Skip to main content
SpringerLink
Log in

A coupled non-convex hybrid regularization and weak \(H^{-1}\) image decomposition model for denoising application

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

Recently, many non-convex variational models have been developed to reduce staircase artifacts in the smooth regions of restored images. Among these models, the \(L^2\)-norm is often used to capture high-frequency oscillation information in the image. However, this modeling method makes it difficult to separate the image structure component from the oscillation component. To address this problem, we propose a coupled non-convex hybrid regularization and weak \(H^{-1}\) decomposition model for image denoising in this paper, which leverages non-convex TV and fractional-order TV regularizers to measure the structure and textural information of the image, respectively. Additionally, we employ weak \(H^{-1}\)-space to model the oscillatory noisy component. By these modeling techniques, the proposed model effectively separates the high-frequency oscillation component and alleviates the staircase effect. To solve this non-convex minimization, an alternating direction method of multipliers combined with the majorization–minimization algorithm is introduced. Furthermore, we provide a detailed discussion on the convergence conditions of the proposed algorithm. Numerical experimental results demonstrate the feasibility of our decomposition model in denoising applications. Moreover, compared with other variational models, our proposed model exhibits excellent performance in terms of peak signal-to-noise ratio and structural similarity index values.

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

Similar content being viewed by others

References

  1. Belyaev, A., Fayolle, P.: Adaptive curvature-guided image filtering for structure + texture image decomposition. IEEE Trans. Image Process. 27(10), 5192–5203 (2018)

    ADS  MathSciNet  Google Scholar 

  2. Jobson, D.J., Rahman, Z., Woodell, G.A.: Properties and performance of a center/surround retinex. IEEE Trans. Image Process. 6(3), 451–462 (1997)

    ADS  CAS  PubMed  Google Scholar 

  3. Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)

    ADS  MathSciNet  PubMed  Google Scholar 

  4. Lv, Y.: Total generalized variation denoising of speckled images using a primal-dual algorithm. J. Appl. Math. Comput. 62, 489–509 (2020)

    MathSciNet  Google Scholar 

  5. Aubert, G., Vese, L.: A variational method in image recovery. SIAM J. Numer. Anal. 34(5), 1948–1979 (1997)

    MathSciNet  Google Scholar 

  6. Xu, S., Zhang, J.S., Zhang, C.X.: Hyperspectral image denoising by low-rank models with hyper-laplacian total variation prior. Signal Process. 201, 108733 (2022)

    Google Scholar 

  7. Hansen, P.C.: Analysis of discrete ill-posed problems by means of the l-curve. SIAM Rev. 34(4), 561–580 (1992)

    MathSciNet  Google Scholar 

  8. Chang, S.G., Yu, B., Vetterli, M.: Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. Image Process. 9(9), 1532–1546 (2000)

    ADS  MathSciNet  CAS  PubMed  Google Scholar 

  9. Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Prob. 10(6), 1217–1229 (1997)

    ADS  MathSciNet  Google Scholar 

  10. Tihonov, A.N.: On the solution of ill-posed problems and the method of regularization, Doklady Akademii Nauk. Russ. Acad. Sci. 151, 501–504 (1963)

    Google Scholar 

  11. Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. ACM 9(1), 84–97 (1962)

    MathSciNet  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  13. Shi, B.L., Gu, F., Pang, Z.F., Zeng, Y.H.: Remove the salt and pepper noise based on the high order total variation and the nuclear norm regularization. Appl. Math. Comput. 421, 126925 (2022)

    MathSciNet  Google Scholar 

  14. Lv, X.G., Song, Y.Z., Wang, S.X., Le, J.: Image restoration with a high-order total variation minimization method. Appl. Math. Model. 37(16–17), 8210–8224 (2013)

    MathSciNet  Google Scholar 

  15. Moll, J.S.: The anisotropic total variation flow. Math. Ann. 332(1), 177–218 (2005)

    MathSciNet  Google Scholar 

  16. Guo, J.C., Chen, Q.H.: Image denoising based on nonconvex anisotropic total-variation regularization. Signal Process. 186, 108–124 (2021)

    Google Scholar 

  17. Thanh, D.N.H., Surya Prasath, V.B., Hieu, L.M., Dvoenko, S.: An adaptive method for image restoration based on high-order total variation and inverse gradient. SIViP 14(6), 1189–1197 (2020)

    Google Scholar 

  18. 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)

    MathSciNet  Google Scholar 

  19. Jon, K., Sun, Y., Li, Q.X., Liu, J., Wang, X.F., Zhu, W.S.: Image restoration using overlapping group sparsity on hyper-Laplacian prior of image gradient. Neurocomputing 420, 57–69 (2021)

    Google Scholar 

  20. Bai, J., Feng, X.C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492–2502 (2007)

    ADS  MathSciNet  PubMed  Google Scholar 

  21. Ben-Loghfyry, A., Hakim, A., Laghrib, A.: A denoising model based on the fractional beltrami regularization and its numerical solution. J. Appl. Math. Comput. 69, 1431–1463 (2023)

    MathSciNet  Google Scholar 

  22. Wu, L., Tang, L.M., Li, C.Y.: Hybrid regularization model combining overlapping group sparse second-order total variation and nonconvex total variation. J. Electron. Imaging 31(4), 043012 (2022)

    ADS  Google Scholar 

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

    MathSciNet  Google Scholar 

  24. Tang, L.M., Zhang, H.L., He, C.J., Fang, Z.: Non-convex and non-smooth variational decomposition for image restoration. Appl. Math. Model. 69, 355–377 (2019)

    MathSciNet  Google Scholar 

  25. Wang, J., Xu, G., Li, C., Wang, Z.S., Yan, F.J.: Surface defects detection using non-convex total variation regularized RPCA with kernelization. IEEE Trans. Instrum. Meas. 70(99), 1–13 (2021)

    Google Scholar 

  26. Yuan, G.Z., Ghanem, B.: \( l_{0} \)TV: A sparse optimization method for impulse noise image restoration. IEEE Trans. Pattern Anal. Mach. Intell. 41(2), 352–364 (2019)

    PubMed  Google Scholar 

  27. 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, 69–85 (2018)

    Google Scholar 

  28. Pang, Z.F., Zhang, H.L., Luo, S.S., Zeng, T.Y.: Image denoising based on the adaptive weighted \(TV^p\) regularization. Signal Process. 167, 1–21 (2020)

    Google Scholar 

  29. Tang, L.M., Wu, L., Fang, Z., Li, C.Y.: A non-convex ternary variational decomposition and its application for image denoising. IET Signal Proc. 16(3), 248–266 (2022)

    Google Scholar 

  30. Liu, Q.H., Sun, L.P., Gao, S.: Non-convex fractional-order derivative for single image blind. Appl. Math. Model. 102, 207–227 (2022)

    MathSciNet  Google Scholar 

  31. Adam, T., Paramesran, R.: Hybrid non-convex second-order total variation with applications to non-blind image deblurring. SIViP 14(1), 115–123 (2020)

    Google Scholar 

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

    Google Scholar 

  33. 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(24), 15–31 (2019)

    Google Scholar 

  34. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: the Fifteenth Dean Jacqueline B. American Mathematical Society, Lewis Memorial Lectures (2001)

    Google Scholar 

  35. Gilles, J., Meyer, Y.: Properties of \( BV-G \) structures + textures decomposition models. application to road detection in satellite images. IEEE Trans. Image Process. 19(11), 2793–2800 (2010)

  36. Osher, S., Sole, A., Vese, L.A.: Image decomposition and restoration using total variation minimization and the \( H^{-1} \). SIAM Multisc. Model. Simul. 1(3), 349-170 (2003)

  37. Vese, L.A., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19, 553–572 (2003)

    MathSciNet  Google Scholar 

  38. Vese, L.A., Osher, S.: Image denoising and decomposition with total variation minimization and oscillatory functions. J. Math. Imaging Vis. 20, 7–18 (2004)

    MathSciNet  Google Scholar 

  39. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Now Found. Trends 128, 1 (2011)

    Google Scholar 

  40. Chen, G., Li, G., Zhang, J.S.: Tensor compressed video sensing reconstruction by combination of fractional-order total variation and sparsifying transform. Signal Process.: Image Commun. 55, 146–156 (2017)

    Google Scholar 

  41. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004)

    MathSciNet  Google Scholar 

  42. Lanza, A., Morigi, S., Selesnick, I., Sgallari, F.: Nonconvex nonsmooth optimization via convex-nonconvex majorization-minimization. Numer. Math. 136(2), 1–39 (2017)

    MathSciNet  Google Scholar 

  43. Bayram, I.: On the convergence of the iterative shrinkage/thresholding algorithm with a weakly convex penalty. IEEE Trans. Image Process. 64(6), 1597–1608 (2016)

    ADS  MathSciNet  Google Scholar 

  44. Sun, M., Sun, H.C.: Improved proximal ADMM with partially parallel splitting for multi-block separable convex programming. J. Appl. Math. Comput. 58, 151–181 (2018)

    MathSciNet  Google Scholar 

  45. Chan, T.H., Ma, W.K., Chi, C.Y., Wang, Y.: A Convex Analysis Framework for Blind Separation of Non-Negative Sources. IEEE Trans. Image Process. 56(10), 5120–5134 (2008)

    ADS  MathSciNet  Google Scholar 

  46. Wang, Y., Yin, W., Zeng, J.S.: Global convergence of ADMM in nonconvex nonsmooth optimization. J. Sci. Comput. 78, 29–63 (2019)

    MathSciNet  Google Scholar 

  47. 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)

    ADS  PubMed  Google Scholar 

  48. Zhang, K., Zuo, W., 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 (2017)

    ADS  MathSciNet  PubMed  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the Natural Science Foundation of China under Grant Nos. 62061016, 61561019, and the Doctoral Scientific Fund Project of Hubei Minzu University for under Grant No. MY2015B001.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Hubei Minzu University, Enshi, 445000, Hubei, People’s Republic of China

    Wenjing Lu, Zhuang Fang, Liang Wu, Liming Tang & Hanxin Liu

Authors
  1. Wenjing Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhuang Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Liming Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hanxin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhuang Fang.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this work.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, W., Fang, Z., Wu, L. et al. A coupled non-convex hybrid regularization and weak \(H^{-1}\) image decomposition model for denoising application. J. Appl. Math. Comput. 70, 197–233 (2024). https://doi.org/10.1007/s12190-023-01949-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-023-01949-6

Keywords

Search

Navigation