Skip to main content
Springer Nature Link
Log in

Extended Shearlet HMT Model-Based Image Denoising Using BKF Distribution

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Images are often corrupted by noise in the procedures of image acquisition and transmission. It is a challenging work to design an edge-preserving image denoising scheme. Extended discrete Shearlet transform (extended DST) is an effective multi-scale and multi-direction analysis method; it not only can exactly compute the Shearlet coefficients based on a multiresolution analysis, but also can represent images with very few coefficients. In this paper, we propose a new image denoising approach in extended DST domain, which combines hidden Markov tree (HMT) model and Bessel K Form (BKF) distribution. Firstly, the marginal statistics of extended DST coefficients are studied, and their distribution is analytically calculated by modeling extended DST coefficients with BKF probability density function. Then, an extended Shearlet HMT model is established for capturing the intra-scale, inter-scale, and cross-orientation coefficients dependencies. Finally, an image denoising approach based on the extended Shearlet HMT model is presented. Extensive experimental results demonstrate that our extended Shearlet HMT denoising approach can obtain better performances in terms of both subjective and objective evaluations than other state-of-the-art HMT denoising techniques. Especially, the proposed approach can preserve edges very well while removing noise.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

References

  1. Lee, S., Cho, S.: Recent advances in image deblurring. In: Proceedings ofSIGGRAPH Asia 2013 Courses, New York, vol. 6, pp. 1–108 (2013)

  2. Donoho, D.L.: De-noising by soft-thresholding. IEEE Trans. Inf. Theory 90(432), 1200–1224 (1995)

    MathSciNet  MATH  Google Scholar 

  3. Buades, A., Coll, B., Morel, J.M.: A nonlocal algorithm for image denoising. In: IEEE Computer Vision and Pattern Recognition 2005, San Diego, vol. 2, pp. 60–65 (2005)

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

    Article  MathSciNet  Google Scholar 

  5. Yan, R., Shao, L., Lin, Y.: Nonlocal hierarchical dictionary learning using wavelets for image denoising. IEEE Trans. Image Process. 22(12), 4689–4698 (2013)

    Article  MathSciNet  Google Scholar 

  6. Khashabi, D., Nowozin, S., Jancsary, J.: Joint demosaicing and denoising via learned non-parametric random fields. IEEE Trans. Image Process. 23(12), 4968–4981 (2014)

    Article  MathSciNet  Google Scholar 

  7. Chen, S., Liu, M., Zhang, W.: Edge preserving image denoising with a closed form solution. Pattern Recogn. 46(3), 976–988 (2013)

    Article  MATH  Google Scholar 

  8. Zhong, P., Wang, R.: Multiple-spectral-band CRFs for denoising junk bands of hyperspectral imagery. IEEE Trans. Geosci. Remote Sens. 51(4), 2260–2275 (2013)

    Article  MathSciNet  Google Scholar 

  9. Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color Images. In: Proceedings of the IEEE International Conference on Computer Vision(ICCV), Bombay, pp. 839–846 (1998)

  10. Gunturk, B.K.: Fast bilateral filter with arbitrary range and domain kernels. IEEE Trans. Image Process. 20(9), 2690–2696 (2011)

    Article  MathSciNet  Google Scholar 

  11. Aurich, V., Weule, J.: Non-linear Gaussian Filters Performing Edge Preserving Diffusion. Mustererkennung 1995. Springer, Berlin (1995)

    Book  Google Scholar 

  12. Tsiotsios, C., Petrou, M.: On the choice of the parameters for anisotropic diffusion in image processing. Pattern Recogn. 46(5), 1369–1381 (2013)

    Article  Google Scholar 

  13. Cho, S.I., Kang, S.J., Kim, H.S.: Dictionary-based anisotropic diffusion for noise reduction. Pattern Recogn. Lett. 46, 36–45 (2014)

    Article  Google Scholar 

  14. Wang, Xiang-Yang, Hong-Ying Yang, Yu., Zhang, Zhong-Kai Fu: Image denoising using SVM classification in nonsubsampled contourlet transform domain. Inf. Sci. 246, 155–176 (2013)

    Article  Google Scholar 

  15. Argenti, F., Bianchi, T., Alparone, L.: Multiresolution MAP despeckling of SAR images based on locally adaptive generalized Gaussian pdf modeling. IEEE Trans. Image Process. 15(11), 3385–3399 (2006)

    Article  Google Scholar 

  16. Hashemi, M., Beheshti, S.: Adaptive Bayesian denoising for general Gaussian distributed signals. IEEE Trans. Signal Process. 62(5), 1147–1156 (2014)

    Article  MathSciNet  Google Scholar 

  17. Azzari, L., Foi, A.: Gaussian-Cauchy mixture modeling for robust signal-dependent noise estimation. In: 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, pp. 5357–5361 (2014)

  18. Gao, Q., Lu, Y., Sun, D.: Directionlet-based denoising of SAR images using a Cauchy model. Signal Process. 93(5), 1056–1063 (2013)

    Article  Google Scholar 

  19. Hill, P.R., Achim, A.M., Bull, D.R.: Dual-tree complex wavelet coefficient magnitude modelling using the bivariate Cauchy–Rayleigh distribution for image denoising. Signal Process. 105, 464–472 (2014)

    Article  Google Scholar 

  20. Portilla, J., Strela, V., Wainwright, M.J., Simoncelli, E.P.: Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans. Image Process. 12(11), 1338–1351 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hammond, D.K., Simoncelli, E.P.: Image modeling and denoising with orientation-adapted Gaussian scale mixtures. IEEE Trans. Image Process. 17(11), 2089–2101 (2008)

    Article  MathSciNet  Google Scholar 

  22. Goossens, B., Aelterman, J., Luong, Q.: Complex wavelet joint denoising and demosaicing using Gaussian scale mixtures. In: IEEE International Conference on Image Processing (ICIP 2013), Melbourne, pp. 445–448 (2013)

  23. Crouse, M.S., Nowak, R.D., Baraniuk, R.G.: Wavelet-based statistical signal processing using hidden Markov models. IEEE Trans. Signal Process. 46(4), 886–902 (1998)

    Article  MathSciNet  Google Scholar 

  24. Po, D.D.Y., Do, M.N.: Directional multiscale modeling of images using the contourlet transform. IEEE Trans. Image Process. 15(6), 1610–1620 (2006)

    Article  MathSciNet  Google Scholar 

  25. Yan, F.X., Cheng, L.Z., Peng, S.L.: Dual-tree complex wavelet hidden Markov tree model for image denoising. Electron. Lett. 43(18), 973–975 (2007)

    Article  Google Scholar 

  26. Zhang, W., Wang, S., Liu, F.: Image denoising using Bandelets and hidden Markov tree models. Chin. J. Electron. 19(4), 646–650 (2010)

    Google Scholar 

  27. Choi, H., Romberg, J., Baraniuk, R., Kingsbury, N.: Hidden Markov tree modeling of complex wavelet transforms. In: Proceedings of 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), vol. 1, pp. 133–136 (2000)

  28. Milone, F.H., Di Persia, L.E., Torres, M.E.: Denoising and recognition using hidden Markov models with observation distributions modeled by hidden Markov trees. Pattern Recognit. 43(4), 1577–1589 (2010)

    Article  MATH  Google Scholar 

  29. Li, C., Hao, Q.: Functional MR image statistical restoration for neural activity detection using hidden Markov tree model. Int. J. Comput. Biol. Drug Des. 6(3), 190–209 (2013)

    Article  Google Scholar 

  30. Long, Z., Younan, N.H.: Statistical image modeling in the contourlet domain using contextual hidden Markov models. Signal Process. 89(5), 946–951 (2009)

    Article  MATH  Google Scholar 

  31. Amini, M., Ahmad, M.O., Swamy, M.N.S.: Image denoising in wavelet domain using the vector-based hidden Markov model. In: The 12th International New Circuits and Systems Conference (NEWCAS), Trois-Rivieres, pp. 29–32 (2014)

  32. Sun, Q., Wang, Y., Xu, C.: Fabric image denoising method based on wavelet-domain HMT model. Multimed. Signal Process. 346, 383–388 (2012)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Lim, W.Q.: The discrete shearlet transform: a new directional transform and compactly supported shearlet frames. IEEE Trans. Image Process. 19(5), 1166–1180 (2010)

    Article  MathSciNet  Google Scholar 

  35. Bian, Y., Liang, S.: Locally optimal detection of image watermarks in the wavelet domain using Bessel K form distribution. IEEE Trans. Image Process. 22(6), 2372–2384 (2013)

  36. Jalal, F.M., Larbi, B.: Analytical form for a Bayesian wavelet estimator of images using the Bessel K form densities. IEEE Trans. Image Process. 14(2), 231–240 (2005)

    Article  MathSciNet  Google Scholar 

  37. Pizurica, A., Philips, W.: Estimating the probability of the presence of a signal of interest in multiresolution single-and multiband image denoising. IEEE Trans. Image Process 15(3), 654–665 (2006)

    Article  Google Scholar 

  38. Goossens, B., Pizurica, A., Philips, W.: Removal of correlated noise by modeling the signal of interest in the wavelet domain. IEEE Trans. Image Process. 18(6), 1153–1165 (2009)

    Article  MathSciNet  Google Scholar 

  39. Pizurica, A., Philips, W., Lemahieu, I.: A joint inter-and intrascale statistical model for Bayesian wavelet based image denoising. IEEE Trans. Image Process. 11(5), 545–557 (2002)

    Article  Google Scholar 

  40. Fan, G., Xia, X.G.: Improved hidden Markov models in the wavelet-domain. IEEE Trans. Signal Process. 49(1), 115–120 (2001)

    Article  Google Scholar 

  41. Luisier, F., Blu, T., Unser, M.: A new SURE approach to image denoising: interscale orthonormal wavelet thresholding. IEEE Trans. Image Process. 16(3), 593–606 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61472171 & 61272416, and Liaoning Research Project for Institutions of Higher Education of China under Grant No. L2013407.

Author information

Authors and Affiliations

  1. School of Computer and Information Technology, Liaoning Normal University, Dalian, 116029, People’s Republic of China

    Xiang-Yang Wang, Na Zhang, Hong-Liang Zheng & Yang-Cheng Liu

Authors
  1. Xiang-Yang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Na Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong-Liang Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yang-Cheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiang-Yang Wang.

Ethics declarations

Ethical standard

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Conflict of interest

The authors declare that they have no conflict of interest.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, XY., Zhang, N., Zheng, HL. et al. Extended Shearlet HMT Model-Based Image Denoising Using BKF Distribution. J Math Imaging Vis 54, 301–319 (2016). https://doi.org/10.1007/s10851-015-0605-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-015-0605-9

Keywords