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Image segmentation based on the hybrid total variation model and the K-means clustering strategy
1. | College of Mathematics and Statistic, Henan University, Kaifeng 475004, Henan, China, China |
2. | School of Statistic and Mathematics, Zhejiang Gongshang University, Hangzhou 310012, China |
References:
[1] |
E. Bae, J. Yuan and X.-C. 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. |
[2] |
Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Ananlysis and Machine Intelligence, 23 (2001), 1-18. |
[3] |
K. Bredies, K. Kunisch and T. Pock., Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526.
doi: 10.1137/090769521. |
[4] |
X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of the active contour/snake model, Journal of Mathematical Imaging and Vision, 28 (2007), 151-167.
doi: 10.1007/s10851-007-0002-0. |
[5] |
E. S. Brown, T. F. Chan and X. Bresson, Completely convex formulation of the Chan-Vese image segmentation model, International Journal of Computer Vision, 98 (2012), 103-121.
doi: 10.1007/s11263-011-0499-y. |
[6] |
Y. Boykov, V. Kolmogorov, D. Cremers and A. Delong, An integral solution to surface evolution PDEs via geo-cuts. Proc. ECCV LCNS, 3953 (2006), 409-422.
doi: 10.1007/11744078_32. |
[7] |
Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Analysis and Machine Intelligence, 23 (2001), 1222-1239. |
[8] |
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 Journal on Image Science, 6 (2013), 368-390.
doi: 10.1137/120867068. |
[9] |
J.-F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, SIAM: Multiscale Modeling and Simulation, 8 (2009), 337-369.
doi: 10.1137/090753504. |
[10] |
V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, 22 (1997), 61-79. |
[11] |
A. Chambolle, D. Cremers and T. Pock, A convex approach to minimal partitions, SIAM Journal on Image Science, 5 (2012), 1113-1158.
doi: 10.1137/110856733. |
[12] |
T. Chan, S. Esedoglu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM Journal on Applied Mathematics, 66 (2006), 1632-1648.
doi: 10.1137/040615286. |
[13] |
T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503-516.
doi: 10.1137/S1064827598344169. |
[14] |
T. Chan and L. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277.
doi: 10.1109/83.902291. |
[15] |
C. Chen, B. He and X. Yuan, Matrix completion via an alternating direction method, IMA Journal of Numerical Analysis, 32 (2012), 227-245.
doi: 10.1093/imanum/drq039. |
[16] |
A. Delong, A. Osokin, H. Isack and Y. Boykov, Fast approximate energy minimization with label costs, International Journal of Computer Vision, 96 (2012), 1-27.
doi: 10.1007/s11263-011-0437-z. |
[17] |
S. Esedoglu and Y. Tsai, Threshold dynamics for the piecewise constant Mumford-Shah functional, Journal of Computational Physics, 211 (2006), 367-384.
doi: 10.1016/j.jcp.2005.05.027. |
[18] |
Y. Gu, L.-L. Wang and X.-C. Tai, A direct approach towards global minimization for multiphase labeling and segmentation problems, IEEE Transactions on Image Processing, 21 (2012), 2399-2411.
doi: 10.1109/TIP.2011.2182522. |
[19] |
L. Grady, The piecewise smooth Mumford-Shah functional on an arbitrary graph, IEEE Transactions on Image Processing, 18 (2009), 2547-2561.
doi: 10.1109/TIP.2009.2028258. |
[20] |
M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331.
doi: 10.1007/BF00133570. |
[21] |
J. Lellmann, B. Lellmann, F. Widmann and C. Schnorr, Discrete and continuous models for partitioning problems, International Journal of Computer Vision, 104 (2013), 241-269.
doi: 10.1007/s11263-013-0621-4. |
[22] |
J. Lellmann, J. Kappes, J. Yuan, F. Becker and C. Schnoor, Convex multi-class image labeling by simplex-constrainted total variation, Scale Space and Variational Methods in Computer Vision, 5567 (2009), 150-162. |
[23] |
J. Lellmann and C. Schnoor, Continuous multiclass labeling approaches and algorithms, Journal of Imaging Science, 4 (2011), 1049-1096.
doi: 10.1137/100805844. |
[24] |
F. Li, M. Ng, T. Y. Zeng and C. Shen, A multiphase image segmentation method based on fuzzy region competition, SIAM Journal on Scientific Computing, 3 (2010), 277-299.
doi: 10.1137/080736752. |
[25] |
F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, Journal of Visual Communication and Image Representation, 18 (2007), 322-330.
doi: 10.1016/j.jvcir.2007.04.005. |
[26] |
J. Lie, M. Lysaker and X.-C. Tai, A variant of the level set method and applications to image segmentation, Mathematics of Computation, 75 (2006), 1155-1174.
doi: 10.1090/S0025-5718-06-01835-7. |
[27] |
J. Lie, M. Lysaker and X.-C. Tai, A binary level set model and some applications to Mumford-Shah image segmentation, IEEE Transactions on Image Processing, 15 (2006), 1171-1181.
doi: 10.1109/TIP.2005.863956. |
[28] |
M. Lysaker, A. Lundervold and X. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on Image Processing, 12 (2003), 1579-1590.
doi: 10.1109/TIP.2003.819229. |
[29] |
D. Krishnan, Q. Pham and A. Yip, A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems, Advances in Computational Mathematics, 31 (2009), 237-266.
doi: 10.1007/s10444-008-9101-8. |
[30] |
B. Macqueen, Some methods for classification and analysis of multivariate observations, In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability, 1 (1967), 281-297. |
[31] |
A. Marquina and S. J. Osher, Image super-resolution by TV-regularization and Bregman iteration, Journal of Scientific Computing, 37 (2008), 367-382.
doi: 10.1007/s10915-008-9214-8. |
[32] |
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. |
[33] |
C. Nieuwenhuis, E. Toppe and D. Cremers, A survey and comparison of discrete and continuous multi-label optimization approaches for the Potts model, International Journal of Computer Vision, 104 (2013), 223-240.
doi: 10.1007/s11263-013-0619-y. |
[34] |
K. Papafitsoros and C. Schonlieb, A combined first and second variational approach for image reconstruction, Journal of Mathematical Imaging and Vision, 48 (2014), 308-338.
doi: 10.1007/s10851-013-0445-4. |
[35] |
T. Pock, A. Chambolle, H. Bischof and D. Cremers, A convex relaxation approach for computing minimal partitions, In: IEEE Conference on Computer Vision and Pattern Recognition(CVPR), (2009), 810-817.
doi: 10.1109/CVPR.2009.5206604. |
[36] |
L. 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. |
[37] |
S. Setzer, Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage, In Proceedings of the Second International Conference on Scale Space and VariationalMethods in Computer Vision, 5567 (2009), 464-476.
doi: 10.1007/978-3-642-02256-2_39. |
[38] |
R. Schafer, R. Mersereau and M. Richaards, Constrained iterative restoration algorithms, Proceedings of the IEEE, 69 (1981), 432-450.
doi: 10.1109/PROC.1981.11987. |
[39] |
O. Tobias and R. Seara, Image segmentation by histogram thresholding using fuzzy sets, IEEE Transactions on Image Processing, 11 (2002), 1457-1465.
doi: 10.1109/TIP.2002.806231. |
[40] |
L. Vese and T. Chan, A multiphase level set framework for image segmentation using the Mumford-Shah model, International Journal of Computer Vision, 50 (2002), 271-293. |
[41] |
X. Wang, D. Huang and H. Xu, An efficient local Chan-Vese model for image segmentation, Pattern Recognit, 43 (2010), 603-618.
doi: 10.1016/j.patcog.2009.08.002. |
[42] |
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. |
[43] |
J. Yang and Y. Zhang, Alternating direction algorithms for $l^1$ problems in compressive sensing, SIAM Journal on Scientific Computing, 33 (2011), 250-278.
doi: 10.1137/090777761. |
[44] |
J. Yuan, E. Bae, X.-C. Tai and Y. Boykov, A continuous max-flow approach to Potts model, 11th European Conference on Computer Vision (ECCV), 6316 (2010), 379-392.
doi: 10.1007/978-3-642-15567-3_28. |
[45] |
J. Yuan, E. Bae, X.-C. Tai and Y. Boykov, A study on continuous max-flow and min-cut approaches, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2010), 2217-2224.
doi: 10.1109/CVPR.2010.5539903. |
[46] |
J. Yuan, C. Schnörr and G. Steidl, Total-variation based piecewise affine regularization, Scale Space and Variational Methods in Computer Vision, 5567 (2009), 552-564.
doi: 10.1007/978-3-642-02256-2_46. |
[47] |
C. Zach, D. Gallup, J. Frahm and M. Niethammer, Fast global labeling for real-time stereo using multiple plane sweeps, In: Vision, modeling, and visualization, (2008), 243-252. |
[48] |
R. Zhang, X. Bresson and X.-C. Tai, Four color theorem and convex relaxation for image segmentation with any number of regions, Inverse Problems and Imaging, 7 (2013), 1099-1113.
doi: 10.3934/ipi.2013.7.1099. |
[49] |
S. Zhu and A. Yuille, Region competition: Unifying snakes, region growing, and Bayes/MDL for multi-band image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 18 (1996), 884-900. |
show all references
References:
[1] |
E. Bae, J. Yuan and X.-C. 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. |
[2] |
Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Ananlysis and Machine Intelligence, 23 (2001), 1-18. |
[3] |
K. Bredies, K. Kunisch and T. Pock., Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526.
doi: 10.1137/090769521. |
[4] |
X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of the active contour/snake model, Journal of Mathematical Imaging and Vision, 28 (2007), 151-167.
doi: 10.1007/s10851-007-0002-0. |
[5] |
E. S. Brown, T. F. Chan and X. Bresson, Completely convex formulation of the Chan-Vese image segmentation model, International Journal of Computer Vision, 98 (2012), 103-121.
doi: 10.1007/s11263-011-0499-y. |
[6] |
Y. Boykov, V. Kolmogorov, D. Cremers and A. Delong, An integral solution to surface evolution PDEs via geo-cuts. Proc. ECCV LCNS, 3953 (2006), 409-422.
doi: 10.1007/11744078_32. |
[7] |
Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Analysis and Machine Intelligence, 23 (2001), 1222-1239. |
[8] |
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 Journal on Image Science, 6 (2013), 368-390.
doi: 10.1137/120867068. |
[9] |
J.-F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, SIAM: Multiscale Modeling and Simulation, 8 (2009), 337-369.
doi: 10.1137/090753504. |
[10] |
V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, 22 (1997), 61-79. |
[11] |
A. Chambolle, D. Cremers and T. Pock, A convex approach to minimal partitions, SIAM Journal on Image Science, 5 (2012), 1113-1158.
doi: 10.1137/110856733. |
[12] |
T. Chan, S. Esedoglu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM Journal on Applied Mathematics, 66 (2006), 1632-1648.
doi: 10.1137/040615286. |
[13] |
T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503-516.
doi: 10.1137/S1064827598344169. |
[14] |
T. Chan and L. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277.
doi: 10.1109/83.902291. |
[15] |
C. Chen, B. He and X. Yuan, Matrix completion via an alternating direction method, IMA Journal of Numerical Analysis, 32 (2012), 227-245.
doi: 10.1093/imanum/drq039. |
[16] |
A. Delong, A. Osokin, H. Isack and Y. Boykov, Fast approximate energy minimization with label costs, International Journal of Computer Vision, 96 (2012), 1-27.
doi: 10.1007/s11263-011-0437-z. |
[17] |
S. Esedoglu and Y. Tsai, Threshold dynamics for the piecewise constant Mumford-Shah functional, Journal of Computational Physics, 211 (2006), 367-384.
doi: 10.1016/j.jcp.2005.05.027. |
[18] |
Y. Gu, L.-L. Wang and X.-C. Tai, A direct approach towards global minimization for multiphase labeling and segmentation problems, IEEE Transactions on Image Processing, 21 (2012), 2399-2411.
doi: 10.1109/TIP.2011.2182522. |
[19] |
L. Grady, The piecewise smooth Mumford-Shah functional on an arbitrary graph, IEEE Transactions on Image Processing, 18 (2009), 2547-2561.
doi: 10.1109/TIP.2009.2028258. |
[20] |
M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331.
doi: 10.1007/BF00133570. |
[21] |
J. Lellmann, B. Lellmann, F. Widmann and C. Schnorr, Discrete and continuous models for partitioning problems, International Journal of Computer Vision, 104 (2013), 241-269.
doi: 10.1007/s11263-013-0621-4. |
[22] |
J. Lellmann, J. Kappes, J. Yuan, F. Becker and C. Schnoor, Convex multi-class image labeling by simplex-constrainted total variation, Scale Space and Variational Methods in Computer Vision, 5567 (2009), 150-162. |
[23] |
J. Lellmann and C. Schnoor, Continuous multiclass labeling approaches and algorithms, Journal of Imaging Science, 4 (2011), 1049-1096.
doi: 10.1137/100805844. |
[24] |
F. Li, M. Ng, T. Y. Zeng and C. Shen, A multiphase image segmentation method based on fuzzy region competition, SIAM Journal on Scientific Computing, 3 (2010), 277-299.
doi: 10.1137/080736752. |
[25] |
F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, Journal of Visual Communication and Image Representation, 18 (2007), 322-330.
doi: 10.1016/j.jvcir.2007.04.005. |
[26] |
J. Lie, M. Lysaker and X.-C. Tai, A variant of the level set method and applications to image segmentation, Mathematics of Computation, 75 (2006), 1155-1174.
doi: 10.1090/S0025-5718-06-01835-7. |
[27] |
J. Lie, M. Lysaker and X.-C. Tai, A binary level set model and some applications to Mumford-Shah image segmentation, IEEE Transactions on Image Processing, 15 (2006), 1171-1181.
doi: 10.1109/TIP.2005.863956. |
[28] |
M. Lysaker, A. Lundervold and X. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on Image Processing, 12 (2003), 1579-1590.
doi: 10.1109/TIP.2003.819229. |
[29] |
D. Krishnan, Q. Pham and A. Yip, A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems, Advances in Computational Mathematics, 31 (2009), 237-266.
doi: 10.1007/s10444-008-9101-8. |
[30] |
B. Macqueen, Some methods for classification and analysis of multivariate observations, In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability, 1 (1967), 281-297. |
[31] |
A. Marquina and S. J. Osher, Image super-resolution by TV-regularization and Bregman iteration, Journal of Scientific Computing, 37 (2008), 367-382.
doi: 10.1007/s10915-008-9214-8. |
[32] |
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. |
[33] |
C. Nieuwenhuis, E. Toppe and D. Cremers, A survey and comparison of discrete and continuous multi-label optimization approaches for the Potts model, International Journal of Computer Vision, 104 (2013), 223-240.
doi: 10.1007/s11263-013-0619-y. |
[34] |
K. Papafitsoros and C. Schonlieb, A combined first and second variational approach for image reconstruction, Journal of Mathematical Imaging and Vision, 48 (2014), 308-338.
doi: 10.1007/s10851-013-0445-4. |
[35] |
T. Pock, A. Chambolle, H. Bischof and D. Cremers, A convex relaxation approach for computing minimal partitions, In: IEEE Conference on Computer Vision and Pattern Recognition(CVPR), (2009), 810-817.
doi: 10.1109/CVPR.2009.5206604. |
[36] |
L. 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. |
[37] |
S. Setzer, Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage, In Proceedings of the Second International Conference on Scale Space and VariationalMethods in Computer Vision, 5567 (2009), 464-476.
doi: 10.1007/978-3-642-02256-2_39. |
[38] |
R. Schafer, R. Mersereau and M. Richaards, Constrained iterative restoration algorithms, Proceedings of the IEEE, 69 (1981), 432-450.
doi: 10.1109/PROC.1981.11987. |
[39] |
O. Tobias and R. Seara, Image segmentation by histogram thresholding using fuzzy sets, IEEE Transactions on Image Processing, 11 (2002), 1457-1465.
doi: 10.1109/TIP.2002.806231. |
[40] |
L. Vese and T. Chan, A multiphase level set framework for image segmentation using the Mumford-Shah model, International Journal of Computer Vision, 50 (2002), 271-293. |
[41] |
X. Wang, D. Huang and H. Xu, An efficient local Chan-Vese model for image segmentation, Pattern Recognit, 43 (2010), 603-618.
doi: 10.1016/j.patcog.2009.08.002. |
[42] |
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. |
[43] |
J. Yang and Y. Zhang, Alternating direction algorithms for $l^1$ problems in compressive sensing, SIAM Journal on Scientific Computing, 33 (2011), 250-278.
doi: 10.1137/090777761. |
[44] |
J. Yuan, E. Bae, X.-C. Tai and Y. Boykov, A continuous max-flow approach to Potts model, 11th European Conference on Computer Vision (ECCV), 6316 (2010), 379-392.
doi: 10.1007/978-3-642-15567-3_28. |
[45] |
J. Yuan, E. Bae, X.-C. Tai and Y. Boykov, A study on continuous max-flow and min-cut approaches, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2010), 2217-2224.
doi: 10.1109/CVPR.2010.5539903. |
[46] |
J. Yuan, C. Schnörr and G. Steidl, Total-variation based piecewise affine regularization, Scale Space and Variational Methods in Computer Vision, 5567 (2009), 552-564.
doi: 10.1007/978-3-642-02256-2_46. |
[47] |
C. Zach, D. Gallup, J. Frahm and M. Niethammer, Fast global labeling for real-time stereo using multiple plane sweeps, In: Vision, modeling, and visualization, (2008), 243-252. |
[48] |
R. Zhang, X. Bresson and X.-C. Tai, Four color theorem and convex relaxation for image segmentation with any number of regions, Inverse Problems and Imaging, 7 (2013), 1099-1113.
doi: 10.3934/ipi.2013.7.1099. |
[49] |
S. Zhu and A. Yuille, Region competition: Unifying snakes, region growing, and Bayes/MDL for multi-band image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 18 (1996), 884-900. |
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