Skip to main content
SpringerLink
Log in

Low-Rank Estimation for Image Denoising Using Fractional-Order Gradient-Based Similarity Measure

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The aim of this paper is to introduce a novel similarity measure using fractional-order derivative for patch comparison in low-rank image denoising approach. Recently, several outstanding low-rank image denoising algorithms have been proposed. However, these methods have limitations in the sense that certain irrelevant patches can be selected during patch comparison. These undesired patches affect singular values shrinkage and aggregation phases of these approaches. Thus, the fine details and edges of denoised image may not be well preserved. To address this issue, a novel method is proposed in which gradient information is injected in patch comparison using discretized fractional-order derivatives. The advantages of proposed approach are twofold: firstly, the patch comparison becomes more reliable by combining intensity and gradient information; secondly, the fractional-order gradient provides an additional degree of freedom to quantify the gradient information for patch comparison in an efficient way. In addition, the proposed algorithm estimates noise level using geometric details encoded in the image patches. The noise estimation strategy may help in terminating the iterative low-rank approximation. Experimental results on test images reveal that the proposed method performs better than several outstanding algorithms, specifically, in the presence of severe noise levels.

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

Similar content being viewed by others

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. A. Achim, P. Tsakalides, A. Bezerianos, Sar image denoising via bayesian wavelet shrinkage based on heavy-tailed modeling. IEEE Trans Geosci Remote Sens 41(8), 1773–1784 (2003). https://doi.org/10.1109/TGRS.2003.813488

    Article  Google Scholar 

  2. A. Amer, E. Dubois, Fast and reliable structure-oriented video noise estimation. IEEE Trans Circuits Syst Video Technol 15(1), 113–118 (2005). https://doi.org/10.1109/TCSVT.2004.837017

    Article  Google Scholar 

  3. J. Bai, X. Feng, Fractional-order anisotropic diffusion for image denoising. IEEE Trans Image Process 16(10), 2492–2502 (2007). https://doi.org/10.1109/TIP.2007.904971

    Article  MathSciNet  Google Scholar 

  4. M.J. Black, G. Sapiro, D.H. Marimont, D. Heeger, Robust anisotropic diffusion. IEEE Trans Image Process 7(3), 421–432 (1998). https://doi.org/10.1109/83.661192

    Article  Google Scholar 

  5. A. Buades, B. Coll, JM. Morel A non-local algorithm for image denoising. In: IN CVPR, pp 60–65 (2005)

  6. J.F. Cai, E.J. Candès, Z. Shen, A singular value thresholding algorithm for matrix completion. SIAM J Opt 20(4), 1956–1982 (2010). https://doi.org/10.1137/080738970

    Article  MathSciNet  MATH  Google Scholar 

  7. T.F. Chan, Chiu-Kwong Wong, Total variation blind deconvolution. IEEE Trans Image Process 7(3), 370–375 (1998). https://doi.org/10.1109/83.661187

    Article  Google Scholar 

  8. G. Chen, P. Zhang, Y. Wu, D. Shen, P.T. Yap, Denoising magnetic resonance images using collaborative non-local means. Neurocomputing 177, 215–227 (2016) http://www.sciencedirect.com/science/article/pii/S0925231215017506

  9. S. Chen, L. Gao, Q. Li, Sar image despeckling by using nonlocal sparse coding model. Circuits Syst Signal Process 37, 3023–3045 (2018). https://doi.org/10.1007/s00034-017-0704-5

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering. IEEE Trans Image Process 16(8), 2080–2095 (2007). https://doi.org/10.1109/TIP.2007.901238

    Article  MathSciNet  Google Scholar 

  11. J. Dong, Z. Xue, W. Wang, Robust pca using nonconvex rank approximation and sparse regularizer. Circuits Syst Signal Process 39, 3086–3104 (2020). https://doi.org/10.1007/s00034-019-01310-y

    Article  MATH  Google Scholar 

  12. W. Dong, X. Li, L. Zhang, G. Shi, Sparsity-based image denoising via dictionary learning and structural clustering. CVPR 2011, 457–464 (2011). https://doi.org/10.1109/CVPR.2011.5995478

    Article  Google Scholar 

  13. W. Dong, G. Shi, X. Li, Nonlocal image restoration with bilateral variance estimation: a low-rank approach. IEEE Trans Image Process 22(2), 700–711 (2013). https://doi.org/10.1109/TIP.2012.2221729

    Article  MathSciNet  MATH  Google Scholar 

  14. W. Dong, L. Zhang, G. Shi, X. Li, Nonlocally centralized sparse representation for image restoration. IEEE Trans Image Process 22(4), 1620–1630 (2013). https://doi.org/10.1109/TIP.2012.2235847

    Article  MathSciNet  MATH  Google Scholar 

  15. Dong-Hyuk Shin, Rae-Hong Park, Seungjoon Yang, Jae-Han Jung, Block-based noise estimation using adaptive gaussian filtering. IEEE Trans Consum Electron 51(1), 218–226 (2005). https://doi.org/10.1109/TCE.2005.1405723

    Article  Google Scholar 

  16. D.L. Donoho, J.M. Johnstone, Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3), 425–455 (1994). https://doi.org/10.1093/biomet/81.3.425

    Article  MathSciNet  MATH  Google Scholar 

  17. M. D’Ovidio, R. Garra (2013) Fractional gradient and its application to the fractional advection equation 1305.4400

  18. M. Elad, M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans Image Process 15(12), 3736–3745 (2006). https://doi.org/10.1109/TIP.2006.881969

    Article  MathSciNet  Google Scholar 

  19. Erciliasousa, How to approximate the fractional derivative of order (1 2]. Int J Bifurc Chaos (2012). https://doi.org/10.1142/S0218127412500757

  20. A. Fathi, A.R. Naghsh-Nilchi, Efficient image denoising method based on a new adaptive wavelet packet thresholding function. IEEE Trans Image Process 21(9), 3981–3990 (2012). https://doi.org/10.1109/TIP.2012.2200491

    Article  MathSciNet  MATH  Google Scholar 

  21. M.A.T. Figueiredo, R.D. Nowak, Wavelet-based image estimation: an empirical bayes approach using jeffrey’s noninformative prior. IEEE Trans Image Process 10(9), 1322–1331 (2001). https://doi.org/10.1109/83.941856

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Gilboa, S. Osher, Nonlocal operators with applications to image processing. Multiscale Modell Simul 7(3), 1005–1028 (2009). https://doi.org/10.1137/070698592

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Gu, L. Zhang, W. Zuo, X. Feng Weighted nuclear norm minimization with application to image denoising. In: 2014 IEEE Conference on Computer Vision and Pattern Recognition, pp 2862–2869, (2014) https://doi.org/10.1109/CVPR.2014.366

  24. A. Hore, D. Ziou Image quality metrics: Psnr vs. ssim. In: Pattern Recognition (ICPR), 2010 20th International Conference on, pp 2366–2369, (2010) https://doi.org/10.1109/ICPR.2010.579

  25. J. Jiang, J. Yang, Y. Cui, L. Luo, Mixed noise removal by weighted low rank model. Neurocomputing 151, 817–826 (2015), http://www.sciencedirect.com/science/article/pii/S0925231214013563

  26. AA. Kilbas, HM. Srivastava, JJ. Trujillo theory and applications of fractional differential equations, Volume 204 (2006) (North-Holland Mathematics Studies). Elsevier Science Inc., New York, NY, USA

  27. D.G. Kim, Z.H. Shamsi, Enhanced residual noise estimation of low rank approximation for image denoising. Neurocomputing 293, 1–11 (2018). https://doi.org/10.1016/j.neucom.2018.02.063

    Article  Google Scholar 

  28. C. Li, F. Zhang, A survey on the stability of fractional differential equations. Eur Phys J Spec Topics 193, 27–47 (2011). https://doi.org/10.1140/epjst/e2011-01379-1

    Article  Google Scholar 

  29. G. Liu, Z. Lin, S. Yan, J. Sun, Y. Yu, Y. Ma, Robust recovery of subspace structures by low-rank representation. IEEE Trans Patt Anal Mach Intell 35(1), 171–184 (2013). https://doi.org/10.1109/TPAMI.2012.88

    Article  Google Scholar 

  30. G. Liu, H. Xu, J. Tang, Q. Liu, S. Yan, A deterministic analysis for lrr. IEEE Trans on Pattern Anal Mach Intell 38(3), 417–430 (2016). https://doi.org/10.1109/TPAMI.2015.2453969

    Article  Google Scholar 

  31. G. Liu, Q. Liu, P. Li, Blessing of dimensionality: recovering mixture data via dictionary pursuit. IEEE Trans Pattern Anal Mach Intell 39(1), 47–60 (2017). https://doi.org/10.1109/TPAMI.2016.2539946

    Article  Google Scholar 

  32. X. Liu, M. Tanaka, M. Okutomi Noise level estimation using weak textured patches of a single noisy image. In: 2012 19th IEEE International Conference on Image Processing, pp 665–668, (2012) https://doi.org/10.1109/ICIP.2012.6466947

  33. J. Mairal, F. Bach, J. Ponce, G. Sapiro, A. Zisserman Non-local sparse models for image restoration. In: 2009 IEEE 12th International Conference on Computer Vision, pp 2272–2279, (2009) https://doi.org/10.1109/ICCV.2009.5459452

  34. M. Malfait, D. Roose, Wavelet-based image denoising using a markov random field a priori model. IEEE Trans Image Process 6(4), 549–565 (1997). https://doi.org/10.1109/83.563320

    Article  Google Scholar 

  35. F. Merrikh-Bayat, M. Karimi-Ghartemani, Some properties of three-term fractional order system. Fract Calc Appl Anal 11(3), 317–328, (2008). http://www.math.bas.bg/~fcaa/volume11/fcaa112/contfc11_2._LINKS.pdf

  36. X. Mingliang, L. Pei, L. Mingyuan, F. Hao, Z. Hongling, Z. Bing, L. Yusong, Z. Liwei, Medical image denoising by parallel non-local means. Neurocomputing 195, 117–122 (2016) http://www.sciencedirect.com/science/article/pii/S092523121600120X, learning for Medical Imaging

  37. Olsen SI, Estimation of noise in images: an evaluation. CVGIP Graph Models Image Process 55(4), 319–323 (1993)

    Article  Google Scholar 

  38. P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Mach Intell 12(7), 629–639 (1990). https://doi.org/10.1109/34.56205

    Article  Google Scholar 

  39. I. Petras fractional derivatives, fractional integrals, and fractional differential equations in matlab, pp 240–262. (2011) https://doi.org/10.5772/19412

  40. I. Podlubny, Fractional Differential Equations : an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications (Academic Press, San Diego, 1999)

    MATH  Google Scholar 

  41. A. Rajwade, A. Rangarajan, A. Banerjee, Image denoising using the higher order singular value decomposition. IEEE Trans Pattern Anal Mach Intell 35(4), 849–862 (2013). https://doi.org/10.1109/TPAMI.2012.140

    Article  Google Scholar 

  42. K. Rank, M. Lendl, R. Unbehauen, Estimation of image noise variance. IEEE Proceed Vision Image Signal Process 146(2), 80–84 (1999). https://doi.org/10.1049/ip-vis:19990238

    Article  Google Scholar 

  43. Z.H. Shamsi, D.G. Kim, Multiscale hybrid nonlocal means filtering using modified similarity measure. Math Probl Eng (2015). https://doi.org/10.1155/2015/318341

    Article  MathSciNet  MATH  Google Scholar 

  44. Tai. Shen-Chuan, Shih-Ming Yang A fast method for image noise estimation using laplacian operator and adaptive edge detection. In: 2008 3rd International Symposium on Communications, Control and Signal Processing, pp 1077–1081, (2008) https://doi.org/10.1109/ISCCSP.2008.4537384

  45. S. Sudha, GR. Suresh, R. Sukanesh Wavelet based image denoising using adaptive thresholding. In: International Conference on Computational Intelligence and Multimedia Applications (ICCIMA 2007), vol 3, pp 296–300,(2007) https://doi.org/10.1109/ICCIMA.2007.305

  46. Z. Wang, A. Bovik, H. Sheikh, E. Simoncelli, Image quality assessment: from error visibility to structural similarity. Image Process IEEE Trans 13(4), 600–612 (2004). https://doi.org/10.1109/TIP.2003.819861

    Article  Google Scholar 

  47. Z. Xu, H. Chen, Z. Li, Blind image deblurring via the weighted schatten p-norm minimization prior. Circuits Syst Signal Process (2020). https://doi.org/10.1007/s00034-020-01457-z

    Article  Google Scholar 

  48. H. Yang, Y. Park, J. Yoon, B. Jeong, An improved weighted nuclear norm minimization method for image denoising. IEEE Access 7(97), 919–927 (2019)

    Google Scholar 

  49. Q. Yang, D. Chen, T. Zhao, Y. Chen, Fractional calculus in image processing: A review. Fractional Calculus and Applied Analysis 19 (2016). https://doi.org/10.1515/fca-2016-0063

  50. Y. You, M. Kaveh, Fourth-order partial differential equations for noise removal. IEEE Trans Image Process 9(10), 1723–1730 (2000). https://doi.org/10.1109/83.869184

    Article  MathSciNet  MATH  Google Scholar 

  51. X. Zhu, P. Milanfar, Automatic parameter selection for denoising algorithms using a no-reference measure of image content. IEEE Trans Image Process 19(12), 3116–3132 (2010). https://doi.org/10.1109/TIP.2010.2052820

    Article  MathSciNet  MATH  Google Scholar 

  52. D. Zoran, Y. Weiss From learning models of natural image patches to whole image restoration. In: 2011 International Conference on Computer Vision, pp 479–486, (2011)https://doi.org/10.1109/ICCV.2011.6126278

Download references

Acknowledgements

This work was supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea Government (MSIT) (No. 2020-0-01343, Artificial Intelligence Convergence Research Center (Hanyang University ERICA))

Author information

Authors and Affiliations

  1. Department of Mathematics, University of the Punjab, Lahore, 54590, Pakistan

    Zahid Hussain Shamsi, Mukhtar Hussain & Rana Muhammad Bakhtawar Khan Sajawal

  2. Department of Applied Mathematics, Hanyang University (ERICA), Ansan, 15588, Kyunggi-do, Korea

    Dai-Gyoung Kim

Authors
  1. Zahid Hussain Shamsi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dai-Gyoung Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mukhtar Hussain
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rana Muhammad Bakhtawar Khan Sajawal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zahid Hussain Shamsi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest

Additional information

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

Shamsi, Z.H., Kim, DG., Hussain, M. et al. Low-Rank Estimation for Image Denoising Using Fractional-Order Gradient-Based Similarity Measure. Circuits Syst Signal Process 40, 4946–4968 (2021). https://doi.org/10.1007/s00034-021-01700-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-021-01700-1

Keywords

Search

Navigation