June  2019, 13(3): 653-677. doi: 10.3934/ipi.2019030

A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation

1. 

Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

2. 

Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

3. 

Depart of Mathematics, University of Bergen, P. O. Box 7800, N-5020, Bergen, Norway

* Corresponding author: Jun Liu

Received  May 2018 Revised  October 2018 Published  March 2019

A dual expectation-maximization (EM) algorithm for total variation (TV) regularized Gaussian mixture model (GMM) is proposed in this paper. The algorithm is built upon the EM algorithm with TV regularization (EM-TV) model which combines the statistical and variational methods together for image segmentation. Inspired by the projection algorithm proposed by Chambolle, we give a dual algorithm for the EM-TV model. The related dual problem is smooth and can be easily solved by a projection gradient method, which is stable and fast. Given the parameters of GMM, the proposed algorithm can be seen as a forward-backward splitting method which converges. This method can be easily extended to many other applications. Numerical results show that our algorithm can provide high quality segmentation results with fast computation speed. Compared with the well-known statistics based methods such as hidden Markov random field with EM method (HMRF-EM), the proposed algorithm has a better performance. The proposed method could also be applied to MRI segmentation such as SPM8 software and improve the segmentation results.

Citation: Shi Yan, Jun Liu, Haiyang Huang, Xue-Cheng Tai. A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation. Inverse Problems & Imaging, 2019, 13 (3) : 653-677. doi: 10.3934/ipi.2019030
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P. ArbelaezM. MaireC. Fowlkes and J. Malik, Contour detection and hierarchical image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 898-916.  doi: 10.1109/TPAMI.2010.161.  Google Scholar

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J. Ashburner and K. J. Friston, Unified segmentation, NeuroImage, 26 (2005), 839–851, URL http://www.sciencedirect.com/science/article/pii/S1053811905001102. doi: 10.1016/j.neuroimage.2005.02.018.  Google Scholar

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E. BaeJ. Yuan and X. Tai, Global minimization for continuous multiphase partitioning problems using a dual approach, International Journal of Computer Vision, 92 (2011), 112-129.  doi: 10.1007/s11263-010-0406-y.  Google Scholar

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J. Baillon and G. Haddad, Quelques propriétés des opérateurs angle-bornés etn-cycliquement monotones, Israel Journal of Mathematics, 26 (1977), 137-150.  doi: 10.1007/BF03007664.  Google Scholar

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Y. Boykov and V. Kolmogorov, An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision, Energy Minimization Methods in Computer Vision and Pattern Recognition, (2001), 359–374. doi: 10.1007/3-540-44745-8_24.  Google Scholar

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Y. BoykovO. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Analysis and Machine Intelligence, 23 (2001), 1222-1239.  doi: 10.1109/ICCV.1999.791245.  Google Scholar

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C. CarsonS. BelongieH. Greenspan and J. Malik, Blobworld: Image segmentation using expectation-maximization and its application to image querying, IEEE Transactions on Pattern Analysis and Machine Intelligence, 24 (2002), 1026-1038.  doi: 10.1109/TPAMI.2002.1023800.  Google Scholar

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A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.  doi: 10.1023/B:JMIV.0000011325.36760.1e.  Google Scholar

[11]

T. Chan, S. Esegodlu and A. Yip, Recent developments in total variation image restoration, Mathematical Models of Computer Vision, 17 (2005). Google Scholar

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T. Chan and L. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277.  doi: 10.1109/83.902291.  Google Scholar

[13]

Y. ChiangP. Borbat and J. Freed, Maximum entropy: A complement to tikhonov regularization for determination of pair distance distributions by pulsed esr, Journal of Magnetic Resonance, 177 (2005), 184-196.  doi: 10.1016/j.jmr.2005.07.021.  Google Scholar

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P. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.  doi: 10.1080/02331930412331327157.  Google Scholar

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P. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Modeling and Simulation, 4 (2005), 1168-1200.  doi: 10.1137/050626090.  Google Scholar

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S. Gao and T. Bui, Image segmentation and selective smoothing by using mumford-shah model, IEEE Transactions on Image Processing, 14 (2005), 1537-1549.   Google Scholar

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T. Goldstein and S. Osher, The split bregman method for l1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar

[22]

R. Gonzalez, R. Woods and S. Eddins, Digital Image Publishing Using MATLAB, Prentice Hall, 2004. Google Scholar

[23]

L. Gupta and T. Sortrakul, A gaussian-mixture-based image segmentation algorithm, Pattern Recognition, 31 (1998), 315-325.  doi: 10.1016/S0031-3203(97)00045-9.  Google Scholar

[24]

J. Hiriart-Urruty and C. Lemar chal, Convex Analysis and Minimization Algorithms Ⅰ, Springer, 1993. doi: 10.1007/978-3-662-02796-7.  Google Scholar

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H. Ishikawa, Exact optimization for markov random fields with convex priors, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 1333-1336.  doi: 10.1109/TPAMI.2003.1233908.  Google Scholar

[26]

J. LiuY. Ku and S. Leung, Expectation–maximization algorithm with total variation regularization for vector-valued image segmentation, Journal of Visual Communication and Image Representation, 23 (2012), 1234-1244.  doi: 10.1016/j.jvcir.2012.09.002.  Google Scholar

[27]

J. LiuX. TaiH. Huang and Z. Huan, A weighted dictionary learning model for denoising images corrupted by mixed noise, IEEE Transactions on Image Processing, 22 (2013), 1108-1120.  doi: 10.1109/TIP.2012.2227766.  Google Scholar

[28]

J. LiuX. TaiS. Leung and H. Huang, A new continuous max-flow algorithm for multiphase image segmentation using super-level set functions, Journal of Visual Communication & Image Representation, 25 (2014), 1472-1488.  doi: 10.1016/j.jvcir.2014.04.011.  Google Scholar

[29]

D. Martin, C. Fowlkes, D. Tal and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, in Computer Vision, 2001. ICCV 2001. Proceedings. Eighth IEEE International Conference on, vol. 2, IEEE, 2001,416–423. doi: 10.1109/ICCV.2001.937655.  Google Scholar

[30]

G. McLachlan and T. Krishnan, The EM Algorithm and Extensions, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1997.  Google Scholar

[31]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar

[32]

S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[33]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[34]

M. Teboulle, A unified continuous optimization framework for center-based clustering methods, Journal of Machine Learning Research, 8 (2007), 65-102.   Google Scholar

[35]

L. Vese and T. Chan, A multiphase level set framework for image segmentation using the mumford and shah model, International Journal of Computer Vision, 50 (2002), 271-293.   Google Scholar

[36]

Q. Wang, Hmrf-em-image: Implementation of the hidden markov random field model and its expectation-maximization algorithm, arXiv preprint arXiv: 1207.3510. Google Scholar

[37]

C. Wu and X. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[38]

C. Wu, On the convergence properties of the em algorithm, The Annals of Statistics, 11 (1983), 95-103.  doi: 10.1214/aos/1176346060.  Google Scholar

[39]

G. Yu and G. Sapiro, Statistical compressed sensing of gaussian mixture models, IEEE Transactions on Signal Processing, 59 (2011), 5842-5858.  doi: 10.1109/TSP.2011.2168521.  Google Scholar

[40]

J. Yuan, E. Bae, X. Tai and Y. Boykov, A continuous max-flow approach to potts model, Computer Vision (ECCV 2010), European Conference on, 2010,379–392. doi: 10.1007/978-3-642-15567-3_28.  Google Scholar

[41]

Y. ZhangM. Brady and S. Smith, Segmentation of brain mr images through a hidden markov random field model and the expectation-maximization algorithm, IEEE Transactions on Medical Imaging, 20 (2001), 45-57.  doi: 10.1109/42.906424.  Google Scholar

[42]

M. ZhuS. J. Wright and T. Chan, Duality-based algorithms for total-variation-regularized image restoration, Computational Optimization and Applications, 47 (2010), 377-400.  doi: 10.1007/s10589-008-9225-2.  Google Scholar

show all references

References:
[1]

W. Allard, Total variation regularization for image denoising, i. geometric theory, SIAM Journal on Mathematical Analysis, 39 (2007), 1150-1190.  doi: 10.1137/060662617.  Google Scholar

[2]

U. Amato and W. Hughes, Maximum entropy regularization of fredholm integral equations of the first kind, Inverse Problems, 7 (1991), 793-808.  doi: 10.1088/0266-5611/7/6/004.  Google Scholar

[3]

P. ArbelaezM. MaireC. Fowlkes and J. Malik, Contour detection and hierarchical image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 898-916.  doi: 10.1109/TPAMI.2010.161.  Google Scholar

[4]

J. Ashburner and K. J. Friston, Unified segmentation, NeuroImage, 26 (2005), 839–851, URL http://www.sciencedirect.com/science/article/pii/S1053811905001102. doi: 10.1016/j.neuroimage.2005.02.018.  Google Scholar

[5]

E. BaeJ. Yuan and X. Tai, Global minimization for continuous multiphase partitioning problems using a dual approach, International Journal of Computer Vision, 92 (2011), 112-129.  doi: 10.1007/s11263-010-0406-y.  Google Scholar

[6]

J. Baillon and G. Haddad, Quelques propriétés des opérateurs angle-bornés etn-cycliquement monotones, Israel Journal of Mathematics, 26 (1977), 137-150.  doi: 10.1007/BF03007664.  Google Scholar

[7]

Y. Boykov and V. Kolmogorov, An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision, Energy Minimization Methods in Computer Vision and Pattern Recognition, (2001), 359–374. doi: 10.1007/3-540-44745-8_24.  Google Scholar

[8]

Y. BoykovO. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Analysis and Machine Intelligence, 23 (2001), 1222-1239.  doi: 10.1109/ICCV.1999.791245.  Google Scholar

[9]

C. CarsonS. BelongieH. Greenspan and J. Malik, Blobworld: Image segmentation using expectation-maximization and its application to image querying, IEEE Transactions on Pattern Analysis and Machine Intelligence, 24 (2002), 1026-1038.  doi: 10.1109/TPAMI.2002.1023800.  Google Scholar

[10]

A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.  doi: 10.1023/B:JMIV.0000011325.36760.1e.  Google Scholar

[11]

T. Chan, S. Esegodlu and A. Yip, Recent developments in total variation image restoration, Mathematical Models of Computer Vision, 17 (2005). Google Scholar

[12]

T. Chan and L. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277.  doi: 10.1109/83.902291.  Google Scholar

[13]

Y. ChiangP. Borbat and J. Freed, Maximum entropy: A complement to tikhonov regularization for determination of pair distance distributions by pulsed esr, Journal of Magnetic Resonance, 177 (2005), 184-196.  doi: 10.1016/j.jmr.2005.07.021.  Google Scholar

[14]

P. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.  doi: 10.1080/02331930412331327157.  Google Scholar

[15]

P. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Modeling and Simulation, 4 (2005), 1168-1200.  doi: 10.1137/050626090.  Google Scholar

[16]

A. DempsterN. Laird and D. Rubin, Maximum likelihood from incomplete data via the em algorithm, Journal of the Royal Statistical Society. Series B (methodological), 39 (1977), 1-38.  doi: 10.1111/j.2517-6161.1977.tb01600.x.  Google Scholar

[17]

J. Duarte-Carvajalino, G. Sapiro, G. Yu and L. Carin, Online adaptive statistical compressed sensing of gaussian mixture models, arXiv preprint, arXiv: 1112.5895. Google Scholar

[18]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

[19] L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 2015.   Google Scholar
[20]

S. Gao and T. Bui, Image segmentation and selective smoothing by using mumford-shah model, IEEE Transactions on Image Processing, 14 (2005), 1537-1549.   Google Scholar

[21]

T. Goldstein and S. Osher, The split bregman method for l1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar

[22]

R. Gonzalez, R. Woods and S. Eddins, Digital Image Publishing Using MATLAB, Prentice Hall, 2004. Google Scholar

[23]

L. Gupta and T. Sortrakul, A gaussian-mixture-based image segmentation algorithm, Pattern Recognition, 31 (1998), 315-325.  doi: 10.1016/S0031-3203(97)00045-9.  Google Scholar

[24]

J. Hiriart-Urruty and C. Lemar chal, Convex Analysis and Minimization Algorithms Ⅰ, Springer, 1993. doi: 10.1007/978-3-662-02796-7.  Google Scholar

[25]

H. Ishikawa, Exact optimization for markov random fields with convex priors, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 1333-1336.  doi: 10.1109/TPAMI.2003.1233908.  Google Scholar

[26]

J. LiuY. Ku and S. Leung, Expectation–maximization algorithm with total variation regularization for vector-valued image segmentation, Journal of Visual Communication and Image Representation, 23 (2012), 1234-1244.  doi: 10.1016/j.jvcir.2012.09.002.  Google Scholar

[27]

J. LiuX. TaiH. Huang and Z. Huan, A weighted dictionary learning model for denoising images corrupted by mixed noise, IEEE Transactions on Image Processing, 22 (2013), 1108-1120.  doi: 10.1109/TIP.2012.2227766.  Google Scholar

[28]

J. LiuX. TaiS. Leung and H. Huang, A new continuous max-flow algorithm for multiphase image segmentation using super-level set functions, Journal of Visual Communication & Image Representation, 25 (2014), 1472-1488.  doi: 10.1016/j.jvcir.2014.04.011.  Google Scholar

[29]

D. Martin, C. Fowlkes, D. Tal and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, in Computer Vision, 2001. ICCV 2001. Proceedings. Eighth IEEE International Conference on, vol. 2, IEEE, 2001,416–423. doi: 10.1109/ICCV.2001.937655.  Google Scholar

[30]

G. McLachlan and T. Krishnan, The EM Algorithm and Extensions, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1997.  Google Scholar

[31]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar

[32]

S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[33]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[34]

M. Teboulle, A unified continuous optimization framework for center-based clustering methods, Journal of Machine Learning Research, 8 (2007), 65-102.   Google Scholar

[35]

L. Vese and T. Chan, A multiphase level set framework for image segmentation using the mumford and shah model, International Journal of Computer Vision, 50 (2002), 271-293.   Google Scholar

[36]

Q. Wang, Hmrf-em-image: Implementation of the hidden markov random field model and its expectation-maximization algorithm, arXiv preprint arXiv: 1207.3510. Google Scholar

[37]

C. Wu and X. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[38]

C. Wu, On the convergence properties of the em algorithm, The Annals of Statistics, 11 (1983), 95-103.  doi: 10.1214/aos/1176346060.  Google Scholar

[39]

G. Yu and G. Sapiro, Statistical compressed sensing of gaussian mixture models, IEEE Transactions on Signal Processing, 59 (2011), 5842-5858.  doi: 10.1109/TSP.2011.2168521.  Google Scholar

[40]

J. Yuan, E. Bae, X. Tai and Y. Boykov, A continuous max-flow approach to potts model, Computer Vision (ECCV 2010), European Conference on, 2010,379–392. doi: 10.1007/978-3-642-15567-3_28.  Google Scholar

[41]

Y. ZhangM. Brady and S. Smith, Segmentation of brain mr images through a hidden markov random field model and the expectation-maximization algorithm, IEEE Transactions on Medical Imaging, 20 (2001), 45-57.  doi: 10.1109/42.906424.  Google Scholar

[42]

M. ZhuS. J. Wright and T. Chan, Duality-based algorithms for total-variation-regularized image restoration, Computational Optimization and Applications, 47 (2010), 377-400.  doi: 10.1007/s10589-008-9225-2.  Google Scholar

Figure 1.  Comparison of Dual EM-TV, Original EM-TV [26] and HMRF-EM methods for 2 classes segmentation
Figure 2.  Enlarged red square regions in Figure 1
Figure 3.  Comparison of Dual EM-TV, Original EM-TV [26] and HMRF-EM methods for 4 classes segmentation
Figure 4.  Enlarged red square regions in Figure 3
Figure 5.  An image with two regions, which are the same mean but different variance, and the histogram
Figure 6.  Comparison of segmentation results of GMM-EM, Dual EM-TV and CV model on region with same mean but different variances image
Figure 7.  Comparison between Dual EM-TV and HMRF-EM method
Figure 8.  Segmentation results by Dual EM-TV in color images for two-region images
Figure 9.  Segmentation results by Dual EM-TV in color images for multi-region images
Figure 10.  Segmentation comparison under different noise levels. Noise standard deviation: A: 120/255, B: 240/255, C: 300/255, D: 390/255
Figure 11.  Segmentation comparison of the same multi-region image by different noise levels. Noise standard deviation: A: 50/255, B: 60/255, C: 70/255, D: 80/255
Figure 12.  Results of SPM8 with one slice of MR Images. First row: left image, MRI with 0% RF and 0% noise; right: MRI with 40% RF and 9% noise). Second and third rows: the first and third columns are the newsegment method for left and right images, respectively; the second and fourth columns are the proposed method for left and right images, respectively
Table 1.  Comparison of 2 classes segmentation for Dual EM-TV, Original EM-TV [26] and HMRF-EM method (Heart image with size $ 615\times 615 $)
Methods CPU time (seconds) Accuracy (SAI)
Dual EM-TV 1.233 99.70%
Original EM-TV [26] 9.482 98.02%
HMRF-EM 4n 4.052 91.84%
HMRF-EM 8n 3.052 97.38%
%HMRF-EM 12n 2.653 98.69%
HMRF-EM 20n 1.528 99.24%
Methods CPU time (seconds) Accuracy (SAI)
Dual EM-TV 1.233 99.70%
Original EM-TV [26] 9.482 98.02%
HMRF-EM 4n 4.052 91.84%
HMRF-EM 8n 3.052 97.38%
%HMRF-EM 12n 2.653 98.69%
HMRF-EM 20n 1.528 99.24%
Table 2.  Comparison of 4 classes segmentation for Dual EM-TV, Original EM-TV [26] and HMRF-EM methods (4-color image with size $ 600\times 600 $)
Methods CPU time (seconds) Accuracy (SAI)
Dual EM-TV 1.699 99.90%
Original EM-TV [26] 18.502 99.02%
HMRF-EM 4n 3.480 97.39%
HMRF-EM 8n 3.202 99.81%
%HMRF-EM 12n 2.696 99.87%
HMRF-EM 20n 2.388 99.83%
Methods CPU time (seconds) Accuracy (SAI)
Dual EM-TV 1.699 99.90%
Original EM-TV [26] 18.502 99.02%
HMRF-EM 4n 3.480 97.39%
HMRF-EM 8n 3.202 99.81%
%HMRF-EM 12n 2.696 99.87%
HMRF-EM 20n 2.388 99.83%
Table 3.  Comparison of DM in SPM8 on brain images
noise level 5% 5% 7% 7% 9% 9%
brain part WM GM WM GM WM GM
Dual EM-TV 0.9380 0.9122 0.9218 0.8970 0.9035 0.8818
New Segment 0.9317 0.9099 0.9035 0.8822 0.8759 0.8536
noise level 5% 5% 7% 7% 9% 9%
brain part WM GM WM GM WM GM
Dual EM-TV 0.9380 0.9122 0.9218 0.8970 0.9035 0.8818
New Segment 0.9317 0.9099 0.9035 0.8822 0.8759 0.8536
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