Advertisement

A coupled variational model for image denoising using a duality strategy and split Bregman

  • Jianlou Xu
  • Xiangchu Feng
  • Yan Hao
Article

Abstract

To reduce the staircase effect, high-order diffusion equations are used with high computational cost. Recently, a two-step method with two energy functions has been introduced to alleviate the staircase effect successfully. In the two-step method, firstly, the normal vector of noisy image is smoothed, and then the image is reconstructed from the smoothed normal field. In this paper, we propose a new image restoration model with only one energy function. When the alternating direction method is used, the estimation of the vector field and the reconstruction of the image are interlaced, which makes the new vector field can utilize sufficiently the information of the restored image, thus the constructed vector field is more accurate than that generated by the two-step method. To speed up the computation, the dual approach and split Bregman are employed in our numerical algorithm. The experimental results show that the new model is more effective to filter out the Gaussian noise than the state-of-the-art models.

Keywords

Image denoising Staircase effect Dual formulation Split Bregman 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Afonso M., Bioucas-Dias J., Figueiredo M. (2011) An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Transactions on Image Processing 20(3): 681–695CrossRefMathSciNetGoogle Scholar
  2. Bai J., Feng X. C. (2007) Fractional-order anisotropic diffusion for image denoising. IEEE Transactions on Image Processing 16(10): 2492–2502CrossRefMathSciNetGoogle Scholar
  3. Cai J. F., Osher S., Shen Z. (2009a) Linearized Bregman iterations for compressed sensing. Mathematics of Computation 78: 1515–1536CrossRefMATHMathSciNetGoogle Scholar
  4. Cai J. F., Osher S., Shen Z. (2009b) Linearized Bregman iterations for frame-based image deblurring. SIAM Journal on Imaging Sciences 2(1): 226–252CrossRefMATHMathSciNetGoogle Scholar
  5. Chambolle A. (2004) An algorithm for total variation minimization and applications. Journal Mathematical Imaging Vision 20(1–2): 89–97MathSciNetGoogle Scholar
  6. Chan T., Marquina A., Mulet P. (2000) High-order total variation-based image restoration. SIAM Journal on Scientific Computing 22(2): 503–516CrossRefMATHMathSciNetGoogle Scholar
  7. Chen Y., Ji Z. C., Hua C. J. (2008) Spatial adaptive Bayesian wavelet threshold exploiting scale and space consistency. Multidimensional Systems and Signal Processing 19(1): 157–170CrossRefMATHMathSciNetGoogle Scholar
  8. Daubechies I., Teschke G. (2005) Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising. Applied and Computational Harmonic Analysis 19(1): 1–16CrossRefMATHMathSciNetGoogle Scholar
  9. Didas S., Weickert J., Burgeth B. (2009) Properties of higher order nonlinear diffusion filtering. Journal Mathematical Imaging Vision 35(3): 208–226CrossRefMathSciNetGoogle Scholar
  10. Dong F. F., Liu Z. (2009) A new gradient fidelity term for avoiding staircasing effect. Journal of Computer Science and Technology 24(6): 1162–1170CrossRefMathSciNetGoogle Scholar
  11. Dong F. F., Liu Z., Kong D. X., Liu K. F. (2009) An improved LOT model for image restoration. Journal Mathematical Imaging Vision 34(1): 89–97CrossRefMathSciNetGoogle Scholar
  12. Donoho D. (1995) Denoising by soft-thresholding. IEEE Transactions on Information Theory 41(3): 613–627CrossRefMATHMathSciNetGoogle Scholar
  13. Fadili M., Peyré G. (2011) Total variation projection with first order schemes. IEEE Transactions on Image Processing 20(3): 657–669CrossRefMathSciNetGoogle Scholar
  14. Goldstein T., Osher S. (2009) The split Bregman method for L1 regularized problems. SIAM Journal on Imaging Sciences 2(2): 323–343CrossRefMATHMathSciNetGoogle Scholar
  15. Guidotti P., Lambers J. (2009) Two new nonlinear nonlocal diffusions for noise reduction. Journal Mathematical Imaging Vision 33(1): 25–37CrossRefMathSciNetGoogle Scholar
  16. Hahn J., Tai X. C., Borok S., Bruckstein A. M. (2011) Orientation-matching minimization for image denoising and inpainting. International Journal of Computer Vision 92(3): 308–324CrossRefMATHMathSciNetGoogle Scholar
  17. Jiang L. L., Feng X. C., Yin H. Q. (2008) Variational image restoration and decomposition with curvelet shrinkage. Journal Mathematical Imaging Vision 30(2): 125–132CrossRefMathSciNetGoogle Scholar
  18. Jiang L. L., Yin H. Q. (2012) Bregman iteration algorithm for sparse nonnegative matrix factorizations via alternating l 1-norm minimization. Multidimensional Systems and Signal Processing 23(3): 315–328CrossRefMathSciNetGoogle Scholar
  19. Lysaker M., Lundervold A., Tai X. C. (2003) Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Transactions on Image Processing 12(12): 1579–1590CrossRefGoogle Scholar
  20. Lysaker M., Osher S., Tai X. C. (2004) Noise removal using smoothed normals and surface fitting. IEEE Transactions on Image Processing 13(10): 1345–1357CrossRefMathSciNetGoogle Scholar
  21. Litvinov W., Rahman T., Tai X. C. (2011) A modified TV-Stokes model for image processing. SIAM Journal on Scientific Computing 33(4): 1574–1597CrossRefMATHMathSciNetGoogle Scholar
  22. Nikolova M., Ng M., Chi-Pan T. (2010) Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Transactions on Image Processing 19(12): 3073–3088CrossRefMathSciNetGoogle Scholar
  23. Osher S., Burger M., Goldfarb D., Xu J., Yin W. (2005) An iterative regularization method for total variation-based image restoration. Multiscale Modeling and Simulation 4(2): 460–489CrossRefMATHMathSciNetGoogle Scholar
  24. Osher S., Mao Y., Dong B., Yin W. (2010) Fast linearized Bregman iteration for compressive sensing and sparse denoising. Communications in Mathematical Sciences 8(1): 93–111CrossRefMATHMathSciNetGoogle Scholar
  25. Rudin L., Osher S., Fatemi E. (1992) Nonlinear total variation based noise removal algorithms. Physica D 60: 259–268CrossRefMATHGoogle Scholar
  26. Setzer S. (2009) Split Bregman algorithm, Douglas–Rachford splitting and frame shrinkage. Scale Space and Variational Methods in Computer Vision 5567: 464–476CrossRefGoogle Scholar
  27. Tai X. C., Wu C. (2009) Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. Scale Space and Variational Methods in Computer Vision 5567: 502–513CrossRefGoogle Scholar
  28. Wu C., Tai X. C. (2010) Augmented Lagrangian method, dual methods and split-Bregman iterations for ROF, vectorial TV and higher order models. SIAM Journal on Imaging Science 3(3): 300–339CrossRefMATHMathSciNetGoogle Scholar
  29. Yang Y., Pang Z., Shi B., Wang Z. (2011) Split Bregman method for the modified LOT model in image denoising. Applied Mathematics and Computation 217(12): 5392–5403CrossRefMATHMathSciNetGoogle Scholar
  30. Yin W. (2010) Analysis and generalizations of the linearized Bregman method. SIAM Journal on Imaging Sciences 3(4): 856–877CrossRefMATHMathSciNetGoogle Scholar
  31. You Y., Kaveh M. (2000) Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing 9(10): 1723–1730CrossRefMATHMathSciNetGoogle Scholar
  32. Zhu L. X., Xia D. S. (2008) Staircase effect alleviation by coupling gradient fidelity term. Image Vision Computing 26(8): 1163–1170CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of SciencesXidian UniversityXi’anChina
  2. 2.School of Mathematics and StatisticsHenan University of Science and TechnologyLuoyangChina

Personalised recommendations