Skip to main content
SpringerLink
Log in

Image denoising using combined higher order non-convex total variation with overlapping group sparsity

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

It is widely known that the total variation image restoration suffers from the stair casing artifacts which results in blocky restored images. In this paper, we address this problem by proposing a combined non-convex higher order total variation with overlapping group sparse regularizer. The hybrid scheme of both the overlapping group sparse and the non-convex higher order total variation for blocky artifact removal is complementary. The overlapping group sparse term tends to smoothen out blockiness in the restored image more globally, while the non-convex higher order term tends to smoothen parts that are more local to texture while preserving sharp edges. To solve the proposed image restoration model, we develop an iteratively re-weighted \(\ell _1\) based alternating direction method of multipliers algorithm to deal with the constraints and subproblems. In this study, the images are degraded with different levels of Gaussian noise. A comparative analysis of the proposed method with the overlapping group sparse total variation, the Lysaker, Lundervold and Tai model, the total generalized variation and the non-convex higher order total variation, was carried out for image denoising. The results in terms of peak signal-to-noise ratio and structure similarity index measure show that the proposed method gave better performance than the compared algorithms.

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

Similar content being viewed by others

Notes

  1. MATLAB codes can be downloaded at https://github.com/tarmiziAdam2005.

References

  • Beck, A., & Teboulle, M. (2009). Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Transactions on Image Processing, 18(11), 2419–2434.

    Article  MathSciNet  MATH  Google Scholar 

  • Boyd, S., Parikh, N., Chu, E., Peleato, B., & Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends® in Machine Learning, 3(1), 1–122.

    Article  MATH  Google Scholar 

  • Bredies, K., Kunisch, K., & Pock, T. (2010). Total generalized variation. SIAM Journal on Imaging Sciences, 3(3), 492–526. https://doi.org/10.1137/090769521.

    Article  MathSciNet  MATH  Google Scholar 

  • Candes, E. J., Wakin, M. B., & Boyd, S. P. (2008). Enhancing sparsity by reweighted \(\ell 1\) minimization. Journal of Fourier analysis and applications, 14(5–6), 877–905.

    Article  MathSciNet  MATH  Google Scholar 

  • Chambolle, A., & Pock, T. (2016). An introduction to continuous optimization for imaging. Acta Numerica, 25, 161–319.

    Article  MathSciNet  MATH  Google Scholar 

  • Chan, R. H., Tao, M., & Yuan, X. (2013). Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers. SIAM Journal on imaging Sciences, 6(1), 680–697.

    Article  MathSciNet  MATH  Google Scholar 

  • Chan, T., Marquina, A., & Mulet, P. (2000). High-order total variation-based image restoration. SIAM Journal on Scientific Computing, 22(2), 503–516.

    Article  MathSciNet  MATH  Google Scholar 

  • Chartrand, R. (2007). Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Processing Letters, 14(10), 707–710.

    Article  Google Scholar 

  • Chen, P. Y., & Selesnick, I. W. (2014). Group-sparse signal denoising: Non-convex regularization, convex optimization. IEEE Transactions on Signal Processing, 62(13), 3464–3478.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, X., & Zhou, W. (2014). Convergence of the reweighted \(\ell _1\) minimization algorithm for \(\ell _2\)-\(\ell _p\) minimization. Computational Optimization and Applications, 59(1–2), 47–61.

    Article  MathSciNet  Google Scholar 

  • Condat, L. (2014). A generic proximal algorithm for convex optimization-application to total variation minimization. IEEE Signal Processing Letters, 21(8), 985–989.

    Article  Google Scholar 

  • Figueiredo, M., Dias, J. B., Oliveira, J. P., Nowak, R. D., et al. (2006). On total variation denoising: A new majorization-minimization algorithm and an experimental comparison with wavalet denoising. In IEEE international conference on image processing (pp. 2633–2636), IEEE.

  • Gong, P., Zhang, C., Lu, Z., Huang, J. Z., & Ye, J. (2013). A general iterative shrinkage and thresholding algorithm for non-convex regularized optimization problems. In Proceedings of the international conference on machine learning (Vol. 28, p. 37), NIH Public Access.

  • He, C., Hu, C., Yang, X., He, H., & Zhang, Q. (2014). An adaptive total generalized variation model with augmented Lagrangian method for image denoising. Mathematical Problems in Engineering, 2014, 11.

    Google Scholar 

  • He, C., Hu, C., Li, X., Yang, X., & Zhang, W. (2016). A parallel alternating direction method with application to compound l 1-regularized imaging inverse problems. Information Sciences, 348, 179–197.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, J., Huang, T. Z., Selesnick, I. W., Lv, X. G., & Chen, P. Y. (2015a). Image restoration using total variation with overlapping group sparsity. Information Sciences, 295, 232–246.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, R. W., Wu, D., Wu, C. S., Xu, T., & Xiong, N. (2015b). Constrained nonconvex hybrid variational model for edge-preserving image restoration. In IEEE international conference on systems, man, and cybernetics (pp 1809–1814).

  • Liu, X. (2016). Weighted total generalised variation scheme for image restoration. Image Processing, IET, 10(1), 80–88.

    Article  Google Scholar 

  • Lu, C., Tang, J., Yan, S., & Lin, Z. (2016). Nonconvex nonsmooth low rank minimization via iteratively reweighted nuclear norm. IEEE Transactions on Image Processing, 25(2), 829–839.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Lyu, Q., Lin, Z., She, Y., & Zhang, C. (2013). A comparison of typical \(\ell _p\) minimization algorithms. Neurocomputing, 119, 413–424.

    Article  Google Scholar 

  • Ng, M., Wang, F., & Yuan, X. M. (2011). Fast minimization methods for solving constrained total-variation superresolution image reconstruction. Multidimensional Systems and Signal Processing, 22(1), 259–286.

    Article  MathSciNet  MATH  Google Scholar 

  • Nikolova, M., Ng, M. K., & Tam, C. P. (2010). Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Transactions on Image Processing, 19(12), 3073–3088.

    Article  MathSciNet  MATH  Google Scholar 

  • Papafitsoros, K. (2015). Novel higher order regularisation methods for image reconstruction. Ph.D. thesis, University of Cambridge.

  • Papafitsoros, K., & Schönlieb, C. B. (2014). A combined first and second order variational approach for image reconstruction. Journal of Mathematical Imaging and Vision, 48(2), 308–338.

    Article  MathSciNet  MATH  Google Scholar 

  • Parekh, A., & Selesnick, I. (2015). Convex denoising using non-convex tight frame regularization. IEEE Signal Processing Letters, 22(10), 1786–1790.

    Article  Google Scholar 

  • Peyré, G., & Fadili, J. (2011). Group sparsity with overlapping partition functions. In Signal Processing Conference, 2011 19th European (pp 303–307), IEEE.

  • Powell, M. J. D. (1969). A method for nonlinear constraints in minimization problems. In R. Fletcher (Ed.), Optimization (pp. 283–298). New York: Academic Press.

    Google Scholar 

  • Qin, Z., Goldfarb, D., & Ma, S. (2015). An alternating direction method for total variation denoising. Optimization Methods and Software, 30(3), 594–615.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Selesnick, I. W., & Chen, P. Y. (2013). Total variation denoising with overlapping group sparsity. In IEEE international conference on acoustics, speech and signal processing (ICASSP) (pp. 5696–5700), IEEE.

  • Shi, M., Han, T., & Liu, S. (2016). Total variation image restoration using hyper-laplacian prior with overlapping group sparsity. Signal Processing, 126, 65–76.

    Article  Google Scholar 

  • Sun, Y., Babu, P., & Palomar, D. P. (2016). Majorization-minimization algorithms in signal processing, communications, and machine learning. IEEE Transactions on Signal Processing, 65(3), 794–816.

    Article  MathSciNet  MATH  Google Scholar 

  • Tai, X. C., & Wu, C. (2009). Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. In Scale space and variational methods in computer vision (pp 502–513), Springer.

  • Tao, M., Yang, J., & He, B. (2009). Alternating direction algorithms for total variation deconvolution in image reconstruction. Research report, Rice University.

  • Wang, Y., Yin, W., & Zeng, J. (2015). Global convergence of ADMM in nonconvex nonsmooth optimization. arXiv preprint arXiv:1511.06324.

  • Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. (2004). Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13(4), 600–612.

    Article  Google Scholar 

  • Wen, Z., Yang, C., Liu, X., & Marchesini, S. (2012). Alternating direction methods for classical and ptychographic phase retrieval. Inverse Problems, 28(11), 115,010.

    Article  MathSciNet  MATH  Google Scholar 

  • Wu, C., & Tai, X. C. (2010). Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM Journal on Imaging Sciences, 3(3), 300–339. https://doi.org/10.1137/090767558.

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, J., Feng, X., & Hao, Y. (2014). A coupled variational model for image denoising using a duality strategy and split Bregman. Multidimensional Systems and Signal Processing, 25(1), 83–94.

    Article  MATH  Google Scholar 

  • Xu, Y., Yin, W., Wen, Z., & Zhang, Y. (2012). An alternating direction algorithm for matrix completion with nonnegative factors. Frontiers of Mathematics in China, 7(2), 365–384.

    Article  MathSciNet  MATH  Google Scholar 

  • You, Y. L., & Kaveh, M. (2000). Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing, 9(10), 1723–1730.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, Y. (2010). An alternating direction algorithm for nonnegative matrix factorization. preprint.

Download references

Acknowledgements

We would like to thank the anonymous reviewers for their valuable suggestions for improving this paper. We would also like to thank Dr. Jun Liu of University of Electronic Science and Technology of China for generously sharing the codes for OGSTV with us.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia

    Tarmizi Adam & Raveendran Paramesran

Authors
  1. Tarmizi Adam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raveendran Paramesran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Tarmizi Adam or Raveendran Paramesran.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Adam, T., Paramesran, R. Image denoising using combined higher order non-convex total variation with overlapping group sparsity. Multidim Syst Sign Process 30, 503–527 (2019). https://doi.org/10.1007/s11045-018-0567-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-018-0567-3

Keywords

Search

Navigation