American Institute of Mathematical Sciences

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doi: 10.3934/ipi.2022014
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Robust region-based active contour models via local statistical similarity and local similarity factor for intensity inhomogeneity and high noise image segmentation

Ibrar Hussain 1, , Haider Ali 1,2,3, , Muhammad Shahkar Khan 1,2,3, , Sijie Niu 1,2,3, and  Lavdie Rada 1,2,3,

1. 

Department of Mathematics, University of Peshawar, Peshawar
2. 

School of Information Science and Engineering, University of Jinan, China
3. 

Bahcesehir University, Istanbul, Turkey

Received  April 2021 Revised  February 2022 Early access April 2022

In this paper, we design a novel variational segmentation method for two types of segmentation problems, namely, global segmentation (all objects /features in a given image are aimed to be segmented) and selective/ interactive segmentation (an objects /feature of interest in a given image is aimed to be segmented) for inhomogeneous and severe additive noisy images. The proposed segmentation models implement a local denoising constraint, capable to tackle efficiently noise/outliers and coping with intensity inhomogeneity issues, combined with local similarity factor based on spatial distances and intensity differences in the local region that guides accurately the level set function to distinguish between outliers and minute important details. Furthermore, to exhibit the accuracy of the proposed models, an experimental comparison is inducted and shown comparisons with state-of-art models on synthetic images, outdoor images, and medical images.

Keywords: Local spatial distance, local intensity difference, level set method, partial differential equations, calculus of variations, image segmentation, selective segmentation.
Mathematics Subject Classification: Primary: 65M30, 65M22, 35J05, 35J25, 65M06.
Citation: Ibrar Hussain, Haider Ali, Muhammad Shahkar Khan, Sijie Niu, Lavdie Rada. Robust region-based active contour models via local statistical similarity and local similarity factor for intensity inhomogeneity and high noise image segmentation. Inverse Problems and Imaging, doi: 10.3934/ipi.2022014
References:
[1]

F. Akram, J. H. Kim, H. U. Lim and K. N. Choi, Segmentation of intensity inhomogeneous brain MR images using active contours, Computational and Mathematical Methods in Medicine, (2014), 1-14.  doi: 10.1155/2014/194614.

[2]

H. Ali, L. Rada and N. Badshah, Image segmentation for intensity inhomogeneity in presence of high noise, IEEE Trans. Image Process., 27 (2018), 3729-3738.  doi: 10.1109/TIP.2018.2825101.

[3]

G. Aubert and J. Aujol, A variational approach to remove multiplicative noise, SIAM J. Appl. Math., 68 (2008), 925-946.  doi: 10.1137/060671814.

[4]

V. Badrinarayanan, A. Kendall and R. Cipolla, SegNet: A deep convolutional encoder-decoder architecture for image segmentation, CoRR., 39 (2017), 2481-2495.  doi: 10.1109/TPAMI.2016.2644615.

[5]

N. Badshah and K. Chen, Image selective segmentation under geometrical constraints using an active contour approach, Math. Comput., 7 (2010), 759-778.  doi: 10.4208/cicp.2009.09.026.

[6]

N. Badshah, K. Chen, H. Ali and G. Murtaza, Coefficient of variation based image selective segmentation using active contour, East Asian J. Appl. Math., 2 (2012), 150-169.  doi: 10.4208/eajam.090312.080412a.

[7]

X. Bai and G. Sapiro, A geodesic framework for fast interactive image and video segmentation and matting, IEEE International Conference on Computer Vision, (2007), 1-8. 

[8]

X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran and S. Osher, Fast global minimization of the active contour/snake model, J. Math. Imag. Vis., 28 (2007), 151-167.  doi: 10.1007/s10851-007-0002-0.

[9]

G. J. Brostow, J. Fauqueur and R. Cipolla, Semantic object classes in video: A high-definition ground truth database, Pattern Recogn. Lett., 2 (2009), 88-97. 

[10]

X. Cai, R. Chan and T. Zeng, A two-stage image segmentation method using a convex variant of the Mumford-shah model and thresholding, SIAM J. Imaging Sci., 6 (2013), 368-390.  doi: 10.1137/120867068.

[11]

V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, 22 (1997), 61-79.  doi: 10.1109/ICCV.1995.466871.

[12]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Proc., 10 (2001), 266-277.  doi: 10.1109/83.902291.

[13]

L. C. Chen, G. Papandreou, I. Kokkinos, K. Murphy and A. L. Yuille, DeepLab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected CRFs, IEEE Transactions on Pattern Analysis and Machine Intelligence, 40 (2018), 834-848.  doi: 10.1109/TPAMI.2017.2699184.

[14]

H. Ding, X. Jiang, B. Shuai, A. Q. Liu and G. Wang, Semantic segmentation with context ecoding and multi-path decoding, IEEE Trans. Image Process., 29 (2019), 3520-3533. 

[15]

X. Dong, J. Shen and L. Shao, Submarkov random walk for image segmentation, IEEE Transactions on Image Processing, 25 (2016), 516-527.  doi: 10.1109/TIP.2015.2505184.

[16]

D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, J. Amer. Statist. Assoc., 90 (1995), 1200-1224.  doi: 10.1080/01621459.1995.10476626.

[17]

S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, Readings in Computer Vision, (1987), 564-584.  doi: 10.1016/B978-0-08-051581-6.50057-X.

[18]

T. Goldstein, X. Bresson and S. Osher, Geometric applications of the split Bregman method: Segmentation and surface reconstruction, J. Sci. Comput., 45 (2010), 272-293.  doi: 10.1007/s10915-009-9331-z.

[19]

T. Goldstein, X. Bresson and S. Osher, Active contours with selective local or global segmentation: A new formulation and level set method, Image and Vision Computing, 28, 668–676.

[20]

C. Gout, C. Le Guyader and L. Vese, Segmentation under geometrical conditions with geodesic active contour and interpolation using level set methods, Numer. Algorithms, 39 (2005), 155-173.  doi: 10.1007/s11075-004-3627-8.

[21]

L. Grady, Random walks for image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 28 (2006), 1768-1783.  doi: 10.1109/TPAMI.2006.233.

[22]

Y. Huang, M. Ng and T. Zeng, The convex relaxation method on deconvolution model with multiplicative noise, Commun. Comput. Phys., 13 (2013), 1066-1092.  doi: 10.4208/cicp.310811.090312a.

[23]

M. Kass, A. Witkin and D. Terzopoulos, Active contours models, International Journal of Computer Vision, 22 (1993), 123-135. 

[24]

C. Le Guyader and C. Gout, Geodesic active contour under geometrical conditions theory and 3D applications, Numerical Algorithms, 48 (2008), 105-133.  doi: 10.1007/s11075-008-9174-y.

[25]

C. Li, R. Huang, Z. Ding, J. C. Gatenby, D. Metaxas and J. C. Gore, A level set method for image segmentation in the presence of intensity inhomogeneities with application to MRI, IEEE Trans. Image Process., 20 (2011), 2007-2016.  doi: 10.1109/TIP.2011.2146190.

[26]

C. Li, C.-Y. Kao, J. C. Gore and Z. Ding, Implicit active contours driven by local binary fitting energy, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 37 (2007), 1-7.  doi: 10.1109/CVPR.2007.383014.

[27]

C. Li, C.-Y. Kao, J. C. Gore and Z. Ding, Implicit active contours driven by local binary fitting energy, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. (CVPR)., 42 (2007), 1-7.  doi: 10.1109/CVPR.2007.383014.

[28]

G. Li, H. F. Li, F. X. Shang and H. Guo, Noise image segmentation model with local intensity differnce, Jornal of Computer Applications, 38 (2018), 842-847. 

[29]

L. Li, S. Luo, X. C. Tai and J. Yang, A new variational approach based on level-set function for convex hull problem with outliers, Inverse Problems and Imaging, 15 (2021), 315-338.  doi: 10.3934/ipi.2020070.

[30]

C. Liu, M. K.-P. Ng and T. Zeng, Weighted variational model for selective image segmentation with application to medical images, Math. Comp., 76 (2017), 367-379.  doi: 10.1016/j.patcog.2017.11.019.

[31]

T. Lu, P. Neittaanmaki and X.-C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, RAIRO Modél. Math. Anal. Numér, 26 (1992), 673-708.  doi: 10.1051/m2an/1992260606731.

[32]

L. Mabood, H. Ali, N. Badshah, K. chen and G. A. Khan, Active contours textural and inhomogeneous object extraction, Pattern Recognition, 55 (2016), 87-99.  doi: 10.1016/j.patcog.2016.01.021.

[33]

T. N. A. Nguyen, J. Cai, J. Zhang and J. Zheng, Robust interactive image segmentation using convex active contours, IEEE Trans. Image Process., 21 (2012), 3734-3743.  doi: 10.1109/TIP.2012.2191566.

[34]

S. Niu, Q. Chen, L. D. Sisternes, Z. Ji, Z. Zhou and D. L. Rubin, Robust noise region-based active contour model via local similarity factor for image segmentation, Pattern Recognit., 61 (2017), 104-119.  doi: 10.1016/j.patcog.2016.07.022.

[35]

L. Rada and K. Chen, Improved selective segmentation model using one level-set, Numerical Algorithm, 48 (2008), 105-133.  doi: 10.1260/1748-3018.7.4.509.

[36]

C. Rother, V. Kolmogorov and A. Blake, Grabcut: Interactive foreground extraction using iterated graph cuts, ACM Siggraph, 23 (2004), 1-6. 

[37]

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

[38]

J. Shen, Y. Du and X. Li, Interactive segmentation using constrained Laplacian optimization, IEEE Transactions on Circuits and Systems for Video Technology, 24 (2014), 1088-1100.  doi: 10.1109/TCSVT.2014.2302545.

[39]

W. Tao and X. C. Tai, Multiple piecewise constant with geodesic active contours (MPC-GAC) framework for interactive image segmentation using graph cut optimization, Image and Vision Computing, 29 (2011), 499-508. 

[40]

L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Computer Vision, 50 (2002), 271-293. 

[41]

X.-F. Wang, D. S. Huang and H. Xu, An efficient local Chan-Vese model for image segmentation, Pattern Recognition, 43 (2010), 603-618.  doi: 10.1016/j.patcog.2009.08.002.

[42]

X. Wang, X. Jiang and J. Ren, Blood vessel segmentation from fundus image by a cascade classification framwork, Patttern Recognition, 88 (2019), 331-341. 

[43]

K. Zhang, H. Song and L. Zhang, Active contours driven by local image fitting energy, Pattern Recognition, 43 (2010), 1199-1206.  doi: 10.1016/j.patcog.2009.10.010.

show all references

References:
[1]

F. Akram, J. H. Kim, H. U. Lim and K. N. Choi, Segmentation of intensity inhomogeneous brain MR images using active contours, Computational and Mathematical Methods in Medicine, (2014), 1-14.  doi: 10.1155/2014/194614.

[2]

H. Ali, L. Rada and N. Badshah, Image segmentation for intensity inhomogeneity in presence of high noise, IEEE Trans. Image Process., 27 (2018), 3729-3738.  doi: 10.1109/TIP.2018.2825101.

[3]

G. Aubert and J. Aujol, A variational approach to remove multiplicative noise, SIAM J. Appl. Math., 68 (2008), 925-946.  doi: 10.1137/060671814.

[4]

V. Badrinarayanan, A. Kendall and R. Cipolla, SegNet: A deep convolutional encoder-decoder architecture for image segmentation, CoRR., 39 (2017), 2481-2495.  doi: 10.1109/TPAMI.2016.2644615.

[5]

N. Badshah and K. Chen, Image selective segmentation under geometrical constraints using an active contour approach, Math. Comput., 7 (2010), 759-778.  doi: 10.4208/cicp.2009.09.026.

[6]

N. Badshah, K. Chen, H. Ali and G. Murtaza, Coefficient of variation based image selective segmentation using active contour, East Asian J. Appl. Math., 2 (2012), 150-169.  doi: 10.4208/eajam.090312.080412a.

[7]

X. Bai and G. Sapiro, A geodesic framework for fast interactive image and video segmentation and matting, IEEE International Conference on Computer Vision, (2007), 1-8. 

[8]

X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thiran and S. Osher, Fast global minimization of the active contour/snake model, J. Math. Imag. Vis., 28 (2007), 151-167.  doi: 10.1007/s10851-007-0002-0.

[9]

G. J. Brostow, J. Fauqueur and R. Cipolla, Semantic object classes in video: A high-definition ground truth database, Pattern Recogn. Lett., 2 (2009), 88-97. 

[10]

X. Cai, R. Chan and T. Zeng, A two-stage image segmentation method using a convex variant of the Mumford-shah model and thresholding, SIAM J. Imaging Sci., 6 (2013), 368-390.  doi: 10.1137/120867068.

[11]

V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, 22 (1997), 61-79.  doi: 10.1109/ICCV.1995.466871.

[12]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Proc., 10 (2001), 266-277.  doi: 10.1109/83.902291.

[13]

L. C. Chen, G. Papandreou, I. Kokkinos, K. Murphy and A. L. Yuille, DeepLab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected CRFs, IEEE Transactions on Pattern Analysis and Machine Intelligence, 40 (2018), 834-848.  doi: 10.1109/TPAMI.2017.2699184.

[14]

H. Ding, X. Jiang, B. Shuai, A. Q. Liu and G. Wang, Semantic segmentation with context ecoding and multi-path decoding, IEEE Trans. Image Process., 29 (2019), 3520-3533. 

[15]

X. Dong, J. Shen and L. Shao, Submarkov random walk for image segmentation, IEEE Transactions on Image Processing, 25 (2016), 516-527.  doi: 10.1109/TIP.2015.2505184.

[16]

D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, J. Amer. Statist. Assoc., 90 (1995), 1200-1224.  doi: 10.1080/01621459.1995.10476626.

[17]

S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, Readings in Computer Vision, (1987), 564-584.  doi: 10.1016/B978-0-08-051581-6.50057-X.

[18]

T. Goldstein, X. Bresson and S. Osher, Geometric applications of the split Bregman method: Segmentation and surface reconstruction, J. Sci. Comput., 45 (2010), 272-293.  doi: 10.1007/s10915-009-9331-z.

[19]

T. Goldstein, X. Bresson and S. Osher, Active contours with selective local or global segmentation: A new formulation and level set method, Image and Vision Computing, 28, 668–676.

[20]

C. Gout, C. Le Guyader and L. Vese, Segmentation under geometrical conditions with geodesic active contour and interpolation using level set methods, Numer. Algorithms, 39 (2005), 155-173.  doi: 10.1007/s11075-004-3627-8.

[21]

L. Grady, Random walks for image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 28 (2006), 1768-1783.  doi: 10.1109/TPAMI.2006.233.

[22]

Y. Huang, M. Ng and T. Zeng, The convex relaxation method on deconvolution model with multiplicative noise, Commun. Comput. Phys., 13 (2013), 1066-1092.  doi: 10.4208/cicp.310811.090312a.

[23]

M. Kass, A. Witkin and D. Terzopoulos, Active contours models, International Journal of Computer Vision, 22 (1993), 123-135. 

[24]

C. Le Guyader and C. Gout, Geodesic active contour under geometrical conditions theory and 3D applications, Numerical Algorithms, 48 (2008), 105-133.  doi: 10.1007/s11075-008-9174-y.

[25]

C. Li, R. Huang, Z. Ding, J. C. Gatenby, D. Metaxas and J. C. Gore, A level set method for image segmentation in the presence of intensity inhomogeneities with application to MRI, IEEE Trans. Image Process., 20 (2011), 2007-2016.  doi: 10.1109/TIP.2011.2146190.

[26]

C. Li, C.-Y. Kao, J. C. Gore and Z. Ding, Implicit active contours driven by local binary fitting energy, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 37 (2007), 1-7.  doi: 10.1109/CVPR.2007.383014.

[27]

C. Li, C.-Y. Kao, J. C. Gore and Z. Ding, Implicit active contours driven by local binary fitting energy, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. (CVPR)., 42 (2007), 1-7.  doi: 10.1109/CVPR.2007.383014.

[28]

G. Li, H. F. Li, F. X. Shang and H. Guo, Noise image segmentation model with local intensity differnce, Jornal of Computer Applications, 38 (2018), 842-847. 

[29]

L. Li, S. Luo, X. C. Tai and J. Yang, A new variational approach based on level-set function for convex hull problem with outliers, Inverse Problems and Imaging, 15 (2021), 315-338.  doi: 10.3934/ipi.2020070.

[30]

C. Liu, M. K.-P. Ng and T. Zeng, Weighted variational model for selective image segmentation with application to medical images, Math. Comp., 76 (2017), 367-379.  doi: 10.1016/j.patcog.2017.11.019.

[31]

T. Lu, P. Neittaanmaki and X.-C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, RAIRO Modél. Math. Anal. Numér, 26 (1992), 673-708.  doi: 10.1051/m2an/1992260606731.

[32]

L. Mabood, H. Ali, N. Badshah, K. chen and G. A. Khan, Active contours textural and inhomogeneous object extraction, Pattern Recognition, 55 (2016), 87-99.  doi: 10.1016/j.patcog.2016.01.021.

[33]

T. N. A. Nguyen, J. Cai, J. Zhang and J. Zheng, Robust interactive image segmentation using convex active contours, IEEE Trans. Image Process., 21 (2012), 3734-3743.  doi: 10.1109/TIP.2012.2191566.

[34]

S. Niu, Q. Chen, L. D. Sisternes, Z. Ji, Z. Zhou and D. L. Rubin, Robust noise region-based active contour model via local similarity factor for image segmentation, Pattern Recognit., 61 (2017), 104-119.  doi: 10.1016/j.patcog.2016.07.022.

[35]

L. Rada and K. Chen, Improved selective segmentation model using one level-set, Numerical Algorithm, 48 (2008), 105-133.  doi: 10.1260/1748-3018.7.4.509.

[36]

C. Rother, V. Kolmogorov and A. Blake, Grabcut: Interactive foreground extraction using iterated graph cuts, ACM Siggraph, 23 (2004), 1-6. 

[37]

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

[38]

J. Shen, Y. Du and X. Li, Interactive segmentation using constrained Laplacian optimization, IEEE Transactions on Circuits and Systems for Video Technology, 24 (2014), 1088-1100.  doi: 10.1109/TCSVT.2014.2302545.

[39]

W. Tao and X. C. Tai, Multiple piecewise constant with geodesic active contours (MPC-GAC) framework for interactive image segmentation using graph cut optimization, Image and Vision Computing, 29 (2011), 499-508. 

[40]

L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Computer Vision, 50 (2002), 271-293. 

[41]

X.-F. Wang, D. S. Huang and H. Xu, An efficient local Chan-Vese model for image segmentation, Pattern Recognition, 43 (2010), 603-618.  doi: 10.1016/j.patcog.2009.08.002.

[42]

X. Wang, X. Jiang and J. Ren, Blood vessel segmentation from fundus image by a cascade classification framwork, Patttern Recognition, 88 (2019), 331-341. 

[43]

K. Zhang, H. Song and L. Zhang, Active contours driven by local image fitting energy, Pattern Recognition, 43 (2010), 1199-1206.  doi: 10.1016/j.patcog.2009.10.010.

Figure 1.  Segmentation comparison, on various synthetic images, as shown in the first column, between Ali et al. model (second column after 200 iterations and the third column the final segmentation with this method) and the proposed model (fourth column showing intermediate results after 200 iterations and the last column the final results)
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Figure 2.  Segmentation of different images using the proposed model. In the first row the given images has been shown whereas in the second row segmentation results are indicated with blue outlines
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Figure 3.  Segmentation of an airplane image using various models. (a) The given image, (b) the initial contour on object, (c) segmentation results of LIF model, (d) segmentation results of LBF model, (e) segmentation results of RLSF model, (f) segmentation results of SPGA model, (g) segmentation results of BFE field correction model, and (h) the proposed model
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Figure 4.  Performance of LBF model, SPGA model, RLSF model and the proposed model in second, third, fourth and fifth column, respectively
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Figure 5.  Performance of proposed model for blood vessel image segmentation corrupted by Gaussian noise from left to right respectively. In the second row, segmentation results of proposed model corresponding to images given in first row
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Figure 6.  Windows size choice and the number of iterations for corrupted images segmentation with noise level 0.002 in the first row and 0.02 in the second row
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Figure 7.  Segmentation results of given images with Gaussian noise level 0.002 (first column) with RLSF method in the second column and the proposed method in the last column
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Figure 8.  Quantitative comparisons of segmentation accuracy on histogram (using Jaccard similarity coefficient) of the proposed, the RLSF, the SPGA, the LIF and the LBF model respectively
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Figure 9.  Consecutive denoising and segmentation results for images with speckle noise $ 0.2 $. The segmentation results of pre-processed denoised images by AA model [3] are followed by (a) the LIF segmentation model, (b) the LBF segmentation model, (c) the SPGA segmentation model (d) the RLSF segmentation model, (e) segmentation results of the proposed model
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Figure 10.  Consecutive denoising and segmentation results for images with $ 10 \% $ Gaussian noise. Segmentation results of pre-processed denoised images followed by (a) the LIF segmentation model, (b) the LBF segmentation model, (c) the SPGA segmentation model (d) the RLSF segmentation model, (e) segmentation results of the proposed model without any denoising pre-processing
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Figure 11.  The non-convexity of the model shown through different initial contours produce different results
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Figure 12.  Segmentation performance of Liu et al. [30], Mabood et al. [32], Rada et al. [35] models in comparison with the proposed method with noise level $ \sigma = 0.1 $
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Figure 13.  Segmentation performance comparison between Liu et al [30], Mabood et al. [32] and Rada e al. [35] incapable to complete the selective segmentation task given the initial points as shown in first column. Our proposed successfully can segment the aimed object. The noise level is $ \sigma = 0.2 $
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Table 1.  Speed comparison of AOS cheme with the explicit Time Marching for different image size and number of iterations
Size Explicit AOS
Image Size Iteration CPU Iteration CPU
$ 110\times110 $ 250 52.21 80 45.65
$ 150\times150 $ 400 136.47 100 110.29
$ 200\times200 $ 750 410.33 110 167.44
$ 300\times300 $ 1500 2100.92 130 460.58
$ 500\times500 $ 2000 stuck 250 580.39
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Size Explicit AOS
Image Size Iteration CPU Iteration CPU
$ 110\times110 $ 250 52.21 80 45.65
$ 150\times150 $ 400 136.47 100 110.29
$ 200\times200 $ 750 410.33 110 167.44
$ 300\times300 $ 1500 2100.92 130 460.58
$ 500\times500 $ 2000 stuck 250 580.39
Table 2.  Speed comparison of the Liu et al. [30], Mabood et al. [32], Rada et al. [35] and proposed model of Fig. 12 and 13a, 13b, and 13c
IMG. Liu Mabood Rada Our model
Iter CPU Iter CPU Iter CPU Iter CPU
Fig. 12 200 68.1153 200 21.6026 150 17.8908 20 5.2901
Fig. 13a 200 32.4277 200 25.9021 150 12.0173 20 7.0322
Fig. 13b 200 51.9043 200 33.7350 150 24.5707 50 8.3102
Fig. 13c 200 44.1836 200 27.6316 150 13.0754 30 10.4117
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IMG. Liu Mabood Rada Our model
Iter CPU Iter CPU Iter CPU Iter CPU
Fig. 12 200 68.1153 200 21.6026 150 17.8908 20 5.2901
Fig. 13a 200 32.4277 200 25.9021 150 12.0173 20 7.0322
Fig. 13b 200 51.9043 200 33.7350 150 24.5707 50 8.3102
Fig. 13c 200 44.1836 200 27.6316 150 13.0754 30 10.4117
[1]

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