Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 797239, 14 pages
http://dx.doi.org/10.1155/2013/797239
Research Article

Efficient Algorithm for Isotropic and Anisotropic Total Variation Deblurring and Denoising

1Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
2Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

Received 17 October 2012; Accepted 24 January 2013

Academic Editor: Changbum Chun

Copyright © 2013 Yuying Shi and Qianshun Chang. 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. View at Google Scholar · View at Scopus
  2. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. Birkholz, “A unifying approach to isotropic and anisotropic total variation denoising models,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2502–2514, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. S. Moll, “The anisotropic total variation flow,” Mathematische Annalen, vol. 332, no. 1, pp. 177–218, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. C. A. Z. Barcelos and Y. Chen, “Heat flows and related minimization problem in image restoration,” Computers & Mathematics with Applications, vol. 39, no. 5-6, pp. 81–97, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. W. Zuo and Z. Lin, “A generalized accelerated proximal gradient approach for total-variation-based image restoration,” IEEE Transactions on Image Processing, vol. 20, no. 10, pp. 2748–2759, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. Grasmair and F. Lenzen, “Anisotropic total variation filtering,” Applied Mathematics and Optimization, vol. 62, no. 3, pp. 323–339, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. Li and F. Santosa, “An affine scaling algorithm for minimizing total variation in image enhancement,” Tech. Rep. 12/94, Center for Theory and Simulation in Science and Engineering, Cornell University, 1994. View at Google Scholar
  9. R. Chan, T. Chan, and H. Zhou, “Advanced signal processing algorithms,” in Proceedings of the International Society of Photo-Optical Instrumentation Engineers, F. Luk, Ed., pp. 314–325, SPIE, 1995.
  10. M. E. Oman, “Fast multigrid techniques in total variation-based image reconstruction,” in Proceedings of the Copper Mountain Conference on Multigrid Methods, 1995.
  11. C. R. Vogel, “A multigrid method for total variation-based image denoising,” in Computation and Control, IV, vol. 20 of Progress in Systems and Control Theory, pp. 323–331, Birkhäuser, Boston, Mass, USA, 1995. View at Google Scholar · View at Zentralblatt MATH · 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. C. R. Vogel and M. E. Oman, “Fast, robust total variation-based reconstruction of noisy, blurred images,” IEEE Transactions on Image Processing, vol. 7, no. 6, pp. 813–824, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. K. Ng, R. H. Chan, and W.-C. Tang, “A fast algorithm for deblurring models with Neumann boundary conditions,” SIAM Journal on Scientific Computing, vol. 21, no. 3, pp. 851–866, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. H. Chan, T. F. Chan, and C. K. Wong, “Cosine transform based preconditioned for total variation deblurring,” IEEE Transactions on Image Processing, vol. 8, no. 10, pp. 1472–1478, 1999. View at Google Scholar · View at Scopus
  16. P. Blomgren and T. F. Chan, “Modular solvers for image restoration problems using the discrepancy principle,” Numerical Linear Algebra with Applications, vol. 9, no. 5, pp. 347–358, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. C. A. Micchelli, L. Shen, and Y. Xu, “Proximity algorithms for image models: denoising,” Inverse Problems, vol. 27, no. 4, Article ID 045009, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM Journal on Scientific Computing, vol. 20, no. 6, pp. 1964–1977, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Q. Chang and I.-L. Chern, “Acceleration methods for total variation-based image denoising,” SIAM Journal on Scientific Computing, vol. 25, no. 3, pp. 982–994, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Y. Shi and Q. Chang, “Remark on convergence of algebraic multigrid in the form of matrix decomposition,” Applied Mathematics and Computation, vol. 170, no. 2, pp. 1170–1184, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Y. Shi and Q. Chang, “Acceleration methods for image restoration problem with different boundary conditions,” Applied Numerical Mathematics, vol. 58, no. 5, pp. 602–614, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. 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 · View at MathSciNet
  23. Q. Chang, L. Chien, W. Wang, and J. Xu, “A robust algorithm variation deblurring and denoising,” in Proceedings of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision (SSVM '09), vol. 5567 of Lecture Notes in Computer Science, Springer, 2009.
  24. Q. Chang, Y. S. Wong, and H. Fu, “On the algebraic multigrid method,” Journal of Computational Physics, vol. 125, no. 2, pp. 279–292, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Q. Chang and Z. Huang, “Efficient algebraic multigrid algorithms and their convergence,” SIAM Journal on Scientific Computing, vol. 24, no. 2, pp. 597–618, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J. W. Ruge and K. Stüben, “Algebraic multigrid,” in Multigrid Methods, S. F. McCormick, Ed., vol. 3 of Frontiers Appl. Math., pp. 73–130, SIAM, Philadelphia, Pa, USA, 1987. View at Google Scholar · View at MathSciNet
  27. R.-Q. Jia, H. Zhao, and W. Zhao, “Convergence analysis of the Bregman method for the variational model of image denoising,” Applied and Computational Harmonic Analysis, vol. 27, no. 3, pp. 367–379, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. A. Brandt and V. Mikulinsky, “On recombining iterants in multigrid algorithms and problems with small islands,” SIAM Journal on Scientific Computing, vol. 16, no. 1, pp. 20–28, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. C. W. Oosterlee and T. Washio, “Krylov subspace acceleration of nonlinear multigrid with application to recirculating flows,” SIAM Journal on Scientific Computing, vol. 21, no. 5, pp. 1670–1690, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet