Below the Surface of the Non-local Bayesian Image Denoising Method

  • Pablo AriasEmail author
  • Mila Nikolova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)


The non-local Bayesian (NLB) patch-based approach of Lebrun et al. [12] is considered as a state-of-the-art method for the restoration of (color) images corrupted by white Gaussian noise. It gave rise to numerous ramifications like e.g., possible improvements, processing of various data sets and video. This article is the first attempt to analyse the method in depth in order to understand the main phenomena underlying its effectiveness. Our analysis, corroborated by numerical tests, shows several unexpected facts. In a variational setting, the first-step Bayesian approach to learn the prior for patches is equivalent to a pseudo-Tikhonov regularisation where the regularisation parameters can be positive or negative. Practically very good results in this step are mainly due to the aggregation stage – whose importance needs to be re-evaluated.


Negative Eigenvalue Image Patch Image Denoising Negative Weight Wiener Filter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aguerrebere, C., Almansa, A., Gousseau, Y., Musé, P.: A hyperprior Bayesian approach for solving image inverse problems. HAL (2016)Google Scholar
  2. 2.
    Arias, P., Morel, J.M.: Video denoising via empirical Bayesian estimation of space-time patches, January 2017, preprint.
  3. 3.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  4. 4.
    Buades, A., Lisani, J.L., Miladinović, M.: Patch-based video denoising with optical flow estimation. IEEE Trans. Image Process. 25(6), 2573–2586 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chatterjee, P., Milanfar, P.: Patch-based near-optimal image denoising. IEEE Trans. Image Process. 21(4), 1635–1649 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: BM3D image denoising with shape-adaptive principal component analysis. In: Proceedings of the Workshop on Signal Processing with Adaptive Sparse Structured Representations, pp. 1–7 (2009)Google Scholar
  7. 7.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3D transform-domain collaborative filtering. IEEE Trans. IP 16(8), 2080–2095 (2007)Google Scholar
  8. 8.
    Deledalle, C.A., Salmon, J., Dalalyan, A.: Image denoising with patch based PCA: local versus global. In: Proceedings of the British Machine Vision Conference, pp. 25.1-25.10 (2011)Google Scholar
  9. 9.
    Guillemot, T., Almansa, A., Boubekeur, T.: Covariance trees for 2D and 3D processing. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 556–563 (2014)Google Scholar
  10. 10.
    Laus, F., Nikolova, M., Persch, J., Steidl, G.: A nonlocal denoising algorithm for manifold-valued images using second order statistics. SIAM J. Imaging Sci. 10, 416–448 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lebrun, M., Buades, A., Morel, J.M.: Implementation of the “non-local Bayes” (NL-Bayes) image denoising algorithm. Image Process. Line 3, 1–42 (2013)CrossRefGoogle Scholar
  12. 12.
    Lebrun, M., Buades, A., Morel, J.M.: A nonlocal Bayesian image denoising algorithm. SIAM J. Imaging Sci. 6(3), 1665–1688 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lebrun, M., Colom, M., Buades, A., Morel, J.M.: Secrets of image denoising cuisine. Acta Numerica 21, 475–576 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lebrun, M., Colom, M., Morel, J.M.: The noise clinic: a blind image denoising algorithm. Image Process. Online (IPOL) 5, 1–54 (2015)CrossRefGoogle Scholar
  15. 15.
    Muresan, D.D., Parks, T.W.: Adaptive principal components and image denoising. In: Proceedings of the IEEE International Conference on Image Processing, vol. 1, pp. I-101–104, September 2003Google Scholar
  16. 16.
    Yaroslavsky, L.: Theoretical Foundations of Digital Imaging Using MATLAB. CRC Press, Boca Raton (2013)Google Scholar
  17. 17.
    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)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, L., Dong, W., Zhang, D., Shi, G.: Two-stage image denoising by principal component analysis with local pixel grouping. Pattern Recogn. 43(4), 1531–1549 (2010)CrossRefzbMATHGoogle Scholar
  19. 19.
    Zoran, D., Weiss, Y.: From learning models of natural image patches to whole image restoration. In: IEEE International Conference on Computer Vision (ICCV 2011), pp. 479–486, November 2011Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CMLA, ENS Cachan, Université Paris-SaclayCachanFrance
  2. 2.CMLA, ENS Cachan, CNRS, Université Paris-SaclayCachanFrance

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