SpringerLink
Log in

A novel robust fractional-time anisotropic diffusion for multi-frame image super-resolution

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we propose an image multi-frame Super Resolution (SR) method based on fractional-time Caputo derivative combined with Weickert-type diffusion process idea. We provide the existence and uniqueness results with a detailed discretization using the finite difference scheme. Our approach is based on anisotropic diffusion behavior with coherence enhancing diffusion tensor together with the fractional-time derivative to benefit from its memory effect potential and to control the smoothing process near strong edges and flat regions while avoiding tiny corners destruction. The experimental results confirm the effectiveness of fractional-time derivative and the robustness of the proposed Partial Differential Equation (PDE) compared with some competitive super-resolution methods.

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
EUR 32.99 /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 includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Holland, D., Boyd, D., Marshall, P.: Updating topographic map** in great britain using imagery from high-resolution satellite sensors. ISPRS J. Photogramm. Remote. Sens. 60(3), 212–223 (2006)

    Article  Google Scholar 

  2. Lv, Z., Jia, Y., Zhang, Q.: Joint image registration and point spread function estimation for the super-resolution of satellite images. Signal Processing: Image Communication 58, 199–211 (2017)

    Google Scholar 

  3. Robinson, M.D., Chiu, S.J., Lo, J., Toth, C., Izatt, J., Farsiu, S.: New applications of super-resolution in medical imaging. Super-Resolution Imaging 2010, 384–412 (2010)

    Google Scholar 

  4. Greenspan, H.: Super-resolution in medical imaging. Comput. J. 52(1), 43–63 (2008)

    Article  Google Scholar 

  5. Greenspan, H., Oz, G., Kiryati, N., Peled, S.: Mri inter-slice reconstruction using super-resolution. Magn. Reson. Imaging 20(5), 437–446 (2002)

    Article  Google Scholar 

  6. Park, S.C., Park, M.K., Kang, M.G.: Super-resolution image reconstruction: a technical overview. IEEE Signal Process. Mag. 20(3), 21–36 (2003)

    Article  Google Scholar 

  7. Yao, W., Shen, J., Guo, Z., Sun, J., Wu, B.: A total fractional-order variation model for image super-resolution and its sav algorithm. J. Sci. Comput. 82(3), 1–18 (2020)

    Article  MathSciNet  Google Scholar 

  8. Laghrib, A., Ben-Loghfyry, A., Hadri, A., Hakim, A.: A nonconvex fractional order variational model for multi-frame image super-resolution. Signal Processing: Image Communication 67, 1–11 (2018)

    Google Scholar 

  9. Lou, Y., Yin, P., **n, J.: Point source super-resolution via non-convex \( l_1\) based methods. J. Sci. Comput. 68(3), 1082–1100 (2016)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  11. Zhang, H., Zhang, L., Shen, H.: A super-resolution reconstruction algorithm for hyperspectral images. Signal Process. 92(9), 2082–2096 (2012)

    Article  Google Scholar 

  12. He, Y., Yap, K.-H., Chen, L., Chau, L.-P.: A nonlinear least square technique for simultaneous image registration and super-resolution. IEEE Trans. Image Process. 16(11), 2830–2841 (2007)

    Article  MathSciNet  Google Scholar 

  13. He, Y., Yap, K.-H., Chen, L., Chau, L.-P.: Blind super-resolution image reconstruction using a maximum a posteriori estimation. In: Image Processing, 2006 IEEE International conference on. IEEE, pp. 1729–1732 (2006)

  14. Farsiu, S., Elad, M., Milanfar, P.: Multiframe demosaicing and super-resolution of color images. IEEE Trans. Image Process. 15(1), 141–159 (2006)

    Article  Google Scholar 

  15. Baker, S., Kanade, T.: Limits on super-resolution and how to break them. IEEE Trans. Pattern Anal. Mach. Intell. 24(9), 1167–1183 (2002)

    Article  Google Scholar 

  16. Ben-loghfyry, A., Hakim, A.: Total variable-order variation as a regularizer applied on multi-frame image super-resolution. The visual computer, 1–11 (2023)

  17. Gao, M., Qin, S.: High performance super-resolution reconstruction of multi-frame degraded images with local weighted anisotropy and successive regularization. Optik-International Journal for Light and Electron Optics 126(23), 4219–4227 (2015)

    Article  Google Scholar 

  18. Lu, J., Zhang, H., Sun, Y.: Video super resolution based on non-local regularization and reliable motion estimation. Signal Processing: Image Communication 29(4), 514–529 (2014)

    Google Scholar 

  19. Purkait, P., Chanda, B.: Super resolution image reconstruction through bregman iteration using morphologic regularization. IEEE Trans. Image Process. 21(9), 4029–4039 (2012)

    Article  MathSciNet  Google Scholar 

  20. Zhang, L., Zhang, H., Shen, H., Li, P.: A super-resolution reconstruction algorithm for surveillance images. Signal Process. 90(3), 848–859 (2010)

    Article  Google Scholar 

  21. Yue, L., Shen, H., Li, J., Yuan, Q., Zhang, H., Zhang, L.: Image super-resolution: the techniques, applications, and future. Signal Process. 128, 389–408 (2016)

    Article  Google Scholar 

  22. Shen, H., Peng, L., Yue, L., Yuan, Q., Zhang, L.: Adaptive norm selection for regularized image restoration and super-resolution. IEEE transactions on cybernetics 46(6), 1388–1399 (2016)

    Article  Google Scholar 

  23. Yue, L., Shen, H., Yuan, Q., Zhang, L.: A locally adaptive l 1- l 2 norm for multi-frame super-resolution of images with mixed noise and outliers. Signal Process. 105, 156–174 (2014)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Lysaker, M., Lundervold, A., Tai, X.-C.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)

    Article  Google Scholar 

  26. Pierre, F., Aujol, J.-F., Bugeau, A., Steidl, G., Ta, V.-T.: Variational contrast enhancement of gray-scale and rgb images. Journal of Mathematical Imaging and Vision 57(1), 99–116 (2017)

    Article  MathSciNet  Google Scholar 

  27. Farsiu, S., Robinson, M.D., Elad, M., Milanfar, P.: Fast and robust multiframe super resolution. IEEE Trans. Image Process. 13(10), 1327–1344 (2004)

    Article  Google Scholar 

  28. Zeng, W., Lu, X., Fei, S.: Image super-resolution employing a spatial adaptive prior model. Neurocomputing 162, 218–233 (2015)

    Article  Google Scholar 

  29. Ren, Z., He, C., Zhang, Q.: Fractional order total variation regularization for image super-resolution. Signal Process. 93(9), 2408–2421 (2013)

    Article  Google Scholar 

  30. Ben-loghfyry, A., Hakim, A.: Robust time-fractional diffusion filtering for noise removal. Mathematical Methods in the Applied Sciences 45(16), 9719–9735 (2022)

    Article  MathSciNet  Google Scholar 

  31. Ben-Loghfyry, A., Hakim, A., Laghrib, A.: A denoising model based on the fractional beltrami regularization and its numerical solution. Journal of Applied Mathematics and Computing, 1–33 (2022)

  32. Ben-Loghfyry, A., Hakim, A.: Time-fractional diffusion equation for signal and image smoothing. Math. Modeling and Comput. 9(2), 351–364 (2022)

    Article  Google Scholar 

  33. Weickert, J.: Applications of nonlinear diffusion in image processing and computer vision. Acta Math. Univ. Comenianae 70(1), 33–50 (2001)

    MathSciNet  Google Scholar 

  34. Bella, F.A., Hadri, A., Hakim, A., Laghrib, A.: A nonlocal weickert type pde applied to multi-frame super-resolution. Evolution Equations & Control Theory 10(3), 633 (2021)

    Article  MathSciNet  Google Scholar 

  35. Laghrib, A., Hadri, A., Hakim, A.: An edge preserving high-order pde for multiframe image super-resolution. J. Franklin Inst. 356(11), 5834–5857 (2019)

    Article  MathSciNet  Google Scholar 

  36. Li, H., Yu, Z., Mao, C.: Fractional differential and variational method for image fusion and super-resolution. Neurocomputing 171, 138–148 (2016)

    Article  Google Scholar 

  37. Oliveira, D.S., de Oliveira, E.C.: On a caputo-type fractional derivative. Advances in Pure and Applied Mathematics 10(2), 81–91 (2019)

    Article  MathSciNet  Google Scholar 

  38. Li, C., Qian, D., Chen, Y.: On riemann-liouville and caputo derivatives. Discrete Dynamics in Nature and Society 2011 (2011)

  39. Cuahutenango-Barro, B., Taneco-Hernández, M., Gómez-Aguilar, J.: On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel. Chaos, Solitons & Fractals 115, 283–299 (2018)

    Article  MathSciNet  Google Scholar 

  40. Cuahutenango-Barro, B., Taneco-Hernández, M., Lv, Y.-P., Gómez-Aguilar, J., Osman, M., Jahanshahi, H., Aly, A.A.: Analytical solutions of fractional wave equation with memory effect using the fractional derivative with exponential kernel. Results in Physics 25, 104148 (2021)

    Article  Google Scholar 

  41. Singh, J., Ganbari, B., Kumar, D., Baleanu, D.: Analysis of fractional model of guava for biological pest control with memory effect. J. Adv. Res. 32, 99–108 (2021)

    Article  Google Scholar 

  42. Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198. Elsevier, ??? (1998)

  43. Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)

    Article  MathSciNet  Google Scholar 

  44. Alikhanov, A.: A priori estimates for solutions of boundary value problems for fractional-order equations. Differential Equations 46(5), 660–666 (2010)

    Article  MathSciNet  Google Scholar 

  45. Schwartz, J.T., Karcher, H.: Nonlinear functional analysis. CRC Press, ??? (1969)

  46. Ben-Loghfyry, A., Charkaoui, A.: Regularized perona & malik model involving caputo time-fractional derivative with application to image denoising. Chaos, Solitons & Fractals 175, 113925 (2023)

    Article  MathSciNet  Google Scholar 

  47. Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Springer, ??? (2010)

  48. Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization. SIAM, ??? (2014)

  49. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker guide to the fractional sobolev spaces. Bulletin des sciences mathématiques 136(5), 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  50. Weickert, J.: Scale-space properties of nonlinear diffusion filtering with a diffusion tensor (1994)

  51. Murio, D.A.: Implicit finite difference approximation for time fractional diffusion equations. Computers & Mathematics with Applications 56(4), 1138–1145 (2008)

    Article  MathSciNet  Google Scholar 

  52. Werlberger, M., Trobin, W., Pock, T., Wedel, A., Cremers, D., Bischof, H.: Anisotropic huber-l1 optical flow. In: BMVC, vol. 1, p. 3 (2009)

  53. Weickert, J.: Anisotropic diffusion in image processing, vol. 1. Teubner Stuttgart, ??? (1998)

Download references

Acknowledgements

We would like to thank the editor and anonymous reviewers for their helpful comments and valuable suggestions.

Author information

Authors and Affiliations

  1. LMCMAN, Faculty of Sciences and Technologies Mohammedia, University Hassan II, Casablanca, Morocco

    Anouar Ben-loghfyry

  2. Department of Mathematics, LAMAI FST, University of Cadi Ayyad, Marrakesh, Morocco

    Abdelilah Hakim

Authors
  1. Anouar Ben-loghfyry
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdelilah Hakim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anouar Ben-loghfyry.

Ethics declarations

Consent for publication

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Additional information

Communicated by: Raymond H. Chan

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

Ben-loghfyry, A., Hakim, A. A novel robust fractional-time anisotropic diffusion for multi-frame image super-resolution. Adv Comput Math 49, 79 (2023). https://doi.org/10.1007/s10444-023-10079-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-023-10079-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation