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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 965281, 14 pages
doi:10.1155/2012/965281
Research Article

Poisson Noise Removal Scheme Based on Fourth-Order PDE by Alternating Minimization Algorithm

Weifeng Zhou1,2 and Qingguo Li1

1College of Mathematics and Econometrics, Hunan University, Hunan, Changsha 410082, China
2Department of Mathematics, Chuxiong Normal University, Yunnan, Chuxiong 675000, China

Received 17 September 2011; Revised 23 October 2011; Accepted 19 November 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Weifeng Zhou and Qingguo Li. 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. 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.
  2. Y. Vardi, L. A. Shepp, and L. Kaufman, “A statistical model for positron emission tomography,” Journal of the American Statistical Association, vol. 80, no. 389, pp. 8–37, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. Kido, H. Nakamura, W. Ito, K. Shimura, and H. Kato, “Computerized detection of pulmonary nodules by single-exposure dual-energy computed radiography of the chest—part 1,” European Journal of Radiology, vol. 44, no. 3, pp. 198–204, 2002. View at Publisher · View at Google Scholar
  4. E. Bratsolis and M. Sigelle, “A spatial regularization method preserving local photometry for Richardson-Lucy restoration,” Astronomy and Astrophysics, vol. 375, no. 3, pp. 1120–1128, 2001.
  5. T. Le, R. Chartrand, and T. J. Asaki, “A variational approach to reconstructing images corrupted by Poisson noise,” Journal of Mathematical Imaging and Vision, vol. 27, no. 3, pp. 257–263, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  6. 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 PubMed · View at Zentralblatt MATH · View at MathSciNet
  7. S. Didas, J. Weickert, and B. Burgeth, “Properties of higher order nonlinear diffusion filtering,” Journal of Mathematical Imaging and Vision, vol. 35, no. 3, pp. 208–226, 2009. View at Publisher · View at Google Scholar
  8. 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, vol. 18, no. 4, pp. 322–330, 2007. View at Publisher · View at Google Scholar
  9. P. Guidotti and K. Longo, “Two enhanced fourth order diffusion models for image denoising,” Journal of Mathematical Imaging and Vision, vol. 40, no. 2, pp. 188–198, 2011. View at Publisher · View at Google Scholar
  10. M. Lysaker, A. Lundervold, and X.-C. 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, vol. 12, no. 12, pp. 1579–1589, 2003. View at Publisher · View at Google Scholar · View at PubMed
  11. M. R. Hajiaboli, “A self-governing fourth-order nonlinear diffusion filter for image noise removal,” IPSJ Transactions on Computer Vision and Applications, vol. 2, pp. 94–103, 2010.
  12. M. R. Hajiaboli, “An anisotropic fourth-order partial differential equation for noise removal,” Lecture Notes in Computer Science, vol. 5567, pp. 356–367, 2009.
  13. X. Liu, L. Huang, and Z. Guo, “Adaptive fourth-order partial differential equation filter for image denoising,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1282–1288, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. Kim and H. Lim, “Fourth-order partial differential equations for effective image denosing,” Electronic Journal of Differential Equations: Conference 17, vol. 17, pp. 107–121, 2009.
  15. 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
  16. D. P. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex Analysis and Optimization, Tsinghua University Press, 2006.
  17. 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
  18. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, vol. 147, Springer, New York, NY, USA, 2002.
  19. D. Wang, Y. Hou, and J. Peng, Partial Differential Equation Based Approach to Image Processing, Science Press, Beijing, China, 2008.
  20. 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
  21. 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
  22. M. Bergounioux and L. Piffet, “A second-order model for image denoising,” Set-Valued and Variational Analysis, vol. 18, no. 3-4, pp. 277–306, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 2, pp. 323–343, 2009. View at Publisher · View at Google Scholar
  24. Y. 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 Zentralblatt MATH
  25. Y.-M. Huang, M. K. Ng, and Y.-W. Wen, “A new total variation method for multiplicative noise removal,” SIAM Journal on Imaging Sciences, vol. 2, no. 1, pp. 20–40, 2009. View at Publisher · View at Google Scholar
  26. 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
  27. Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. 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 Zentralblatt MATH