Abstract and Applied Analysis

An Efficient Variational Method for Image Restoration

Jun Liu, Ting-Zhu Huang, Xiao-Guang Lv, and Si Wang

Full-text: Open access

Abstract

Image restoration is one of the most fundamental issues in imaging science. Total variation regularization is widely used in image restoration problems for its capability to preserve edges. In this paper, we consider a constrained minimization problem with double total variation regularization terms. To solve this problem, we employ the split Bregman iteration method and the Chambolle’s algorithm. The convergence property of the algorithm is established. The numerical results demonstrate the effectiveness of the proposed method in terms of peak signal-to-noise ratio (PSNR) and the structure similarity index (SSIM).

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 213536, 11 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393447480

Digital Object Identifier
doi:10.1155/2013/213536

Mathematical Reviews number (MathSciNet)
MR3134164

Zentralblatt MATH identifier
1319.94017

Citation

Liu, Jun; Huang, Ting-Zhu; Lv, Xiao-Guang; Wang, Si. An Efficient Variational Method for Image Restoration. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 213536, 11 pages. doi:10.1155/2013/213536. https://projecteuclid.org/euclid.aaa/1393447480


Export citation

References

  • M. R. Banham and A. K. Kataggelos, “Digital image restoration,” IEEE Signal Processing Magazine, vol. 14, pp. 24–41, 1997.
  • 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.
  • J. G. Nagy, M. K. Ng, and L. Perrone, “Kronecker product approximations for image restoration with reflexive boundary conditions,” SIAM Journal on Matrix Analysis and Applications, vol. 25, no. 3, pp. 829–841, 2003.
  • J. Huang, T.-Z. Huang, X.-L. Zhao, and Z.-B. Xu, “Image restoration with shifting reflective boundary conditions,” Science China Information Sciences, vol. 56, no. 6, pp. 1–15, 2013.
  • K. T. Lay and A. K. Katsaggelos, “Identification and restoration based on the expectationmaximization algorithm,” Optical Engineering, vol. 29, pp. 436–445, 1990.
  • P. C. Hansen, J. G. Nagy, and D. P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, vol. 3, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2006.
  • P. C. Hansen, Rank-deficient and Discrete Ill-Posed Problems, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1998.
  • X.-G. Lv, T.-Z. Huang, Z.-B. Xu, and X.-L. Zhao, “Kronecker product approximations for image restoration with whole-sample symmetric boundary conditions,” Information Sciences, vol. 186, pp. 150–163, 2012.
  • A. Tikhonov and V. Arsenin, Solution of Ill-Poised Problems, Winston, Washington, DC, USA, 1977.
  • V. Agarwal, A. V. Gribok, and M. A. Abidi, “Image restoration using ${L}_{1}$ norm penalty function,” Inverse Problems in Science and Engineering, vol. 15, no. 8, pp. 785–809, 2007.
  • L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Journal of Physics D, vol. 60, pp. 259–268, 1992.
  • 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, pp. 17–31, Springer, New York, NY, USA, 2006.
  • P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, pp. 629–639, 1990.
  • Z. J. Xiang and P. J. Ramadge, “Edge-preserving image regularization based on morphological wavelets and dyadic trees,” IEEE Transactions on Image Processing, vol. 21, no. 4, pp. 1548–1560, 2012.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • M. Hintermüller and K. Kunisch, “Total bounded variation regularization as a bilaterally constrained optimization problem,” SIAM Journal on Applied Mathematics, vol. 64, no. 4, pp. 1311–1333, 2004.
  • 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.
  • L. Bregman, “The relaxation method of finnding the common points of convex sets and its application to the solution of problems in convex optimization,” USSR Computational Mathematics and Mathematical Physics, vol. 7, pp. 200–217, 1967.
  • S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” SIAM Multiscale Modeling and Simulation, vol. 4, no. 2, pp. 460–489, 2005.
  • W. T. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman iterative algorithms for ${l}_{1}$-minimization with applications to compressed sensing,” SIAM Journal on Imaging Sciences, vol. 1, no. 1, pp. 143–168, 2008.
  • J.-F. Cai, S. Osher, and Z. Shen, “Split Bregman methods and frame based image restoration,” SIAM Multiscale Modeling and Simulation, vol. 8, no. 2, pp. 337–369, 2009.
  • W. H. Li, Q. L. Li, W. Gong, and S. Tang, “Total variation blind deconvolution employing split Bregman iteration,” Journal of Visual Communication and Image Representation, vol. 23, pp. 409–417, 2012.
  • M. Zhu and T. F. Chan, “An efficient primal-dual hybrid gradient algorithm for total variation image restoration,” CAM Report 08-34, Mathematics Department, UCLA, 2008.
  • M. K. Ng, L. Qi, Y.-F. Yang, and Y.-M. Huang, “On semismooth Newton's methods for total variation minimization,” Journal of Mathematical Imaging and Vision, vol. 27, no. 3, pp. 265–276, 2007.
  • C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
  • P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” SIAM Multiscale Modeling & Simulation, vol. 4, no. 4, pp. 1168–1200, 2005.
  • 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.
  • A. Horé and D. Ziou, “Image quality metrics: PSNR vs. SSIM,” in Proceedings of the IEEE International Conference on Pattern Recognition, pp. 2366–2369, Istanbul, Turkey, 2010.