Skip to main content
SpringerLink
Log in

On the convexification of the Perona–Malik diffusion model

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

Of the image denoising models, the Perona–Malik one has captured the interest of most scholars for its ability to recover semantically important image features. However, this model emanates from an energy functional that is not entirely convex, a drawback that may cause undesirable solutions. In this work, we have attempted to convexify the functional by replacing its non-convex portion by the Charbonnier potential, which is strictly convex and has been reported by authors that it generates stable solutions. Extensive range of experiments have been conducted to demonstrate that our method produces compelling results.

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

Similar content being viewed by others

Notes

  1. https://www.mathworks.com/matlabcentral/fileexchange/62274-modified-perona-malik-model.

References

  1. Abdallah, M.B., Malek, J., Azar, A.T., Belmabrouk, H., Monreal, J.E., Krissian, K.: Adaptive noise-reducing anisotropic diffusion filter. Neural Comput. Appl. 27(5), 1273–1300 (2016)

    Google Scholar 

  2. Ahmed, F., Das, S.: Removal of high-density salt-and-pepper noise in images with an iterative adaptive fuzzy filter using alpha-trimmed mean. IEEE Trans. Fuzzy Syst. 22(5), 1352–1358 (2014)

    Google Scholar 

  3. Bhateja, V., Sharma, A., Tripathi, A., Satapathy, S.C., Le, D.-N.: An optimized anisotropic diffusion approach for despeckling of SAR images. In: Subramanian, S., Nadarajan, R., Rao, S., Sheen, S. (eds.) Digital Connectivity—Social Impact, pp. 134–140. Springer, Singaore (2016)

    Google Scholar 

  4. Boscaini, D., Masci, J., Rodolà, E., Bronstein, M.M., Cremers, D.: Anisotropic diffusion descriptors. In: Computer Graphics Forum, vol. 35, pp. 431–441. Wiley Online Library, Hoboken (2016)

  5. Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process. 6(2), 298–311 (1997)

    Google Scholar 

  6. Chen, G., Bui, T.D., Quach, K.G., Qian, S.-E.: Denoising hyperspectral imagery using principal component analysis and block-matching 4d filtering. Can. J. Remote Sens. 40(1), 60–66 (2014)

    Google Scholar 

  7. Dong, J., Han, Z., Zhao, Y., Wang, W., Prochazka, A., Chambers, J.: Sparse analysis model based multiplicative noise removal with enhanced regularization. Signal Process. 137, 160–176 (2017)

    Google Scholar 

  8. Dosselmann, R., Dong Yang, X.: A comprehensive assessment of the structural similarity index. SIViP 5(1), 81–91 (2011)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  10. Eslami, R., Radha, H.: The contourlet transform for image denoising using cycle spinning. In: Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, 2004, vol. 2, pp. 1982–1986. IEEE (2003)

  11. Gao, Z., Li, Q., Zhai, R., Shan, M., Lin, F.: Adaptive and robust sparse coding for laser range data denoising and inpainting. IEEE Trans. Circuits Syst. Video Technol. 26(12), 2165–2175 (2016)

    Google Scholar 

  12. Gastal, E.S.L.: Efficient high-dimensional, edge-aware filtering. IEEE Comput. Graph. Appl. 36(6), 86–95 (2016)

    Google Scholar 

  13. Getreuer, P.: Rudin-osher-fatemi total variation denoising using split Bregman. Image Process. On Line 2, 74–95 (2012)

    Google Scholar 

  14. Goldstein, T., Osher, S.: The split bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Ham, B., Cho, M., Ponce, J.: Robust guided image filtering using nonconvex potentials. IEEE Trans. Pattern Anal. Mach. Intell. 40, 192–207 (2017)

    Google Scholar 

  16. He, K., Sun, J., Tang, X.: Guided image filtering. IEEE Trans. Pattern Anal. Mach. Intell. 35(6), 1397–1409 (2013)

    Google Scholar 

  17. Heydari, M., Karami, M.-R., Babakhani, A.: A new adaptive coupled diffusion PDE for MRI Rician noise. SIViP 10(7), 1211–1218 (2016)

    Google Scholar 

  18. Jain, S.K., Ray, R.K.: Edge detectors based telegraph total variational model for image filtering. In: Satapathy, S., Mandal, J., Udgata, S., Bhateja, V. (eds.) Information Systems Design and Intelligent Applications, pp. 119–126. Springer, New Delhi (2016)

    Google Scholar 

  19. Khan, T.M., Bailey, D.G., Khan, M.A., Kong, Y.: Efficient hardware implementation for fingerprint image enhancement using anisotropic Gaussian filter. IEEE Trans. Image Process. 26, 2116–2126 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Kuhn, A., Hirschmüller, H., Scharstein, D., Mayer, H.: A tv prior for high-quality scalable multi-view stereo reconstruction. Int. J. Comput. Vis. 124, 1–16 (2016)

    MathSciNet  Google Scholar 

  21. Kumar, B.K.S.: Image denoising based on non-local means filter and its method noise thresholding. SIViP 7(6), 1211–1227 (2013)

    Google Scholar 

  22. Lahmiri, S.: An iterative denoising system based on wiener filtering with application to biomedical images. Opt. Laser Technol. 90, 128–132 (2017)

    Google Scholar 

  23. Li, W., Li, Q., Gong, W., Tang, S.: Total variation blind deconvolution employing split Bregman iteration. J. Vis. Commun. Image Represent. 23(3), 409–417 (2012)

    Google Scholar 

  24. Li, Z., Leng, S., Yu, L., Manduca, A., McCollough, C.H.: An effective noise reduction method for multi-energy CT images that exploits spatio-spectral features. Med. Phys. 44, 1610–1623 (2017)

    Google Scholar 

  25. Lien, C.-Y., Huang, C.-C., Chen, P.-Y., Lin, Y.-F.: An efficient denoising architecture for removal of impulse noise in images. IEEE Trans. Comput. 62(4), 631–643 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Liu, Q., Zhang, C., Guo, Q., Hui, X., Zhou, Y.: Adaptive sparse coding on PCA dictionary for image denoising. Vis. Comput. 32(4), 535–549 (2016)

    Google Scholar 

  27. Ma, J., Plonka, G.: Combined curvelet shrinkage and nonlinear anisotropic diffusion. IEEE Trans. Image Process. 16(9), 2198–2206 (2007)

    MathSciNet  Google Scholar 

  28. Maiseli, B., Elisha, O., Mei, J., Gao, H.: Edge preservation image enlargement and enhancement method based on the adaptive Perona–Malik non-linear diffusion model. IET Image Process. 8(12), 753–760 (2014)

    Google Scholar 

  29. Maiseli, B., Chuan, W., Mei, J., Liu, Q., Gao, H.: A robust super-resolution method with improved high-frequency components estimation and aliasing correction capabilities. J. Frankl. Inst. 351(1), 513–527 (2014)

    MATH  Google Scholar 

  30. Maiseli, B.J., Liu, Q., Elisha, O.A., Gao, H.: Adaptive Charbonnier superresolution method with robust edge preservation capabilities. J. Electron. Imaging 22(4), 043027–043027 (2013)

    Google Scholar 

  31. Maiseli, B.J., Ally, N., Gao, H.: A noise-suppressing and edge-preserving multiframe super-resolution image reconstruction method. Signal Process. Image Commun. 34, 1–13 (2015)

    Google Scholar 

  32. Maiseli, B.J., Elisha, O.A., Gao, H.: A multi-frame super-resolution method based on the variable-exponent nonlinear diffusion regularizer. EURASIP J. Image Video Process. 2015(1), 22 (2015)

    Google Scholar 

  33. Maiseli, B.J., Gao, H.: Robust edge detector based on anisotropic diffusion-driven process. Inf. Process. Lett. 116(5), 373–378 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Ogada, E.A., Guo, Z., Wu, B.: An alternative variational framework for image denoising. In: Dugelay, J-L. (ed.) Abstract and Applied Analysis, vol. 2014. Hindawi Publishing Corporation, London (2014)

  35. Pandit, D.: Random motion of image sensor based super-resolution technique. IJAR 2(9), 08–12 (2016)

    Google Scholar 

  36. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Google Scholar 

  37. Poornima, D., Karegowda, A.G., Bharathi, P.T.: A pragmatic review of denoising techniques applied for medical images. Int. J. Appl. Res. Inf. Technol. Comput. 8(1), 81–104 (2017)

    Google Scholar 

  38. Poungponsri, S., Xiao-Hua, Y.: An adaptive filtering approach for electrocardiogram (ECG) signal noise reduction using neural networks. Neurocomputing 117, 206–213 (2013)

    Google Scholar 

  39. Pyatykh, S., Hesser, J., Zheng, L.: Image noise level estimation by principal component analysis. IEEE Trans. Image Process. 22(2), 687–699 (2013)

    MathSciNet  MATH  Google Scholar 

  40. Rafsanjani, H.K., Sedaaghi, M.H., Saryazdi, S.: An adaptive diffusion coefficient selection for image denoising. Digit. Signal Process. 64, 71–82 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  42. Sivakumar, R.: Denoising of computer tomography images using curvelet transform. ARPN J. Eng. Appl. Sci. 2(1), 21–26 (2007)

    MathSciNet  Google Scholar 

  43. Starck, J.-L., Candès, E.J., Donoho, D.L.: The curvelet transform for image denoising. IEEE Trans. Image Process. 11(6), 670–684 (2002)

    MathSciNet  MATH  Google Scholar 

  44. Tang, Y., Chen, Y., Ning, X., Jiang, A., Zhou, L.: Image denoising via sparse coding using eigenvectors of graph Laplacian. Digit. Signal Process. 50, 114–122 (2016)

    Google Scholar 

  45. Tasdizen, T.: Principal components for non-local means image denoising. In: 15th IEEE International Conference on Image Processing, 2008. ICIP 2008, pp. 1728–1731. IEEE (2008)

  46. Wang, Y., Yang, Y., Chen, T.: Spectral-spatial adaptive and well-balanced flow-based anisotropic diffusion for multispectral image denoising. J. Vis. Commun. Image Represent. 43, 185–197 (2017)

    Google Scholar 

  47. Wang, Z., Bovik, A.C.: Mean squared error: love it or leave it? A new look at signal fidelity measures. IEEE Signal Process. Mag. 26(1), 98–117 (2009)

    Google Scholar 

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

    Google Scholar 

  49. Weickert, J.: Anisotropic Diffusion in Image Processing, vol. 1. Teubner, Stuttgart (1998)

    MATH  Google Scholar 

  50. Wu, B., Achieng Ogada, E., Sun, J., Guo, Z.: A total variation model based on the strictly convex modification for image denoising. In: Sharma, G. (ed.) Abstract and Applied Analysis, vol. 2014. Hindawi Publishing Corporation, London (2014)

  51. Xu, J., Jia, Y., Shi, Z., Pang, K.: An improved anisotropic diffusion filter with semi-adaptive threshold for edge preservation. Signal Process. 119, 80–91 (2016)

    Google Scholar 

  52. Xu, L., Li, J., Shu, Y., Peng, J.: SAR image denoising via clustering-based principal component analysis. IEEE Trans. Geosci. Remote Sens. 52(11), 6858–6869 (2014)

    Google Scholar 

  53. Zhang, L., Dong, W., Zhang, D., Shi, G.: Two-stage image denoising by principal component analysis with local pixel grouping. Pattern Recogn. 43(4), 1531–1549 (2010)

    MATH  Google Scholar 

  54. Zhu, L., Gao, F., Wang, W., Wang, Q., Qin, J., Zhao, Y., Zhou, F., Zhang, H., Heng, P.-A.: Feature asymmetry anisotropic diffusion for speckle reduction. J. Med. Imaging Health Inform. 7(1), 197–202 (2017)

    Google Scholar 

  55. Zuo, W., Zhang, L., Song, C., Zhang, D., Gao, H.: Gradient histogram estimation and preservation for texture enhanced image denoising. IEEE Trans. Image Process. 23(6), 2459–2472 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electronics and Telecommunications Engineering, College of Information and Communication Technologies, University of Dar es Salaam, 16103, Dar es Salaam, Tanzania

    Baraka Jacob Maiseli

Authors
  1. Baraka Jacob Maiseli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Baraka Jacob Maiseli.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

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

Maiseli, B.J. On the convexification of the Perona–Malik diffusion model. SIViP 14, 1283–1291 (2020). https://doi.org/10.1007/s11760-020-01663-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-020-01663-x

Keywords

Search

Navigation