Directional Total Generalized Variation Regularization for Impulse Noise Removal

  • Rasmus Dalgas KongskovEmail author
  • Yiqiu Dong
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)


A recently suggested regularization method, which combines directional information with total generalized variation (TGV), has been shown to be successful for restoring Gaussian noise corrupted images. We extend the use of this regularizer to impulse noise removal and demonstrate that using this regularizer for directional images is highly advantageous. In order to estimate directions in impulse noise corrupted images, which is much more challenging compared to Gaussian noise corrupted images, we introduce a new Fourier transform-based method. Numerical experiments show that this method is more robust with respect to noise and also more efficient than other direction estimation methods.


Directional total generalized variation Impulse noise Variational methods Regularization Image restoration 



The authors would like to thank the reviewers for their comments and suggestions, which has helped to improve this article. The work was supported by Advanced Grant 291405 from the European Research Council.


  1. 1.
    Bayram, I., Kamasak, M.E.: A directional total variation. Eur. Signal Process. Conf. 19(12), 265–269 (2012)Google Scholar
  2. 2.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cai, J.F., Chan, R.H., Nikolova, M.: Fast two-phase image deblurring under impulse noise. J. Math. Imaging Vis. 36(1), 46–53 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chan, R.H., Dong, Y., Hintermuller, M.: An efficient two-phase \(L^1\)-TV method for restoring blurred images with impulse noise. IEEE Trans. Image Process. 19(7), 1731–1739 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chan, R.H., Ho, C.W., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization. IEEE Trans. Image Process. 14(10), 1479–1485 (2005)Google Scholar
  7. 7.
    Dong, Y., Hintermüller, M., Neri, M.: An efficient primal-dual method for \(L^1\)TV image restoration. SIAM J. Imaging Sci. 2(4), 1168–1189 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Easley, G.R., Labate, D., Colonna, F.: Shearlet-based total variation diffusion for denoising. IEEE Trans. Image Process. 18(2), 260–268 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Estellers, V., Soatto, S., Bresson, X.: Adaptive regularization with the structure tensor. IEEE Trans. Image Process. 24(6), 1777–1790 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fei, X., Wei, Z., Xiao, L.: Iterative directional total variation refinement for compressive sensing image reconstruction. Signal Process. Lett. IEEE 20(11), 1070–1073 (2013)CrossRefGoogle Scholar
  11. 11.
    Ferstl, D., Reinbacher, C., Ranftl, R., Ruether, M., Bischof, H.: Image guided depth upsampling using anisotropic total generalized variation. Proc. IEEE International Conference on Computer Vision, pp. 993–1000 (2013)Google Scholar
  12. 12.
    Hafner, D., Schroers, C., Weickert, J.: Introducing maximal anisotropy into second order coupling models. Ger. Conf. Pattern Recognit. 9358, 79–90 (2015)MathSciNetGoogle Scholar
  13. 13.
    Jespersen, K.M., Zangenberg, J., Lowe, T., Withers, P.J., Mikkelsen, L.P.: Fatigue damage assessment of uni-directional non-crimp fabric reinforced polyester composite using X-ray computed tomography. Compos. Sci. Technol. 136, 94–103 (2016)CrossRefGoogle Scholar
  14. 14.
    Kongskov, R.D., Dong, Y., Knudsen, K.: Directional Total Generalized Variation Regularization. submitted (2017).
  15. 15.
    Lefkimmiatis, S., Roussos, A., Maragos, P., Unser, M.: Structure tensor total variation. SIAM J. Imaging Sci. 8(2), 1090–1122 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ma, L., Ng, M., Yu, J., Zeng, T.: Efficient box-constrained TV-type-l1 algorithms for restoring images with impulse noise. J. Comput. Math. 31(3), 249–270 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ma, L., Yu, J., Zeng, T.: Sparse representation prior and total variation-based image deblurring under impulse noise. SIAM J. Imaging Sci. 6(4), 2258–2284 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1–2), 99–120 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nikolova, M., Chan, R.H., Cai, J.F.: Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise. Inverse Probl. Imaging 2(2), 187–204 (2008)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ranftl, R., Gehrig, S., Pock, T., Bischof, H.: Pushing the limits of stereo using variational stereo estimation. IEEE Intell. Veh. Symp. Proc. 1, 401–407 (2012)Google Scholar
  23. 23.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 60(1–4), 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sandoghchi, S.R., Jasion, G.T., Wheeler, N.V., Jain, S., Lian, Z., Wooler, J.P., Boardman, R.P., Baddela, N.K., Chen, Y., Hayes, J.R., Fokoua, E.N., Bradley, T., Gray, D.R., Mousavi, S.M., Petrovich, M.N., Poletti, F., Richardson, D.J.: X-ray tomography for structural analysis of microstructured and multimaterial optical fibers and preforms. Opt. Express 22(21), 26181 (2014)CrossRefGoogle Scholar
  25. 25.
    Tikhonov, A.N., Arsenin, V.I.: Solutions of ill-posed problmes. Winston, Philadelphia (1977)zbMATHGoogle Scholar
  26. 26.
    Turgay, E., Akar, G.B.: Directionally adaptive super-resolution. In: 2009 16th IEEE International Conference on Image Processing vol. 1, 1201–1204 (2009)Google Scholar
  27. 27.
    Zhang, Z., Hartwig, G.: Relation of damping and fatigue damage of unidirectional fibre composites. Int. J. Fatigue 24(7), 713–718 (2002)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Applied Mathematics and Computer Science at Technical University of DenmarkKongens LyngbyDenmark

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