Figure 1.
Comparisons of three different clean images without noise. Column 1: the original clean images; Column 2: detected edges; Column 3: restored images.
-
Previous Article
A linear sampling method for inverse acoustic scattering by a locally rough interface
- IPI Home
- This Issue
-
Next Article
An efficient multi-grid method for TV minimization problems
October
2021, 15(5): 1223-1245.
doi: 10.3934/ipi.2021035
Edge detection with mixed noise based on maximum a posteriori approach
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China |
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.
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 and Imaging,
2021, 15
(5)
: 1223-1245.
doi: 10.3934/ipi.2021035
![]()
References:
| [1] |
S. Alex, (nonlocal) Total variation in medical imaging, PhD Thesis, Univeristy of Muenster, Germany. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
J. Canny,
A computational approach to edge detection, IEEE T. Pattern Anal., PAMI-8 (1986), 679-698.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
M. Kass, A. Witkin and D. Terzopoulos,
Snakes: Active contour models, Int. J. Comput. Vis., 1 (1988), 321-331.
doi: 10.1007/BF00133570. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
S. Smith, Edge thinning used in the SUSAN edge detector, Technical Report, TR95SMS5. |
| [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. |
| [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. |
show all references
References:
| [1] |
S. Alex, (nonlocal) Total variation in medical imaging, PhD Thesis, Univeristy of Muenster, Germany. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
J. Canny,
A computational approach to edge detection, IEEE T. Pattern Anal., PAMI-8 (1986), 679-698.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
M. Kass, A. Witkin and D. Terzopoulos,
Snakes: Active contour models, Int. J. Comput. Vis., 1 (1988), 321-331.
doi: 10.1007/BF00133570. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
S. Smith, Edge thinning used in the SUSAN edge detector, Technical Report, TR95SMS5. |
| [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. |
| [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. |
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.
Figure 3.
Comparisons of two images with Laplace noise and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.
Figure 4.
Comparisons of two images with RVIN and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.
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.
Figure 6.
Comparisons of color image with Laplace noise and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.
Figure 7.
Comparisons of color images with RVIN and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.
Figure 8.
Comparisons of the two models with G1 and SP1. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.
Figure 11.
Results with different λ and λ1.
Figure 12.
Results with different α and β.
Figure 13.
Results with different r.
Figure 9.
Comparisons of the two models with G1 and L1. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.
Figure 10.
Comparisons of the two models with G1 and R1. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.
Figure 14.
ROC curves using the two models (28) and (29) with three different mixed noise.
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 |
| 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 |
| 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 |
| 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 |
| [1] |
Li Shen, Eric Todd Quinto, Shiqiang Wang, Ming Jiang. Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography. Inverse Problems and Imaging, 2018, 12 (6) : 1343-1364. doi: 10.3934/ipi.2018056 |
| [2] |
Esther Klann, Ronny Ramlau, Wolfgang Ring. A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. Inverse Problems and Imaging, 2011, 5 (1) : 137-166. doi: 10.3934/ipi.2011.5.137 |
| [3] |
Antonin Chambolle, Francesco Doveri. Minimizing movements of the Mumford and Shah energy. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 153-174. doi: 10.3934/dcds.1997.3.153 |
| [4] |
Monika Muszkieta. A variational approach to edge detection. Inverse Problems and Imaging, 2016, 10 (2) : 499-517. doi: 10.3934/ipi.2016009 |
| [5] |
Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems and Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51 |
| [6] |
Giovanna Citti, Maria Manfredini, Alessandro Sarti. Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space. Communications on Pure and Applied Analysis, 2010, 9 (4) : 905-927. doi: 10.3934/cpaa.2010.9.905 |
| [7] |
Liming Zhang, Tao Qian, Qingye Zeng. Edge detection by using rotational wavelets. Communications on Pure and Applied Analysis, 2007, 6 (3) : 899-915. doi: 10.3934/cpaa.2007.6.899 |
| [8] |
Zhenhua Zhao, Yining Zhu, Jiansheng Yang, Ming Jiang. Mumford-Shah-TV functional with application in X-ray interior tomography. Inverse Problems and Imaging, 2018, 12 (2) : 331-348. doi: 10.3934/ipi.2018015 |
| [9] |
Yuying Shi, Ying Gu, Li-Lian Wang, Xue-Cheng Tai. A fast edge detection algorithm using binary labels. Inverse Problems and Imaging, 2015, 9 (2) : 551-578. doi: 10.3934/ipi.2015.9.551 |
| [10] |
Hirotada Honda. On a model of target detection in molecular communication networks. Networks and Heterogeneous Media, 2019, 14 (4) : 633-657. doi: 10.3934/nhm.2019025 |
| [11] |
H. Thomas Banks, Shuhua Hu, Zackary R. Kenz, Carola Kruse, Simon Shaw, John Whiteman, Mark P. Brewin, Stephen E. Greenwald, Malcolm J. Birch. Model validation for a noninvasive arterial stenosis detection problem. Mathematical Biosciences & Engineering, 2014, 11 (3) : 427-448. doi: 10.3934/mbe.2014.11.427 |
| [12] |
Linfei Wang, Dapeng Tao, Ruonan Wang, Ruxin Wang, Hao Li. Big Map R-CNN for object detection in large-scale remote sensing images. Mathematical Foundations of Computing, 2019, 2 (4) : 299-314. doi: 10.3934/mfc.2019019 |
| [13] |
Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control and Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012 |
| [14] |
Frederik Riis Mikkelsen. A model based rule for selecting spiking thresholds in neuron models. Mathematical Biosciences & Engineering, 2016, 13 (3) : 569-578. doi: 10.3934/mbe.2016008 |
| [15] |
Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems and Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399 |
| [16] |
Edward S. Canepa, Alexandre M. Bayen, Christian G. Claudel. Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming. Networks and Heterogeneous Media, 2013, 8 (3) : 783-802. doi: 10.3934/nhm.2013.8.783 |
| [17] |
Yuan Gao, Jian-Guo Liu, Tao Luo, Yang Xiang. Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3177-3207. doi: 10.3934/dcdsb.2020224 |
| [18] |
Sudeb Majee, Subit K. Jain, Rajendra K. Ray, Ananta K. Majee. A fuzzy edge detector driven telegraph total variation model for image despeckling. Inverse Problems and Imaging, 2022, 16 (2) : 367-396. doi: 10.3934/ipi.2021054 |
| [19] |
Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems and Imaging, 2021, 15 (2) : 339-366. doi: 10.3934/ipi.2020071 |
| [20] |
Jianbin Yang, Cong Wang. A wavelet frame approach for removal of mixed Gaussian and impulse noise on surfaces. Inverse Problems and Imaging, 2017, 11 (5) : 783-798. doi: 10.3934/ipi.2017037 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]