Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 238561, 11 pages
http://dx.doi.org/10.1155/2013/238561
Research Article

Split Bregman Iteration Algorithm for Image Deblurring Using Fourth-Order Total Bounded Variation Regularization Model

School of Mathematical Sciences/Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Received 28 December 2012; Accepted 7 April 2013

Academic Editor: Ke Chen

Copyright © 2013 Yi Xu et al. 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. C. R. Vogel, Computational Methods for Inverse Problems, vol. 23 of Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, Bristol, UK, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. Andrews and B. Hunt, Digital Image Restoration, Prentice Hall, Englewood Cliffs, NJ, USA, 1977.
  4. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, V. H. Winston & Sons, Washington, DC, USA, 1977. View at MathSciNet
  5. 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 Zentralblatt MATH · View at Scopus
  6. Y. Li and F. Santosa, “An affine scaling algorithm for minimizing total variation in image enhancement,” Tech. Rep., Cornell University, Ithaca, NY, USA, 1994. View at Google Scholar
  7. R. Acar and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Problems, vol. 10, no. 6, pp. 1217–1229, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numerische Mathematik, vol. 76, no. 2, pp. 167–188, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. T. F. Chan and P. Mulet, “On the convergence of the lagged diffusivity fixed point method in total variation image restoration,” SIAM Journal on Numerical Analysis, vol. 36, no. 2, pp. 354–367, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  10. C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM Journal on Scientific Computing, vol. 17, no. 1, pp. 227–238, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Zhang, K. Chen, and B. Yu, “An iterative Lagrange multiplier method for constrained total-variation-based image denoising,” SIAM Journal on Numerical Analysis, vol. 50, no. 3, pp. 983–1003, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  12. C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM Journal on Scientific Computing, vol. 17, no. 1, pp. 227–238, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4203–4215, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  14. D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. T. Chan and J. Shen, Theory and Computation of Variational Image Deblurring, IMS Lecture Notes, 2007.
  16. T. F. Chan, S. Esedo\=glu, and M. Nikolova, “Algorithms for finding global minimizers of image segmentation and denoising models,” SIAM Journal on Applied Mathematics, vol. 66, no. 5, pp. 1632–1648, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. X. Liu and L. Huang, “Split Bregman iteration algorithm for total bounded variation regularization based image deblurring,” Journal of Mathematical Analysis and Applications, vol. 372, no. 2, pp. 486–495, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. Chavent and K. Kunisch, “Regularization of linear least squares problems by total bounded variation,” ESAIM Control, Optimisation and Calculus of Variations, vol. 2, pp. 359–376, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM Journal on Scientific Computing, vol. 22, no. 2, pp. 503–516, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. T. F. Chan, S. Esedoglu, and F. E. Park, “A fourth order dual method for staircase reduction in texture extraction and image restoration problems,” CAM Report, UCLA, 2005. View at Google Scholar
  21. Y.-L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Transactions on Image Processing, vol. 9, no. 10, pp. 1723–1730, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. H.-z. Chen, J.-p. Song, and X.-C. Tai, “A dual algorithm for minimization of the LLT model,” Advances in Computational Mathematics, vol. 31, no. 1–3, pp. 115–130, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Z.-F. Pang and Y.-F. Yang, “Semismooth Newton method for minimization of the LLT model,” Inverse Problems and Imaging, vol. 3, no. 4, pp. 677–691, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. L. M. Brègman, “A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming,” USSR Computational Mathematics and Mathematical Physics, vol. 7, no. 3, pp. 200–217, 1967. View at Google Scholar · View at MathSciNet
  25. S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Modeling & Simulation, vol. 4, no. 2, pp. 460–489, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. T. Goldstein and S. Osher, “The split Bregman method for 11 regularized problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 2, pp. 323–343, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  27. W. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman iterative algorithms for 11-minimization with applications to compressed sensing,” SIAM Journal on Imaging Sciences, vol. 1, no. 1, pp. 143–168, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  28. J. Darbon and S. Osher, Fast Discrete Optimization for Sparse Approximations and Deconvolutions, UCLA, 2007.
  29. Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Journal on Imaging Sciences, vol. 1, no. 3, pp. 248–272, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Y. Wang, W. Yin, and Y. Zhang, “A fast algorithm for image deblurring with total variation regularization,” CAAM Technical Report TR07-10, Rice University, Houston, Tex, USA, 2007. View at Google Scholar
  31. J.-F. Cai, B. Dong, S. Osher, and Z. Shen, “Image restoration: total variation, wavelet frames, and beyond,” Journal of the American Mathematical Society, vol. 25, no. 4, pp. 1033–1089, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  32. L. Moisan, “How to discretize the Total Variation of an image,” Proceedings in Applied Mathematics and Mechanics, vol. 7, no. 1, pp. 1041907–1041908, 2007. View at Publisher · View at Google Scholar
  33. J.-F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” Multiscale Modeling & Simulation, vol. 8, no. 2, pp. 337–369, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  34. S. Setzer, “Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage,” Computer Science, vol. 5567, pp. 464–476, 2009. View at Google Scholar