. In Fig. 24, the IEs of the FISB method, 9.2% and 5.8%, are still smallest compared with other popular methods. The results indicate that the developed solver is competent in handling different imaging tasks.

6. Conclusions In this study, a new objective functional and an effective numerical method are proposed to improve reconstruct accuracy. The major research conclusions are summarized as follows: (1) With the focus on model deviation and measurement noises, the proposed L1 norm-based cost function can model the ECT image reconstruction task, which may improve the solution

34

quality. (2) The FIST method and the soft thresholding method, as the sub-problem solvers, are embedded into the ISB framework to generate an effective numerical method to solve the presented objective functional more effectively, which lowers the costly computation and improves the numerical performances. (3) Numerical simulation results imply the efficacy of the proposed method in dealing with ECT image reconstruction problems, and this study takes an important step towards improving ECT imaging reconstruction. (4) The experiment results indicate that the proposed method is competent in practical reconstruction tasks and leads to the improved imaging quality.

Nomenclature ART

Algebraic reconstruction technique

CGM

Conjugate gradient method

ECT

Electrical capacitance tomography

FIST

Fast-iterative shrinkage thresholding

FISB

Fast iterative split Bregman

IEs

Image errors

ILA

Iteration Landweber algorithm

ISB

Iterative split Bregman

LBPM

Linear back projection method

SIRT

Simultaneous iterative reconstruction technique

35

TRT

Tikhonov regularization technique

TSVDM

Truncated singular value decomposition method

Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities [grant numbers JB2019117, 2017MS012 and 2017MS073]; the National Natural Science Foundation of China [grant numbers 51206048, 51576196 and 61571189,]; the National Key Research and Development Program of China [grant number 2017YFB0903601]; the State Administration of Foreign Experts Affairs for supporting the project 'Overseas Expertise Introduction Program for Disciplines Innovation in Universities' [grant number B13009].

Contributors Hongbo Guo received the B.E. Degree from Shanxi University, Shanxi,China, in 2013. She is currently pursuing the Ph.D. Degree in control science and engineering at North China Electric Power University, Beijing, China. Her current research interests include image reconstruction, convex optimization problems in engineering and multi-phase flow measurement.

Shi Liu received the B.E. and M.E. degrees in thermal engineering from Chongqing University, Chongqing, China, in 1982 and 1984, respectively, and the Ph.D. in chemical engineering at University of Cambridge, Cambridge, U.K., in 1992. He is currently a Professor with the School of Energy, Power and Mechanical Engineering, North China Electric Power University, Beijing, China. He has published more than 200 research papers. His current research interests include

36

multiphase flow measurement, electrical capacitance tomography, numerical optimization, and computational fluid dynamics.

Hongyan Chen received the B.E. degree in detection technology and automatic equipment from North China Electric Power University, Changping District, Beijing, China, in 2017. She is currently pursuing the M.E. degree in detection technology and automatic equipment at the School of Control and Computer Engineering, North China Electric Power University. Her current research interests include multiphase flow measurement and numerical optimization.

Shanxun Sun received the B.E. degree in Measurement and Instrumentation from North China Electric Power University (NCEPU) in 2015. Currently she is a PhD candidate in control and measurement in NCEPU. Her research interests include Measurement Technology, Numerical Optimization and Renewable Energy.

Jiankang Ding received the B.E. degree in Qufu Normal University in 2015 and the M.E. degree in automation and pattern recognition and intelligent systems from North China Electric Power University, Changping District, Beijing, China, in 2018. During his master's degree, he majored in control engineering, and his main research direction was pattern recognition. After graduation, he worked in North China Electric Power Design Institute and engaged in electric power design.

Hongqi Guo received the B.E. degree from China University of Mining and Technology,

37

Jiangsu, China, in 2008, and the M.E. degree from North China University of Water Resources and Electric Power, Henan, in 2018. She is currently a teacher with the school of electrical engineering, Shanxi Yanbei Coal Industrial School, Shanxi, China. Her current research interests include image recognition and numerical optimization.

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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