Journal of Mathematical Imaging and Vision

, Volume 59, Issue 3, pp 456–480 | Cite as

Demystifying the Asymptotic Behavior of Global Denoising

  • Antoine Houdard
  • Andrés Almansa
  • Julie Delon


In this work, we revisit the global denoising framework recently introduced by Talebi and Milanfar. We analyze the asymptotic behavior of its mean-squared error restoration performance in the oracle case when the image size tends to infinity. We introduce precise conditions on both the image and the global filter to ensure and quantify this convergence. We also make a clear distinction between two different levels of oracle that are used in that framework. By reformulating global denoising with the classical formalism of diagonal estimation, we conclude that the second-level oracle can be avoided by using Donoho and Johnstone’s theorem, whereas the first-level oracle is mostly required in the sequel. We also discuss open issues concerning the most challenging aspect, namely the extension of these results to the case where neither oracle is required.


Diagonal estimation Global denoising Wiener filtering Asymptotic study 


  1. 1.
    Alvarez, L., Gousseau, Y., Morel, J.M.: The Size of Objects in Natural and Artificial Images, vol. 111. Elsevier, Amsterdam (1999). doi: 10.1016/S1076-5670(08)70218-0 Google Scholar
  2. 2.
    Awate, S., Whitaker, R.: Image denoising with unsupervised information-theoretic adaptive filtering. In: International Conference on Computer Vision and Pattern Recognition (CVPR 2005), pp. 44–51 (2004)Google Scholar
  3. 3.
    Brand, M.: Fast low-rank modifications of the thin singular value decomposition. Linear Algebra Appl. 415(1), 20–30 (2006). doi: 10.1016/j.laa.2005.07.021.
  4. 4.
    Brodatz, P.: Textures: A Photographic Album for Artists and Designers. Dover Pubns, New York (1966)Google Scholar
  5. 5.
    Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005. CVPR 2005, vol. 2, pp. 60–65. IEEE (2005)Google Scholar
  6. 6.
    Chan, S.H., Zickler, T., Lu, Y.M.: Demystifying symmetric smoothing filters. arXiv preprint arXiv:1601.00088 (2016)
  7. 7.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-d transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Donoho, D.L., Johnstone, I.M., et al.: Ideal denoising in an orthonormal basis chosen from a library of bases. Comp. Rendus l’Acad. Sci. Ser. Math. 319(12), 1317–1322 (1994)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Donoho, D.L., Johnstone, J.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3), 425–455 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duval, V., Aujol, J.F., Gousseau, Y.: A bias-variance approach for the nonlocal means. SIAM J. Imaging Sci. 4(2), 760–788 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Facciolo, G., Almansa, A., Aujol, J.F., Caselles, V.: Irregular to regular sampling, denoising, and deconvolution. SIAM MMS 7(4), 1574–1608 (2009). doi: 10.1137/080719443 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guichard, F., Moisan, L., Morel, J.M.: A review of PDE models in image processing and image analysis. In: Journal de Physique IV (Proceedings), vol. 12, pp. 137–154. EDP sciences (2002)Google Scholar
  13. 13.
    Halko, N., Martinsson, P.G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011). doi: 10.1137/090771806 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lebrun, M., Buades, A., Morel, J.M.: A nonlocal bayesian image denoising algorithm. SIAM J. Imaging Sci. 6(3), 1665–1688 (2013). doi: 10.1137/120874989 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Levin, A., Nadler, B., Durand, F., Freeman, W.T.: Patch complexity, finite pixel correlations and optimal denoising. In: ECCV 2012, LNCS 7576 LNCS(PART 5), 73–86 (2012). doi: 10.1007/978-3-642-33715-4_6
  16. 16.
    Mallat, S.: A Wavelet Tour of Signal Processing: The Sparse Way. Academic press, London (2008)zbMATHGoogle Scholar
  17. 17.
    Milanfar, P.: A tour of modern image filtering: New insights and methods, both practical and theoretical. Signal Process. Mag. IEEE 30(1), 106–128 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Milanfar, P.: Symmetrizing smoothing filters. SIAM J. Imaging Sci. 6(1), 263–284 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ordentlich, E., Seroussi, G., Verdu, S., Weinberger, M., Weissman, T.: A discrete universal denoiser and its application to binary images. In: Proceedings of 2003 International Conference on Image Processing, 2003. ICIP 2003, vol. 1, pp. I–117. IEEE (2003)Google Scholar
  20. 20.
    Pati, Y.C., Rezaiifar, R., Krishnaprasad, P.: Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition. In: 1993 Conference Record of the Twenty-Seventh Asilomar Conference on Signals, Systems and Computers, 1993. pp. 40–44. IEEE (1993)Google Scholar
  21. 21.
    Pierazzo, N., Rais, M., Morel, J.M., Facciolo, G.: DA3D: fast and data adaptive dual domain denoising. In: 2015 IEEE International Conference on Image Processing (ICIP), pp. 432–436. IEEE (2015)Google Scholar
  22. 22.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 60(1), 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Talebi, H., Milanfar, P.: Global denoising is asymptotically optimal. In: International Conference on Image Processing (ICIP), pp. 818–822. IEEE. Paris (2014)Google Scholar
  24. 24.
    Talebi, H., Milanfar, P.: Global image denoising. IEEE Trans. Image Process. 23(2), 755–768 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Talebi, H., Milanfar, P.: Asymptotic performance of global denoising. SIAM J. Imaging Sci. 9(2), 665–683 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Weissman, T., Ordentlich, E., Seroussi, G., Verdú, S., Weinberger, M.J.: Universal discrete denoising: known channel. IEEE Trans. Inf Theory 51(1), 5–28 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yaroslavsky, L.: Digital Picture Processing: An Introduction. Springer, New York (2012)Google Scholar
  28. 28.
    Yu, G., Sapiro, G., Mallat, S.: Solving inverse problems with piecewise linear estimators: from Gaussian mixture models to structured sparsity. IEEE Trans. Image Process. 21(5), 2481–2499 (2012). doi: 10.1109/TIP.2011.2176743.

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.LTCITélécom ParisTechParisFrance
  2. 2.CNRS MAP5Université Paris DescartesParisFrance

Personalised recommendations