Journal of Computer Science and Technology

, Volume 29, Issue 6, pp 1026–1037 | Cite as

A Two-Step Regularization Framework for Non-Local Means

Regular Paper

Abstract

As an effective patch-based denoising method, non-local means (NLM) method achieves favorable denoising performance over its local counterparts and has drawn wide attention in image processing community. The implementation of NLM can formally be decomposed into two sequential steps, i.e., computing the weights and using the weights to compute the weighted means. In the first step, the weights can be obtained by solving a regularized optimization. And in the second step, the means can be obtained by solving a weighted least squares problem. Motivated by such observations, we establish a two-step regularization framework for NLM in this paper. Meanwhile, using the framework, we reinterpret several non-local filters in the unified view. Further, taking the framework as a design platform, we develop a novel non-local median filter for removing salt-pepper noise with encouraging experimental results.

Keywords

non-local means non-local median framework image denoising regularization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

11390_2014_1487_MOESM1_ESM.pdf (114 kb)
ESM 1(PDF 114 kb)

References

  1. [1]
    Buades A, Coll B, Morel J M. A non-local algorithm for image denoising. In Proc. IEEE Computer Society Conference Computer Vision and Pattern Recognition (CVPR), June 2005, Vol.2, pp.60–65.Google Scholar
  2. [2]
    Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1990, 12(7): 629–639.CrossRefGoogle Scholar
  3. [3]
    Yaroslavsky L P. Digital Picture Processing: An Introduction (1st edition). Springer-Verlag, 1985.Google Scholar
  4. [4]
    Smith S M, Brady J M. SUSAN | A new approach to low level image processing. Int. J. Computer Vision, 1997, 23(1): 45–78.CrossRefGoogle Scholar
  5. [5]
    Tomasi C, Manduch R. Bilateral filtering for gray and color images. In Proc. the 6th International Conference on Computer Vision (ICCV), Jan. 1998, pp.839–846.Google Scholar
  6. [6]
    Efros A A, Leung T K. Texture synthesis by non-parametric sampling. In Proc. the 7th International Conference on Computer Vision (ICCV), Sept. 1999, pp.1033–1038.Google Scholar
  7. [7]
    Buades A, Coll B, Morel J M. Image denoising methods. A new nonlocal principle. SIAM Review: Multiscale Modeling and Simulation, 2010, 52(1): 113–147.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    Awate S P, Whitaker R T. Unsupervised, information-theoretic, adaptive image filtering for image restoration.IEEE Transactions on Pattern Analysis and Machine Intelligence, 2006, 28(3): 364–376.CrossRefGoogle Scholar
  9. [9]
    Dabov K, Foi A, Katkovnik V, Egiazarian K. Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Transactions on Image Processing, 2007, 16(8): 2080–2095.CrossRefMathSciNetGoogle Scholar
  10. [10]
    Aharon M, Elad M, Bruckstein A. K-SVD: An algorithm for designing of overcomplete dictionaries for sparse representation. IEEE Trans. Signal Processing, 2006, 54(11): 4311–4322.CrossRefGoogle Scholar
  11. [11]
    Peyré G, Bougleux S, Cohen L. Non-local regularization of inverse problems. Inverse Problems and Imaging, 2011, 5(2): 511–530.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    Facciolo G, Arias P, Caselles V, Sapiro G. Exemplar-based interpolation of sparsely sampled images. In Proc. the 7th Int. Conf. Energy Minimization Methods in Computer Vision and Pattern Recognition, Aug. 2009, pp.331–344.Google Scholar
  13. [13]
    Arias P, Caselles V, Sapiro G. A variational framework for non-local image inpainting. In Proc. the 7th Int. Conf. Energy Minimization Methods in Computer Vision and Pattern Recognition, Aug. 2009, pp.345–358.Google Scholar
  14. [14]
    Peyré G, Bougleux S, Cohen L. Non-local regularization of inverse problems. In Lecture Notes in Computer Science 5304, Forsyth D, Torr P, Zisserman A (eds.), Springer-Verlag, 2008, pp.57–68.Google Scholar
  15. [15]
    Chaudhury K N, Singer A. Non-local Euclidean medians. IEEE Signal Processing Letters, 2012, 19(11): 745–748.CrossRefGoogle Scholar
  16. [16]
    Zhang L, Qiao L, Chen S. Graph-optimized locality preserving projections. Pattern Recognition, 2010, 43(6): 1993–2002.CrossRefMATHGoogle Scholar
  17. [17]
    Dowson N, Salvado O. Hashed non-local means for rapid image filtering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2011, 33(3): 485–499.CrossRefGoogle Scholar
  18. [18]
    Vignesh R, Byung T O, Kuo C C J. Fast non-local means (NLM) computation with probabilistic early termination. IEEE Signal Processing Letters, 2010, 17(3): 277–280.CrossRefGoogle Scholar
  19. [19]
    Sun Z, Chen S. Modifying NL-means to a universal filter. Optics Communications, 2012, 285(24): 4918–4926.CrossRefGoogle Scholar
  20. [20]
    Malerba D, Esposito F, Gioviale V, Tamma V. Comparing dissimilarity measures for symbolic data analysis. In Proc. Techniques and Technologies for Statistics - Exchange of Technology and Know-How, June 2001, pp.473–481.Google Scholar
  21. [21]
    Liu H, Song D, Stefan R, Hu R, Victoria U. Comparing dissimilarity measures for content-based image retrieval. In Lecture Notes in Computer Science 4993, Li H, Liu T, Ma W Y et al. (eds.), Springer-Verlag, 2008, pp.44–50.Google Scholar
  22. [22]
    Sun J, Zhao W, Xue J, Shen Z, Shen Y. Clustering with feature order preferences. In Lecture Notes in Computer Science 5351, Ho T B, Zhou Z H (eds.), Springer-Verlag, 2008, pp.382–393.Google Scholar
  23. [23]
    Luo P, Zhan G, He Q, Shi Z, Lu K. On defining partition entropy by inequalities. IEEE Transactions on Information Theory, 2007, 53(9): 3233–3239.CrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    Sun J, Zhao W, Xue J, Shen Z, Shen Y. Clustering with feature order preferences. Intelligent Data Analysis, 2010, 14: 479–495.Google Scholar
  25. [25]
    Huber P J, Ronchetti E M. Robust Statistics (2nd edition). New Jersey: John Wiley & Sons, 2009.Google Scholar
  26. [26]
    Sun Z, Chen S. Analysis of non-local Euclidean medians and its improvement. IEEE Signal Processing Letters, 2013, 20(4): 303–306.CrossRefGoogle Scholar
  27. [27]
    Cai J F, Chan R H, Nikolova M. Fast two-phase image deblurring under impulse noise. Journal of Mathematical Imaging and Vision, 2010, 36(1): 46–53.CrossRefMathSciNetGoogle Scholar
  28. [28]
    Brownrigg D R K. The weighted median filter. Communications of the ACM, 1984, 27(8): 807–818.CrossRefGoogle Scholar
  29. [29]
    Hwang H, Haddad R A. Adaptive median filters: New algorithms and results. IEEE Transactions on Image Processing, 1995, 4(4): 499–502.CrossRefGoogle Scholar
  30. [30]
    Bovik A. Handbook of Image and Video Processing. Academic Press, 2000.Google Scholar
  31. [31]
    Rudin L I, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D Nonlinear Phenomena, 1992, 60(1/2/3/4): 259–268.Google Scholar
  32. [32]
    Yang R, Yin L, Gabbouj M, Astola J, Neuvo T. Optimal weighted median filtering under structural constraints. IEEE Transactions on Signal Processing, 1995, 43(3): 591–604.CrossRefGoogle Scholar
  33. [33]
    Wang G, Qi J. Penalized likelihood PET image reconstruction using patch-based edge-preserving regularization. IEEE Transactions on Medical Imaging, 2012, 31(12): 2194–2204.CrossRefGoogle Scholar
  34. [34]
    Yang Z, Jacob M. Nonlocal regularization of inverse problems: A unified variational framework. IEEE Transactions on Image Processing, 2013, 22(8): 3192–3203.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics ScienceLiaocheng UniversityLiaochengChina
  2. 2.College of Computer Science and TechnologyNanjing University of Aeronautics & AstronauticsNanjingChina

Personalised recommendations