Edge detection with mixed noise based on maximum a posteriori approach

Yuying Shi; Zijin Liu; Xiaoying Wang; Jinping Zhang

doi:10.3934/ipi.2021035

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October 2021, 15(5): 1223-1245. doi: 10.3934/ipi.2021035

Edge detection with mixed noise based on maximum a posteriori approach

Yuying Shi , Zijin Liu , Xiaoying Wang ^, and Jinping Zhang

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

^* Corresponding author: wangxiaoying@ncepu.edu.cn

Received September 2020 Revised February 2021 Published October 2021 Early access April 2021

Fund Project: The first author is supported by Beijing Natural Science Foundation (No. Z200001)

Edge detection is an important problem in image processing, especially for mixed noise. In this work, we propose a variational edge detection model with mixed noise by using Maximum A-Posteriori (MAP) approach. The novel model is formed with the regularization terms and the data fidelity terms that feature different mixed noise. Furthermore, we adopt the alternating direction method of multipliers (ADMM) to solve the proposed model. Numerical experiments on a variety of gray and color images demonstrate the efficiency of the proposed model.

Keywords: Edge detection, MAP, mixed noise, Bayes' rule, Mumford-Shah model.

Mathematics Subject Classification: Primary: 68U10.

Citation: Yuying Shi, Zijin Liu, Xiaoying Wang, Jinping Zhang. Edge detection with mixed noise based on maximum a posteriori approach. Inverse Problems & Imaging, 2021, 15 (5) : 1223-1245. doi: 10.3934/ipi.2021035

References:

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[2]	L. Alvarez, P.-L. Lions and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion Ⅱ, SIAM J. Numer. Anal., 29 (1992), 845-866. doi: 10.1137/0729052. Google Scholar
[3]	L. Ambrosio and V. Tortorelli, Approximation of functions depending on jumps by elliptic functions via $\Gamma$-convergence, Comm. Pure Appl. Math., 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805. Google Scholar
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[7]	A. Brook, R. Kimmel and N. A. Sochen, Variational restoration and edge detection for color images, J. Math. Imaging Vision, 18 (2003), 247-268. doi: 10.1023/A:1022895410391. Google Scholar
[8]	J.-F. Cai, R. H. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Probl. Imaging, 2 (2008), 187-204. doi: 10.3934/ipi.2008.2.187. Google Scholar
[9]	L. Calatroni, J. C. De Los Reyes and C.-B. Schönlieb, Infimal convolution of data discrepancies for mixed noise removal, SIAM J. Imaging Sci., 10 (2017), 1196-1233. doi: 10.1137/16M1101684. Google Scholar
[10]	L. Calatroni and K. Papafitsoros, Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal, Inverse Problems, 35 (2019), 114001, 37 pp. doi: 10.1088/1361-6420/ab291a. Google Scholar
[11]	J. Canny, A computational approach to edge detection, IEEE T. Pattern Anal., PAMI-8 (1986), 679-698. Google Scholar
[12]	V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Proceedings of IEEE International Conference on Computer Vision, Cambridge, MA, USA, (1995), 694–699. doi: 10.1109/ICCV.1995.466871. Google Scholar
[13]	F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193. doi: 10.1137/0729012. Google Scholar
[14]	R. H. Chan and J. Ma, A multiplicative iterative algorithm for box-constrained penalized likelihood image restoration, IEEE Trans. Image Process., 21 (2012), 3168-3181. doi: 10.1109/TIP.2012.2188811. Google Scholar
[15]	E. Chouzenoux, A. Jezierska, J.-C. Pesquet and H. Talbot, A convex approach for image restoration with exact Poisson-Gaussian likelihood, SIAM J. Imaging Sci., 8 (2015), 2662-2682. doi: 10.1137/15M1014395. Google Scholar
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[17]	M. Hintermüller and A. Langer, Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed $l^1/l^2$ data-fidelity in image processing, SIAM J. Imaging Sci., 6 (2013), 2134-2173. doi: 10.1137/120894130. Google Scholar
[18]	T. Jia, Y. Shi, Y. Zhu and L. Wang, An image restoration model combining mixed l¹/l² fidelity terms, J. Vis. Commun. Image. R., 38 (2016), 461-473. doi: 10.1016/j.jvcir.2016.03.022. Google Scholar
[19]	M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, Int. J. Comput. Vis., 1 (1988), 321-331. doi: 10.1007/BF00133570. Google Scholar
[20]	E. Lćpez-Rubio, Restoration of images corrupted by gaussian and uniform impulsive noise, Pattern Recogn., 43 (2010), 1835–1846, http://www.sciencedirect.com/science/article/pii/S0031320309004361. Google Scholar
[21]	B. Llanas and S. Lantarón, Edge detection by adaptive splitting, J. Sci. Comput., 46 (2011), 486-518. doi: 10.1007/s10915-010-9416-8. Google Scholar
[22]	R. J. Marks, G. L. Wise, D. H. Haldeman and J. L. Whited, Detection in Laplace noise, IEEE Transactions on Aerospace and Electronic Systems, 14 (1978), 866-872. doi: 10.1109/TAES.1978.308550. Google Scholar
[23]	E. Meinhardt, E. Zacur, A. F. Frangi and V. Caselles, 3D edge detection by selection of level surface patches, J. Math. Imaging Vis., 34 (2009), 1-16. doi: 10.1007/s10851-008-0118-x. Google Scholar
[24]	D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. Google Scholar
[25]	P. Perona and J. Malik, Scale-space and edge-detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: 10.1109/34.56205. Google Scholar
[26]	T. Pock, D. Cremers, H. Bischof and A. Chambolle, An algorithm for minimizing the Mumford-Shah functional, in 2009 IEEE 12th International Conference on Computer Vision, Kyoto, Japan, (2009), 1133–1140. doi: 10.1109/ICCV.2009.5459348. Google Scholar
[27]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D., 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. Google Scholar
[28]	Y. Shi and Q. Chang, Acceleration methods for image restoration problem with different boundary conditions, Appl. Numer. Math., 58 (2008), 602-614. doi: 10.1016/j.apnum.2007.01.007. Google Scholar
[29]	Y. Shi, Y. Gu, L.-L. Wang and X.-C. Tai, A fast edge detection algorithm using binary labels, Inverse Probl. Imaging, 9 (2015), 551-578. doi: 10.3934/ipi.2015.9.551. Google Scholar
[30]	Y. Shi, Z. Huo, J. Qin and Y. Li, Automatic prior shape selection for image edge detection with modified Mumford-Shah model, Comput. Math. Appl., 79 (2020), 1644-1660. doi: 10.1016/j.camwa.2019.09.021. Google Scholar
[31]	S. Smith, Edge thinning used in the SUSAN edge detector, Technical Report, TR95SMS5. Google Scholar
[32]	W. Tao, F. Chang, L. Liu, H. Jin and T. Wang, Interactively multiphase image segmentation based on variational formulation and graph cuts, Pattern Recogn., 43 (2010), 3208-3218. doi: 10.1016/j.patcog.2010.04.014. Google Scholar
[33]	L.-L. Wang, Y. Shi and X.-C. Tai, Robust edge detection using Mumford-Shah model and binary level set method, the Third International Conference on Scale Space and Variational Methods in Computer Vision (SSVM2011), Springer, Berlin, Heidelberg, 6667 (2012), 291–301. doi: 10.1007/978-3-642-24785-9_25. Google Scholar

show all references

References:

[1]	S. Alex, (nonlocal) Total variation in medical imaging, PhD Thesis, Univeristy of Muenster, Germany. Google Scholar
[2]	L. Alvarez, P.-L. Lions and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion Ⅱ, SIAM J. Numer. Anal., 29 (1992), 845-866. doi: 10.1137/0729052. Google Scholar
[3]	L. Ambrosio and V. Tortorelli, Approximation of functions depending on jumps by elliptic functions via $\Gamma$-convergence, Comm. Pure Appl. Math., 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805. Google Scholar
[4]	L. Ambrosio and V. Tortorelli, On the approximation of functionals depending on jumps by quadratic, elliptic functions, Boll. Un. Mat. Ital., 6 (1992), 105-123. Google Scholar
[5]	N. Badshah and K. Chen, Image selective segmentation under geometrical constraints using an active contour approach, Commun. Compu. Phys., 7 (2010), 759-778. doi: 10.4208/cicp.2009.09.026. Google Scholar
[6]	K. Bowyer, C. Kranenburg and S. Dougherty, Edge detector evaluation using empirical ROC curves, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149), Fort Collins, CO, USA, 1 (1999), 354–359. doi: 10.1109/CVPR.1999.786963. Google Scholar
[7]	A. Brook, R. Kimmel and N. A. Sochen, Variational restoration and edge detection for color images, J. Math. Imaging Vision, 18 (2003), 247-268. doi: 10.1023/A:1022895410391. Google Scholar
[8]	J.-F. Cai, R. H. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Probl. Imaging, 2 (2008), 187-204. doi: 10.3934/ipi.2008.2.187. Google Scholar
[9]	L. Calatroni, J. C. De Los Reyes and C.-B. Schönlieb, Infimal convolution of data discrepancies for mixed noise removal, SIAM J. Imaging Sci., 10 (2017), 1196-1233. doi: 10.1137/16M1101684. Google Scholar
[10]	L. Calatroni and K. Papafitsoros, Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal, Inverse Problems, 35 (2019), 114001, 37 pp. doi: 10.1088/1361-6420/ab291a. Google Scholar
[11]	J. Canny, A computational approach to edge detection, IEEE T. Pattern Anal., PAMI-8 (1986), 679-698. Google Scholar
[12]	V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Proceedings of IEEE International Conference on Computer Vision, Cambridge, MA, USA, (1995), 694–699. doi: 10.1109/ICCV.1995.466871. Google Scholar
[13]	F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193. doi: 10.1137/0729012. Google Scholar
[14]	R. H. Chan and J. Ma, A multiplicative iterative algorithm for box-constrained penalized likelihood image restoration, IEEE Trans. Image Process., 21 (2012), 3168-3181. doi: 10.1109/TIP.2012.2188811. Google Scholar
[15]	E. Chouzenoux, A. Jezierska, J.-C. Pesquet and H. Talbot, A convex approach for image restoration with exact Poisson-Gaussian likelihood, SIAM J. Imaging Sci., 8 (2015), 2662-2682. doi: 10.1137/15M1014395. Google Scholar
[16]	R. Deriche, Using Canny's criteria to derive a recursively implemented optimal edge detector, Int. J. Comput. Vis., 1 (1987), 167-187. doi: 10.1007/BF00123164. Google Scholar
[17]	M. Hintermüller and A. Langer, Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed $l^1/l^2$ data-fidelity in image processing, SIAM J. Imaging Sci., 6 (2013), 2134-2173. doi: 10.1137/120894130. Google Scholar
[18]	T. Jia, Y. Shi, Y. Zhu and L. Wang, An image restoration model combining mixed l¹/l² fidelity terms, J. Vis. Commun. Image. R., 38 (2016), 461-473. doi: 10.1016/j.jvcir.2016.03.022. Google Scholar
[19]	M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, Int. J. Comput. Vis., 1 (1988), 321-331. doi: 10.1007/BF00133570. Google Scholar
[20]	E. Lćpez-Rubio, Restoration of images corrupted by gaussian and uniform impulsive noise, Pattern Recogn., 43 (2010), 1835–1846, http://www.sciencedirect.com/science/article/pii/S0031320309004361. Google Scholar
[21]	B. Llanas and S. Lantarón, Edge detection by adaptive splitting, J. Sci. Comput., 46 (2011), 486-518. doi: 10.1007/s10915-010-9416-8. Google Scholar
[22]	R. J. Marks, G. L. Wise, D. H. Haldeman and J. L. Whited, Detection in Laplace noise, IEEE Transactions on Aerospace and Electronic Systems, 14 (1978), 866-872. doi: 10.1109/TAES.1978.308550. Google Scholar
[23]	E. Meinhardt, E. Zacur, A. F. Frangi and V. Caselles, 3D edge detection by selection of level surface patches, J. Math. Imaging Vis., 34 (2009), 1-16. doi: 10.1007/s10851-008-0118-x. Google Scholar
[24]	D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. Google Scholar
[25]	P. Perona and J. Malik, Scale-space and edge-detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: 10.1109/34.56205. Google Scholar
[26]	T. Pock, D. Cremers, H. Bischof and A. Chambolle, An algorithm for minimizing the Mumford-Shah functional, in 2009 IEEE 12th International Conference on Computer Vision, Kyoto, Japan, (2009), 1133–1140. doi: 10.1109/ICCV.2009.5459348. Google Scholar
[27]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D., 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. Google Scholar
[28]	Y. Shi and Q. Chang, Acceleration methods for image restoration problem with different boundary conditions, Appl. Numer. Math., 58 (2008), 602-614. doi: 10.1016/j.apnum.2007.01.007. Google Scholar
[29]	Y. Shi, Y. Gu, L.-L. Wang and X.-C. Tai, A fast edge detection algorithm using binary labels, Inverse Probl. Imaging, 9 (2015), 551-578. doi: 10.3934/ipi.2015.9.551. Google Scholar
[30]	Y. Shi, Z. Huo, J. Qin and Y. Li, Automatic prior shape selection for image edge detection with modified Mumford-Shah model, Comput. Math. Appl., 79 (2020), 1644-1660. doi: 10.1016/j.camwa.2019.09.021. Google Scholar
[31]	S. Smith, Edge thinning used in the SUSAN edge detector, Technical Report, TR95SMS5. Google Scholar
[32]	W. Tao, F. Chang, L. Liu, H. Jin and T. Wang, Interactively multiphase image segmentation based on variational formulation and graph cuts, Pattern Recogn., 43 (2010), 3208-3218. doi: 10.1016/j.patcog.2010.04.014. Google Scholar
[33]	L.-L. Wang, Y. Shi and X.-C. Tai, Robust edge detection using Mumford-Shah model and binary level set method, the Third International Conference on Scale Space and Variational Methods in Computer Vision (SSVM2011), Springer, Berlin, Heidelberg, 6667 (2012), 291–301. doi: 10.1007/978-3-642-24785-9_25. Google Scholar

Figure 1. Comparisons of three different clean images without noise. Column 1: the original clean images; Column 2: detected edges; Column 3: restored images.

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Figure 2. Comparisons of two images with Gaussian noise and salt & pepper noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 3. Comparisons of two images with Laplace noise and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 4. Comparisons of two images with RVIN and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 5. Comparisons of color image with Gaussian noise and salt & pepper noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 6. Comparisons of color image with Laplace noise and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 7. Comparisons of color images with RVIN and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 8. Comparisons of the two models with G1 and SP1. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 11. Results with different λ and λ₁.

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Figure 12. Results with different α and β.

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Figure 13. Results with different r.

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Figure 9. Comparisons of the two models with G1 and L1. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 10. Comparisons of the two models with G1 and R1. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 14. ROC curves using the two models (28) and (29) with three different mixed noise.

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Table 1. AUC of two models with different mixed noise

noise	G1 and S1	G1 and L1	G1 and R1	G2 and S2	G2 and L2	G2 and R2
model (28)	0.8095	0.8872	0.8025	0.8415	0.8503	0.8305
model (29)	0.8254	0.8897	0.8273	0.8844	0.8801	0.8683

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noise	G1 and S1	G1 and L1	G1 and R1	G2 and S2	G2 and L2	G2 and R2
model (28)	0.8095	0.8872	0.8025	0.8415	0.8503	0.8305
model (29)	0.8254	0.8897	0.8273	0.8844	0.8801	0.8683

Table 2. Running times of two models with different mixed noise

noise	G1 and S1	G1 and L1	G1 and R1	G2 and S2	G2 and L2	G2 and R2
model (28)	2.83	2.87	2.88	3.02	2.85	2.87
model (29)	2.61	2.37	2.40	2.27	2.25	2.32

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noise	G1 and S1	G1 and L1	G1 and R1	G2 and S2	G2 and L2	G2 and R2
model (28)	2.83	2.87	2.88	3.02	2.85	2.87
model (29)	2.61	2.37	2.40	2.27	2.25	2.32

Table 3. Iteration numbers of two models with different mixed noise

noise	G1 and S1	G1 and L1	G1 and R1	G2 and S2	G2 and L2	G2 and R2
model (28)	88	70	93	75	73	91
model (29)	74	65	81	68	60	80

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noise	G1 and S1	G1 and L1	G1 and R1	G2 and S2	G2 and L2	G2 and R2
model (28)	88	70	93	75	73	91
model (29)	74	65	81	68	60	80

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Edge detection with mixed noise based on maximum a posteriori approach

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