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August  2022, 16(4): 753-769. doi: 10.3934/ipi.2021074

A wavelet frame constrained total generalized variation model for imaging conductivity distribution

1. 

College of Electronic and Electrical Engineering, Henan Normal University, Henan Key Laboratory of Optoelectronic Sensing Integrated Application, Xinxiang, 453007, China

2. 

School of Biomedical Engineering, Fourth Military Medical University, Xi'an, 710032, China

*Corresponding author: Meng wang, Feng Fu

Y. Shi and Z. Tian contributed equally to this work

Received  July 2021 Revised  August 2021 Published  August 2022 Early access  December 2021

Electrical impedance tomography (EIT) is a sensing technique with which conductivity distribution can be reconstructed. It should be mentioned that the reconstruction is a highly ill-posed inverse problem. Currently, the regularization method has been an effective approach to deal with this problem. Especially, total variation regularization method is advantageous over Tikhonov method as the edge information can be well preserved. Nevertheless, the reconstructed image shows severe staircase effect. In this work, to enhance the quality of reconstruction, a novel hybrid regularization model which combines a total generalized variation method with a wavelet frame approach (TGV-WF) is proposed. An efficient mean doubly augmented Lagrangian algorithm has been developed to solve the TGV-WF model. To demonstrate the effectiveness of the proposed method, numerical simulation and experimental validation are conducted for imaging conductivity distribution. Furthermore, some comparisons are made with typical regularization methods. From the results, it can be found that the proposed method shows better performance in the reconstruction since the edge of the inclusion can be well preserved and the staircase effect is effectively relieved.

Citation: Zhiwei Tian, Yanyan Shi, Meng Wang, Xiaolong Kong, Lei Li, Feng Fu. A wavelet frame constrained total generalized variation model for imaging conductivity distribution. Inverse Problems and Imaging, 2022, 16 (4) : 753-769. doi: 10.3934/ipi.2021074
References:
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J. CaiB. DongS. Osher and Z. Shen, Image restoration: Total variation, wavelet frames, and beyond, J. Amer. Math. Soc., 25 (2012), 1033-1089.  doi: 10.1090/S0894-0347-2012-00740-1.

[2]

B. Dong and Y. Zhang, An efficient algorithm for L$_0$ minimization in wavelet frame based image restoration, J. Sci. Comput., 54 (2013), 350-368. 

[3]

M. GoharianM. BruwerA. JegatheesanG. Moran and J. MacGregor, A novel approach for EIT regularization via spatial and spectrai principal component analysis, Physiological Measurement, 28 (2018), 1001-1016. 

[4]

S. Hamilton and A. Hauptmann, Deep D-Bar: Real-time electrical impedance tomography imaging with deep neural networks, IEEE Transactions on Medical Imaging, 37 (2018), 2367-2377.  doi: 10.1109/TMI.2018.2828303.

[5]

A. Hauptmann, Open 2D electrical impedance tomography data archive, arXiv: 1704.01178, 2017.

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N. HyvonenH. Hakula and S. Pursiainen, Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography, Inverse Probl. Imaging, 1 (2007), 299-317.  doi: 10.3934/ipi.2007.1.299.

[7]

L. JardineJ. Chen and J. Hough, Chest wall mesenchymal hamartoma in an infant: Evaluation with electrical impedance tomography, Pediatric Pulmonology, 54 (2019), 14-16.  doi: 10.1002/ppul.24483.

[8]

B. Kim and K. Kim, Resistivity imaging of binary mixture using weighted Landweber method in electrical impedance tomography, Flow Measurement and Instrumentation, 53 (2017), 39-48.  doi: 10.1016/j.flowmeasinst.2016.05.002.

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F. LiJ. AbascalM. Desco and M. Soleimani, Total variation regularization with split bregman-based method in magnetic induction tomography using experimental data, IEEE Sensors Journal, 17 (2017), 976-985.  doi: 10.1109/JSEN.2016.2637411.

[11]

S. LiuM. HuangH. WuC. Tan and J. Jia, Efficient multi-task structure-aware sparse bayesian learning for frequency-difference electrical impedance tomography, IEEE Transactions on Industrial Informatics, 17 (2021), 463-472. 

[12]

D. LiuV. KolehmainenS. SiltanenA. M. Laukkanen and A. Seppanen, Estimation of conductivity changes in a region of interest with electrical impedance tomography, Inverse Probl. Imaging, 9 (2015), 211-229.  doi: 10.3934/ipi.2015.9.211.

[13]

S. LiuH. WuY. HuangY. Yang and J. Jia, Accelerated structure-aware sparse Bayesian learning for 3D electrical impedance tomography, IEEE Transactions on Industrial Informatics, 15 (2019), 5033-5041. 

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H. LiuS. ZhaoC. Tan and F. Dong, A bilateral constrained image reconstruction method using electrical impedance tomography and ultrasonic measurement, IEEE Sensors Journal, 19 (2019), 9883-9895.  doi: 10.1109/JSEN.2019.2928022.

[15]

X. LvY. Z. Song and F. Li, An efficient nonconvex regularization for wavelet frame and total variation based image restoration, J. Comput. Appl. Math., 290 (2015), 553-566.  doi: 10.1016/j.cam.2015.06.006.

[16]

G. MaZ. HaoX. Wu. and X. Wang, An optimal electrical impedance tomography drive pattern for human-computer interaction applications, IEEE Transactions on Biomedical Circuits and Systems, 14 (2020), 402-411.  doi: 10.1109/TBCAS.2020.2967785.

[17]

S. Martin and C. T. M. Choi, Fast and accurate solution of the inverse problem for image reconstruction using electrical impedance tomography, IEEE Transactions on Magnetics, 55 (2019), 7203004.  doi: 10.1109/TMAG.2019.2900349.

[18]

M. Melenthin, The ACE1 electrical impedance tomography system for thoracic imaging, IEEE Transactions on Instrumentation and Measurement, 68 (2019), 3137-3150.  doi: 10.1109/TIM.2018.2874127.

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P. MullerJ. L. Mueller and M. M. Mellenthin, Real-time implementation of Calderón's method on subject-specific domains, IEEE Transactions on Medical Imaging, 36 (2017), 1868-1875.  doi: 10.1109/TMI.2017.2695893.

[20]

P. PadV. Uhlmann and M. Unser, Maximally localized radial profiles for tight steerable wavelet frames, IEEE Trans. Image Process, 25 (2016), 2275-2287.  doi: 10.1109/TIP.2016.2545301.

[21]

S. RenY. WangG. Liang and F. Dong, A robust inclusion boundary reconstructor for electrical impedance tomography with geometric constraints, IEEE Transactions on Instrumentation and Measurement, 68 (2019), 762-773.  doi: 10.1109/TIM.2018.2853358.

[22]

S. RenK. SunC. Tan and F. Dong, A two-stage deep learning method for robust shape reconstruction with electrical impedance tomography, IEEE Transactions on Instrumentation and Measurement, 69 (2020), 4887-4897.  doi: 10.1109/TIM.2019.2954722.

[23]

A. Ron and Z. Shen, Affine systems in L2(Rd): The analysis of the analysis operator, J. Funct. Anal., 148 (1997), 408-447. 

[24]

D. SbarbaroM. Vauhkonen and T. A. Johansen, State estimation and inverse problems in electrical impedance tomography: Observability, convergence and regularization, Inverse Problems, 31 (2015), 045004.  doi: 10.1088/0266-5611/31/4/045004.

[25]

Y. Shi, Total variation regularization based on iteratively reweighted least squares method for electrical resistance tomography, IEEE Transactions on Instrumentation and Measurement, 69 (2020), 3576-3586.  doi: 10.1109/TIM.2019.2938640.

[26]

Y. Shi, Reduction of staircase effect with total generalized variation regularization for electrical impedance tomography, IEEE Sensors Journal, 19 (2019), 9850-9858.  doi: 10.1109/JSEN.2019.2926232.

[27]

Y. ShiM. Wang and M. Shen, Characterization of oil-water two-phase flow in a horizontal pipe with multi-electrode conductance sensor, Journal of Petroleum Science and Engineering, 146 (2016), 584-590.  doi: 10.1016/j.petrol.2016.07.020.

[28]

X. SongY. Xu and F. Dong, A hybrid regularization method combining Tikhonov with total variation for electrical resistance tomography, Flow Measurement and Instrumentation, 46 (2015), 268-275.  doi: 10.1016/j.flowmeasinst.2015.07.001.

[29]

M. TakhtiY. Teng and K. Odame, A 10 MHz read-out chain for electrical impedance tomography, IEEE Transactions on Biomedical Circuits and Systems, 12 (2018), 222-230.  doi: 10.1109/TBCAS.2017.2778288.

[30]

J. TehraniA. McEwanC. Jin and A. Schaik, L$_1$ regularization method in electrical impedance tomography by using the L$_1$-curve (Pareto frontier curve), Applied Mathematical Modelling, 36 (2012), 1095-1105. 

[31]

M. VauhkonenW. R. B. LionheartL. M. HeikkinenP. J. Vauhkonen and J. P. Kaipio, A MATLAB package for the EIDORS project to reconstruct two-dimensional EIT images, Physiological Measurement, 22 (2001), 107-111.  doi: 10.1088/0967-3334/22/1/314.

[32]

Z. WangJ. XiongY. Yang and H. Li, A flexible and robust threshold selection method, IEEE Transactions on Circuits and Systems for Video Technology, 28 (2018), 2220-2232.  doi: 10.1109/TCSVT.2017.2719122.

[33]

Y. WuD. JiangX. LiuR. Bayford and A. Demosthenous, A human-machine interface using electrical impedance tomography for hand prosthesis control, IEEE Transactions on Biomedical Circuits and Systems, 12 (2018), 1322-1333.  doi: 10.1109/TBCAS.2018.2878395.

[34]

Z. Xu, Development of a portable electrical impedance tomography system for biomedical applications, IEEE Sensors Journal, 18 (2018), 8117-8124.  doi: 10.1109/JSEN.2018.2864539.

[35]

Y. XuB. Han and F. Dong, A new regularization algorithm based on the neighborhood method for electrical impedance tomography, Measurement Science and Technology, 29 (2018), 085401.  doi: 10.1088/1361-6501/aac8b6.

[36]

X. Zhang, A numerical computation forward problem model of electrical impedance tomography based on generalized finite element method, IEEE Transactions on Magnetics, 50 (2014), 1045-1048.  doi: 10.1109/TMAG.2013.2285161.

[37]

Y. ZhangB. Dong and Z. Lu, L$_0$ minimization of wavelet frame based image restoration, Mathematics of Computation, 82 (2013), 995-1015. 

[38]

K. ZhangM. LiF. YangS. Xu and A. Abubakar, Three-Dimensional electrical impedance tomography with multiplicative regularization, IEEE Transactions on Biomedical Engineering, 66 (2019), 2470-2480. 

[39]

L. ZhouB. Harrach and J. K. Seo, Monotonicity-based electrical impedance tomography for lung imaging, Inverse Problems, 34 (2018), 045005.  doi: 10.1088/1361-6420/aaaf84.

[40]

D. ZhuA. McEwan and C. Eiber, Microelectrode array electrical impedance tomography for fast functional imaging in the thalamus, Neuroimage, 198 (2019), 44-52.  doi: 10.1016/j.neuroimage.2019.05.023.

show all references

References:
[1]

J. CaiB. DongS. Osher and Z. Shen, Image restoration: Total variation, wavelet frames, and beyond, J. Amer. Math. Soc., 25 (2012), 1033-1089.  doi: 10.1090/S0894-0347-2012-00740-1.

[2]

B. Dong and Y. Zhang, An efficient algorithm for L$_0$ minimization in wavelet frame based image restoration, J. Sci. Comput., 54 (2013), 350-368. 

[3]

M. GoharianM. BruwerA. JegatheesanG. Moran and J. MacGregor, A novel approach for EIT regularization via spatial and spectrai principal component analysis, Physiological Measurement, 28 (2018), 1001-1016. 

[4]

S. Hamilton and A. Hauptmann, Deep D-Bar: Real-time electrical impedance tomography imaging with deep neural networks, IEEE Transactions on Medical Imaging, 37 (2018), 2367-2377.  doi: 10.1109/TMI.2018.2828303.

[5]

A. Hauptmann, Open 2D electrical impedance tomography data archive, arXiv: 1704.01178, 2017.

[6]

N. HyvonenH. Hakula and S. Pursiainen, Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography, Inverse Probl. Imaging, 1 (2007), 299-317.  doi: 10.3934/ipi.2007.1.299.

[7]

L. JardineJ. Chen and J. Hough, Chest wall mesenchymal hamartoma in an infant: Evaluation with electrical impedance tomography, Pediatric Pulmonology, 54 (2019), 14-16.  doi: 10.1002/ppul.24483.

[8]

B. Kim and K. Kim, Resistivity imaging of binary mixture using weighted Landweber method in electrical impedance tomography, Flow Measurement and Instrumentation, 53 (2017), 39-48.  doi: 10.1016/j.flowmeasinst.2016.05.002.

[9]

J. Kourunen, T. Savolainen, A. Lehikoinen, M. Vauhkonen and L. Heikkinen, Measurement Science and Technology, 20 (2008), 015503.

[10]

F. LiJ. AbascalM. Desco and M. Soleimani, Total variation regularization with split bregman-based method in magnetic induction tomography using experimental data, IEEE Sensors Journal, 17 (2017), 976-985.  doi: 10.1109/JSEN.2016.2637411.

[11]

S. LiuM. HuangH. WuC. Tan and J. Jia, Efficient multi-task structure-aware sparse bayesian learning for frequency-difference electrical impedance tomography, IEEE Transactions on Industrial Informatics, 17 (2021), 463-472. 

[12]

D. LiuV. KolehmainenS. SiltanenA. M. Laukkanen and A. Seppanen, Estimation of conductivity changes in a region of interest with electrical impedance tomography, Inverse Probl. Imaging, 9 (2015), 211-229.  doi: 10.3934/ipi.2015.9.211.

[13]

S. LiuH. WuY. HuangY. Yang and J. Jia, Accelerated structure-aware sparse Bayesian learning for 3D electrical impedance tomography, IEEE Transactions on Industrial Informatics, 15 (2019), 5033-5041. 

[14]

H. LiuS. ZhaoC. Tan and F. Dong, A bilateral constrained image reconstruction method using electrical impedance tomography and ultrasonic measurement, IEEE Sensors Journal, 19 (2019), 9883-9895.  doi: 10.1109/JSEN.2019.2928022.

[15]

X. LvY. Z. Song and F. Li, An efficient nonconvex regularization for wavelet frame and total variation based image restoration, J. Comput. Appl. Math., 290 (2015), 553-566.  doi: 10.1016/j.cam.2015.06.006.

[16]

G. MaZ. HaoX. Wu. and X. Wang, An optimal electrical impedance tomography drive pattern for human-computer interaction applications, IEEE Transactions on Biomedical Circuits and Systems, 14 (2020), 402-411.  doi: 10.1109/TBCAS.2020.2967785.

[17]

S. Martin and C. T. M. Choi, Fast and accurate solution of the inverse problem for image reconstruction using electrical impedance tomography, IEEE Transactions on Magnetics, 55 (2019), 7203004.  doi: 10.1109/TMAG.2019.2900349.

[18]

M. Melenthin, The ACE1 electrical impedance tomography system for thoracic imaging, IEEE Transactions on Instrumentation and Measurement, 68 (2019), 3137-3150.  doi: 10.1109/TIM.2018.2874127.

[19]

P. MullerJ. L. Mueller and M. M. Mellenthin, Real-time implementation of Calderón's method on subject-specific domains, IEEE Transactions on Medical Imaging, 36 (2017), 1868-1875.  doi: 10.1109/TMI.2017.2695893.

[20]

P. PadV. Uhlmann and M. Unser, Maximally localized radial profiles for tight steerable wavelet frames, IEEE Trans. Image Process, 25 (2016), 2275-2287.  doi: 10.1109/TIP.2016.2545301.

[21]

S. RenY. WangG. Liang and F. Dong, A robust inclusion boundary reconstructor for electrical impedance tomography with geometric constraints, IEEE Transactions on Instrumentation and Measurement, 68 (2019), 762-773.  doi: 10.1109/TIM.2018.2853358.

[22]

S. RenK. SunC. Tan and F. Dong, A two-stage deep learning method for robust shape reconstruction with electrical impedance tomography, IEEE Transactions on Instrumentation and Measurement, 69 (2020), 4887-4897.  doi: 10.1109/TIM.2019.2954722.

[23]

A. Ron and Z. Shen, Affine systems in L2(Rd): The analysis of the analysis operator, J. Funct. Anal., 148 (1997), 408-447. 

[24]

D. SbarbaroM. Vauhkonen and T. A. Johansen, State estimation and inverse problems in electrical impedance tomography: Observability, convergence and regularization, Inverse Problems, 31 (2015), 045004.  doi: 10.1088/0266-5611/31/4/045004.

[25]

Y. Shi, Total variation regularization based on iteratively reweighted least squares method for electrical resistance tomography, IEEE Transactions on Instrumentation and Measurement, 69 (2020), 3576-3586.  doi: 10.1109/TIM.2019.2938640.

[26]

Y. Shi, Reduction of staircase effect with total generalized variation regularization for electrical impedance tomography, IEEE Sensors Journal, 19 (2019), 9850-9858.  doi: 10.1109/JSEN.2019.2926232.

[27]

Y. ShiM. Wang and M. Shen, Characterization of oil-water two-phase flow in a horizontal pipe with multi-electrode conductance sensor, Journal of Petroleum Science and Engineering, 146 (2016), 584-590.  doi: 10.1016/j.petrol.2016.07.020.

[28]

X. SongY. Xu and F. Dong, A hybrid regularization method combining Tikhonov with total variation for electrical resistance tomography, Flow Measurement and Instrumentation, 46 (2015), 268-275.  doi: 10.1016/j.flowmeasinst.2015.07.001.

[29]

M. TakhtiY. Teng and K. Odame, A 10 MHz read-out chain for electrical impedance tomography, IEEE Transactions on Biomedical Circuits and Systems, 12 (2018), 222-230.  doi: 10.1109/TBCAS.2017.2778288.

[30]

J. TehraniA. McEwanC. Jin and A. Schaik, L$_1$ regularization method in electrical impedance tomography by using the L$_1$-curve (Pareto frontier curve), Applied Mathematical Modelling, 36 (2012), 1095-1105. 

[31]

M. VauhkonenW. R. B. LionheartL. M. HeikkinenP. J. Vauhkonen and J. P. Kaipio, A MATLAB package for the EIDORS project to reconstruct two-dimensional EIT images, Physiological Measurement, 22 (2001), 107-111.  doi: 10.1088/0967-3334/22/1/314.

[32]

Z. WangJ. XiongY. Yang and H. Li, A flexible and robust threshold selection method, IEEE Transactions on Circuits and Systems for Video Technology, 28 (2018), 2220-2232.  doi: 10.1109/TCSVT.2017.2719122.

[33]

Y. WuD. JiangX. LiuR. Bayford and A. Demosthenous, A human-machine interface using electrical impedance tomography for hand prosthesis control, IEEE Transactions on Biomedical Circuits and Systems, 12 (2018), 1322-1333.  doi: 10.1109/TBCAS.2018.2878395.

[34]

Z. Xu, Development of a portable electrical impedance tomography system for biomedical applications, IEEE Sensors Journal, 18 (2018), 8117-8124.  doi: 10.1109/JSEN.2018.2864539.

[35]

Y. XuB. Han and F. Dong, A new regularization algorithm based on the neighborhood method for electrical impedance tomography, Measurement Science and Technology, 29 (2018), 085401.  doi: 10.1088/1361-6501/aac8b6.

[36]

X. Zhang, A numerical computation forward problem model of electrical impedance tomography based on generalized finite element method, IEEE Transactions on Magnetics, 50 (2014), 1045-1048.  doi: 10.1109/TMAG.2013.2285161.

[37]

Y. ZhangB. Dong and Z. Lu, L$_0$ minimization of wavelet frame based image restoration, Mathematics of Computation, 82 (2013), 995-1015. 

[38]

K. ZhangM. LiF. YangS. Xu and A. Abubakar, Three-Dimensional electrical impedance tomography with multiplicative regularization, IEEE Transactions on Biomedical Engineering, 66 (2019), 2470-2480. 

[39]

L. ZhouB. Harrach and J. K. Seo, Monotonicity-based electrical impedance tomography for lung imaging, Inverse Problems, 34 (2018), 045005.  doi: 10.1088/1361-6420/aaaf84.

[40]

D. ZhuA. McEwan and C. Eiber, Microelectrode array electrical impedance tomography for fast functional imaging in the thalamus, Neuroimage, 198 (2019), 44-52.  doi: 10.1016/j.neuroimage.2019.05.023.

Figure 1.  Configuration of the EIT system
Figure 2.  Comparison of the convergence
Figure 3.  Reconstructed images for inclusions with the same conductivity
Figure 4.  Reconstructed images for inclusions with different conductivities
Figure 5.  Comparison of different methods in suppressing the staircase effect
Figure 6.  Reconstructed images for inclusions under the noise level of 5%
Figure 7.  Reconstructed images for inclusions under the noise level of 10%
Figure 8.  Reconstructed images for experimental phantom
Table 1.  Comparisons of Ree and Coc for inclusions with the same conductivity
Model TV TGV TGV-WF
Ree/Coc Ree/Coc Ree/Coc
1 0.3513/0.8868 0.3428/0.8962 0.3380/0.9126
2 0.3798/0.8748 0.3698/0.8804 0.3502/0.8978
3 0.4071/0.8765 0.3814/0.8963 0.3754/0.9103
Model TV TGV TGV-WF
Ree/Coc Ree/Coc Ree/Coc
1 0.3513/0.8868 0.3428/0.8962 0.3380/0.9126
2 0.3798/0.8748 0.3698/0.8804 0.3502/0.8978
3 0.4071/0.8765 0.3814/0.8963 0.3754/0.9103
Table 2.  Comparisons of Ree and Coc for inclusions with different conductivities
Model TV TGV TGV-WF
Ree/Coc Ree/Coc Ree/Coc
4 0.3824/0.8619 0.3726/0.8735 0.3681/0.8904
5 0.4326/0.8022 0.4158/0.8202 0.4006/0.8356
Model TV TGV TGV-WF
Ree/Coc Ree/Coc Ree/Coc
4 0.3824/0.8619 0.3726/0.8735 0.3681/0.8904
5 0.4326/0.8022 0.4158/0.8202 0.4006/0.8356
Table 3.  The values of Ree and Coc under 5% noise level
Model TV TGV TGV-WF
Ree/Coc Ree/Coc Ree/Coc
1 0.3756/0.8753 0.3548/0.8850 0.3426/0.9026
2 0.3863/0.8648 0.3759/0.8688 0.3652/0.8864
3 0.4129/0.8653 0.3958/0.8806 0.3884/0.9021
4 0.3901/0.8493 0.3786/0.8652 0.3712/0.8865
5 0.4504/0.7899 0.4296/0.8057 0.4153/0.8286
Model TV TGV TGV-WF
Ree/Coc Ree/Coc Ree/Coc
1 0.3756/0.8753 0.3548/0.8850 0.3426/0.9026
2 0.3863/0.8648 0.3759/0.8688 0.3652/0.8864
3 0.4129/0.8653 0.3958/0.8806 0.3884/0.9021
4 0.3901/0.8493 0.3786/0.8652 0.3712/0.8865
5 0.4504/0.7899 0.4296/0.8057 0.4153/0.8286
Table 4.  The values of Ree and Coc under 10% noise level
Model TV TGV TGV-WF
Ree/Coc Ree/Coc Ree/Coc
1 0.3865/0.8693 0.3627/0.8768 0.3528/0.8934
2 0.3923/0.8521 0.3806/0.8602 0.3714/0.8813
3 0.4315/0.8523 0.4021/0.8712 0.3996/0.8923
4 0.4082/0.8203 0.3864/0.8486 0.3794/0.8746
5 0.4712/0.7396 0.4355/0.7863 0.4251/0.8093
Model TV TGV TGV-WF
Ree/Coc Ree/Coc Ree/Coc
1 0.3865/0.8693 0.3627/0.8768 0.3528/0.8934
2 0.3923/0.8521 0.3806/0.8602 0.3714/0.8813
3 0.4315/0.8523 0.4021/0.8712 0.3996/0.8923
4 0.4082/0.8203 0.3864/0.8486 0.3794/0.8746
5 0.4712/0.7396 0.4355/0.7863 0.4251/0.8093
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