Skip to main content
SpringerLink
Log in

Two efficient three-term conjugate gradient methods for impulse noise removal from medical images

  • Published:
Multimedia Tools and Applications Aims and scope Submit manuscript

Abstract

In this paper, we discuss two efficient three-term conjugate gradient methods (ECG) for impulse noise removal. The directions of ECG are first the direction of steepest descent and then spanned by the three terms: The steepest descent direction, the previous direction, and the gradient differences at the previous and current points. The second and third terms are scaled by two different step sizes called conjugate gradient parameters. Our goal is to generate and control these parameters such that they do not jointly dominate while preserving the effect of all terms, except near the optimizer where the first term dominates the other two terms. They are independent of the line search method and useful for finite precision arithmetic. The global convergence of ECG is proved. The efficiency (the lowest relative cost of function evaluations) and robustness (highest number of solved problems ) of ECG compared to known conjugate gradient methods are shown in terms of PSNR (peak signal noise ratio) and time in seconds.

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. Anisha K, Wilscy M (2011) Impulse noise removal from medical images using fuzzy genetic algorithm. Int J Multimed Appl 3:93–106

    Google Scholar 

  2. L. Arman Y, Xu M, Rostami F (2020) Rahpeymaii, Some three-term conjugate gradient methods for solving unconstrained optimization problems, Pacific Journal of Optimization

  3. Beale EML (1972) A derivative of conjugate gradients In: Lootsma, F.A (ed.) Numerical Methods for Nonlinear Optimization, Academic, London. 39–43

  4. Black MA (1996) Rangarajan, On the unification of line processes, outlier rejection, and robust statistics with applications to early vision. Int J Comput Vis 19:57–91

  5. Bouman C, Sauer K (1995) On discontinuity-adaptive smoothness priors in computer vision. IEEE Trans Pattern Anal Mach Intell 17:576–586

    Article  Google Scholar 

  6. Cai JF, Chan RH, Morini B (2007) Minimization of an edge-preserving regularization functional by conjugate gradient type methods, image processing based on partial differential equations. In: Mathematics and Visualization, Springer, Berlin Heidelberg, pp. 109–122. https://doi.org/10.1007/978-3-540-33267-1_7

  7. Chan RH, Ho CW, Nikolova M (2005) Salt-and-pepper noise removal by median-type noise detector and edge-preserving regularization. IEEE Trans Image Process 14:1479–1485

    Article  Google Scholar 

  8. Chan R, Hu C, Nikolova M (2004) Iterative procedure for removing random-valued impulse noise. IEEE Signal Process. Lett 11(12):921–924. https://doi.org/10.1109/lsp.2004.838190

    Article  Google Scholar 

  9. Chan TF, Shen J, Zhou H (2006) Total variation wavelet inpainting. J Math Imaging Vision 25:107–125. https://doi.org/10.1007/s10851-006-5257-3

    Article  MathSciNet  Google Scholar 

  10. Charbonnier P, Blanc-Féraud L, Aubert G, Barlaud M (1997) Deterministic edge-preserving regularization in computed imaging. IEEE Trans Image Process 6:298–311

    Article  Google Scholar 

  11. Chen J, Zhan Y, Cao H (2020) Iterative deviation filter for fixed-valued impulse noise removal. Multimed Tools Appl 79:23695–23710

    Article  Google Scholar 

  12. Dai YH, Yuan Y (1999) A nonlinear conjugate gradient method with a strong global convergence property. SIAM J Optim 10:177–182

    Article  MathSciNet  Google Scholar 

  13. Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91:201–213

    Article  MathSciNet  Google Scholar 

  14. Fletcher R, Reeves C (1964) Function minimization by conjugate gradients. Comput J 7(2):149–154

    Article  MathSciNet  Google Scholar 

  15. Green PJ (1990) Bayesian reconstructions from emission tomography data using a modified EM algorithm, IEEE Transactions on Medical Imaging, MI-9, 84–93

  16. Hager WW, Zhang H (2005) A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J Optim 16:170–192

    Article  MathSciNet  Google Scholar 

  17. Halder A, Choudhuri R (2022) Impulse Noise Removal Algorithm Using Modified Adaptive Distance-Related Weighted Mean Filter. In: Das A.K., Nayak J., Naik B., Dutta S., Pelusi D. (eds) Computational Intelligence in Pattern Recognition. Advances in Intelligent Systems and Computing, vol 1349. Springer, Singapore

  18. Hestenes MR, Stiefel EL (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49:409–436

    Article  MathSciNet  Google Scholar 

  19. Hwang H, Haddad RA (1995) Adaptive medianfilters: new algorithms and results. IEEE Trans Image Process 4:499–502

    Article  Google Scholar 

  20. Karthikeyan K, Chandrasekar C (2011) Speckle Noise Reduction of Medical Ultrasound Images using Bayesshrink Wavelet Threshold. Int J Comput Appl 22:8–14

    Google Scholar 

  21. Kimiaei M, Rahpeymaii F (2019) Impulse noise removal by an adaptive trust-region method. Soft Comput 23:11901–11923

    Article  Google Scholar 

  22. Kimiaei M, Rostami M (2016) Impulse noise removal based on new hybrid spectral conjugate gradient approach. KYBERNETIKA 52(5):791–823

    MathSciNet  Google Scholar 

  23. Liu J, Cao H, Zhao Y, Zhang L (2020) A gradient-type iterative method for impulse noise removal. Numer Linear Algebra Appl 28(4)

  24. Nadeem M, Hussain A, Munir A, Habib M, Tahir Naseem M (2020) Removal of random valued impulse noise from grayscale images using quadrant based spatially adaptive fuzzy filter. Signal Process 169:107403

    Article  Google Scholar 

  25. Nikolova M (2004) A variational approach to remove outliers and impulse noise. Journal of Math Imaging Vis 20:99–120

    Article  MathSciNet  Google Scholar 

  26. Polak E, Ribière G (1969) Note sur la convergence de directions conjugées. Rev Francaise Informat Recherche Opertionelle 3e Année. 16:35–43

  27. Russo F, Ramponi F (2000) A Fuzzy filter for images corrupted by impulse noise. IEEE Trans Image Process 3:168–170

    Google Scholar 

  28. Rytsar YB, Ivasenko IB (1997) Application of (alpha, beta)-trimmed mean filtering for removal of additive noise from images. SPIE Proceeding. Optoelectronic and Hybrid Optical/Digital Systems for Image Processing 45–52

  29. Shah A, Bangash JI, Khan AW, Ahmed I, Khan A, Khan A, Khan A (2020) Comparative analysis of median filter and its variants for removal of impulse noise from gray scale images. Journal of King Saud University - Computer and Information Sciences

  30. Shukla HS, Kumar N, Tripathi RP (2014) Median Filter based Wavelet Transform for Multilevel Noise. Int J Comput Appl 107(14):11–14

    Google Scholar 

  31. Wang L, Lu J, Li Y, Yahagi T, Okamoto T (2005) Noise reduction using wavelet with application to medical X-ray image. Int Conf Ind Technol 20:33–38

    Google Scholar 

  32. Wang L, Xiao D, Hou WS, Wu XY, Chen L (2021) Weighted Schatten p-norm minimization for impulse noise removal with TV regularization and its application to medical images. Biomed Signal Process Control 66:102123

    Article  Google Scholar 

  33. P. Wolfe (1971) Convergence conditions for ascent methods. II: some corrections. SIAM Rev 13(2):185–188

  34. Yu G, Huang J, Zhou Y (2010) A descent spectral conjugate gradient method for impulse noise removal. Appl Math Lett 23:555–560

    Article  MathSciNet  Google Scholar 

  35. Yu G, Qi L, Sun Y, Zhou Y (2010) Impulse noise removal by a nonmonotone adaptive gradient method. Signal Process 90:2891–2897

    Article  Google Scholar 

  36. Zhang L, Zhou W, Li DH (2006) A descent modified Polak-Ribiére-Polyak conjugate gradient method and its global convergence. IMA J Numer Anal 26(4):629–640

    Article  MathSciNet  Google Scholar 

  37. Zhou EL, Xia BY, Li E, Wang TT (2022) An efficient algorithm for impulsive active noise control using maximum correntropy with conjugate gradient. Appl Acoust 188:108511

    Article  Google Scholar 

  38. Zoutendijk G (1970) Nonlinear programming, computational methods. In: Abadie J (ed) Integer and nonlinear programming. North-holland, Amsterdam, pp 37–86

    Google Scholar 

Download references

Funding

There is no any Funding

Author information

Authors and Affiliations

  1. Department of Computer Engineering, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran

    Ali Mousavi & Amir Sheikhahmadi

  2. Computer Engineering Department, Hamedan Branch, Islamic Azad University, Hamedan, Iran

    Mansour Esmaeilpour

Authors
  1. Ali Mousavi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mansour Esmaeilpour
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amir Sheikhahmadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mansour Esmaeilpour.

Ethics declarations

Conflicts of interests

None

Competing interests

None

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

Mousavi, A., Esmaeilpour, M. & Sheikhahmadi, A. Two efficient three-term conjugate gradient methods for impulse noise removal from medical images. Multimed Tools Appl 83, 43685–43703 (2024). https://doi.org/10.1007/s11042-023-17352-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11042-023-17352-z

Keywords

Mathematics Subject Classification

Search

Navigation