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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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): 681–695
Bai J., Feng X. C. (2007) Fractional-order anisotropic diffusion for image denoising. IEEE Transactions on Image Processing 16(10): 2492–2502
Cai J. F., Osher S., Shen Z. (2009a) Linearized Bregman iterations for compressed sensing. Mathematics of Computation 78: 1515–1536
Cai J. F., Osher S., Shen Z. (2009b) Linearized Bregman iterations for frame-based image deblurring. SIAM Journal on Imaging Sciences 2(1): 226–252
Chambolle A. (2004) An algorithm for total variation minimization and applications. Journal Mathematical Imaging Vision 20(1–2): 89–97
Chan T., Marquina A., Mulet P. (2000) High-order total variation-based image restoration. SIAM Journal on Scientific Computing 22(2): 503–516
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): 157–170
Daubechies I., Teschke G. (2005) Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising. Applied and Computational Harmonic Analysis 19(1): 1–16
Didas S., Weickert J., Burgeth B. (2009) Properties of higher order nonlinear diffusion filtering. Journal Mathematical Imaging Vision 35(3): 208–226
Dong F. F., Liu Z. (2009) A new gradient fidelity term for avoiding staircasing effect. Journal of Computer Science and Technology 24(6): 1162–1170
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): 89–97
Donoho D. (1995) Denoising by soft-thresholding. IEEE Transactions on Information Theory 41(3): 613–627
Fadili M., Peyré G. (2011) Total variation projection with first order schemes. IEEE Transactions on Image Processing 20(3): 657–669
Goldstein T., Osher S. (2009) The split Bregman method for L1 regularized problems. SIAM Journal on Imaging Sciences 2(2): 323–343
Guidotti P., Lambers J. (2009) Two new nonlinear nonlocal diffusions for noise reduction. Journal Mathematical Imaging Vision 33(1): 25–37
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): 308–324
Jiang L. L., Feng X. C., Yin H. Q. (2008) Variational image restoration and decomposition with curvelet shrinkage. Journal Mathematical Imaging Vision 30(2): 125–132
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): 315–328
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): 1579–1590
Lysaker M., Osher S., Tai X. C. (2004) Noise removal using smoothed normals and surface fitting. IEEE Transactions on Image Processing 13(10): 1345–1357
Litvinov W., Rahman T., Tai X. C. (2011) A modified TV-Stokes model for image processing. SIAM Journal on Scientific Computing 33(4): 1574–1597
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): 3073–3088
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): 460–489
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): 93–111
Rudin L., Osher S., Fatemi E. (1992) Nonlinear total variation based noise removal algorithms. Physica D 60: 259–268
Setzer S. (2009) Split Bregman algorithm, Douglas–Rachford splitting and frame shrinkage. Scale Space and Variational Methods in Computer Vision 5567: 464–476
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: 502–513
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): 300–339
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): 5392–5403
Yin W. (2010) Analysis and generalizations of the linearized Bregman method. SIAM Journal on Imaging Sciences 3(4): 856–877
You Y., Kaveh M. (2000) Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing 9(10): 1723–1730
Zhu L. X., Xia D. S. (2008) Staircase effect alleviation by coupling gradient fidelity term. Image Vision Computing 26(8): 1163–1170
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xu, J., Feng, X. & Hao, Y. A coupled variational model for image denoising using a duality strategy and split Bregman. Multidim Syst Sign Process 25, 83–94 (2014). https://doi.org/10.1007/s11045-012-0190-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-012-0190-7