Skip to main content
SpringerLink
Log in

Data-Driven Tight Frame Learning Scheme Based on Local and Non-local Sparsity with Application to Image Recovery

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Sparse representation has shown to be a successful technique in the fields of image restoration and other image processing tasks over the past few years. The key of the approach with application to image recovery is how to sparsely represent the true-image data, which seriously affects its recovery quality. One such representative method is the widely used wavelet tight frame. However, the filters of the classical wavelet frame are fixed and unchanged for different input images. To overcome the limitation, a data-driven tight frame was proposed to learn a good sparse representation from the image itself very recently. However, the filters of the tight frame are only learned from the selected local image patches, and it doesn’t utilize the non-local similarity of image patches. In this paper, we further propose a 3D data-driven tight frame learning scheme which considers both the local and non-local sparse representation of image patches, and establish a variational model for image recovery based on the new sparse representation method. Numerical experiments demonstrate that our approach is competitive with the recently proposed state-of-the-art methods.

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

References

  1. Steidl, G.: A note on the dual treatment of higher-order regularization functionals. Computing 76(1–2), 135–148 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, D.Q., Cheng, L.Z., Su, F.: A new TV-Stokes model with augmented Lagrangian method for image denoising and deconvolution. J. Sci. Comput. 51(3), 505–526 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lefkimmiatis, S., Ward, J.P., Unser, M.: Hessian Schatten-norm regularization for linear inverse problems. IEEE Trans. Image Process. 22(5), 1873 (2013)

    Article  MathSciNet  Google Scholar 

  5. Candès, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities. Commun. Pure Appl. Math. 57(2), 219–266 (2004)

    Article  MATH  Google Scholar 

  6. Le Pennec, E., Mallat, S.: Sparse geometric image representations with bandelets. IEEE Trans. Image Process. 14(4), 423–438 (2005)

    Article  MathSciNet  Google Scholar 

  7. Easley, G., Labate, D., Lim, W.Q.: Sparse directional image representations using the discrete shearlet transform. Appl. Comput. Harmon. Anal. 25(1), 25–46 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Le Pennec, E., Mallat, S.: Bandelet image approximation and compression. Multiscale Model. Simul. 4(3), 992–1039 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Aharon, M., Elad, M., Bruckstein, A.: K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54(11), 4311–4322 (2006)

    Article  Google Scholar 

  10. Elad, M., Aharon, M.: Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736–3745 (2006)

    Article  MathSciNet  Google Scholar 

  11. Mairal, J., Sapiro, G., Elad, M.: Learning multiscale sparse representations for image and video restoration. Multiscale Model. Simul. 7(1), 214–241 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cai, J.F., Ji, H., Shen, Z., et al.: Data-driven tight frame construction and image denoising. Appl. Comput. Harmon. Anal. 37(1), 89–105 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, D.Q.: Image denoising based on improved data-driven sparse representation. arXiv preprint arXiv:1502.03273 (2015)

  14. Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4(4), 1091–1115 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, X., Burger, M., Bresson, X., et al.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3(3), 253–276 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Deledalle, C.A., Denis, L., Tupin, F.: How to compare noisy patches? Patch similarity beyond Gaussian noise. Int. J. Comput. Vis. 99(1), 86–102 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ram, I., Elad, M., Cohen, I.: Image processing using smooth ordering of its patches. IEEE Trans. Image Process. 22(7), 2764–2774 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image restoration by sparse 3D transform-domain collaborative filtering. In: Proceedings of SPIE, the International Society for Optical Engineering, pp. 681207-1–681207-12. Bellingham, Wash, Society of Photo-Optical Instrumentation Engineers (2008)

  21. Chatterjee, P., Milanfar, P.: Clustering-based denoising with locally learned dictionaries. IEEE Trans. Image Process. 18(7), 1438–1451 (2009)

    Article  MathSciNet  Google Scholar 

  22. Danielyan, A., Katkovnik, V., Egiazarian, K.: BM3D frames and variational image deblurring. IEEE Trans. Image Process. 21(4), 1715–1728 (2012)

    Article  MathSciNet  Google Scholar 

  23. Dong, W., Li, X., Zhang, D., et al.: Sparsity-based image denoising via dictionary learning and structural clustering. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2011, pp. 457–464

  24. Mairal, J., Bach, F., Ponce, J., et al.: Non-local sparse models for image restoration. In: IEEE International Conference on Computer Vision (ICCV), 2009, pp. 2272–2279

  25. Dong, W., Shi, G., Li, X.: Nonlocal image restoration with bilateral variance estimation: a low-rank approach. IEEE Trans. Image Process. 22(2), 700–711 (2013)

    Article  MathSciNet  Google Scholar 

  26. Quan, Y., Ji, H., Shen, Z.: Data-driven multi-scale non-local wavelet frame construction and image recovery. J. Sci. Comput. 63(2), 307–329 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chen, D.Q., Zhou, Y.: Wavelet frame based image restoration via combined sparsity and nonlocal prior of coefficients. J. Sci. Comput. 66(1), 196–224 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ram, I., Elad, M., Cohen, I.: Generalized tree-based wavelet transform. IEEE Trans. Signal Process. 59(9), 4199–4209 (2011)

    Article  MathSciNet  Google Scholar 

  29. Daubechies, I., Han, B., Ron, A., et al.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14(1), 1–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dong, B., Shen, Z.: MRA based wavelet frames and applications. In: IAS Lecture Notes Series, Summer Program on The Mathematics of Image Processing, Park City Mathematics Institute, 2010

  31. Wipf, D., Nagarajan, S.: Iterative reweighted l1 and l2 methods for finding sparse solutions. IEEE J. Sel. Top. Signal Process. 4(2), 317–329 (2010)

    Article  Google Scholar 

  32. Zhang, Y., Dong, B., Lu, Z.: \(l_0\) Minimization for wavelet frame based image restoration. Math. Comput. 82(282), 995–1015 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Dong, B., Zhang, Y.: An efficient algorithm for \(l_0\) minimization in wavelet frame based image restoration. J. Sci. Comput. 54(2–3), 350–368 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ram, I., Elad, M., Cohen, I.: Redundant wavelets on graphs and high dimensional data clouds. IEEE Signal Process. Lett. 19(5), 291–294 (2012)

    Article  Google Scholar 

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

    Article  Google Scholar 

  36. Tikhonov, A.N., Arsenin, V.I.A.: Solutions of Ill-Posed Problems. Wiley, New York (1977)

    MATH  Google Scholar 

  37. Eckstein, J., Bertsekas, D.P.: On the Douglas Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1–3), 293–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The work was supported in part by the National Natural Science Foundation of China under Grant 61401473, and the Social Livelihood Science and Technology Innovation Special Project of CSTC (cstc2015shmszx120002). We appreciate the constructive comments of the anonymous reviewers, which led to great improvements in this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Biomedical Engineering, Third Military Medical University, Chongqing, 400038, People’s Republic of China

    Dai-Qiang Chen

Authors
  1. Dai-Qiang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dai-Qiang Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, DQ. Data-Driven Tight Frame Learning Scheme Based on Local and Non-local Sparsity with Application to Image Recovery. J Sci Comput 69, 461–486 (2016). https://doi.org/10.1007/s10915-016-0205-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0205-x

Keywords

Search

Navigation