Noise is regarded as an unavoidable component of digital image acquisition. Hence, noise removal has been considered as one of the fundamental tasks in the field of image processing. Accordingly, excellent results have been achieved by using second-order models. However, these outcomes are affected by the staircase effect. To eliminate this anomaly and maintaining the balance of removing noise and preserving edges, a fourth-order model is proposed. The existence and uniqueness of the entropy solution for this model are established. Besides, to to verify the effectiveness of the model in noise removal, we carried out numerical experiment and presented our results. Indeed, the experimental results show that our model is superior to PM model and ROF model in terms of removing noise and preserving edges.
Citation: Abdelgader Siddig, Zhichang Guo, Zhenyu Zhou, Boying Wu. Entropy solutions for an adaptive fourth-order nonlinear degenerate problem for noise removal[J]. AIMS Mathematics, 2021, 6(4): 3974-3995. doi: 10.3934/math.2021236
Noise is regarded as an unavoidable component of digital image acquisition. Hence, noise removal has been considered as one of the fundamental tasks in the field of image processing. Accordingly, excellent results have been achieved by using second-order models. However, these outcomes are affected by the staircase effect. To eliminate this anomaly and maintaining the balance of removing noise and preserving edges, a fourth-order model is proposed. The existence and uniqueness of the entropy solution for this model are established. Besides, to to verify the effectiveness of the model in noise removal, we carried out numerical experiment and presented our results. Indeed, the experimental results show that our model is superior to PM model and ROF model in terms of removing noise and preserving edges.
[1] | T. Barbu, PDE-based restoration model using nonlinear second and fourth order diffusions, Proc. Romanian Acad. Ser. A, 16 (2015), 138–146. |
[2] | T. Barbu, I. Munteanu, A nonlinear fourth-order diffusion-based model for image denoising and restoration, Proc. Romanian Acad. Ser. A, 18 (2017), 108–115. |
[3] | C. Brito-Loeza, K. Chen, V. Uc-Cetina, Image denoising using the gaussian curvature of the image surface, Numer. Meth. Part. Differ. Equ., 32 (2016), 1066–1089. doi: 10.1002/num.22042 |
[4] | L. Deng, H. Zhu, Z. Yang, Y. Li, Hessian matrix-based fourth-order anisotropic diffusion filter for image denoising, Opt. Laser Technol., 110 (2019), 184–190. doi: 10.1016/j.optlastec.2018.08.043 |
[5] | I. Fonseca, G. Leoni, J. Malỳ, R. Paroni, A note on meyers' theorem in $W^{k, 1}$, Trans. Am. Math. Soc., 354 (2002), 3723–3741. doi: 10.1090/S0002-9947-02-03027-1 |
[6] | P. Guidotti, K. Longo, Well-posedness for a class of fourth order diffusions for image processing, Nonlinear Differ. Equ. Appl. NoDEA, 18 (2011), 407–425. doi: 10.1007/s00030-011-0101-x |
[7] | Z. Guo, J. Yin, Q. Liu, On a reaction-diffusion system applied to image decomposition and restoration, Math. Comput. Modell., 53 (2011), 1336–1350. doi: 10.1016/j.mcm.2010.12.031 |
[8] | W. Hinterberger, O. Scherzer, Variational methods on the space of functions of bounded hessian for convexification and denoising, Computing, 76 (2006), 109–133. doi: 10.1007/s00607-005-0119-1 |
[9] | A. Laghrib, A. Chakib, A. Hadri, A. Hakim, A nonlinear fourth-order pde for multi-frame image super-resolution enhancement, Discrete Cont. Dyn. Syst.-B, 25 (2020), 415. |
[10] | A. Laghrib, A. Hadri, A. Hakim, An edge preserving high-order pde for multiframe image super-resolution, J. Franklin Inst., 356 (2019), 5834–5857. doi: 10.1016/j.jfranklin.2019.02.032 |
[11] | F. Li, C. Shen, J. Fan, C. Shen, Image restoration combining a total variational filter and a fourth-order filter, J. Vis. Commun. Image R., 18 (2007), 322–330. doi: 10.1016/j.jvcir.2007.04.005 |
[12] | G. Liu, T. Z. Huang, J. Liu, High-order TVL1-based images restoration and spatially adapted regularization parameter selection, Comput. Math. Appl., 67 (2014), 2015–2026. doi: 10.1016/j.camwa.2014.04.008 |
[13] | Q. Liu, Z. Yao, Y. Ke, Entropy solutions for a fourth-order nonlinear degenerate problem for noise removal, Nonlinear Anal., 67 (2007), 1908–1918. doi: 10.1016/j.na.2006.08.016 |
[14] | X. Liu, L. Huang, Z. Guo, Adaptive fourth-order partial differential equation filter for image denoising, Appl. Math. Lett., 24 (2011), 1282–1288. doi: 10.1016/j.aml.2011.01.028 |
[15] | X. Liu, C. H. Lai, K. Pericleous, A fourth-order partial differential equation denoising model with an adaptive relaxation method, Int. J. Comput. Math., 92 (2015), 608–622. doi: 10.1080/00207160.2014.904854 |
[16] | M. Lysaker, A. Lundervold, X. C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Proc., 12 (2003), 1579–1590. doi: 10.1109/TIP.2003.819229 |
[17] | P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629–639. doi: 10.1109/34.56205 |
[18] | L. I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D: Nonlinear Phenom., 60 (1992), 259–268. doi: 10.1016/0167-2789(92)90242-F |
[19] | D. M. Strong, Adaptive total variation minimizing image restoration, Department of Mathematics, University of California, Los Angeles, 1997. |
[20] | L. Sun, K. Chen, A new iterative algorithm for mean curvature-based variational image denoising, BIT Numer. Math., 54 (2014), 523–553. doi: 10.1007/s10543-013-0448-y |
[21] | Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Proc., 13 (2004), 600–612. doi: 10.1109/TIP.2003.819861 |
[22] | Y. Wen, J. Sun, Z. Guo, A new anisotropic fourth-order diffusion equation model based on image feature for image denoising, CAM report, 2020. |
[23] | Z. Wu, J. Yin, C. Wang, Elliptic & parabolic equations, World Scientific Publishing Company Incorporated, 2006. |
[24] | X. Y. Liu, C. H. Lai, K. A. Pericleous, A fourth-order partial differential equation denoiding model with an adaptive relaxation method, Int. J. Comut. Math., 92 (2015), 608–622. doi: 10.1080/00207160.2014.904854 |
[25] | Q. Yang, Image denoising combining the P-M model and the LLT model, J. Comput. Commun., 3 (2015), 22–30. |
[26] | Y. L. You, M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Proc., 9 (2000), 1723–1730. doi: 10.1109/83.869184 |
[27] | X. Zhang, W. Ye, An adaptive fourth-order partial differential equation for image denoising, Comput. Math. Appl., 74 (2017), 2529–2545. doi: 10.1016/j.camwa.2017.07.036 |
[28] | W. Zhu, T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sci., 5 (2012), 1–32. doi: 10.1137/110822268 |