Abstract
The total variation model proposed by Rudin, Osher, and Fatemi performs very well for removing noise while preserving edges. However, it favors a piecewise constant solution in BV space which often leads to the staircase effect, and small details such as textures are often filtered out with noise in the process of denoising. In this paper, we propose a fractional-order multi-scale variational model which can better preserve the textural information and eliminate the staircase effect. This is accomplished by replacing the first-order derivative with the fractional-order derivative in the regularization term, and substituting a kind of multi-scale norm in negative Sobolev space for the L 2 norm in the fidelity term of the ROF model. To improve the results, we propose an adaptive parameter selection method for the proposed model by using the local variance measures and the wavelet based estimation of the singularity. Using the operator splitting technique, we develop a simple alternating projection algorithm to solve the new model. Numerical results show that our method can not only remove noise and eliminate the staircase effect efficiently in the non-textured region, but also preserve the small details such as textures well in the textured region. It is for this reason that our adaptive method can improve the result both visually and in terms of the peak signal to noise ratio efficiently.
Similar content being viewed by others
References
Ruding, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)
Chan, T.F., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)
Chan, T.F., Esedoglu, S., Park, F.E.: A fourth order dual method for staircase reduction in texture extraction and image restoration problems. UCLA CAM Report, 05-28 (2005)
You, Y.L., Kaveh, M.: Fourth-order partial differential equations for noise removal. IEEE Trans. Image Process. 9(10), 1723–1730 (2000)
Lysaker, M., Lundervold, A., Tai, X.C.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)
Bai, J., Feng, X.C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492–2502 (2007)
Zhang, J., Wei, Z.H.: Fractional variational model and algorithm for image denoising. In: Fourth International Conference on Natural Computation (ICNC2008), vol. 5, pp. 524–528. IEEE Press, New York (2008)
Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures. Am. Math. Soc., Boston (2001)
Lieu, L., Vese, L.: Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev space. UCLA CAM Report, 05-33 (2005)
Aujol, J.F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005)
Zhang, J., Wei, Z.H.: A class of multi-scale models for image denoising in negative Hilbert-Sobolev spaces. In: International Conference on Emerging Intelligent Computing and Applications. Lecture Notes in Control and Information Science, vol. 345, pp. 584–592. Springer, Berlin (2006)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Aujol, J.F., Gilboa, G., Chan, T.F., Osher, S.: Structure-texture image decomposition: modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67(1), 111–136 (2006)
Gilboa, G., Sochen, N., Zeevi, Y.: Variational denoising of partly-textured images by spatially varying constraints. IEEE Trans. Image Process. 15(8), 2281–2289 (2006)
Gilles, J.: Noisy image decomposition: a new structure, texture and noise model based on local adaptively. J. Math. Imaging Vis. 28(3), 285–295 (2007)
Li, F., Ng, M.K., Shen, C.: Multiplicative noise removal with spatially varying regularization parameters. SIAM J. Imaging Sci. 3(1), 1–20 (2010)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1), 89–97 (2004)
Chen, G.H.G., Rockafellar, R.T.: Convergence rates in forward-backward splitting [J]. SIAM J. Optim. 7(2), 421–444 (1997)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, J., Wei, Z. & Xiao, L. Adaptive Fractional-order Multi-scale Method for Image Denoising. J Math Imaging Vis 43, 39–49 (2012). https://doi.org/10.1007/s10851-011-0285-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-011-0285-z