Machine Vision and Applications

, Volume 27, Issue 4, pp 437–446 | Cite as

Normalized filter pool for prior modeling of nature images

  • Yangang WangEmail author
  • Jinli Suo
  • Qionghai Dai
Original Paper


Markov random field (MRF), as one of special undirected graphs, is widely used in modeling priors of natural images. Targeting to learn better prior models from a given database, we explore the natural image statistics at different scales and build normalized filter pool, a kind of high-order MRF, for prior learning of nature images. The main contribution of the proposed model is that we construct a multi-scale MRF model through constraining the norms of filters in kernel space and integrate all the filtering responses in a unified framework. We formulate both learning and inference as constrained optimization problems and solve them using augmented Lagrange method. The experiment results demonstrate that the normalization of filters at different scales helps to achieve fast convergence in learning stage and obtain superior performance in image restoration, e.g., image denoising and image inpainting.


High-order Markov random field Multi-scale Normalized filter pool Image denoising Image inpainting 



We would like thank all the reviewers. This work was supported in part by NSFC (No. 61305026).


  1. 1.
    Baker, S., Scharstein, D., Lewis, J.P., Roth, S., Black, M.J., Szeliski, R.: A database and evaluation methodology for optical flow. Int. J. Comput. Vis. 92(1), 1–31 (2011)CrossRefGoogle Scholar
  2. 2.
    Black, M.J., Rangarajan, A.: On the unification of line processes, outlier rejection, and robust statistics with applications in early vision. Int. J. Comput. Vis. 19(1), 57–92 (1996)CrossRefGoogle Scholar
  3. 3.
    Blunsden, S., Atallah, L.: Investigating the effects of scale in MRF texture classification. In: VIE (2005)Google Scholar
  4. 4.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23(11), 1222–1239 (2001)CrossRefGoogle Scholar
  5. 5.
    Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: IEEE Conference on Computer Vision and Pattern Recognition (2005)Google Scholar
  6. 6.
    Chen, J., Nunez-Yanez, J., Achim, A.: Video super-resolution using generalized gaussian Markov random fields. IEEE Signal Process. Lett. 19(2), 63–66 (2012)CrossRefGoogle Scholar
  7. 7.
    Cho, T.S., Zitnick, C.L., Joshi, N., Kang, S.B., Szeliski, R., Freeman, W.T.: Image restoration by matching gradient distributions. IEEE Trans. Pattern Anal. Mach. Intell. 34(4), 683–694 (2012)CrossRefGoogle Scholar
  8. 8.
    Freeman, W.T., Pasztor, E.C., Carmichael, O.T.: Learning low-level vision. Int. J. Comput. Vis. 40(1), 25–47 (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Geman, S., Geman, D.: Stochastic relaxation, Gibbs distribution and Bayesian restoration of images. IEEE Trans Pattern Anal. Mach. Intell. 9(9), 721–741 (1984)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hinton, G.: Product of experts. In: ICANN (1999)Google Scholar
  11. 11.
    Huang, J.: Statistics of Nature Images and Models. PhD thesis, Brown University (2000)Google Scholar
  12. 12.
    Ishikawa, H.: Higher-order clique reduction in binary graph cut. In: IEEE Conference on Computer Vision and Pattern Recognition (2009)Google Scholar
  13. 13.
    Kohli, P., Kumar, M.P.: Energy minimization for linear envelope MRFs. In: IEEE Conference on Computer Vision and Pattern Recognition (2010)Google Scholar
  14. 14.
    Komodakis, N., Paragios, N.: (2009) Beyond pairwise energies: efficient optimization for higher-order MRFs. In: IEEE Conference on Computer Vision and Pattern RecognitionGoogle Scholar
  15. 15.
    Köster, U., Lindgren, J., Hyvärinen, A.: Estimating Markov random field potentials for natural images. In: ICA (2009)Google Scholar
  16. 16.
    Li, S.Z.: Markov Random Field Modeling in Image Analysis, 3rd edn. Springer, New York (2009)Google Scholar
  17. 17.
    Liu, C., Pizer, S.M., Joshi, S.: A Markov random field approach to multi-scale shape analysis. In: SSVM (2003)Google Scholar
  18. 18.
    Lyu, S., Simoncelli, E.P., Hughes, H.: Statistical modeling of images with fields of Gaussian scale mixtures. In: NIPS (2006)Google Scholar
  19. 19.
    Martin, D., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: International Conference on Computer Vision (2001)Google Scholar
  20. 20.
    Murphy, K.: (2014). Accessed 17 Apr 2014
  21. 21.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)Google Scholar
  22. 22.
    Paget, R., Longstaff, I.D.: Texture synthesis via a noncausal nonparametric multiscale Markov random field. IEEE Trans. Image Process. 7(6), 925–931 (1998)CrossRefGoogle Scholar
  23. 23.
    Paulsen, R.R., Brentzen, J.A., Larsen, R.: Markov random field surface reconstruction. IEEE Trans. Vis. Comput. Graph. 16, 636–646 (2010). doi: 10.1109/TVCG.2009.208 CrossRefGoogle Scholar
  24. 24.
    Portilla, J., Strela, V., Wainwright, M.J., Simoncelli, E.P.: Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans. Image Process. 12(11), 1338–1351 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Roth, S., Black, M.J.: On the spatial statistics of optical flow. In: International Conference on Computer Vision (2005)Google Scholar
  26. 26.
    Roth, S., Black, M.J.: Steerable random fields. In: International Conference on Computer Vision (2007)Google Scholar
  27. 27.
    Roth, S., Black, M.J.: Fields of experts. Int. J. Comput. Vis. 82(2), 205–229 (2009)CrossRefGoogle Scholar
  28. 28.
    Rother, C., Kohli, P., Feng, W., Jia, J.: (2009) Minimizing sparse higher order energy functions of discrete variables. In: IEEE Conference on Computer Vision and Pattern RecognitionGoogle Scholar
  29. 29.
    Ruderman, D.L.: The statistics of natural images. Netw. Comput. Neural Syst. 5, 517–548 (1994)CrossRefzbMATHGoogle Scholar
  30. 30.
    Ruderman, D.L.: Origins of scaling in natural images. Vis. Res. 37(23), 3385–3398 (1997)CrossRefGoogle Scholar
  31. 31.
    Schmidt, U., Gao, Q., Roth, S.: A generative perspective on MRFs in low-level vision. In: IEEE Conference on Computer Vision and Pattern Recognition (2010)Google Scholar
  32. 32.
    Tappen, M.F., Russell, B.C., Freeman, W.T.: Exploiting the sparse derivative prior for super-resolution and image demosaicing. In: International Workshop on Statistical and Computational Theories of Vision at ICCV (2003)Google Scholar
  33. 33.
    Teh, Y.W., Welling, M., Osindero, S., Hinton, G.E.: Energy-based models for sparse overcomplete representations. J. Mach. Learn. Res. 4, 1235–1260 (2003)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Wang, Y., Zhu, S.: Perceptual scale-space and its applications. Int. J. Comput. Vis. 80(1), 143–165 (2008)CrossRefGoogle Scholar
  35. 35.
    Weiss, Y., Freeman, W.T.: What makes a good model of natural images? In: IEEE Conference on Computer Vision and Pattern Recognition (2007)Google Scholar
  36. 36.
    Woodford, O.J., Reid, I.D., Torr, P.H.S., Fitzgibbon, A.W.: Fields of experts for image-based rendering. In: British Conference on Machine and Computer Vision (2006)Google Scholar
  37. 37.
    Zhu, S.C., Mumford, D.: Prior learning and Gibbs reaction–diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 19(11), 1236–1250 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Microsoft ResearchBeijingChina
  2. 2.Tsinghua UniversityBeijingChina

Personalised recommendations