Advertisement

Journal of Mathematical Imaging and Vision

, Volume 61, Issue 4, pp 458–481 | Cite as

RNLp: Mixing Nonlocal and TV-Lp Methods to Remove Impulse Noise from Images

  • Julie Delon
  • Agnès DesolneuxEmail author
  • Camille Sutour
  • Agathe Viano
Article
  • 144 Downloads

Abstract

We propose a new variational framework to remove random-valued impulse noise from images. This framework combines, in the same energy, a nonlocal \(L^p\) data term and a total variation regularization term. The nonlocal \(L^p\) term is a weighted \(L^p\) distance between pixels, where the weights depend on a robust distance between patches centered at the pixels. In a first part, we study the theoretical properties of the proposed energy, and we show how it is related to classical denoising models for extreme choices of the parameters. In a second part, after having explained how to numerically find a minimizer of the energy thanks to primal-dual approaches, we show extensive denoising experiments on various images and noise intensities. The denoising performance of the proposed methods is on par with state-of-the-art approaches, and the remarkable fact is that, unlike other successful variational approaches for impulse noise removal, they do not rely on a noise detector.

Keywords

Image denoising Impulse noise Variational methods Patch-based methods Convex optimization 

Notes

References

  1. 1.
    Alliney, S.: Digital filters as absolute norm regularizers. IEEE Trans. Signal Process. 40(6), 1548–1562 (1992)zbMATHGoogle Scholar
  2. 2.
    Awate, S., Whitaker, R.: Unsupervised, information-theoretic, adaptive image filtering for image restoration. IEEE Trans. Pattern Anal. Mach. Intell. 28(3), 364–376 (2006)Google Scholar
  3. 3.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J., et al.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends\({\textregistered }\) Mach. Learn. 3(1), 1–122 (2011)Google Scholar
  4. 4.
    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
  5. 5.
    Candes, E.J., Wakin, M.B., Boyd, S.P.: Enhancing sparsity by reweighted l1 minimization. J. Fourier Anal. Appl. 14(5), 877–905 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chambolle, A., Cremers, D., Pock, T.: A convex approach to minimal partitions. SIAM J. Imaging Sci. 5(4), 1113–1158 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numer. 25, 161–319 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chan, R., Hu, C., Nikolova, M.: An iterative procedure for removing random-valued impulse noise. IEEE Signal Process. Lett. 11(12), 921–924 (2004)Google Scholar
  10. 10.
    Chaudhury, K.N., Singer, A.: Non-local Euclidean medians. IEEE Signal Process. Lett. 19(11), 745–748 (2012)Google Scholar
  11. 11.
    Chen, S., Donoho, D., Saunders, M.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chen, T., Wu, H.: Adaptive impulse detection using center-weighted median filters. IEEE Signal Process. Lett. 8(1), 1–3 (2001)Google Scholar
  13. 13.
    Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Fixed-point algorithms for inverse problems in science and engineering, pp. 185–212. Springer (2011)Google Scholar
  14. 14.
    Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158(2), 460–479 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Condat, L.: Discrete total variation: new definition and minimization. SIAM J. Imaging Sci. 10(3), 1258–1290 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Deledalle, C.A., Denis, L., Tupin, F.: Iterative weighted maximum likelihood denoising with probabilistic patch-based weights. IEEE Trans. Image Process. 18(12), 2661–2672 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Deledalle, C.A., Tupin, F., Denis, L.: Poisson NL means: unsupervised non local means for Poisson noise. In: 2010 IEEE Int. Conf. Image Process. (ICIP). IEEE Signal Process Soc (2010)Google Scholar
  18. 18.
    Delon, J., Desolneux, A.: A patch-based approach for removing mixed Gaussian-impulse noise. SIAM J. Imaging Sci. 6(2), 1140–1174 (2013).  https://doi.org/10.1137/120885000 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Delon, J., Desolneux, A., Guillemot, T.: PARIGI: a patch-based approach to remove impulse-Gaussian noise from images. Image Process. On Line 6, 130–154 (2016).  https://doi.org/10.5201/ipol.2016.161 MathSciNetGoogle Scholar
  20. 20.
    Dong, Y., Chan, R., Xu, S.: A detection statistic for random-valued impulse noise. IEEE Trans. Image Process. 16(4), 1112–1120 (2007)MathSciNetGoogle Scholar
  21. 21.
    Dong, Y., Xu, S.: A new directional weighted median filter for removal of random-valued impulse noise. IEEE Signal Process. Lett. 14(3), 193–196 (2007).  https://doi.org/10.1109/LSP.2006.884014 Google Scholar
  22. 22.
    Duval, V.: Variational and non-local methods in image processing: a geometric study. Ph.D. thesis, Télécom ParisTech (2011)Google Scholar
  23. 23.
    Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348–1360 (2001)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Foucart, S., Lai, M.J.: Sparsest solutions of underdetermined linear systems via lq-minimization for \(0< q <1\). Appl. Comput. Harmon. Anal. 26(3), 395–407 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Garnett, R., Huegerich, T., Chui, C., He, W.: A universal noise removal algorithm with an impulse detector. IEEE Trans. Image Process. 14(11), 1747–1754 (2005)Google Scholar
  26. 26.
    Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. Multiscale Model. Simul. 6(2), 595–630 (2007)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Glowinski, R., Marroco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de dirichlet non linéaires. Revue française d’automatique, informatique, recherche opérationnelle. Analyse numérique 9(2), 41–76 (1975)zbMATHGoogle Scholar
  29. 29.
    Goldstein, T., Osher, S.: The split bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Holler, M., Kunisch, K.: On infimal convolution of tv-type functionals and applications to video and image reconstruction. SIAM J. Imaging Sci. 7(4), 2258–2300 (2014)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Hu, H., Li, B., Liu, Q.: Removing mixture of gaussian and impulse noise by patch-based weighted means. J. Sci. Comput. 67(1), 103–129 (2016).  https://doi.org/10.1007/s10915-015-0073-9 MathSciNetzbMATHGoogle Scholar
  32. 32.
    Huang, T., Dong, W., Xie, X., Shi, G., Bai, X.: Mixed noise removal via laplacian scale mixture modeling and nonlocal low-rank approximation. IEEE Trans. Image Process. 26(7), 3171–3186 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Huang, Y., Ng, M., Wen, Y.: Fast image restoration methods for impulse and Gaussian noises removal. IEEE Signal Process. Lett. 16(6), 457–460 (2009)Google Scholar
  34. 34.
    Ishikawa, H.: Exact optimization for Markov random fields with convex priors. IEEE Trans. Pattern Anal. Mach. Intell. 25(10), 1333–1336 (2003)Google Scholar
  35. 35.
    Kervrann, C., Boulanger, J.: Optimal spatial adaptation for patch-based image denoising. IEEE Trans. Image Process. 15(10), 2866–2878 (2006)Google Scholar
  36. 36.
    Kervrann, C., Boulanger, J., Coupé, P.: Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal. In: Scale Space and Variational Methods in Computer Vision (SSVM), pp. 520–532. Springer (2007)Google Scholar
  37. 37.
    Ko, S.J., Lee, Y.H.: Center weighted median filters and their applications to image enhancement. IEEE Trans. Circuits Syst. 38(9), 984–993 (1991)Google Scholar
  38. 38.
    Kolmogorov, V., Zabin, R.: What energy functions can be minimized via graph cuts? IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 147–159 (2004)Google Scholar
  39. 39.
    Lebrun, M., Buades, A., Morel, J.M.: A nonlocal Bayesian image denoising algorithm. SIAM J. Imaging Sci. 6(3), 1665–1688 (2013).  https://doi.org/10.1137/120874989 MathSciNetzbMATHGoogle Scholar
  40. 40.
    Lebrun, M., Colom, M., Buades, A., Morel, J.M.: Secrets of image denoising cuisine. Acta Numer. 21, 475–576 (2012)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Li, B., Liu, Q., Xu, J., Luo, X.: A new method for removing mixed noises. Sci. China Inf. Sci. 54(1), 51–59 (2011)zbMATHGoogle Scholar
  42. 42.
    Li, Y., Osher, S.: A new median formula with applications to PDE based denoising. Commun. Math. Sci. 7(3), 741–753 (2009)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)zbMATHGoogle Scholar
  44. 44.
    Motta, G., Ordentlich, E., Ramírez, I., Seroussi, G., Weinberger, M.J.: The dude framework for continuous tone image denoising. In: IEEE International Conference on Image Processing (ICIP) 2005, vol. 3, pp. III–345. IEEE (2005)Google Scholar
  45. 45.
    Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24(2), 227–234 (1995)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers. SIAM J. Numer. Anal. 40(3), 965–994 (2002).  https://doi.org/10.1137/s0036142901389165 MathSciNetzbMATHGoogle Scholar
  47. 47.
    Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1), 99–120 (2004)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: 2011 IEEE International Conference on Computer Vision (ICCV), pp. 1762–1769. IEEE (2011)Google Scholar
  49. 49.
    Pock, T., Cremers, D., Bischof, H., Chambolle, A.: Global solutions of variational models with convex regularization. SIAM J. Imaging Sci. 3(4), 1122–1145 (2010)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Pock, T., Schoenemann, T., Graber, G., Bischof, H., Cremers, D.: A convex formulation of continuous multi-label problems. Comput. Vis. ECCV 2008, 792–805 (2008)Google Scholar
  51. 51.
    Pratt, W.K.: Median filtering. Technical report, Image Proc. Inst., Univ. Southern California (1975)Google Scholar
  52. 52.
    Sutour, C., Deledalle, C.A., Aujol, J.F.: Adaptive regularization of the NL-means: application to image and video denoising. IEEE Trans. Image Process. 23(8), 3506–3521 (2014)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 58, 267–288 (1996)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Wang, Y.Q., Morel, J.M.: SURE guided Gaussian mixture image denoising. SIAM J. Imaging Sci. 6(2), 999–1034 (2013).  https://doi.org/10.1137/120901131 MathSciNetzbMATHGoogle Scholar
  55. 55.
    Xiao, Y., Zeng, T., Yu, J., Ng, M.K.: Restoration of images corrupted by mixed Gaussian-impulse noise via \(\text{ l }_{{1}}-\text{ l }_{{0}}\) minimization. Pattern Recognit. 44(8), 1708–1720 (2011).  https://doi.org/10.1016/j.patcog.2011.02.002 zbMATHGoogle Scholar
  56. 56.
    Xiong, B., Yin, Z.: A universal denoising framework with a new impulse detector and nonlocal means. IEEE Trans. Image Process. 21(4), 1663–1675 (2012).  https://doi.org/10.1109/TIP.2011.2172804 MathSciNetzbMATHGoogle Scholar
  57. 57.
    Yan, M.: Restoration of images corrupted by impulse noise and mixed Gaussian impulse noise using blind inpainting. SIAM J. Imaging Sci. 6(3), 1227–1245 (2013)MathSciNetzbMATHGoogle Scholar
  58. 58.
    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), 2481–99 (2012).  https://doi.org/10.1109/TIP.2011.2176743 MathSciNetzbMATHGoogle Scholar
  59. 59.
    Yuan, G., Ghanem, B.: l0tv: a new method for image restoration in the presence of impulse noise. In: 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2015)Google Scholar
  60. 60.
    Zhou, Y., Ye, Z., Xiao, Y.: A restoration algorithm for images contaminated by mixed Gaussian plus random-valued impulse noise. J. Vis. Commun. Image Represent. 24(3), 283–294 (2013)Google Scholar
  61. 61.
    Zoran, D., Weiss, Y.: From learning models of natural image patches to whole image restoration. In: 2011 Int. Conf. Comput. Vis., pp. 479–486. IEEE (2011).  https://doi.org/10.1109/ICCV.2011.6126278

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.MAP5Université Paris DescartesParisFrance
  2. 2.CMLA, CNRSENS Paris-SaclayCachanFrance
  3. 3.Deloitte FranceParisFrance

Personalised recommendations