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Mathematical Problems in Engineering
Volume 2016, Article ID 3151303, 16 pages
http://dx.doi.org/10.1155/2016/3151303
Research Article

Variable Splitting Based Method for Image Restoration with Impulse Plus Gaussian Noise

College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

Received 9 June 2016; Revised 7 October 2016; Accepted 23 October 2016

Academic Editor: Bogdan Dumitrescu

Copyright © 2016 Tingting Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems, John Wiley & Sons, New York, NY, USA, 1977.
  2. D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Communications on Pure and Applied Mathematics, vol. 42, no. 5, pp. 577–685, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  3. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, no. 1–4, pp. 259–268, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, 1970. View at MathSciNet
  5. Y. M. Huang, M. K. Ng, and Y.-W. Wen, “A fast total variation minimization method for image restoration,” Multiscale Modeling & Simulation, vol. 7, no. 2, pp. 774–795, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. Y.-M. Huang, M. K. Ng, and Y.-W. Wen, “Fast image restoration methods for impulse and Gaussian noises removal,” IEEE Signal Processing Letters, vol. 16, no. 6, pp. 457–460, 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. D. R. Han, X. M. Yuan, and W. X. Zhang, “An augmented Lagrangian based parallel splitting method for separable convex minimization with applications to image processing,” Mathematics of Computation, vol. 83, no. 289, pp. 2263–2291, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. R. H. Chan, C.-W. Ho, and M. Nikolova, “Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization,” IEEE Transactions on Image Processing, vol. 14, no. 10, pp. 1479–1485, 2005. View at Publisher · View at Google Scholar · View at Scopus
  9. Y. Q. Dong, R. H. Chan, and S. F. Xu, “A detection statistic for random-valued impulse noise,” IEEE Transactions on Image Processing, vol. 16, no. 4, pp. 1112–1120, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. H. Hwang and R. A. Haddad, “Adaptive median filters: new algorithms and results,” IEEE Transactions on Image Processing, vol. 4, no. 4, pp. 499–502, 1995. View at Publisher · View at Google Scholar · View at Scopus
  11. T. Chen and H. R. Wu, “Space variant median filters for the restoration of impulse noise corrupted images,” IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 48, no. 8, pp. 784–789, 2001. View at Publisher · View at Google Scholar · View at Scopus
  12. J.-F. Cai, R. H. Chan, and M. Nikolova, “Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise,” Inverse Problems and Imaging, vol. 2, no. 2, pp. 187–204, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S.-J. Ko and Y. H. Lee, “Center weighted median filters and their applications to image enhancement,” IEEE Transactions on Circuits and Systems, vol. 38, no. 9, pp. 984–993, 1991. View at Publisher · View at Google Scholar · View at Scopus
  14. C. Cocianu and A. Stan, “Neural architectures for correlated noise removal in image processing,” Mathematical Problems in Engineering, vol. 2016, Article ID 6153749, 14 pages, 2016. View at Publisher · View at Google Scholar
  15. S. Morillas, V. Gregori, and A. Hervás, “Fuzzy peer groups for reducing mixed Gaussian-impulse noise from color images,” IEEE Transactions on Image Processing, vol. 18, no. 7, pp. 1452–1466, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. M.-G. Shama, T.-Z. Huang, J. Liu, and S. Wang, “A convex total generalized variation regularized model for multiplicative noise and blur removal,” Applied Mathematics and Computation, vol. 276, pp. 109–121, 2016. View at Publisher · View at Google Scholar · View at Scopus
  17. A. Chambolle, “An algorithm for total variation minimization and applications,” Journal of Mathematical Imaging and Vision, vol. 20, no. 1-2, pp. 89–97, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Modeling & Simulation, vol. 4, no. 4, pp. 1168–1200, 2005. View at Publisher · View at Google Scholar · View at Scopus
  19. J.-J. Moreau, “Fonctions convexes duales et points proximaux dans un espace hilbertien,” Comptes Rendus de l'Académie des Sciences, vol. 255, pp. 2897–2899, 1962. View at Google Scholar · View at MathSciNet
  20. M. J. Buckley, “Fast computation of a discretized thin-plate smoothing spline for image data,” Biometrika, vol. 81, no. 2, pp. 247–258, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. P. C. Hansen, J. G. Nagy, and D. P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, vol. 3 of Fundamentals of Algorithms, SIAM, Philadelphia, Pa, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  22. D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, NY, USA, 1982. View at MathSciNet
  23. D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Computers & Mathematics with Applications, vol. 2, no. 1, pp. 17–40, 1976. View at Publisher · View at Google Scholar · View at Scopus
  24. M. Fukushima, “Application of the alternating direction method of multipliers to separable convex programming problems,” Computational Optimization and Applications, vol. 1, no. 1, pp. 93–111, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, vol. 9 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  26. D. R. Han, H. J. He, H. Yang, and X. M. Yuan, “A customized Douglas-Rachford splitting algorithm for separable convex minimization with linear constraints,” Numerische Mathematik, vol. 127, no. 1, pp. 167–200, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. D. R. Han and X. M. Yuan, “Local linear convergence of the alternating direction method of multipliers for quadratic programs,” SIAM Journal on Numerical Analysis, vol. 51, no. 6, pp. 3446–3457, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. B. S. He, L.-Z. Liao, D. R. Han, and H. Yang, “A new inexact alternating directions method for monotone variational inequalities,” Mathematical Programming, vol. 92, no. 1, pp. 103–118, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine Learning, vol. 3, no. 1, pp. 1–122, 2011. View at Google Scholar
  30. E. T. Hale, W. Yin, and Y. Zhang, “Fixed-point continuation for l1-minimization: methodology and convergence,” SIAM Journal on Optimization, vol. 19, no. 3, pp. 1107–1130, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. M. K. Ng, P. Weiss, and X. Yuan, “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods,” SIAM Journal on Scientific Computing, vol. 32, no. 5, pp. 2710–2736, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. Z. N. Zhu, G. Ch. Cai, and Y.-W. Wen, “Adaptive box-constrained total variation image restoration using iterative regularization parameter adjustment method,” International Journal of Pattern Recognition and Artificial Intelligence, vol. 29, no. 7, Article ID 1554003, 20 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. C. Chen, R. H. Chan, S. Ma, and J. Yang, “Inertial proximal ADMM for linearly constrained separable convex optimization,” SIAM Journal on Imaging Sciences, vol. 8, no. 4, pp. 2239–2267, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. C. H. Chen, M. Li, X. Liu, and Y. Y. Ye, “On the convergence of multi-block alternating direction method of multipliers and block coordinate descent method,” https://arxiv.org/abs/1508.00193.
  35. W. H. Yang and D. R. Han, “Linear convergence of the alternating direction method of multipliers for a class of convex optimization problems,” SIAM Journal on Numerical Analysis, vol. 54, no. 2, pp. 625–640, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  36. C. H. Chen, B. S. He, Y. Y. Ye, and X. M. Yuan, “The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent,” Mathematical Programming, vol. 155, no. 1-2, pp. 57–79, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. D. R. Han and X. M. Yuan, “A note on the alternating direction method of multipliers,” Journal of Optimization Theory and Applications, vol. 155, no. 1, pp. 227–238, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. X. J. Cai, D. R. Han, and X. M. Yuan, “On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function,” Computational Optimization and Applications, 2016. View at Publisher · View at Google Scholar
  39. M. Li, D. F. Sun, and K.-C. Toh, “A convergent 3-block semi-proximal ADMM for convex minimization problems with one strongly convex block,” Asia-Pacific Journal of Operational Research, vol. 32, no. 4, Article ID 1550024, 19 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. B. He, M. Tao, and X. Yuan, “Alternating direction method with Gaussian back substitution for separable convex programming,” SIAM Journal on Optimization, vol. 22, no. 2, pp. 313–340, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus