A two-stage image segmentation method for blurry images with poisson or multiplicative gamma noise

SIAM Journal on Imaging Sciences

Volumn 7, Issue 1, 2014, Pages 98-127

  Chan, Raymond ^a   Yang, Hongfei ^a   Zeng, Tieyong ^b  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

Author keywords

Convexity; Gamma noise; Image segmentation; Multiplicative noise; Primal dual algorithm; Total variation

Indexed keywords

ALGORITHMS; MAGNETIC RESONANCE IMAGING;

CONVEXITY; GAMMA NOISE; MULTIPLICATIVE NOISE; PRIMAL DUAL ALGORITHMS; TOTAL VARIATION;

IMAGE SEGMENTATION;

EID: 84897477522     PISSN: None     EISSN: 19364954     Source Type: Journal    
DOI: 10.1137/130920241     Document Type: Article

References (68)

1. L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via G-convergence, Comm. Pure Appl. Math., 43 (1990), pp. 999-1036.
2. L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7), 6 (1992), pp. 105-123.
3. F. J. Anscombe, The transformation of Poisson, binomial and negative-binomial data, Biometrika, 35 (1948), pp. 246-254.
4. G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), pp. 925-946.
5. E. Bae, J. Yuan, and X.-C. Tai, Simultaneous convex optimization of regions and region parameters in image segmentation models, in Innovations for Shape Analysis, Math. Vis., Springer-Verlag, Berlin, Heidelberg, 2013, pp. 421-438.
6. Z. Bao, J. I. Murray, T. Boyle, S. L. Ooi, M. J. Sandel, and R. H. Waterston, Automated cell lineage tracing in Caenorhabditis elegans, Proc. Natl. Acad. Sci. USA, 103 (2006), pp. 2707-2712.
7. L. Bar, T. Chan, G. Chung, M. Jung, N. Kiryati, R. Mohieddine, N. Sochen, and L. Vese, Mumford and Shah model and its applications to image segmentation and image restoration, in Handbook of Mathematical Methods in Imaging, Springer-Verlag, Berlin, Heidelberg, 2011, pp. 1095-1157.
8. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), pp. 1-122.
9. X. Bresson, S. Esedoḡlu, P. Vandergheynst, J. Thiran, and S. Osher, Fast global minimization of the active contour/snake model, J. Math. Imaging Vision, 28 (2007), pp. 151-167.
10. E. S. Brown, T. F. 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), UCLA, Los Angeles, CA, 2010.
11. 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), pp. 368-390.
12. V. Caselles, A. Chambolle, and M. Novaga, The discontinuity set of solutions of the TV denoising problem and some extensions, Multiscale Model. Simul., 6 (2007), pp. 879-894.
13. A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. Appl. Math., 55 (1995), pp. 827-863.
14. A. Chambolle, Finite-differences discretization of the Mumford-Shah functional, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 261-288.
15. A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), pp. 89-97.
16. A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford-Shah functional in dimension two, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 651-672.
17. A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), pp. 120-145.
18. T. F. Chan, S. Esedoḡlu, and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., 66 (2006), pp. 1632-1648.
19. T. F. Chan, G. H. Golub, and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), pp. 1964-1977.
20. T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Process., 10 (2001), pp. 266-277.
21. C. Chesnaud, P. Réfrégier, and V. Boulet, Statistical region snake-based segmentation adapted to different physical noise models, IEEE Trans. Pattern Anal. Mach. Intell., 21 (1999), pp. 1145-1157.
22. G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progr. Math. 233, Birkhäuser Verlag, Basel, 2005.
23. B. Dong, A. Chien, and Z. Shen, Frame based segmentation for medical images, Commun. Math. Sci., 9 (2011), pp. 551-559.
24. S. Durand, J. Fadili, and M. Nikolova, Multiplicative noise removal using L1 fidelity on frame coefficients, J. Math. Imaging Vision, 36 (2010), pp. 201-226.
25. I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics Appl. Math. 28, SIAM, Philadelphia, 1999.
26. S. Esedoḡlu and Y. Tsai, Threshold dynamics for the piecewise constant Mumford-Shah functional, J. Comput. Phys., 211 (2006), pp. 367-384.
27. L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
28. M. Feigin and N. Sochen, Segmentation and denoising via an adaptive threshold Mumford-Shah-like functional, in Proceedings of the 17th International Conference on Pattern Recognition, Vol. 2, IEEE Computer Society, Washington, DC, 2004, pp. 98-101.
29. M. Figueiredo and J. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Trans. Image Process., 19 (2010), pp. 3133-3145.
30. F. Galland, N. Bertaux, and P. Réfrégier, Multi-component image segmentation in homogeneous regions based on description length minimization: Application to speckle, Poisson and Bernoulli noise, Pattern Recogn., 38 (2005), pp. 1926-1936.
31. M. Gobbino, Finite difference approximation of the Mumford-Shah functional, Comm. Pure Appl. Math., 51 (1998), pp. 197-228.
32. T. Goldstein and S. Osher, The split Bregman algorithm for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), pp. 323-343.
33. L. Grady and C. Alvino, Reformulating and optimizing the Mumford-Shah functional on a graph- A faster, lower energy solution, in Proceedings of the European Conference on Computer Vision (ECCV), Springer-Verlag, Berlin, Heidelberg, 2008, pp. 248-261.
34. J. A. Hartigan and M. A. Wong, Algorithm AS 136: A K-means clustering algorithm, J. Roy. Statist. Soc. Ser. C. Appl. Stat., 28 (1979), pp. 100-108.
35. M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM J. Sci. Comput., 28 (2006), pp. 1-23.
36. T. Kanungo, D. Mount, N. Netanyahu, C. Piatko, R. Silverman, and A. Wu, An efficient Kmeans clustering algorithm: Analysis and implementation, IEEE Trans. Pattern Anal. Mach. Intell., 24 (2002), pp. 881-892.
37. Y. Kim, Strictly Convex Realization in Two-Phase Image Segmentation, UCLA CAM Report (13-20), UCLA, Los Angeles, CA, 2013.
38. G. Koepfler, C. Lopez, and J. M. Morel, A multiscale algorithm for image segmentation by variational method, SIAM J. Numer. Anal., 31 (1994), pp. 282-299.
39. T. Le, R. Chartrand, and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27 (2007), pp. 257-263.
40. J. Lellmann and C. Schnörr, Continuous multiclass labeling approaches and algorithms, SIAM J. Imaging Sci., 4 (2011), pp. 1049-1096.
41. F. Li, M. K. Ng, T. Y. Zeng, and C. Shen, A multiphase image segmentation method based on fuzzy region competition, SIAM J. Imaging Sci., 3 (2010), pp. 277-299.
42. F. Li, C. Shen, and C. Li, Multiphase soft segmentation with total variation and H1 regularization, J. Math. Imaging Vision, 37 (2010), pp. 98-111.
43. J. Lie, M. Lysaker, and X. Tai, A binary level set model and some applications to Mumford-Shah image segmentation, IEEE Trans. Image Process., 15 (2006), pp. 1171-1181.
44. L. Ma, M. K. Ng, J. Yu, and T. Zeng, Efficient box-constrained TV-type-l1 algorithms for restoring images with impulse noise, J. Comput. Math., 31 (2013), pp. 249-270.
45. J. MacQueen, Some methods for classification and analysis of multivariate observations, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, Los Angeles, 1967, pp. 281-297.
46. M. Makitalo and A. Foi, Optimal inversion of the Anscombe transformation in low-count Poisson image denoising, IEEE Trans. Image Process., 20 (2011), pp. 99-109.
47. D. Martin, C. Fowlkes, D. Tal, and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, in Proceedings of the Eighth IEEE International Conference on Computer Vision, Vol. 2, 2001, pp. 416-423.
48. P. Martin, P. Réfrégier, F. Goudail, and F. Guérault, Influence of the noise model on level set active contour segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 26 (2004), pp. 799-803.
49. M. Morini and M. Negri, Mumford-Shah functional as G-limit of discrete Perona-Malik energies, Math. Models Methods Appl. Sci., 13 (2003), pp. 785-805.
50. D. Mumford and J. Shah, Boundary detection by minimizing functionals, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 1985, pp. 22-26.
51. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), pp. 577-685.
52. T. Pock, A. Chambolle, D. Cremers, and H. Bischof, A convex relaxation approach for computing minimal partitions, in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), 2009, pp. 810-817.
53. T. Pock, D. Cremers, H. Bischof, and A. Chambolle, An algorithm for minimizing the Mumford-Shah functional, in Proceedings of the 2009 IEEE 12th International Conference on Computer Vision, 2009, pp. 1133-1140.
54. O. Ruch and P. Réfrégier, Minimal-complexity segmentation with a polygonal snake adapted to different optical noise models, Opt. Lett., 26 (2001), pp. 977-979.
55. L. Rudin, P.-L. Lions, and S. Osher, Multiplicative denoising and deblurring: Theory and algorithms, in Geometric Level Set Methods in Imaging, Vision, and Graphics, Springer, New York, 2003, pp. 103-119.
56. L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259-268.
57. B. Sandberg, S. Kang, and T. Chan, Unsupervised multiphase segmentation: A phase balancing model, IEEE Trans. Image Process., 19 (2010), pp. 119-130.
58. A. Sawatzky, D. Tenbrinck, X. Jiang, and M. Burger, A variational framework for region-based segmentation incorporating physical noise models, J. Math. Imaging Vision, 47 (2013), pp. 179-209.
59. B. Shafei and G. Steidl, Segmentation of images with separating layers by fuzzy c-means and convex optimization, J. Vis. Commun. Image Rep., 23 (2012), pp. 611-621.
60. J. Shi and S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise model, SIAM J. Imaging Sci., 1 (2008), pp. 294-321.
61. G. Steidl and T. Teuber, Removing multiplicative noise by Douglas-Rachford splitting methods, J. Math. Imaging Vision, 36 (2010), pp. 168-184.
62. A. Tsai, A. Yezzi, and A. Willsky, Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification, IEEE Trans. Image Process., 10 (2001), pp. 1169-1186.
63. L. Vese and T. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Comput. Vis., 50 (2002), pp. 271-293.
64. Y.-W. Wen and R. H. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Trans. Image Process., 21 (2012), pp. 1770-1781.
65. J. Yuan, E. Bae, and X.-C. Tai, A study on continuous max-flow and min-cut approaches, in Proceedings of the Twenty-Third IEEE Conference on Computer Vision and Pattern Recognition (CVPR), San Francisco, CA, 2010, pp. 2217-2224.
66. J. Yuan, E. Bae, X.-C. Tai, and Y. Boykov, A continuous max-flow approach to Potts model, in Computer Vision-ECCV 2010, Springer, Berlin, Heidelberg, 2010, pp. 379-392.
67. J. Yuan, C. Schnörr, and G. Steidl, Simultaneous higher-order optical flow estimation and decomposition, SIAM J. Sci. Comput., 29 (2007), pp. 2283-2304.
68. M. Zhu and T. Chan, An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration, UCLA CAM Report (08-34), UCLA, Los Angeles, CA, 2008.