Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2016, Article ID 6132356, 11 pages
http://dx.doi.org/10.1155/2016/6132356
Research Article

Solving Adaptive Image Restoration Problems via a Modified Projection Algorithm

1School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
2Sichuan Provincial Key Laboratory of Digital Media, Chengdu, Sichuan 611731, China
3College of Computer Science and Technology, Southwest University for Nationalities, Chengdu, Sichuan 610041, China

Received 26 January 2016; Accepted 9 March 2016

Academic Editor: Daniel Zaldivar

Copyright © 2016 Hao Yang 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. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena, vol. 60, no. 1–4, pp. 259–268, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. J.-F. Aujol, G. Gilboa, T. Chan, and S. Osher, “Structure-texture image decomposition—modeling, algorithms, and parameter selection,” International Journal of Computer Vision, vol. 67, no. 1, pp. 111–136, 2006. View at Publisher · View at Google Scholar · View at Scopus
  3. I. Bayram and M. E. Kamasak, “A directional total variation,” in Proceedings of the 20th European Signal Processing Conference (EUSIPCO '12), pp. 265–269, IEEE, Piscataway, NJ, USA, August 2012. View at Scopus
  4. B. Berkels, M. Burger, M. Droske, O. Nemitz, and M. Rumpf, “Cartoon extraction based on anisotropic image classification,” in Vision, Modeling, and Visualization Proceedings, pp. 293–300, IOS, Amsterdam, The Netherlands, 2006. View at Google Scholar
  5. A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, vol. 4, no. 2, pp. 490–530, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Modeling & Simulation, vol. 7, no. 3, pp. 1005–1028, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. K. Bredies, K. Kunisch, and T. Pock, “Total generalized variation,” SIAM Journal on Imaging Sciences, vol. 3, no. 3, pp. 492–526, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. S. Setzer, G. Steidl, and T. Teuber, “Infimal convolution regularizations with discrete l1-type functionals,” Communications in Mathematical Sciences, vol. 9, no. 3, pp. 797–827, 2011. View at Publisher · View at Google Scholar
  9. F. Lenzen, F. Becker, and J. Lellmann, “Adaptive second-order total variation: an approach aware of slope discontinuities,” in Scale Space and Variational Methods in Computer Vision: 4th International Conference, SSVM 2013, Schloss Seggau, Leibnitz, Austria, June 2–6, 2013. Proceedings, vol. 7893 of Lecture Notes in Computer Science, pp. 61–73, Springer, New York, NY, USA, 2013. View at Publisher · View at Google Scholar
  10. 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
  11. T. Chan, S. Esedoglu, F. Park, and A. Yip, “Total variation image restoration: overview and recent developments,” in Handbook of Mathematical Models in Computer Vision, N. Paragios, Y. Chen, and O. D. Faugeras, Eds., vol. 17, chapter 2, pp. 17–31, Springer, New York, NY, USA, 2006. View at Google Scholar
  12. J. P. Oliveira, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Adaptive total variation image deblurring: a majorization-minimization approach,” Signal Processing, vol. 89, no. 9, pp. 1683–1693, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, NY, USA, 2003.
  14. X.-P. Luo, “Tikhonov regularization methods for inverse variational inequalities,” Optimization Letters, vol. 8, no. 3, pp. 877–887, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. X.-P. Luo and J. Yang, “Regularization and iterative methods for monotone inverse variational inequalities,” Optimization Letters, vol. 8, no. 4, pp. 1261–1272, 2014. View at Publisher · View at Google Scholar · View at Scopus
  16. X.-p. Luo and H. Yang, “Regularization and iterative methods for inverse mixed variational inequalities,” Nonlinear Analysis Forum, vol. 19, pp. 53–63, 2014. View at Google Scholar · View at MathSciNet
  17. X.-P. Luo and H. Yang, “Tikhonov regularization methods for inverse mixed variational inequalities,” Advances in Nonlinear Variational Inequalities, vol. 17, no. 2, pp. 13–25, 2014. View at Google Scholar · View at MathSciNet
  18. D. Chan and J.-S. Pang, “The generalized quasi-variational inequality problem,” Mathematics of Operations Research, vol. 7, no. 2, pp. 211–222, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  19. F. Lenzen, J. Lellmann, F. Becker, and C. Schnörr, “Solving quasi-variational inequalities for image restoration with adaptive constraint sets,” SIAM Journal on Imaging Sciences, vol. 7, no. 4, pp. 2139–2174, 2014. View at Publisher · View at Google Scholar · View at Scopus
  20. F. Lenzen, F. Becker, J. Lellmann, S. Petra, and C. Schnörr, “A class of quasi-variational inequalities for adaptive image denoising and decomposition,” Computational Optimization and Applications, vol. 54, no. 2, pp. 371–398, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. F. Lenzen, F. Becker, J. Lellmann, S. Petra, and C. Schnörr, “Variational image denoising with adaptive constraint sets,” in Scale Space and Variational Methods in Computer Vision: Third International Conference, SSVM 2011, Ein-Gedi, Israel, May 29–June 2, 2011, Revised Selected Papers, vol. 6667 of Lecture Notes in Computer Science, pp. 206–217, Springer, New York, NY, USA, 2012. View at Publisher · View at Google Scholar
  22. M. A. Noor and K. I. Noor, “Some new classes of extended general mixed quasi-variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 962978, 9 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  23. L. Alvarez and J. M. Morel, “Formalization and computational aspects of image analysis,” Acta Numerica, vol. 3, pp. 1–59, 1994. View at Google Scholar
  24. 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 · View at Scopus
  25. X. Li, X. S. Li, and N. J. Huang, “A generalized f-projection algorithm for inverse mixed variational inequalities,” Optimization Letters, vol. 8, no. 3, pp. 1063–1076, 2014. View at Google Scholar
  26. A. Nagurney, Network Economics. A Variational Inequality Approach, Kluwer Academics Publishers, Boston, Mass, USA, 1999.
  27. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Transactions on Image Processing, vol. 13, no. 4, pp. 600–612, 2004. View at Publisher · View at Google Scholar · View at Scopus
  28. T. Goldstein and S. Osher, “The split Bregman algorithm for L1 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
  29. X. W. Liu and L. H. Huang, “A new nonlocal total variation regularization algorithm for image denoising,” Mathematics and Computers in Simulation, vol. 97, pp. 224–233, 2014. View at Publisher · View at Google Scholar · View at Scopus
  30. Z. W. Qin, D. Goldfarb, and S. Q. Ma, “An alternating direction method for total variation denoising,” Optimization Methods & Software, vol. 30, no. 3, pp. 594–615, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. L. Gómez-Chova, J. Muñoz-Marí, V. Laparra, J. Malo-López, and G. Camps-Valls, “A review of kernel methods in remote sensing data analysis,” in Optical Remote Sensing, vol. 3, pp. 171–206, Springer, Berlin, Germany, 2011. View at Google Scholar