An image reconstruction algorithm based on the regularized total least squares method for electrical capacitance tomography

Flow Measurement and Instrumentation 19 (2008) 325–330

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An image reconstruction algorithm based on the regularized total least squares method for electrical capacitance tomography Jing Lei a,∗ , Shi Liu b , Zhihong Li b , H. Iñaki Schlaberg c , Meng Sun a a Institute of Engineering Thermophysics, Chinese Academy of Sciences, P.O. Box 2706, Beijing 100080, China b Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, Ministry of Education, North China Electric Power University, Changping District,

Beijing 102206, China c Institute of Particle Science and Engineering, University of Leeds, Leeds, LS2 9JT, UK

article

info

Article history: Received 16 December 2007 Received in revised form 18 March 2008 Accepted 8 April 2008 Keywords: Electrical capacitance tomography Image reconstruction Regularized total least squares method Regularization technique

a b s t r a c t Electrical capacitance tomography (ECT) is a non-invasive imaging technology that is aimed at the visualization of the cross-sectional permittivity distribution of a dielectric object based on the measured capacitance data. ECT image reconstruction is a typical ill-posed problem and its solution is unstable, that is, the solution is highly sensitive to noise in the input data. Methods that ensure the stability of a solution while enhancing the quality of the reconstructed images should be used to obtain a meaningful reconstruction result. In this paper, an image reconstruction algorithm that considers the errors in both sensitivity field matrix and capacitance values based on the regularized total least squares method for electrical capacitance tomography is presented. The image reconstruction problem is transformed into an optimization problem. In addition, the Newton algorithm is employed to solve the objective function. Numerical simulations indicate that the algorithm is efficient and overcomes the numerical instability of ECT image reconstruction. For the cases of the reconstructed objects considered in this paper the spatial resolution of the reconstructed images obtained using the proposed algorithm is enhanced; and the artifacts in the reconstructed images can be eliminated effectively. As a result an efficient method for ECT image reconstruction is introduced. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Electrical capacitance tomography (ECT) is a non-invasive imaging technology that aims at the visualization of the crosssectional permittivity distribution of a dielectric object based on the measured capacitance data. ECT is considered a promising process tomography technology due to its advantages such as high speed, low cost, and non-intrusive sensors. In recent years, it has been widely used in a number of research investigations for imaging two-phase or multi-phase processes [1–10]. Successful applications of ECT depend greatly on the precision and the speed of the image reconstruction algorithm. At present, algorithms that are often applied to ECT are the linear backprojection (LBP), the Tikhonov regularization method [11] and the Landweber iteration algorithm [12]. The advantages of LBP are that it is numerically simple and computationally fast because it only involves a single matrix-vector multiplication; however, the quality of the reconstructed images is relatively low for complicated reconstruction objects and in some aspects it can only

∗ Corresponding author. Tel.: +86 10 82543063; fax: +86 10 62575913. E-mail address: [email protected] (J. Lei). 0955-5986/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.flowmeasinst.2008.04.001

be considered as a qualitative algorithm. The Landweber iteration algorithm has in recent years been widely adopted, however, it is only the “steepest descent” algorithm from the viewpoint of numerical optimization, and the rate of convergence is relatively slow. The standard Tikhonov regularization method is an effective method to solve inverse problems; however, the quality of the reconstructed images using this algorithm is not perfect due to the particularities of ECT. One of the disadvantages is its excessive smoothness; as a result, some information about the detail in the reconstructed images is lost. In addition, other image reconstruction algorithms such as the conjugate gradient algorithm [13], the generalized vector sampled pattern matching method [14], the total variation regularization method [15], the simulated annealing algorithm [16], and algorithms based on the neural networks [17] and multi-objective optimization [18] have been introduced to ECT image reconstruction. Reports on the performances of other algorithms such as the algebraic reconstruction technique (ART), the simultaneous iterative reconstruction technique (SIRT) can be found in [19]. In general, every algorithm has its advantage and disadvantage when applied to different reconstruction objects; in practice, we always select the algorithms according to the characteristics of the different reconstruction objects.

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This paper presents an image reconstruction algorithm that considers the errors in both the sensitivity field matrix and the capacitance data based on the regularized total least squares method for ECT. The image reconstruction problem is transformed into an optimization problem. In addition, the Newton algorithm is employed to solve the objective function. Numerical simulations indicate that the algorithm is efficient and overcomes the numerical instability in the ECT image reconstruction. For the cases of the reconstructed objects considered in this paper, the spatial resolution of reconstructed images obtained using the proposed algorithm is enhanced. As a result an efficient method for ECT image reconstruction is introduced. This paper is organized as follows. In Section 2 the model of ECT image reconstruction is briefly described. In Section 3 an image reconstruction algorithm based on the regularized total least squares method is described in detail. In Section 4 results of image reconstruction obtained using different image reconstruction algorithms are shown. Section 5 presents a summary and conclusion.

capacitance data is inexact. In practice, the sensitivity field matrix may be inexact due to the linearization approximation of the image reconstruction model [21]. More details on the perturbation analysis of Eq. (2) can be found in [22,23]. In this work, a main motivation is to consider the solving of the image reconstruction problem when the capacitance data and the sensitivity field matrix are inexact. The total least squares (TLS) method considers the solving of the image reconstruction problem when the capacitance data and the sensitivity field matrix are inexact. In other words, the total least squares method solves the following equation [24]:

(S + E )G = C + r

(3)

where E is a perturbation matrix of dimension m × n; r is a noise vector of dimension n × 1. From the viewpoint of numerical optimization, the TLS problem can be formulated as [25]: n o min kEk2 + krk2 : (S + E)G = C + r . (4) G,E,r

2. ECT model There are two major computational aspects in the image reconstruction for ECT: the forward problem and the inverse problem. The forward problem in ECT determines the interelectrode capacitances from the permittivity distribution. The inverse problem aims to determine the permittivity distribution from the measured capacitance data. The result is usually presented as a visual image, and hence this process is called image reconstruction. The relationship between the capacitance and the permittivity distribution is governed by [19]: ZZ Q 1 C= =− ε(x, y)∇ φ(x, y)dΓ (1) V

V

Γ

Problem (4) can be rewritten as a double minimization problem: n o min min kEk2 + krk2 : (S + E)G = C + r . (5) G

E,r

Consider the inner minimization problem: n o min kEk2 + krk2 : (S + E)G = C + r .

(6)

E ,r

The Lagrangian function of problem (6) can be expressed by: min L(E, r, λ) = kEk2 + krk2 + 2λT ((S + E)G − (C + r)).

(7)

Applying the KKT conditions to Eq. (7) yields: 2E + 2λGT = 0 (∇E L = 0)

(8)

where ε(x, y) is a permittivity distribution; Q is the electric charge; V is the potential difference between two electrodes forming the capacitance; φ(x, y) is the potential distribution along the electrode and Γ is the electrode surface. In ECT applications, the image reconstruction model can be expressed as [19,20]:

We can obtain λ = r from Eq. (9). Submitting this into Eq. (8) yields:

SG = C

E = −rGT .

(2)

where C is an m × 1 dimensional vector indicating the normalized capacitance values; G is an n × 1 dimensional vector standing for the normalized permittivity distribution, and in the reconstructed image it denotes the gray level values; S is a matrix of dimension m × n, and it is called the sensitivity field matrix. The task of ECT inverse problem is finding G rapidly and efficiently from the known S and C parameters. There are three major difficulties with image reconstruction in ECT [19]: (1) the relationship between the permittivity distribution and the capacitance is usually non-linear and the electric field is distorted by the material that is present, this is the so-called “soft field” effect; (2) the number of independent measurement data is fewer than that of the pixels of the reconstructed image; (3) the inverse problem is ill-posed and ill-conditioned. An ECT solution is often unstable, that is, the solution is highly sensitive to noise in the input data. For ECT image reconstruction, some prior information or constraints should be imposed on the solution to obtain a meaningful reconstruction result.

2r − 2λ = 0 (∇r L = 0)

(9)

(S + E)G = C + r (feasibility).

(10)

(11)

Submitting Eq. (11) to Eq. (10) yields:

(S − rGT )G = C + r.

(12)

So, r=

SG − C . kGk2 + 1

(13)

Submitting Eq. (13) to Eq. (11), we obtain: E=−

(SG − C )GT kGk2 + 1

.

(14)

Finally, submitting Eqs. (13) and (14) into the objective function of problem (6) we can obtain that the value of problem (6) is equal to kSG−C k2 . kGk2 +1

min G

Consequently, the TLS problem can be reduced to [24]:

kSG − C k2 kGk2 + 1

.

(15)

3. Algorithm analysis

In fact, the total least squares method can be considered as generalized least squares method, namely, the weighted least squares methods [24]. As a result, Eq. (15) can be rewritten as:

3.1. Total least squares method

min kW (SG − C )k2 G

For conventional ECT image reconstruction algorithms, which often consider the solving of the image reconstruction problem when the sensitivity field matrix is considered as exact and the

where W = diag( √

1

kGk2 +1

diagonal matrix operator.

,√

1

kGk2 +1

,..., √

1

kGk2 +1

); diag(·) is a

J. Lei et al. / Flow Measurement and Instrumentation 19 (2008) 325–330

3.2. Regularized total least squares method ECT image reconstruction is a typical ill-posed problem; some prior information or constraints should be imposed on a solution to obtain a meaningful reconstruction result. The regularization technique is an effective method to solve ill-posed problems [11]. According to the regularization technique, it is possible to obtain a model for the regularized total least squares method [25]: n o min kEk2 + krk2 + α kLGk2 : (S + E)G = C + r (16) E,r,G

where α > 0, is called the regularization parameter; kLGk2 is a stabilizing item; k·k is the 2-norm; L is a constrained matrix. Optimization problem (16) can be rewritten as a double minimization problem [25]: n o min min kEk2 + krk2 + α kLGk2 : (S + E)G = C + r . (17) G

E,r

For ease of computation, as described in Section 3.1, optimization problem (17) can be reduced to one involving only the G variables, more details can be found in [25]. It can be expressed by: min J(G) =

kSG − C k2 kGk2 + 1

+ α kLGk . 2

(18)

Similarly, Eq. (18) can be rewritten as: min J(G) =

m X

dj2 + α kLGk2

(19)

j=1

S G −C where dj = √ j 2 j , Sj is the jth row of matrix S.

kGk +1

Since the absolute value function is not differentiable at points where its value is zero, the following expression is used to approximate it [29]: 1

|x| ≈ (x2 + ξ) 2

(22)

where ξ > 0, is a predetermined small parameter. Analyzing formula (22), it can be seen that the approximation is not reasonable when ξ is a large value, which will cause a large approximation error; however, when ξ is small, for example the value is smaller than the computer’s accuracy, it may make the result inaccurate due to the influence of the truncated errors in the process of computation [30]. In general, the range of ξ should be sufficiently small to maintain a reasonable approximation, however it should not be smaller than the computer’s accuracy. In this paper, ξ has been given a value of 10−10 . Therefore, for ease of computation Eq. (21) can be approximated by: min J(G) = (1 − δ)

m X Sj G − Cj min ρ(G) = (1 − δ) kSG − C k2 + δ

kSG − C k2 2

kGk + 1

+δ

m X

1

(dj2 + ξ) 2 + α kLGk2 .

(23)

j=1

Studies indicate that the key issues of the regularization technique are the choice of the regularization parameter and the design of the stabilizing item [11,31]. From the viewpoint of the penalty function, their differences lie in using different stabilizing items to stabilize the numerical solution [31]. In general, stabilizing items can be designed flexibly according to the particularities of the reconstructed objects. It is obvious that different reconstruction results will be obtained when using different stabilizing items. Therefore, according to the principle of the regularization technique Eq. (23) can be expressed by a generalized formula:

3.3. Extended objective function It can be seen that the first item of the function uses the function of sum of squares to measure the goodness of fit to the measured data by analyzing Eq. (19). In practice, however, the solution that the function of sum of squares obtains is not optimal when the measured input data C holds large gross errors [26]. From the viewpoint of robust estimation, the reason is that the increase of the function of sum of squares is relatively fast, which leads to the estimation being sensitive to the gross errors. Therefore a function ρ(·) with a slower rate of increase can be used to replace the function of sum of squares [26]. In [27], authors proposed the robust total least squares method. The 1-norm method is known to have significant robustness advantages over the least squares method in many applications; however its computation efficiency is lower than the 2-norm. In [28], the authors proposed the combination estimation of the 1-norm and the 2-norm in the linear regression field to utilize the advantages of both the estimation methods, which can be expressed by:

327

min J(G) = (1 − δ)

kSG − C k2 2

kGk + 1

+δ

m X

1

(dj2 + ξ) 2 + αΩ (G)

(24)

j=1

where Ω (G) is a stabilizing item. In [32], the authors proposed to use the following function to design the stabilizing item: Ω (G) =

n X

2

(−e−Gi + 1).

(25)

i =1

However, applications indicate that the quality of the reconstructed images using the stabilizing item is not perfect due to the particularities of ECT image reconstruction. It is well known that the reconstruction objects in ECT are complicated, especially if multi-phase flow is considered. Therefore, the stabilizing item is extended to increase the adaptability for complicated reconstruction objects based on the idea in [33]. It can be expressed by: Ω (G) =

n X

p

(−e−|Gi | i + 1).

(26)

i =1

(20)

j=1

where 0 ≤ δ ≤ 1; | · | is the absolute value operator. Therefore, considering the randomness and complexity of the gross errors in the measured capacitance data and the computational efficiency, the function of sum of squares in Eq. (18) has been replaced with the combination of the 1-norm and the 2-norm, and can be expressed by: m Sj G − Cj X kSG − C k2 q min J(G) = (1 − δ) + δ + α kLGk2 (21) 2 kG k2 + 1 j=1 kGk + 1 where 0 ≤ δ ≤ 1; analyzing Eq. (21) it can be found that Eq. (21) is equivalent to Eq. (18) when δ = 0. It is obvious that Eq. (18) is a special case of Eq. (21).

From the viewpoint of a penalty function, the difference between Eqs. (25) and (26) lies in imposing different penalties on the unknown variables. Note that the stability item will become a constant when p = 0, which does not achieve a stability solution according to the principle of the regularization technique. From the viewpoint of the penalty function, pi > 0 is feasible; however in practice it is found that we often obtain a good result when 0 < pi < 2. It is obvious that Eq. (26) approximates Eq. (25) when pi → 2. For ease of computation, Eq. (26) can be approximated by: Ω (G) ≈

n X

pi

(−e−(Gi +ξ) 2 + 1) 2

(27)

i =1

where ξ > 0, is a predetermined small parameter, and in this paper it has been given a value of 10−10 .

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328

Fig. 1. Simulated test objects.

According to the above analysis, an extended objective function for ECT image reconstruction can be obtained, and it can be expressed as: min J(G) = (1 − δ) G

+α

n X

kSG − C k 2

2

kG k + 1

+δ

m X

1

(dj2 + ξ) 2

(28)

i =1

Sj G−Cj

where dj = √

kGk2 +1

Computation parameters

(1.a)

(1.b)

(1.c)

(1.d)

Number of iteration Relaxation factor

200 1

200 1

200 1

200 1

j=1 pi

(−e−(Gi +ξ) 2 + 1) 2

Table 1 Selected computation parameters for PLI algorithm

, Sj is the jth row of matrix S; 0 ≤ δ ≤ 1.

It is obvious that the proposed objective function (28) is in essence equivalent to Eq. (18), which is a result of balancing accuracy and stabilization of a solution. The differences are that the proposed objective function considers the robustness of estimation and uses a different stabilizing item to stabilize the numerical solution. 3.4. Solving of objective function Objective function (28) is an unconstrained optimization problem. For the descent algorithms using a linear search, the convergence rate of the Newton iteration algorithm is fast. Therefore this algorithm is used to minimize problem (28). According to the algorithm, an iteration scheme can be obtained [34]: Gk+1 = Gk − [∇ 2 J(Gk )]−1 ∇ J(Gk )

(29)

where ∇ J(·) and ∇ J(·) are the Hessian matrix and gradient vector of objective function (28) respectively. Note that spatial resolution is important in practical applications. We know in advance that the range of the solution is between 0 and 1, therefore the iteration scheme is modified according to this prior information of the solution. As a result a projected operator is introduced after each iteration. The definition of the operator is:  (k)  0 if Gi < 0 (k) (k) (k) Gi = Gi (30) if 0 ≤ Gi ≤ 1   (k) 1 if Gi > 1. 2

A good initial value facilitates faster convergence of the algorithm, so the initial value is obtained using the standard Tikhonov regularization method, that is, G(0) = (ST S + αI )−1 ST C. In the case of ECT image reconstruction, the value of the expression (ST S + αI )−1 ST can be obtained in advance. In fact, the initial values also can be computed using other algorithms such as the SIRT algorithm, the ART algorithm and so on. Note that the choice of the initial value will influence the convergence rate because the Newton algorithm is a local convergence algorithm; other algorithms that the final solution does not greatly depend on the initial values can be adopted. In practice, in order to avoid finding a local minimum solution, other global optimization methods can be adopted. For the described algorithm, the choice of the regularization parameter is important. Accurately computing the regularization

parameter is challenging for reasons such as the linearization approximation of the image reconstruction model, accumulation and diffusion of errors in the process of numerical computation, and error fluctuations of the measured input data and so on. Therefore, in this work, the regularization parameter was chosen empirically as described in [19], that is, the range of the parameter is established by solving typical cases, which makes the computation simpler and faster. 4. Numerical simulation Numerical simulations were used to validate the proposed algorithm (Robust Regularized Total Least Squares Algorithm, for short RRTLS), and the quality of the reconstructed images is compared to the projected Landweber iteration (for short PLI) algorithm. Four typical permittivity distributions were chosen for the simulations. A 12-electrode square sensor was chosen for the simulations; and an image using 32 × 32 pixels is presented. The algorithms are implemented using Matlab on a PC with a Pentium IV 2.4 GHz CPU, and 512 Mbytes memory. Fig. 1 shows the original static objects, the black part stands for high permittivity materials with a value of 2.6; and the white part denotes low permittivity materials with a value of 1.0. Figs. 2 and 3 show the results of the simulations. The initial value of the projected Landweber algorithm was obtained using the LBP algorithm. Table 1 shows the selection of computation parameters for the projected Landweber algorithm. The initial value of the RRTLS algorithm was obtained using the standard Tikhonov regularization method. In addition, in this work, we set pi = p(i = 1, 2, . . . , n) for ease of computation; and the value of p is selected according to the reconstructed objects. Table 2 shows the selected computation parameters for the RRTLS algorithm. The image error is used to evaluate the quality of the reconstructed images; Table 3 shows a comparison of the results. The image error is defined by [19]:

Gtrue − Gcomp image error = (31) kGture k where Gtrue is the true permittivity distribution; Gcomp is the reconstructed permittivity distribution. Figs. 2 and 3 show the reconstructed images of the algorithms that are being compared. Fig. 2 is the result obtained using the projected Landweber iteration algorithm. It can be seen that for the reconstruction objects (1.b) and (1.c) the quality of the reconstructed images obtained using the algorithm is relatively high; however, for the reconstruction objects (1.a) and (1.d) the distortion of the reconstructed images is relatively large.

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Fig. 2. Reconstructed images using projected Landweber iteration algorithm.

Fig. 3. Reconstructed images using RRTLS algorithm. Table 2 Selected computation parameters for RRTLS algorithm Computation parameters

(1.a)

(1.b)

(1.c)

(1.d)

Number of iteration Regularization parameter

5 0.2 0.1 0.5

3 0.02 0.2 0.5

4 0.6 0.1 0.5

3 0.2 0.1 0.5

p

δ

Table 3 Image error (%) Algorithm

(1.a)

(1.b)

(1.c)

(1.d)

RRTLS PLI

18.69 32.51

22.76 23.33

7.09 25.86

22.87 47.68

Fig. 3 is the result obtained using the RRTLS algorithm. It is found that the spatial resolution of the reconstructed images obtained using the RRTLS algorithm is enhanced, the artifacts in the restructured images can be effectively eliminated, and distortion of the reconstructed images is small. Tables 1 and 2 show the selection of computation parameter for the projected Landweber iteration algorithm and the RRTLS algorithm respectively. It can be seen that the selected computation parameters for the projected Landweber iteration algorithm are simpler than those for the RRTLS algorithm. Numerical simulations indicate that the quality of the reconstructed images depends on the selections of the regularization parameter α and p. In this work the computation parameters were chosen empirically, namely, the ranges of computation parameters were established by solving typical cases. In addition, we can also simplify Eq. (28) by selecting special δ (δ = 0 or δ = 1) according to the characteristics of the reconstructed objects. Table 3 shows the image error. It can be seen that for the reconstructed objects considered in this work the image error obtained using the RRTLS algorithm is smaller than the projected Landweber iteration algorithm. 5. Conclusions Electrical capacitance tomography (ECT) is considered as one of the promising process tomography technologies due to its advantages such as high speed, low cost, and non-intrusive sensors.

Its successful application depends greatly on the precision and speed of the image reconstruction algorithm. In this paper, an image reconstruction algorithm that considers the errors in both the sensitivity field matrix and the capacitance data based on the regularized total least square method for electrical capacitance tomography was presented. The image reconstruction problem was transformed into an optimization problem. The Newton algorithm was employed to solve the objective function. In addition, the smooth approximation of the objective function makes the computation easier. Numerical simulations indicate that the proposed algorithm is efficient and overcomes the numerical instability of ECT image reconstruction. For the cases of the reconstructed objects considered in this paper, the spatial resolution of the reconstructed image obtained using the proposed algorithm is enhanced, and the artifacts in the restructured images can be effectively eliminated. However the algorithm cannot reconstruct the data in “real time”, future work should focus on how to solve the objective function rapidly and efficiently; in addition, more work should be done on validating the proposed algorithm, improving the computation of the sensitivity field matrix, and reducing the errors in the capacitance data. In general, the spatial resolution of complicated reconstruction objects is low. Its applications are distinctively appliancedependent, i.e. the sensor design, calibration, and data interpretation etc., depend heavily on the understanding of the specific dielectric and geometric properties of the objects. To enable ECT technology to be used in a real industry environment, more work on the hardware and software of ECT systems should be carried out and the image reconstruction algorithms should be further developed. Acknowledgement The authors wish to thank the National Natural Science Foundation of China for supporting this research under the contracts No. 50736002, No. 60532020 and No. 60672151. References [1] Yang WQ, Liu S. Role of tomography in gas/solids flow measurement. Flow Measurement and Instrumentation 2000;11:237–44. [2] Jaworski AJ, Dyakowski T. Application of electrical capacitance tomography for measurement of gas–solids flow characteristics in a pneumatic conveying system. Measurement Science & Technology 2001;12:1109–19.

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