Skip to main content
SpringerLink
Log in

The ℓ2,p regularized total variation with overlapping group sparsity prior for image restoration with impulse noise

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we consider ℓ2,p(0 < p < 1) regularized total variation with overlapping group sparsity prior for image restoration with impulse noise. The proposed prior inherits the advantages of a total variation regularizer to preserve edges and reduce the staircase effect, meanwhile promoting group-level sparseness. Since the new model is nonsmooth and nonconvex, we utilize the proximal alternate minimization method to solve it by drawing support from the half-quadratic scheme to deal with the constituted ℓ2-ℓ2,p subproblems. In addition, we also provide the convergence analysis for the used numerical methods. Experimental results demonstrate that the new approach outperforms representative gradient-based methods in terms of both visual perception and numerical indexes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Chan, R., Tao, M., Yuan, X.-M.: Constrained total variational deblurring models fast algorithm based on alternating direction methods of multipliers. SIAM J. Imaging Sci. 6(1), 680–697 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Liu, G., Huang, T.-Z., Liu, J., Lv, X.-G.: Total variation with overlapping group sparsity for image deblurring under impulse noise. PLoS One 10 (4), 0122562 (2015)

    Google Scholar 

  3. Wu, C., Zhang, J., Tai, X.-C.: Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Probl. Imaging 5(1), 237–261 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Zeng, C., Wu, C.-L., Jia, R.: Non-Lipschitz models for image restoration with impulse noise removal. SIAM J. Imaging Sci. 12(1), 420–458 (2019)

    Article  MATH  MathSciNet  Google Scholar 

  5. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)

    Article  Google Scholar 

  6. Chan, R., Liang, H.: Half-quadratic algorithm for ℓq-ℓp problems with application to TV-ℓ1 image restoration and compressive sensing. Global Optim. Methods, 78–103 (2014)

  7. Geman, D., Yang, C.: Nonlinear image recovery with half-quadratic regularization. IEEE Trans. Image Process. 4(7), 932–946 (1995)

    Article  Google Scholar 

  8. Nikolova, M., Chan, T.: The equivalence of the half-quadratic minimization and the gradient linearization iteration. IEEE Trans. Image Process. 16 (6), 1623–1627 (2007)

    Article  MathSciNet  Google Scholar 

  9. Chan, T., Zhou, H., Chan, R.: Continuation method for total variation denoising problem. International Symposium on Optical Science, Engineering, and Instrumentation (1995)

  10. Pang, Z.-F., Zhang, H.-L., Luo, S.-S., Zeng, T.-Y.: Image denoising based on the adaptive weighted TVp regularization. Signal Process. 167, 107325 (2020)

    Article  Google Scholar 

  11. Ng, M.N., Weiss, P., Yuan, X.-M.: Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods. SIAM J. Sci. Comput. 32(5), 2710–2736 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. He, B.-S., Yang, H.: Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. Operations Research Letters 23, 151–161 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Glowinski, R.: Lectures on Numerical Methods for Nonlinear Variational Problems. Springer, New York (2008)

    MATH  Google Scholar 

  14. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite-element approximations. Comput. Math. Appl. 2(1), 17–40 (1976)

    Article  MATH  Google Scholar 

  15. Attouch, H., Bolte, J., Svaiter, F.B.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program 137(1-2), 91–129 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program 146, 459–494 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Deng, L.-J., Guo, W.-H., Huang, T.-Z.: Single-image super-resolution via an iterative reproducing kernel Hilbert space method. IEEE Trans. Circ. Syst. Video Technol. 26(11), 2001–2014 (2016)

    Article  Google Scholar 

  18. Tappen, M.F., Russell, B.C., Freeman, W.T.: Exploiting the sparse derivative prior for super-resolution and image demosaicing. In: Proceedings of the IEEE Workshop on Statistical and Computational Theories of Vision (2003)

  19. Burger, M., He, L., Schönlieb, C.B.: Cahn-Hilliard inpainting and a generalization for grayvalue images. SIAM J. Imaging Sci. 2(4), 1129–1167 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Li, F., Lv, X.-G.: A decoupled method for image inpainting with patch-based low rank regulariztion. Appl. Math. Comput. 314, 334–348 (2017)

    MATH  MathSciNet  Google Scholar 

  21. Wang, S., Guo, W.-H., Huang, T.-Z., Raskuttic, G.: Image inpainting using reproducing kernel Hilbert space and Heaviside functions. J. Comput. Appl. Math 311, 551–564 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ji, T.-Y., Huang, T.-Z., Zhao, X.-L., Ma, T.-H., Liu, G.: Tensor completion using total variation and low-rank matrix factorization. Inf. Sci. 326, 243–257 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  23. Krishnan, D., Fergus, R.: Fast image deconvolution using hyper-Laplacian priors. In: Neural Information Processing Systems (2009)

  24. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom 60(1-4), 259–268 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Steidl, G.: A note on the dual treatment of higher-order regularization functionals. Computing 76(1), 135–148 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wang, Y.-L., Yang, J.-F., Yin, W.-T., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math 76, 167–188 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. Chan, T.F., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lysaker, M., Lundervold, A., Tai, X.-C.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)

    Article  MATH  Google Scholar 

  30. Lysaker, M., Tai, X.-C.: Iterative image restoration combining total variation minimization and a second-order functional. Int. J. Comput. Vis. 66(1), 5–18 (2006)

    Article  MATH  Google Scholar 

  31. Wang, S., Huang, T.-Z., Zhao, X.-L., Mei, J.-J., Huang, J.: Speckle noise removal in ultrasound images by first- and second-order total variation. Numer. Algorithms 78(2), 513–533 (2018)

    Article  MATH  MathSciNet  Google Scholar 

  32. Cai, J.-F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zhang, K., Zuo, W.-M., Chen, Y.-J., Meng, D.-Y., Zhang, L.: Beyond a gaussian denoiser: residual learning of deep CNN for image denoising. IEEE Trans. Image Process. 26, 3142–3155 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  34. Selesnick, I.W., Chen, P.Y.: Total variation denoising with overlapping group sparsity. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, pp 5696–5700 (2013)

  35. Liu, J., Huang, T.-Z., Selesnick, I.W., Lv, X.-G., Chen, P.Y.: Image restoration using total variation with overlapping group sparsity. Inf. Sci. 295, 232–246 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ding, M., Huang, T.-Z., Wang, S., Mei, J.-J., Zhao, X.-L.: Total variation with overlapping group sparsity for deblurring images under Cauchy noise. Appl. Math. Comput. 341, 128–147 (2019)

    MATH  MathSciNet  Google Scholar 

  37. Nikolova, M., Ng, M.K., Zhang, S.: Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 1(1), 2–25 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  38. Nikolova, M., Ng, M.K., Tam, C.P.: Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Trans. Image Process. 19(12), 3073–3088 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  39. Yang, J., Wright, J., Huang, T.-S.: Image super-resolution via sparse representation. IEEE Trans. Image Process. 19(11), 2861–2873 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  40. Jing, X.-Y., Zhu, X., Wu, F.: Super-resolution person re-identification with semi-coupled low-rank discriminant dictionary learning. IEEE Trans. Image Process. 26(3), 1363–1378 (2017)

    Article  MathSciNet  Google Scholar 

  41. Jung, M., Kang, M.: Efficient nonsmooth nonconvex optimization for image restoration and segmentation. J. Sci. Comput. 62(2), 336–370 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  42. Oh, S., Woo, H., Yun, S.: Non-convex hybrid total variation for image denoising. J. Vis. Commun. Image Represent 24(3), 332–344 (2013)

    Article  Google Scholar 

  43. Lu, C., Tang, J., Yan, S.: Nonconvex nonsmooth low-rank minimization via iteratively reweighted nuclear norm. IEEE Trans. Image Process. 25(2), 829–839 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  44. Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6(6), 721–741 (1984)

    Article  MATH  Google Scholar 

  45. Feng, X., Yan, S., Wu, C.-L.: The ℓ2,q regularized group sparse optimization: lower bound theory, recovery bound and algorithms. Appl. Comput. Harmon. Anal. 49, 381–414 (2020)

    Article  MATH  MathSciNet  Google Scholar 

  46. Chen, P.-Y., Selesnick, I.W.: Group sparse signal denoising: non-convex regularization, convex optimization. IEEE Trans. Signal Process. 62 (13), 3464–3478 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  47. Xu, Y.-Y., Yin, W.-Tao.: A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM J. Imaging Sci. 6(3), 1758–1789 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  48. Alliney, S.: Digital filters as absolute norm regularizers. IEEE Trans. Signal Process. 40(6), 1548–1562 (1992)

    Article  MATH  Google Scholar 

  49. Eng, H.L., Ma, K.K.: Noise adaptive soft-switching median filter. IEEE Trans. Image Process. 10(2), 242–251 (2001)

    Article  MATH  Google Scholar 

  50. Hwang, H., Haddad, R.A.: Adaptive median filters: new algorithms and results. IEEE Trans. Image Process. 4(4), 499–502 (1995)

    Article  Google Scholar 

  51. Tang, L.-M., Ren, Y.-J., Fang, Z., He, C.-J.: A generalized hybrid nonconvex variational regularization model for staircase reduction in image restoration. Neurocomputing 359, 15–3 (2019)

    Article  Google Scholar 

  52. Zhang, H.-L., Tang, L.-M., Fang, Z., Xiang, C.-C., Li, C.-Y.: Nonconvex and nonsmooth total generalized variation model for image restoration. Signal Process. 143, 68–85 (2018)

    Article  Google Scholar 

  53. Liu, J.-J., Ni, A.-Q., Ni, G.-X.: A nonconvex ℓ1(ℓ1 − ℓ2) model for image restoration with impulse noise. J. Comput. Appl. Math 378, 112934 (2020)

    Article  MATH  MathSciNet  Google Scholar 

  54. 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)

    Article  MATH  MathSciNet  Google Scholar 

  55. Lai, M.-J., Xu, Y., Yin, W.: Improved iteratively reweighted least squares for unconstrained smoothed ℓq minimization. SIAM J. Numer. Anal. 51(2), 927–957 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Funding

This work is supported by the National Natural Science Foundation of China under grant 12071196.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Rong Li & Bing Zheng

Authors
  1. Rong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bing Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bing Zheng.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Availability of data and material

Not applicable.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, R., Zheng, B. The ℓ2,p regularized total variation with overlapping group sparsity prior for image restoration with impulse noise. Numer Algor 91, 1779–1814 (2022). https://doi.org/10.1007/s11075-022-01322-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-022-01322-x

Keywords

Search

Navigation