3D ECT reconstruction by an improved Landweber iteration algorithm

Flow Measurement and Instrumentation 37 (2014) 92–98

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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst

3D ECT reconstruction by an improved Landweber iteration algorithm Hua Yan n, Yi Fan Wang, Ying Gang Zhou, Yan Hui Sun School of Information Science & Engineering, Shenyang University of Technology, Shenyang 110870, China

art ic l e i nf o

a b s t r a c t

Article history: Received 11 June 2013 Received in revised form 1 January 2014 Accepted 6 March 2014 Available online 20 March 2014

Electrical capacitance tomography (ECT) is a relatively mature non-invasive imaging technique that attempts to map dielectric permittivity of materials. Recently, 3D ECT has gained interest because of its potential to generate volumetric images. The study of a fast and accurate image reconstruction algorithm is a challenge task, especially for 3D reconstruction. In this paper, we propose an improved Landweber iteration algorithm. We incorporate an additional acceleration term into the cost function and apply an adaptive threshold operation to the image obtained in each iteration for reducing artefacts. The algorithm proposed is tested by the noise-free and noise-contaminated capacitance data. Sensitivity matrixes and capacitance data of a 3D ECT sensor are obtained by using the finite element (FE) method. Extensive simulations in 3D reconstruction are carried out. The results verify the effectiveness of these improvements. Both the reconstruction time and the artefacts in the reconstructed image are reduced obviously. The experimental results of 3D reconstruction of objects in the shape of letters U and L confirm the effectiveness of the proposed algorithm further. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Electrical capacitance tomography 3D reconstruction Reconstruction algorithm Reducing image artefacts Convergence rate

1. Introduction Electrical capacitance tomography (ECT) is a relatively mature imaging method among many different industrial tomography modalities. ECT is based on measuring capacitances of a multielectrode sensor surrounding an industrial vessel or pipe containing two materials of different permittivities. These measurements are used to reconstruct a cross-section image or 3D image representing the permittivity distribution inside the vessel or pipe [1–3]. Applications of ECT include the monitoring of oil–gas flows in pipelines, gas–solids flows in pneumatic conveying, and fluidized bed dryers, etc. [1,4–6]. ECT offers some advantages over other tomography modalities, such as no radiation, rapid response, low-cost, non-intrusive and non-invasive. ECT has been rapidly developing. A conventional ECT system is based on the measurements between electrodes in a single measurement plane, and the reconstructed image is given in the form of a 2D cross-sectional image. Research of ECT technology so far has focused on 2D reconstruction, but in past few years, 3D ECT has gained interest because of its potential to generate volumetric images. A 3D ECT sensor usually has at least two-plane measuring electrodes in axial direction. The capacitance measurements are taken between any two measuring electrodes; no matter they are in the same plane or in different planes. 3D images are

n

Corresponding author. E-mail address: [email protected] (H. Yan).

http://dx.doi.org/10.1016/j.flowmeasinst.2014.03.006 0955-5986/& 2014 Elsevier Ltd. All rights reserved.

reconstructed directly from these measured capacitance data by using a suitable reconstruction algorithm [7–10]. The 3D ECT is also called direct 3D ECT or electrical capacitance volume tomography (ECVT). There are three major difficulties associated with ECT image reconstruction. (1) The sensitivity distribution of the ECT sensor is non-uniform and would change along with the permittivity distribution to be imaged. (2) It is under-determined since the number of independent capacitance measurements (i.e. the number of projection data) is much smaller than the number of desired pixels. (3) The inverse problem is ill-conditioned, making the solution sensitive to measurement errors and noise [3,11]. In the case of 3D ECT, these difficulties become more serious since the uniformity of sensitivity distribution is much worse and the number of pixels increases many times while the independent measurements are unchanged. Three major image reconstruction techniques for 3D ECT are the single step linear back projection (LBP), the Landweber iterative reconstruction, and the optimization reconstruction. Li et al. examined the challenge of 3D image reconstruction using the LBP algorithm and a 24-electrode ECT sensor [12]. Banasiak et al. presented a nonlinear Landweber iterative reconstruction algorithm with the sensitivity matrix updated during the iterative process. Using the algorithm and a 32-electrode ECT sensor, they reconstructed very complex geometry with objects in the shape of letters H, A, L and T. Banasiak's algorithm requires very long time due to very intensive processing of large-scale matrices and iterated 3D FEM modeling [8,13]. Warsito et al. proposed a 3D neural-network optimization reconstruction technique

H. Yan et al. / Flow Measurement and Instrumentation 37 (2014) 92–98

(3D-NNMOIRT) which simultaneously optimizes four objective functions related to the measured capacitances and the reconstructed image. Compared with the Landweber iterative reconstruction, the 3D-NNMOIRT provides better image resolution and less image artifacts, but also, it is more complex to implement and requires more computational power [7,10,14]. The Landweber iteration algorithm [3,11,15,16] is known as the most widely used iterative algorithm for ECT. In most cases, it generates good quality images. However, it is only a variation of the steepest descent algorithm from the viewpoint of numerical optimization. Its rate of convergence is relatively slow. Usually it needs a lot of iterations to obtain a good quality image [15–17]. Another drawback of the Landweber iteration algorithm is that it often produces scattered artefacts, especially when it is used for 3D image reconstruction. These scattered artefacts usually emerge in the site with stronger sensitivity. As consequence, the generated image seems to be directed toward the stronger sites of the sensitivity [10]. Liu et al. obtained the step length of Landweber iteration algorithm in an optimal manner [16]. Jang et al. added an additional acceleration term to the conventional Landweber iteration algorithm [15]. These methods accelerate the convergence rate significantly, but obvious artefacts still exist in the reconstructed images, especially in 3D reconstructed images. In this paper, an improved Landweber iterative algorithm is proposed. An additional acceleration term is incorporated into the cost function for accelerating the convergence rate; and an adaptive threshold operation is applied to the image reconstructed in each iteration for reducing the low grey-level artefacts. The proposed algorithm is tested in 3D ECT reconstruction. Both simulation and experiment results demonstrate the effectiveness of the above two measures.

2. The improved Landweber iteration algorithm In ECT, the relationship between the measured interelectrode capacitance and the permittivity distribution of the region to be reconstructed is nonlinear but, for simplicity, can be simplified to a linear form [15,18]: C ¼ SG

ð1Þ

where C A RM1 is the normalized capacitance vector, G A RN1 is the normalized permittivity vector, and S A RMN is the Jacobian matrix of normalized capacitance with respect to the normalized permittivity, i.e. the normalized sensitivity matrix. The independent number of capacitance measurements (M) is equal to L(L1)/2, where L is the number of electrodes. The normalized permittivity is often called normalized grey level, and G is often called grey vector too. The Landweber iteration algorithm is one of the variations of the steepest gradient descent method and its cost function to be minimized can be written as f ðGk Þ ¼ 12 ωk J C  SGk J 2

ð2Þ

The iterative procedure for the Landweber iteration algorithm is Gk þ 1 ¼ Gk þ ∇f ðGk Þ ¼ Gk þ ωk ST ðC  SGk Þ; k Z 0

ð3Þ

In order to accelerate the convergent rate, we add an additional acceleration term into the cost function to be minimized. The new cost function formed can be written as
 FðGk Þ ¼ 12 μk J Gk  Gk  1 J 2 þ ωk J C  SGk J 2

ð4Þ

93

The gradient of F(Gk) with respect to Gk can be simply calculated as ∇FðGk Þ ¼ μk ðGk  Gk  1 Þ þ ωk ST ðC  SGk Þ

ð5Þ

So the iteration procedure becomes Gk þ 1 ¼ Gk þ ∇FðGk Þ ¼ Gk þ μk ðGk  Gk  1 Þ þ ωk ST ðC  SGk Þ; k Z0

ð6Þ

where Gk  Gk  1 represents the difference between current and previous permittivity distributions, C  SGk represents the residual between calculated and measured capacitances, the step lengths μk and ωk can be considered as weighting coefficient for them, respectively. We calculate μk and ωk by using v-method [19,20]. 8 μ0 ¼ 0 > > > 4v þ 2 > > < ω0 ¼ 4v þ 1 kð2k  1Þð2k þ 2v þ 1Þ ð7Þ μk ¼ ðk þ 2vÞð2k ; kZ1 > þ 4v þ 1Þð2k þ 2v  1Þ > > > > ð2k þ 2v þ 1Þðk þ vÞ : ωk ¼ 4 ; kZ1 ðk þ 2vÞð2k þ 4v þ 1Þ The parameter v (v 40) in the v-method is given by experience. In this paper, the value of v is six. As seen in (3) and (6), the image vector is corrected iteratively by ωkST(C  SGk). When the number of capacitance data is much smaller than the number of pixels, C  SGk becomes insignificant, and the image is basically corrected by the sensitivity ST, producing the so-called “sensitivity-caused artefacts”. As a consequence, the generated image seems to be directed toward the stronger sites of the sensitivity [10]. In order to avoid enhancing these artefacts iteratively, we apply a threshold operation to the image reconstructed in each iteration. The formula of the improved Landweber iteration algorithm proposed in this paper can be written as follows: 8 G1 ¼ 0 > > > > > G0 ¼ LBP > > > >G T < k þ 1 ¼ P½Gk þ μk ðGk Gk  1 Þ þ ωk S ðC  SGk Þ ; k Z 0 8 ð8Þ > > < 0 if x o ηk > > > > ð0 r ηk o 1Þ > PðxÞ ¼ x if ηk rx r1 > > > > : : 1 if x 4 1 where G0 ¼ LBP means that G0 is obtained by LBP algorithm, ηk is the threshold value for the image reconstructed in kth iteration. It has been observed that the threshold value is dependent upon the permittivity. So an adaptive threshold value is required. Numerical experiments have suggested that following formula is suitable for the calculation of ηk.

ηk ¼

Cl Ch

ð1  C m Þg mk

ð9Þ

where C l and C h are the average values of the M capacitance measurements when the sensor is filled with low and high permittivity materials, respectively; C m is the average value of the M normalized capacitances resulting from the permittivity distribution to be reconstructed; and gmk is calculated as follows: ( g mk ¼ AVGðGk ðiÞ; fGk ðiÞ 4mkgÞ ð10Þ mk ¼ MEDðGk ðiÞ; fGk ðiÞ a 0gÞ where AVG (  ) denotes an average operator, MED (  ) denotes an median operator, {⋯}denotes a conditional statement, and Gk(i) denotes the ith element of vector Gk. If ηk  0 in (8), that is, the adaptive threshold value is fixed at zero in iteration, the improved Landweber iteration (abbreviated as ILI) algorithm becomes accelerated Landweber iteration (abbreviated as ALI) algorithm. And if ηk  0, uk  0, ωk  2/λmax (λmax is the maximum eigenvalue of STS) in (8), the improved Landweber

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becomes the well-known conventional Landweber iteration (abbreviated as LI) algorithm propose by Yang et al. In real ECT application, materials in the imaging regions sometimes have a continuum of concentration. However, the LBP algorithm with threshold operation is still widely used as an online reconstruction algorithm. Analogously, the ILI algorithm proposed in this paper does not require the objects are clearly defined since the threshold operation is used for reducing lowgray artefacts.

Fig. 1 gives the structure diagram of the 3D ECT sensor used in simulation. The sensor has 12 electrodes mounted symmetrically on the outside of an insulating pipe, gives 66 (L¼12, M ¼66) independent capacitance measurements. The electrodes are short wide, arranged in three planes and every four electrodes on each plane are rotated 451 with reference to the previous one. The FE method [21] was used to calculate the sensitivity matrix and capacitance simulation data. The 3D mesh for the sensor has 12,288 pentahedral elements and 4800 inside the pipe.

3.2. Calculation of normalized sensitivity matrix

1 μðkÞ; εh  εl

i ¼ 1; 2; …; L  1; j ¼ i þ 1; …; L; k ¼ 1; 2; …; N

ð11Þ

where εl and εh are the low and high permittivity values, respectively;cij ðkÞ is the capacitance of electrode pair i–j when the kth element has permittivity εh and all other elements have permittivity εl ; clij and chij are the capacitances of electrode pair i–j when the imaging region is filled with material of permittivity εl and εh , respectively; in 3D reconstruction, μðkÞ ¼ V max =V k , V k and V max are the volumes of the kth and maximum elements. The sensitivity matrix s and its transpose matrix sT can be written as 2 3 s12 ð2Þ ⋯ s12 ðNÞ s12 ð1Þ 6 s ð1Þ s13 ð2Þ ⋯ s13 ðNÞ 7 6 13 7 7 ð12Þ s¼6 6 7 ⋮ ⋮ ⋱ ⋮ 4 5 sðL  1ÞL ð1Þ sðL  1ÞL ð2Þ ⋯ sðL  1ÞL ðNÞ

electrode 11 insulating pipe electrode 6

earthed screen

electrode 3

electrode 1 Fig. 1. The 3D ECT sensor.

s13 ð1Þ

⋯

s13 ð2Þ

⋯

⋮

⋮

s13 ðNÞ

⋯

sðL  1ÞL ð1Þ

3

sðL  1ÞL ð2Þ 7 7 7 7 ⋮ 5 sðL  1ÞL ðNÞ

ð13Þ

The normalized sensitivity matrix S in Section 2 can be calculated from matrix s as follows: Smn ¼

smn ∑N n ¼ 1 smn

ð14Þ

sTmn T ∑M n ¼ 1 smn

ð15Þ

where Smn ; smn ; STmn and sTmn denote the elements in the mth row and the nth column of M  N matrix S, N  M matrix ST, M  N matrix s and N  M matrix sT. In order to make C ¼ SG and G ¼ ST C hold when the sensor is filled with high permittivity material, above normalization procedures are needed. 3.3. Criteria for evaluating reconstruction quality In addition to visual observation, spatial error (SIE) and relative image error (RE) are used to evaluate the reconstruction quality quantitatively [3,15,23,24]. ∑N k ¼ 1 jGt ðkÞ  Gr ðkÞj ∑N G ðkÞ k¼1 t 8 1; if the kth element in the true distribution > < is occupied by high permittivity material Gt ðkÞ ¼ > : 0; otherwise SIE ¼

The sensitivity distribution (map) can be obtained by subdividing the imaging region into N elements and determining the change in capacitance of each electrode pair due to a small perturbation of the permittivity in each element. The sensitivity of kth element of electrode pair i–j can be written as [22]

chij clij

6 s ð2Þ 6 12 sT ¼ 6 6 ⋮ 4 s12 ðNÞ

STmn ¼

3.1. The sensors used in evaluation

cij ðkÞ clij

s12 ð1Þ

Similarly, the matrix ST in Section 2 can be calculated from matrix sT as follows:

3. Evaluation by simulation

sij ðkÞ ¼

2

8 > < 1; if the grey level of the kth element in the reconstruction is non  zero Gr ðkÞ ¼ > : 0; otherwise RE ¼

^ GJ JG JGJ

ð16Þ

ð17Þ

^ are the true and the reconstructed normalized where G and G permittivity vectors, respectively; N is the total number of the elements in the imaging region. Smaller SIE and smaller RE give better reconstruction quality. The reconstructed object may differ from the true object in grey level, volume, position and shape. The difference in grey level is a permittivity error, and other differences are spatial errors. SIE represents the total spatial error information as one would see on two images. Whilst RE depends on the combined effect of spatial errors and permittivity error. A small RE does not necessarily mean that the spatial errors and permittivity error are small because their contributions may cancel each other, and vice versa. 3.4. The permittivity distributions used for the evaluation of algorithms Extensive computer simulations were carried out to test the reconstruction performances of the ILI, ALI and LI algorithms for 3D reconstruction. Seven cases are given in this paper. Case 1, case 2 and case 3 were obtained by placing a cylinder object at three different axial positions inside the sensor; case 4 was obtained by placing two cylinder objects inside the sensor; case 5, case 6 and case 7 were obtained by placing an object in the shape of letters H, U and L inside the sensor. The low (background) and the high (object) permittivity materials chosen were air and perspex with relative permittivity values of 1.0 and 2.6 respectively. Fig. 2 shows the top and the main

H. Yan et al. / Flow Measurement and Instrumentation 37 (2014) 92–98

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Fig. 2. The main and the top views of cases 1–7. (a)–(g) are the main views of cases 1–7; (h) is the top view of cases 1–4; (i) is the top view of cases 5–7.

Fig. 3. The true images and the reconstructed images.

views of these seven cases. The black bold lines on the right side of the main views represent the axial positions of the electrode planes. For each given 3D permittivity distribution, a set of capacitance simulation data was calculated by using the FE method. The influence of noise in capacitance data on image reconstruction was investigated, by adding 3% and 6% random noise into the normalized capacitance data. The definition of noise level can be found in [3]. In order to compare the reconstruction performance more quantitatively, we calculated the SIE and RE for all cases. The images obtained by the LI, ALI and ILI algorithms are not binary. However, in order to observe the artefacts clearly, all the reconstructed images given in this paper are binary images. These

binary images are generated in the following way. If the normalized permittivity value (i.e. normalized grey level) of the kth element is greater than zero, fill the kth element with green, otherwise with white. In fact, we expect a binary image when we want to distinguish two different materials. The ratio of the computation times per iteration consumed by the LI, ALI and ILI is 1:1.01:1.3. 3.5. Reconstruction images and reconstruction errors Fig. 3 gives the reconstruction images obtained by 400th LI, 100th ALI and 100th ILI algorithms for cases 1–7 when the capacitance data are contaminated with 0%, 3% and 6% noise.

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Table 1 The reconstruction errors for cases 1–7. Case

1 2 3 4 5 6 7

400th LI

400th LI

400th LI

100th ALI

100th ALI

100th ALI

100th ILI

100th ILI

100th ILI

0% noise

3% noise

6% noise

0% noise

3% noise

6% noise

0% noise

3% noise

6% noise

SIE ¼2.06 RE ¼ 0.70 SIE ¼2.38 RE ¼0.64 SIE ¼2.13 RE ¼0.66 SIE ¼1.75 RE ¼0.68 SIE ¼ 1.73 RE¼ 0.73 SIE ¼2.26 RE¼ 0.73 SIE ¼2.32 RE ¼0.80

SIE¼ 2.04 RE ¼0.70 SIE¼ 2.31 RE¼ 0.64 SIE ¼2.30 RE ¼0.67 SIE¼ 1.83 RE¼ 0.68 SIE¼ 1.86 RE¼ 0.74 SIE ¼2.29 RE¼ 0.74 SIE ¼2.43 RE ¼0.81

SIE ¼ 2.06 RE ¼ 0.71 SIE ¼2.27 RE¼ 0.65 SIE¼ 2.38 RE ¼0.67 SIE ¼ 1.86 RE¼ 0.68 SIE ¼ 1.84 RE¼ 0.76 SIE ¼ 2.16 RE¼ 0.75 SIE¼ 2.32 RE¼ 0.82

SIE ¼1.50 RE ¼ 0.66 SIE ¼1.56 RE¼ 0.58 SIE ¼1.88 RE¼ 0.62 SIE ¼1.25 RE¼ 0.63 SIE ¼ 1.35 RE ¼0.73 SIE ¼1.95 RE ¼0.73 SIE ¼1.86 RE¼ 0.81

SIE ¼1.63 RE ¼0.66 SIE ¼1.70 RE ¼0.59 SIE ¼1.77 RE ¼0.63 SIE¼ 1.21 RE ¼0.63 SIE ¼1.52 RE¼ 0.74 SIE ¼1.92 RE¼ 0.74 SIE ¼1.96 RE ¼0.81

SIE¼ 1.59 RE ¼0.67 SIE¼ 1.79 RE ¼0.60 SIE¼ 1.77 RE¼ 0.64 SIE ¼1.27 RE ¼0.65 SIE¼ 1.53 RE¼ 0.76 SIE¼ 1.78 RE¼ 0.76 SIE¼ 1.98 RE ¼0.82

SIE¼ 0.63 RE¼ 0.63 SIE¼ 0.69 RE¼ 0.55 SIE¼ 0.69 RE¼ 0.62 SIE ¼ 0.44 RE¼ 0.60 SIE¼ 0.90 RE ¼ 0.71 SIE ¼0.97 RE¼ 0.75 SIE ¼ 1.05 RE¼ 0.82

SIE ¼0.61 RE ¼ 0.63 SIE¼ 0.73 RE ¼ 0.56 SIE ¼ 0.94 RE ¼ 0.65 SIE ¼ 0.41 RE ¼ 0.60 SIE ¼ 0.87 RE¼ 0.74 SIE¼ 1.06 RE ¼ 0.78 SIE ¼1.09 RE ¼ 0.83

SIE¼ 0.60 RE¼ 0.65 SIE ¼ 0.72 RE¼ 0.58 SIE¼ 0.94 RE¼ 0.66 SIE¼ 0.38 RE¼ 0.61 SIE ¼ 0.91 RE ¼0.77 SIE ¼ 1.06 RE¼ 0.80 SIE ¼ 1.16 RE¼ 0.84

15

1 Landweber

Landweber 0.9 10

0.8

RE 0.7

SIE 5

0.6 0.5

0

0

100

200

300

400

0.4

0

iteration number

100

200

300

400

iteration number

15

1 Accelerated Landweber Improved Landweber

0.9

10

0.8

RE 0.7

SIE 5

0.6 Accelerated Landweber Improved Landweber

0.5 0

0

50

100

iteration number

0.4

0

50

100

iteration number

Fig. 4. Comparison of the convergence rate for reconstructing case 4 when the capacitance data are with 3% noise.

The true images are also shown in Fig. 3 for comparison. In order to differentiate cases 1–3 easily, the base of the sensor is marked in blue and the axial positions of the electrode planes are given in the images of cases 1–3. Detailed reconstruction errors of the 400th LI, 100th ALI and 100th ILI algorithms for cases 1–7 are summarized in Table 1. For cases 1–7, we calculated the SIE and RE against iterations when the capacitance data were noise-free and contaminated with 3% and 6% noises. However, only the SIE and RE against iteration number for case 4 when the capacitance data are contaminated with 3% noise and for case 5 when the capacitance data are with 6% noise are given in Figs. 4 and 5, respectively, as examples. Following can be seen from Figs. 3–5 and Table 1. (1) The additional acceleration term can accelerate the convergence rate effectively.

(2) The adaptive threshold operation can reduce scattered artefacts obviously, and thus the convergence rate could speed up further. (3) The convergence rate of the ILI algorithm is much faster than that of the LI algorithm. The ILI algorithm with 100 iterations can produce similar quality images to the LI algorithm with 400 iterations, but in one third of the computation time consumed by the latter. (4) Similar to the other linear iterative algorithm, the ILI, ALI and LI have a semi-convergence characteristic, i.e., image error often starts to increase gradually after reaching a local minimum. Even so, fairly good images can be obtained by the LI algorithm with 400 iterations or the ILI algorithm with 100 iterations in 3D reconstruction. (5) The ILI algorithm has a significant advantage in reducing scattered artefacts over the ALI and LI algorithms. The image reconstructed by the ILI algorithm has the smallest SIE. The

H. Yan et al. / Flow Measurement and Instrumentation 37 (2014) 92–98

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1

8 Landweber

Landweber 0.9

6 0.8

RE 0.7

SIE 4

0.6 2 0.5 0

0

100

200

300

400

0.4

0

100

200

300

400

iteration number

iteration number 8

1 Accelerated Landweber Improved Landweber

0.9

6 0.8

SIE 4

RE 0.7 0.6

2 Accelerated Landweber Improved Landweber

0.5 0

0

50

100

iteration number

0.4

0

50

100

iteration number

Fig. 5. Comparison of the convergence rate for reconstructing case 5 when the capacitance data are with 6% noise.

Fig. 6. The photos of the sensor with half of the metal screen, the U-shape object inside the sensor, the L-shape object inside the sensor and the layout of electrodes.

object reconstructed by the ILI algorithm is most like the true object in shape and position. (6) Both the LI and ILI algorithms hold a fairly good robustness to the noise in capacitance data. The distortion of the reconstructed images and the change of reconstruction errors caused by the noise are small when the noise levels are 3% and 6%.

4. Evaluation by experiment To further evaluate and compare the LI, ALI and ILI algorithms, experiments of 3D reconstruction were carried out using a homemade ECT system based on WK65120B impendence analyzer. Two letter-shaped test objects (U-shaped and L-shaped) have been reconstructed using the 400th LI, 100th ALI and 100th ILI algorithms.

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H. Yan et al. / Flow Measurement and Instrumentation 37 (2014) 92–98

Fig. 7. The experimental results of 3D reconstruction of the U-shape and L-shape objects using the 400th LI, 100th ALI and 100th ILI algorithms.

The letter-shaped objects are formed by plastic tubes filled with quartz sands. The photos of the 12-electrode sensor with half of the metal screen, the U-shape object inside the sensor, the L-shape object inside the sensor and the layout of electrodes are given in Fig. 6. The sensitivity matrix was calculated with εh ¼3.7 and εl ¼1. For high permittivity calibration, quartz sands were used. The images reconstructed by 400th LI algorithm, 100th ALI algorithm and 100th ILI algorithm are given in Fig. 7. Similar to the simulation results, good images can be obtained by the 400th LI or the 100th ILI algorithms; the ILI algorithm has an advantage in reducing scattered artefacts over the LI and ALI algorithms.

5. Conclusion ECT image reconstruction is a nonlinear, under-determined and ill-conditioned inverse problem, which makes high quality image reconstructions difficult, especially in 3D reconstruction. The Landweber iteration (abbreviated as LI) algorithm is known as the most widely used iterative algorithm for ECT. However it has two drawbacks. One is that it usually needs a lot of iterations to obtain a good quality image; another is that it often produces scattered artefacts. The most important issues in the development of the ECT reconstruction algorithms have been focused on the improvement of the quality of reconstruction while reducing the reconstruction time. In this paper, an improved Landweber iterative algorithm (abbreviated as ILI) is proposed. Two measures are taken to improve the performance of the LI algorithm. That is, an additional acceleration term is incorporated into the cost function for accelerating the convergence rate; and an adaptive threshold operation is applied to the image reconstructed in each iteration for reducing the low grey-level artefacts. Both simulation and experiment results demonstrate the effectiveness of the above two measures. In the cases of the reconstructed objects in this paper, the ILI algorithm with 100 iterations can produce similar quality images to the LI algorithm with 400 iterations, but in one-third of the computation time consumed by the latter. As a result, an efficient algorithm for direct 3D ECT image reconstruction is introduced.

Acknowledgment The authors wish to thank the National Natural Science Foundation of China (No. 61071141), the Program for Liaoning Excellent Talents in University (No. LR2013005) for supporting this research.

References [1] Yang WQ. Design of electrical capacitance tomography sensors. Meas Sci Technol 2010;21(042001):1–13. [2] Soleimani M, Mitchell CN, Banasiak R. Four-dimensional electrical capacitance tomography image using experimental data. Prog Electromagn Res 2009;90: 171–86. [3] Yang WQ, Peng LH. Image reconstruction algorithms for electrical capacitance tomography. Meas Sci Technol 2003;14:R1–14. [4] Ismail I, Gamio JC, Bukhari SFA, Yang WQ. Tomography for multi-phase flow measurement in oil industry. Flow Meas Instrum 2005;16:145–55. [5] Xu LJ, Zhou HL, Cao Z, Yang WQ. A digital switching demodulator for electrical capacitance tomography. IEEE Trans Instrum Meas 2013;62:1025–33. [6] Jaworski AJ, Dyakowski T. Application of electrical capacitance tomography for measurement of gas–solids flow characteristics in a pneumatic conveying system. Meas Sci Technol 2001;12:1109–19. [7] Wang F, Marashdeh Q, Fan LS, Warsito W. Electrical capacitance volume tomography: design and applications. Sensors 2010;10:1890–917. [8] Banasiak R, Wajman R, Sankowski D. Three-dimensional nonlinear inversion of electrical capacitance tomography data using a complete sensor model. Prog Electromagn Res 2010;100:219–34. [9] Wajman R, Banasiak R, Mazurkiewicz L, Dyakowski T, Sankowski D. Spatial imaging with 3D capacitance measurements. Meas Sci Technol 2006;17:2113–8. [10] Warsito W, Marashdeh Q, Fan LS. Electrical capacitance volume tomography. IEEE Sensors J 2007;7(4):525–35. [11] Yang WQ, Spink DM, York TA, McCann H. An image-reconstruction algorithm based on Landweber's iteration method for electrical-capacitance tomography. Meas Sci Technol 1999;10:1065–9. [12] Li Y, Holland DJ. Fast and robust 3D electrical capacitance tomography. Meas Sci Technol 2013;24(105406):1–10. [13] Banasiak R, Wajman R, Jaworski T, Fiderek P, Fidos H, Nowakowski J, et al. Study on two-phase flow regime visualization and identification using 3D electrical capacitance tomography and fuzzy-logic classification. Int J Multiph Flow 2014;58:1–4. [14] Weber JM, Mei JS. Bubbling fluidized bed characterization using electrical capacitance volume tomography (ECVT). Powder Technol 2013;242:40–50. [15] Jang JD, Lee SH, Kim KY, Choi BY. Modified iterative Landweber method in electrical capacitance tomography. Meas Sci Technol 2006;17:1909–17. [16] Liu S, Fu L, Yang WQ. Optimization of an iterative image reconstruction algorithm for electrical capacitance tomography. Meas Sci Technol 1999;10: L37–L39. [17] Lei J, Liu S, Li ZH, Sun M. An image reconstruction algorithm based on the extended Tikhonov regularization method for electrical capacitance tomography. Measurement 2009;42:368–76. [18] Li Y, Yang WQ. Image reconstruction by nonlinear Landweber iteration for complicated distributions. Meas Sci Technol 2008;19(094014):1–8. [19] Hanke M. Accelerated Landweber iterations for the solution of ill-posed equations. Numer Math 1991;60:341–73. [20] Engl HW, Hanke M, Neubauer A. Regularization of Inverse Problems. London: Kluwer Academic Publishers; 1996; 160–76. [21] Olmos AM, Primicia JA, Marron JL. Simulation design of electrical capacitance tomography sensors. IET Sci Meas Technol 2007;1:216–23. [22] Yan H, Shao FQ, Xu H, Wang S. Three-dimensional analysis of electrical capacitance tomography sensing fields. Meas Sci Technol 1999;10:717–25. [23] Yan H, Liu LJ, Xu H, Shao FQ. Image reconstruction in electrical capacitance tomography using multiple linear regression and regularization. Meas Sci Technol 2001;12:575–81. [24] Xie CG, Huang SM, Lenn C, Stott AL, Beck MS. Experimental evaluation of capacitance tomographic flow imaging systems using physical models. IEE Proc G 1994;141:357–68.