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A theoretical study of a bilateral term with a tensor-based fourth-order PDE for image super-resolution

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Abstract

In this paper, we study a fourth-order partial differential equation (PDE), where a diffusion tensor is combined with a bilateral total variation (BTV) operator. This choice uses the benefit from the classical high-order diffusion model of Perona-Malik type in the homogeneous regions, the Weickert model near sharp edges and also the BTV term in reducing blur. Since the super-resolution (SR) approaches are always considered as ill-posed problem, a mathematical study of the existence and uniqueness of the solution is then ensured in a convenient Sobolev space. Also, an alternative splitting scheme is adopted to reduce the high-order of the proposed PDE. Based on the numerical experiments, we can conclude that the proposed PDE can efficiently improve the quality of the HR image. Particularly, the apparition of blur is less compared to the other methods. In addition to visual evaluation, PSNR- and SSIM-based metrics ensure the quantitative improvements realized by the proposed PDE.

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References

  1. Zhang, K., Gao, X., Tao, D., Li, X.: Single image super-resolution with non-local means and steering kernel regression. IEEE Trans. Image Process. 21(11), 4544–4556 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ma, C., Yang, C.-Y., Yang, X., Yang, M. -H.: Learning a no-reference quality metric for single-image super-resolution. Comput. Vis. Image Underst. 158, 1–16 (2017)

    Article  Google Scholar 

  3. Tsai, R. Y., Huang, T. S.: Multiframe image restoration and registration. In: Huang, T.S. (ed.) Advances in computer vision and image processing, Greenwich, CT, JAI Press. (1984)

  4. Hakim, M., Ghazdali, A., Laghrib, A.: A multi-frame super-resolution based on new variational data fidelity term. Appl. Math. Model. 87, 446–467 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hadri, A., Khalfi, H., Laghrib, A., Nachaoui, M.: An improved spatially controlled reaction–diffusion equation with a non-linear second order operator for image super-resolution. Nonlinear Anal.: Real World Appl. 62, 103352 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dong, C., Loy, C. C., He, K., Tang, X.: Image super-resolution using deep convolutional networks. IEEE Trans. Patt. Anal. Mach. Intell. 38 (2), 295–307 (2015)

    Article  Google Scholar 

  7. Dong, C., Loy, C. C., Tang, X.: Accelerating the super-resolution convolutional neural network. In: European conference on computer vision, pp. 391–407. Springer (2016 )

  8. Kong, X., Zhao, H., Qiao, Y., Dong, C.: Classsr: a general framework to accelerate super-resolution networks by data characteristic. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 12016–12025 (2021)

  9. Ho, M. M., Zhou, J., He, G.: Rr-dncnn v2. 0: enhanced restoration-reconstruction deep neural network for down-sampling-based video coding. IEEE Trans. Image Process. 30, 1702–1715 (2021)

    Article  Google Scholar 

  10. Tseng, C. W., Su, H.-R., Lai, S.-H., Liu, J.: Depth image super-resolution via multi-frame registration and deep learning. In: 2016 Asia-Pacific signal and information processing association annual summit and conference (APSIPA), pp. 1–8. IEEE (2016)

  11. Ma, Z., Liao, R., Tao, X., Xu, L., Jia, J., Wu, E.: Handling motion blur in multi-frame super-resolution. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 5224–5232 (2015)

  12. Youm, G.-Y., Bae, S.-H., Kim, M.: Image super-resolution based on convolution neural networks using multi-channel input. In: 2016 IEEE 12Th image, video, and multidimensional signal processing workshop (IVMSP), pp. 1–5,IEEE (2016)

  13. 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  MATH  Google Scholar 

  14. Laghrib, A., Hadri, A., Hakim, A., Raghay, S.: A new multiframe super-resolution based on nonlinear registration and a spatially weighted regularization. Inform. Sci. 493, 34–56 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Laghrib, A., Hakim, A., Raghay, S.: A combined total variation and bilateral filter approach for image robust super resolution. EURASIP J. Image Video Process. 2015(1), 1–10 (2015)

    Article  Google Scholar 

  17. 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 

  18. Laghrib, A., Ezzaki, M., El Rhabi, M., Hakim, A., Monasse, P., Raghay, S.: Simultaneous deconvolution and denoising using a second order variational approach applied to image super resolution. Comput. Vis. Image Underst. 168, 50–63 (2018)

    Article  Google Scholar 

  19. Zeng, X., Yang, L.: A robust multiframe super-resolution algorithm based on half-quadratic estimation with modified btv regularization. Digital Signal Process. 23(1), 98–109 (2013)

    Article  MathSciNet  Google Scholar 

  20. Marquina, A., Osher, S. J.: Image super-resolution by tv-regularization and bregman iteration. J. Sci. Comput. 37(3), 367–382 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yuan, Q., Zhang, L., Shen, H.: Multiframe super-resolution employing a spatially weighted total variation model. IEEE Trans. Circuits Syst. Video Technol. 22(3), 379–392 (2012)

    Article  Google Scholar 

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

    Article  Google Scholar 

  23. Laghrib, A., Hakim, A., Raghay, S.: An iterative image super-resolution approach based on bregman distance. Signal Process. Image Commun. 58, 24–34 (2017)

    Article  Google Scholar 

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

    Article  Google Scholar 

  25. Nachaoui, M., Afraites, L., Laghrib, A.: A regularization by denoising super-resolution method based on genetic algorithms. Signal Process.: Image Commun. 99, 116505 (2021)

    Google Scholar 

  26. 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)

    Article  Google Scholar 

  27. El Mourabit, I., El Rhabi, M., Hakim, A., Laghrib, A., Moreau, E.: A new denoising model for multi-frame super-resolution image reconstruction. Signal Process. 132, 51–65 (2017)

    Article  Google Scholar 

  28. Laghrib, A., Chakib, A., Hadri, A., Hakim, A.: A nonlinear fourth-order pde for multi-frame image super-resolution enhancement. Discrete Contin. Dynamic. Syst.-B 22(11), 0 (2019)

    MATH  Google Scholar 

  29. Bella, F. A., Hadri, A., Hakim, A., Laghrib, A.: A nonlocal weickert type pde applied to multi-frame super-resolution. Evol. Equ. Cont. Theory 10(3), 633 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hakim, A., Laghrib, A., et al: On the well-posedness of a tensor-based second order pde with bilateral term for image super-resolution, Evol. Equ. and Cont. Theory (2022)

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

  32. Laghrib, A., Ghazdali, A., Hakim, A., Raghay, S.: A multi-frame super-resolution using diffusion registration and a nonlocal variational image restoration. Comput. Math. Appl. 72(9), 2535–2548 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gilbarg, D., Trudinger, N. S., Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Springer (1977)

  34. Brezis, H., Brézis, H.: Functional analysis, Sobolev spaces and partial differential equations. Springer (2011)

  35. Zeidler, E.: Nonlinear functional analysis and its applications: III: variational methods and optimization. Springer science business media (2013)

  36. Simon, J.: Compact sets in the space l p (o, t; b). Annali di Matematica pura ed applicata 146(1), 65–96 (1986)

    Article  Google Scholar 

  37. Afraites, L., Hadri, A., Laghrib, A., Nachaoui, M.: A high order pde-constrained optimization for the image denoising problem. Inverse Problems Sci. Eng. 29(12), 1821–1863 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. You, Y.-L., Xu, W., Tannenbaum, A., Kaveh, M.: Behavioral analysis of anisotropic diffusion in image processing. IEEE Trans. Image Process. 5(11), 1539–1553 (1996)

    Article  Google Scholar 

  39. Tian, W., Ma, T., Zheng, Y., Wang, X., Tian, Y., Al-Dhelaan, A., Al-Rodhaan, M.: Weighted curvature-preserving pde image filtering method. Comput. Math. Appl. 70(6), 1336–1344 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhang, X., Ye, W.: An adaptive fourth-order partial differential equation for image denoising. Comput. Math. Appl. 74(10), 2529–2545 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yi, D., Lee, S.: Fourth-order partial differential equations for image enhancement. Appl. Math. Comput. 175(1), 430–440 (2006)

    MathSciNet  MATH  Google Scholar 

  42. Xiao, J., Pang, G., Zhang, Y., Kuang, Y., Yan, Y., Wang, Y.: Adaptive shock filter for image super-resolution and enhancement. J. Vis. Commun. Image Represent. 40, 168–177 (2016)

    Article  Google Scholar 

  43. Köhler, T., Huang, X., Schebesch, F., Aichert, A., Maier, A., Hornegger, J.: Robust multiframe super-resolution employing iteratively re-weighted minimization. IEEE Trans. Computational Imaging 2(1), 42–58 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We are grateful to the anonymous referees for the corrections and useful suggestions that have increased the quality of this article.

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Authors and Affiliations

  1. LAMAI, FST Marrakech, Université Cadi Ayyad, Marrakesh, Morocco

    El Mourabit Idriss & Abdelilah Hakim

  2. EMI FST Béni-Mellal, Université Sultan Moulay Slimane, Beni-Mellal, Morocco

    Amine Laghrib

  3. Laboratoire SIE, Université IBN ZOHR Agadir, Agadir, Morocco

    Aissam Hadri

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  1. El Mourabit Idriss
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  2. Amine Laghrib
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  3. Aissam Hadri
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  4. Abdelilah Hakim
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Correspondence to Amine Laghrib.

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Communicated by: Raymond H. Chan

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Idriss, E.M., Laghrib, A., Hadri, A. et al. A theoretical study of a bilateral term with a tensor-based fourth-order PDE for image super-resolution. Adv Comput Math 48, 83 (2022). https://doi.org/10.1007/s10444-022-09996-6

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