Total generalized variation and wavelet frame-based adaptive image restoration algorithm

  • Xinwu LiuEmail author
Original Article


To achieve superior image reconstruction, this paper investigates a hybrid regularizers model for image denoising and deblurring. This approach closely incorporates the advantages of the total generalized variation and wavelet frame-based methods. Computationally, a highly efficient alternating minimization algorithm containing no inner iterations is introduced in detail, which synchronously restores the degraded image and automatically estimates the regularization parameter based on Morozov’s discrepancy principle. Illustrationally, we demonstrate that our proposed strategy significantly outperforms several current state-of-the-art numerical methods and closely matches the performance of human vision in solving the image deconvolution problem, with respect to restoration accuracy, staircase artifacts suppression and features preservation.


Image restoration Total generalized variation Wavelet frame Alternating minimization method Discrepancy principle 



The author would like to thank the editors and anonymous reviewers for their constructive comments and valuable suggestions.


This work was supported by National Natural Science Foundation of China (61402166) and Hunan Provincial Natural Science Foundation of China (14JJ3105).

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.


  1. 1.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1), 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aujol, J.F., Gilboa, G.: Constrained and SNR-based solutions for TV-Hilbert space image denoising. J. Math. Imaging Vis. 26(1), 217–237 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Weiss, P., Blanc-Féraud, L., Aubert, G.: Efficient schemes for total variation minimization under constraints in image processing. SIAM J. Sci. Comput. 31(3), 2047–2080 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ng, M.K., Weiss, P., Yuan, X.: Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods. SIAM J. Sci. Comput. 32(5), 2710–2736 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Wen, Y.W., Chan, R.H.: Parameter selection for total-variation-based image restoration using discrepancy principle. IEEE Trans. Image Process. 21(4), 1770–1781 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    He, C., Hu, C., Zhang, W., Shi, B.: A fast adaptive parameter estimation for total variation image restoration. IEEE Trans. Image Process. 23(12), 4954–4967 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21(2), 215–223 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Galatsanos, N.P., Katsaggelos, A.K.: Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE Trans. Image Process. 1(3), 322–336 (1992)CrossRefGoogle Scholar
  9. 9.
    Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34(4), 561–580 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Babacan, S.D., Molina, R., Katsaggelos, A.K.: Parameter estimation in TV image restoration using variational distribution approximation. IEEE Trans. Image Process. 17(3), 326–339 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Babacan, S.D., Molina, R., Katsaggelos, A.K.: Variational Bayesian blind deconvolution using a total variation prior. IEEE Trans. Image Process. 18(1), 12–26 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation (TGV) for MRI. Magn. Reson. Med. 65(2), 480–491 (2011)CrossRefGoogle Scholar
  14. 14.
    Bredies, K., Dong, Y., Hintermüller, M.: Spatially dependent regularization parameter selection in total generalized variation models for image restoration. Int. J. Comput. Math. 90(1), 109–123 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bredies, K., Holler, M.: A TGV regularized wavelet based zooming model. Lect. Notes Comput. Sci. 7893, 149–160 (2013)CrossRefGoogle Scholar
  16. 16.
    Valkonen, T., Bredies, K., Knoll, F.: Total generalized variation in diffusion tensor imaging. SIAM J. Imaging Sci. 6(1), 487–525 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guo, W., Qin, J., Yin, W.: A new detail-preserving regularity scheme. SIAM J. Imaging Sci. 7(2), 1309–1334 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    He, C., Hu, C., Yang, X., He, H., Zhang, Q.: An adaptive total generalized variation model with augmented Lagrangian method for image denoising. Math. Probl. Eng. 2014, 157893 (2014)Google Scholar
  19. 19.
    Liu, X.: Augmented Lagrangian method for total generalized variation based Poissonian image restoration. Comput. Math. Appl. 71(8), 1694–1705 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cai, J.F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cai, J.F., Dong, B., Osher, S., Shen, Z.: Image restoration: total variation, wavelet frames, and beyond. J. Am. Math. Soc. 25(4), 1033–1089 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, Y., Kingsbury, N.: Improved bounds for subband-adaptive iterative shrinkage/thresholding algorithms. IEEE Trans. Image Process. 22(4), 1373–1381 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    He, L., Wang, Y., Xiang, Z.: Wavelet frame-based image restoration using sparsity, nonlocal, and support prior of frame coefficients. Vis. Comput. (2017). Google Scholar
  24. 24.
    Wang, C., Yang, J.: Poisson noise removal of images on graphs using tight wavelet frames. Vis. Comput. (2017). Google Scholar
  25. 25.
    Bredies, K., Valkonen, T.: Inverse problems with second-order total generalized variation constraints. In: Proceedings of SampTA 2011, 9th International Conference on Sampling Theory and Applications (2011)Google Scholar
  26. 26.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)CrossRefzbMATHGoogle Scholar
  27. 27.
    Chen, D.-Q., Cheng, L.-Z.: Deconvolving Poissonian images by a novel hybrid variational model. J. Vis. Commun. Image R. 22(7), 643–652 (2011)CrossRefGoogle Scholar
  28. 28.
    Goldstein, T., Osher, S.: The split Bregman algorithm for L1 regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Setzer, S.: Operator splittings, Bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis. 92(3), 265–280 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Liu, X.: Alternating minimization method for image restoration corrupted by impulse noise. Multimed. Tools Appl. 76(10), 12505–12516 (2017)CrossRefGoogle Scholar
  31. 31.
    Zha, Z., Liu, X., Zhang, X., Chen, Y., Tang, L., Bai, Y., Wang, Q., Shang, Z.: Compressed sensing image reconstruction via adaptive sparse nonlocal regularization. Vis. Comput. 34(1), 117–137 (2018)CrossRefGoogle Scholar
  32. 32.
    Ng, M.K., Chan, R.H., Tang, W.-C.: A fast algorithm for deblurring models with Neumann boundary conditions. SIAM J. Sci. Comput. 21(3), 851–866 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang, Y., Yin, W., Zhang, Y.: A fast algorithm for image deblurring with total variation regularization. CAAM Technical Report TR07-10 (2007)Google Scholar
  34. 34.
    Bertsekas, D., Tsitsiklis, J.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont (1997)zbMATHGoogle Scholar
  35. 35.
    He, B., Liao, L.Z., Han, D., Yang, H.: A new inexact alternating directions method for monotone variational inequalities. Math. Program. 92(1), 103–118 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)CrossRefzbMATHGoogle Scholar
  37. 37.
    Hajiaboli, M.R.: A self-governing fourth-order nonlinear diffusion filter for image noise removal. IPSJ Trans. Comput. Vision Appl. 2, 94–103 (2010)CrossRefGoogle Scholar
  38. 38.
    Zhang, L., Zhang, L., Mou, X., Zhang, D.: FSIM: a feature similarity index for image qualtiy assessment. IEEE Trans. Image Process. 20(8), 2378–2386 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Selesnick, I.W., Abdelnour, A.F.: Symmetric wavelet tight frames with two generators. Appl. Comput. Harmon. Anal. 17(2), 211–225 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Burger, H.C., Schuler, C.J., Harmeling, S.: Image denoising: can plain neural networks compete with BM3D? IEEE Conf. Comput. Vis. Pattern Recogn. 157(10), 2392–2399 (2012)Google Scholar
  41. 41.
    Zhang, K., Chen, Y., Chen, Y., Meng, D., Zhang, L.: Beyond a Gaussian denoiser: residual learning of deep CNN for image denoising. IEEE Trans. Image Process. 26(7), 3142–3155 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Computational ScienceHunan University of Science and TechnologyXiangtanChina

Personalised recommendations