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A fast modified Newton's method for curvature based denoising of 1D signals
Four color theorem and convex relaxation for image segmentation with any number of regions
1. | Hong Kong University of Science and Technology, Hong Kong, China, China |
2. | City University of Hong Kong, Department of Computer Science, Hong Kong |
3. | University of Bergen, University of Bergen Bergen |
References:
[1] |
K. Appel and W. Haken, Every planar map is four colorable, Illinois Journal of Mathematics, 21 (1977), 429-567. |
[2] |
E. Bae and X. Tai, Efficient global minimization for the multiphase chan-vese model of image segmentation, Energy Minimization Methods in Computer Vision and Pattern Recognition, 5681 (2009), 28-41.
doi: 10.1007/978-3-642-03641-5_3. |
[3] |
E. Bae and X. Tai, Efficient global minimization methods for image segmentation models with four regions, UCLA CAM Report, 11 (2011), p. 82. |
[4] |
E. Bae, J. Yuan and X. Tai, Simultaneous convex optimization of regions and region parameters in image segmentation models, UCLA CAM Report 11-83, (2011).
doi: 10.1007/978-3-642-34141-0_19. |
[5] |
E. Bae, J. Yuan, X. Tai and Y. Boykov, A study on continuous max-flow and min-cut approaches part ii: Multiple linearly ordered labels, UCLA CAM Report, (2010), pp. 10-62. |
[6] |
E. Bae, J. Yuan, X. Tai and Y. Boykov, A fast continuous max-flow approach to non-convex multilabeling problems, Efficient global minimization methods for variational problems in imaging and vision, (2011). |
[7] |
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 (2009), 112-129.
doi: 10.1007/s11263-010-0406-y. |
[8] |
Y. Boykov and V. Kolmogorov, An experimental comparison of Min-Cut/Max-Flow algorithms for energy minimization in vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (2004), 1124-1137. |
[9] |
X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of the active contour/snake models, Journal of Mathematical Imaging and Vision, 28 (2007), 151-167.
doi: 10.1007/s10851-007-0002-0. |
[10] |
E. Brown, T. Chan and X. Bresson, A convex relaxation method for a class of Vector-valued minimization problems with applications to Mumford-Shah segmentation, UCLA CAM Report 10-43, (2010). |
[11] |
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. |
[12] |
V. Caselles, R. Kimmel and G. Sapiro, Geodesic Active Contours, International Journal of Computer Vision, 22 (1997), 61-79. |
[13] |
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. |
[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] |
D. Donoho, De-Noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627.
doi: 10.1109/18.382009. |
[16] |
V. Estellers, D. Zosso, R. Lai, J. Thiran, S. Osher and X. Bresson, An efficient algorithm for level set method preserving distance function, UCLA CAM Report 21 (2011), 4722-4734.
doi: 10.1109/TIP.2012.2202674. |
[17] |
S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721-741. |
[18] |
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. |
[19] |
E. Hodneland, X.-C. Tai and H.-H. Gerdes, Four-color theorem and level set methods for watershed segmentation, International Journal of Computer Vision, 82 (2009), 264-283.
doi: 10.1007/s11263-008-0199-4. |
[20] |
J. Z. Huang, M. K. Ng, H. Rong and Z. Li, Automated variable weighting in k-means type clustering, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 27 (2005), 657-668.
doi: 10.1109/TPAMI.2005.95. |
[21] |
S. Kang, B. Sandberg and A. Yip, A regularized k-means and multiphase scale segmentation, Inverse Problems and Imaging, 5 (2011), 407-429.
doi: 10.3934/ipi.2011.5.407. |
[22] |
S. Kim and M. Kang, Multiple-region segmentation without supervision by adaptive global maximum clustering, Image Processing, IEEE Transactions on, 21 (2012), 1600-1612.
doi: 10.1109/TIP.2011.2179058. |
[23] |
F. T. Leighton, A graph coloring algorithm for large scheduling problems, Journal of Research of the National Bureau of Standards, 84 (1979), 489-506.
doi: 10.6028/jres.084.024. |
[24] |
J. Lellmann, J. Kappes, J. Yuan, F. Becker and C. Schnörr, Convex multi-class image labeling by simplex-constrained total variation, in "Scale Space and Variational Methods in Computer Vision" (SSVM 2009), X.-C. Tai, K. Mórken, M. Lysaker, and K.-A. Lie, eds., Springer, 5567 (2009), 150-162.
doi: 10.1007/978-3-642-02256-2_13. |
[25] |
J. Lellmann and C. Schnörr, Continuous multiclass labeling approaches and algorithms, SIAM J. Imaging Sci., 4 (2011), 1049-1096.
doi: 10.1137/100805844. |
[26] |
J. Lie, M. Lysaker and X. 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] |
L. Liu and W. Tao, Image segmentation by iterative optimization of multiphase multiple piecewise constant model and Four-Color relabeling, Pattern Recognition, 44 (2011), 2819-2833.
doi: 10.1016/j.patcog.2011.04.031. |
[28] |
T. Lu, P. Neittaanmäki and X. Tai, A parallel splitting up method and its application to Navier-Stokes equations, Applied Mathematics Letters, 4 (1991), 25-29.
doi: 10.1016/0893-9659(91)90161-N. |
[29] |
C. Michelot, A finite algorithm for finding the projection of a point onto the canonical simplex of Rn, Journal of Optimization Theory and Applications, 50 (1986), 195-200.
doi: 10.1007/BF00938486. |
[30] |
D. Mumford and J. Shah, Optimal approximations of piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.
doi: 10.1002/cpa.3160420503. |
[31] |
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. |
[32] |
T. Pock, A. Chambolle, D. Cremers and H. Bischof, A convex approach for computing minimal partitions, in IEEE Conference on Computer Vision and Pattern Recognition, 2009, pp. 810-817.
doi: 10.1109/CVPR.2009.5206604. |
[33] |
R. Potts and C. Domb, Some generalized order-disorder transformations, Mathematical Proceedings of the Cambridge Philosophical Society, 48 (1952), 106-109.
doi: 10.1017/S0305004100027419. |
[34] |
G. Rosman, L. Dascal, X. Tai and R. Kimmel, On semi-implicit splitting schemes for the beltrami color image filtering, Journal of Mathematical Imaging and Vision, 40 (2011), 199-213.
doi: 10.1007/s10851-010-0254-y. |
[35] |
B. Shafei and G. Steidl, Segmentation of images with separating layers by fuzzy c-means and convex optimization, Journal of Visual Communication and Image Representation, 23 (2012), 611-621.
doi: 10.1016/j.jvcir.2012.02.006. |
[36] |
P. Strandmark, F. Kahl and N. Overgaard, Optimizing Parametric Total Variation Models, in International Conference on Computer Vision, (2009), pp. 2240-2247. |
[37] |
G. Strang, Maximal Flow Through A Domain, Mathematical Programming, 26 (1983), 123-143.
doi: 10.1007/BF02592050. |
[38] |
W. Tao and X. 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.
doi: 10.1016/j.imavis.2011.03.002. |
[39] |
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. |
[40] |
Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272.
doi: 10.1137/080724265. |
[41] |
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. |
[42] |
J. Yuan, E. Bae, Y. Boykov and X.-C. Tai, A continuous max-flow approach to minimal partitions with label cost prior, Scale Space and Variational Methods in Computer Vision, (2012), pp. 279-290.
doi: 10.1007/978-3-642-24785-9_24. |
[43] |
J. Yuan, E. Bae and X. Tai, A study on continuous max-flow and min-cut approaches, in "Computer Vision and Pattern Recognition" (CVPR), 2010 IEEE Conference on, IEEE, 2010, pp. 2217-2224.
doi: 10.1109/CVPR.2010.5539903. |
[44] |
J. Yuan, E. Bae, X. Tai and Y. Boykov, A continuous Max-Flow approach to potts model, Computer Vision-ECCV 2010, 6316 (2010), 379-392.
doi: 10.1007/978-3-642-15567-3_28. |
[45] |
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, pp. 243-252. |
show all references
References:
[1] |
K. Appel and W. Haken, Every planar map is four colorable, Illinois Journal of Mathematics, 21 (1977), 429-567. |
[2] |
E. Bae and X. Tai, Efficient global minimization for the multiphase chan-vese model of image segmentation, Energy Minimization Methods in Computer Vision and Pattern Recognition, 5681 (2009), 28-41.
doi: 10.1007/978-3-642-03641-5_3. |
[3] |
E. Bae and X. Tai, Efficient global minimization methods for image segmentation models with four regions, UCLA CAM Report, 11 (2011), p. 82. |
[4] |
E. Bae, J. Yuan and X. Tai, Simultaneous convex optimization of regions and region parameters in image segmentation models, UCLA CAM Report 11-83, (2011).
doi: 10.1007/978-3-642-34141-0_19. |
[5] |
E. Bae, J. Yuan, X. Tai and Y. Boykov, A study on continuous max-flow and min-cut approaches part ii: Multiple linearly ordered labels, UCLA CAM Report, (2010), pp. 10-62. |
[6] |
E. Bae, J. Yuan, X. Tai and Y. Boykov, A fast continuous max-flow approach to non-convex multilabeling problems, Efficient global minimization methods for variational problems in imaging and vision, (2011). |
[7] |
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 (2009), 112-129.
doi: 10.1007/s11263-010-0406-y. |
[8] |
Y. Boykov and V. Kolmogorov, An experimental comparison of Min-Cut/Max-Flow algorithms for energy minimization in vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (2004), 1124-1137. |
[9] |
X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of the active contour/snake models, Journal of Mathematical Imaging and Vision, 28 (2007), 151-167.
doi: 10.1007/s10851-007-0002-0. |
[10] |
E. Brown, T. Chan and X. Bresson, A convex relaxation method for a class of Vector-valued minimization problems with applications to Mumford-Shah segmentation, UCLA CAM Report 10-43, (2010). |
[11] |
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. |
[12] |
V. Caselles, R. Kimmel and G. Sapiro, Geodesic Active Contours, International Journal of Computer Vision, 22 (1997), 61-79. |
[13] |
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. |
[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] |
D. Donoho, De-Noising by soft-thresholding, IEEE Transactions on Information Theory, 41 (1995), 613-627.
doi: 10.1109/18.382009. |
[16] |
V. Estellers, D. Zosso, R. Lai, J. Thiran, S. Osher and X. Bresson, An efficient algorithm for level set method preserving distance function, UCLA CAM Report 21 (2011), 4722-4734.
doi: 10.1109/TIP.2012.2202674. |
[17] |
S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721-741. |
[18] |
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. |
[19] |
E. Hodneland, X.-C. Tai and H.-H. Gerdes, Four-color theorem and level set methods for watershed segmentation, International Journal of Computer Vision, 82 (2009), 264-283.
doi: 10.1007/s11263-008-0199-4. |
[20] |
J. Z. Huang, M. K. Ng, H. Rong and Z. Li, Automated variable weighting in k-means type clustering, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 27 (2005), 657-668.
doi: 10.1109/TPAMI.2005.95. |
[21] |
S. Kang, B. Sandberg and A. Yip, A regularized k-means and multiphase scale segmentation, Inverse Problems and Imaging, 5 (2011), 407-429.
doi: 10.3934/ipi.2011.5.407. |
[22] |
S. Kim and M. Kang, Multiple-region segmentation without supervision by adaptive global maximum clustering, Image Processing, IEEE Transactions on, 21 (2012), 1600-1612.
doi: 10.1109/TIP.2011.2179058. |
[23] |
F. T. Leighton, A graph coloring algorithm for large scheduling problems, Journal of Research of the National Bureau of Standards, 84 (1979), 489-506.
doi: 10.6028/jres.084.024. |
[24] |
J. Lellmann, J. Kappes, J. Yuan, F. Becker and C. Schnörr, Convex multi-class image labeling by simplex-constrained total variation, in "Scale Space and Variational Methods in Computer Vision" (SSVM 2009), X.-C. Tai, K. Mórken, M. Lysaker, and K.-A. Lie, eds., Springer, 5567 (2009), 150-162.
doi: 10.1007/978-3-642-02256-2_13. |
[25] |
J. Lellmann and C. Schnörr, Continuous multiclass labeling approaches and algorithms, SIAM J. Imaging Sci., 4 (2011), 1049-1096.
doi: 10.1137/100805844. |
[26] |
J. Lie, M. Lysaker and X. 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] |
L. Liu and W. Tao, Image segmentation by iterative optimization of multiphase multiple piecewise constant model and Four-Color relabeling, Pattern Recognition, 44 (2011), 2819-2833.
doi: 10.1016/j.patcog.2011.04.031. |
[28] |
T. Lu, P. Neittaanmäki and X. Tai, A parallel splitting up method and its application to Navier-Stokes equations, Applied Mathematics Letters, 4 (1991), 25-29.
doi: 10.1016/0893-9659(91)90161-N. |
[29] |
C. Michelot, A finite algorithm for finding the projection of a point onto the canonical simplex of Rn, Journal of Optimization Theory and Applications, 50 (1986), 195-200.
doi: 10.1007/BF00938486. |
[30] |
D. Mumford and J. Shah, Optimal approximations of piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685.
doi: 10.1002/cpa.3160420503. |
[31] |
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. |
[32] |
T. Pock, A. Chambolle, D. Cremers and H. Bischof, A convex approach for computing minimal partitions, in IEEE Conference on Computer Vision and Pattern Recognition, 2009, pp. 810-817.
doi: 10.1109/CVPR.2009.5206604. |
[33] |
R. Potts and C. Domb, Some generalized order-disorder transformations, Mathematical Proceedings of the Cambridge Philosophical Society, 48 (1952), 106-109.
doi: 10.1017/S0305004100027419. |
[34] |
G. Rosman, L. Dascal, X. Tai and R. Kimmel, On semi-implicit splitting schemes for the beltrami color image filtering, Journal of Mathematical Imaging and Vision, 40 (2011), 199-213.
doi: 10.1007/s10851-010-0254-y. |
[35] |
B. Shafei and G. Steidl, Segmentation of images with separating layers by fuzzy c-means and convex optimization, Journal of Visual Communication and Image Representation, 23 (2012), 611-621.
doi: 10.1016/j.jvcir.2012.02.006. |
[36] |
P. Strandmark, F. Kahl and N. Overgaard, Optimizing Parametric Total Variation Models, in International Conference on Computer Vision, (2009), pp. 2240-2247. |
[37] |
G. Strang, Maximal Flow Through A Domain, Mathematical Programming, 26 (1983), 123-143.
doi: 10.1007/BF02592050. |
[38] |
W. Tao and X. 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.
doi: 10.1016/j.imavis.2011.03.002. |
[39] |
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. |
[40] |
Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272.
doi: 10.1137/080724265. |
[41] |
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. |
[42] |
J. Yuan, E. Bae, Y. Boykov and X.-C. Tai, A continuous max-flow approach to minimal partitions with label cost prior, Scale Space and Variational Methods in Computer Vision, (2012), pp. 279-290.
doi: 10.1007/978-3-642-24785-9_24. |
[43] |
J. Yuan, E. Bae and X. Tai, A study on continuous max-flow and min-cut approaches, in "Computer Vision and Pattern Recognition" (CVPR), 2010 IEEE Conference on, IEEE, 2010, pp. 2217-2224.
doi: 10.1109/CVPR.2010.5539903. |
[44] |
J. Yuan, E. Bae, X. Tai and Y. Boykov, A continuous Max-Flow approach to potts model, Computer Vision-ECCV 2010, 6316 (2010), 379-392.
doi: 10.1007/978-3-642-15567-3_28. |
[45] |
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, pp. 243-252. |
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