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Hybrid Equation Resulting from the Combination of a Second-Order Equation with a Fourth-Order Equation for the Restoration of Noisy Images

Simo Thierry*, Ntsama Eloundou Pascal**, Noura Alexendre***

* - *** Department of Physics, University of Ngaoundere, Ngaoundere, Cameroon.

Periodicity:January - March'2020
DOI : https://doi.org/10.26634/jip.7.1.17038

Abstract

The filtering of images is an important task in the research area of image processing. The diffusion equations are frequently used in image processing to solve the problem of staircase effect introduced by the second order diffusion equations. Different equations dissemination of fourth order is proposed to improve the filtering performance. However, the diffusion equations of order four suffer from the problems of on-smoothing of the staircase effect and a very slow convergence towards the solution. In this paper, a hybrid equation that uses a convex combination to associate the second-order equation of Wang in the equation of the fourth-order Lysaker is proposed to reap the benefits of two equations combined for restoring noisy images. In this hybrid equation, a diffusion coefficient which uses the Z-shaped fuzzy logic membership function is proposed to limit the diffusion of strong gradients produced by the equation of the second order Wang et al for better preserve the contours of the image. The experimental results illustrate the effectiveness of the proposed model in image restoration.

Keywords

Partial Differential Equation (PDE), Gradient Vector Convolution (GVC), Restoration, Diffusion Coefficient, Noise.

How to Cite this Article?

Thierry, S., Pascal, N. E., and Alexendre, N. (2020). Hybrid Equation Resulting from the Combination of a Second-Order Equation with a Fourth-Order Equation for the Restoration Noisy Images. i-manager's Journal on Image Processing , 7(1), 24-34. https://doi.org/10.26634/jip.7.1.17038

References

[1]. Alvarez, L., Lions, P. L., & Morel, J. M. (1992). Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM Journal on Numerical Analysis, 29 (3), 845- 866. https://doi.org/10.1137/0729052

[2]. Buades, A., Coll, B., & Morel, J. M. (2005). A review of image denoising algorithms, with a new one. Multiscale Modeling & Simulation, 4(2), 490-530. https://doi.org/ 10.1137/040616024

[3]. Chang, S. G., Yu, B., & Vetterli, M. (2000a). Adaptive wavelet thresholding for image denoising and compression. IEEE Transactions on Image Processing, 9(9), 1532-1546. https://doi.org/10.1109/83.862633

[4]. Chang, S. G., Yu, B., & Vetterli, M. (2000b). Spatially adaptive wavelet thresholding with context modeling for image denoising. IEEE Transactions on Image Processing, 9(9), 1522-1531. https://doi.org/10.1109/83.862630

[5]. Chen, Y., Vemuri, B. C., & Wang, L. (2000). Image denoising and segmentation via nonlinear diffusion. Computers & Mathematics with Applications, 39(5-6), 131-149. https://doi.org/10.1016/S0898-1221(00) 00050-X

[6]. Deng, G., & Cahill, L. W. (1993, October). An adaptive Gaussian filter for noise reduction and edge detection. In 1993 IEEE Conference Record Nuclear S c i e n c e S y m p o s i u m a n d M e d i c a l I m a g i n g Conference (pp. 1615-1619). IEEE. https://doi.org/ 10.1109/nssmic.1993.373563

[7]. Deng, L., Zhu, H., Yang, Z., & Li, Y. (2019). Hessian matrix-based fourth-order anisotropic diffusion filter for image denoising. Optics & Laser Technology, 110, 184- 190. https://doi.org/10.1016/j.optlastec.2018.08.043

[8]. Fu, S., Ruan, Q., Wang, W., & Chen, J. (2006, August). Region-based shock-diffusion equation for adaptive image enhancement. In International Workshop on Intelligent Computing in Pattern Analysis and Synthesis (pp. 387-395). Springer, Berlin, Heidelberg. https://doi.org/ 10.1007/11821045_41

[9]. Gilboa, G., Sochen, N., & Zeevi, Y. Y. (2004). Image enhancement and denoising by complex diffusion processes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(8), 1020-1036. https:// doi.org/10.1109/TPAMI.2004.47

[10]. Gilboa, G., Sochen, N., & Zeevi, Y. Y. (2002). For ward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Transactions on Image Processing, 11(7), 689-703. https://doi.org/10.1109/ TIP.2002.800883

[11]. Hajiaboli, M. R. (2009, January). A self-governing hybrid model for noise removal. In Pacific-Rim Symposium on Image and Video Technology (pp. 295- 305). Springer, Berlin, Heidelberg. https://doi.org/ 10.1007/978-3-540-92957-4_26

[12]. Jansen, M., & Bultheel, A. (2001). Empirical bayes approach to improve wavelet thresholding for image noise reduction. Journal of the American Statistical Association, 96(454), 629-639. https://doi.org/10.1198/ 016214 501753168307

[13]. Jiang, X. (2011). Iterative truncated arithmetic mean filter and its properties. IEEE Transactions on Image Processing, 21(4), 1537-1547. https://doi.org/ 10.1109/ TIP.2011.2172805

[14]. Liu, T., & Xiang, Z. (2013). Image restoration combining the second-order and fourth-order PDEs. Mathematical Problems in Engineering, 1–7. https:// doi.org/10.1155/2013/743891

[15]. Lysaker, M., Lundervold, A., & Tai, X. C. (2003). Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Transactions on Image Processing, 12(12), 1579-1590. https:// doi.org/10.1109/TIP.2003. 819229

[16]. Perona, P., & Malik, J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7), 629 - 639. https://doi.org/10.1109/34.56205

[17]. Pitas, I., & Venetsanopoulos, A. N. (1990). Median filters. In Nonlinear Digital Filters (vol. 84, pp. 63-116), Boston, MA: Springer. https://doi.org/10.1007/978-1- 4757-6017-0_4

[18]. Rafsanjani, H. K., Sedaaghi, M. H., & Saryazdi, S. (2016). Efficient diffusion coefficient for image denoising. Computers & Mathematics with Applications, 72(4), 893-903. https://doi.org/10. 1016/j.camwa.2016.06.005

[19]. Rajan, J., Kannan, K., & Kaimal, M. R. (2008). An improved hybrid model for molecular image denoising. Journal of Mathematical Imaging and Vision, 31(1), 73- 79. https://doi.org/10.1007/s10851-008-0067-4

[20]. Rudin, L. I., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1-4), 259-268. https://doi.org/10.1016/0167-2789(92)90242-F

[21]. Senel, H. G., Peters, R. A., & Dawant, B. (2002). Topological median filters. IEEE Transactions on Image processing, 11(2), 89-104. https://doi.org/10.1109/ 83.98 2817

[22]. Tebini, S., Mbarki, Z., Seddik, H., & Braiek, E. B. (2016a). Rapid and efficient image restoration technique based on new adaptive anisotropic diffusion function. Digital Signal Processing, 48, 201-215. https://doi.org/10.1016/j.dsp.2015.09.013

[23]. Tebini, S., Seddik, H., & Braiek, E. B. (2016b). An advanced and adaptive mathematical function for an efficient anisotropic image filtering. Computers & Mathematics with Applications, 72(5), 1369-1385. https://doi.org/10.1016/j.camwa.2016.07.004

[24]. Wang, H., Wang, Y., & Ren, W. (2012). Image denoising using anisotropic second and fourth order diffusions based on gradient vector convolution. Computer Science and Information Systems, 9(4), 1493-1511. https://doi.org/10.2298/CSIS120219060W

[25]. Wang, Y., & Jia, Y. (2008, December). External force for active contours: gradient vector convolution. In Pacific Rim International Conference on Artificial Intelligence (pp. 466-472). Springer, Berlin, Heidelberg. https://doi.org/ 10.1007/978-3-540-89197-0_43

[26]. Wei, G. W. (1999). Generalized Perona-Malik equation for image restoration. IEEE Signal Processing Letters, 6(7), 165-167. https://doi.org/10.1109/ 97.769359

[27]. Whitaker, R. T., & Pizer, S. M. (1993). A multi-scale approach to nonuniform diffusion. CVGIP: Image Understanding, 57(1), 99-110. https://doi.org/10.1006/ ciun.1993.1006

[28]. You, Y. L., & Kaveh, M. (2000). Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing, 9(10), 1723-1730. https://doi.org/10.1109/83.869184

[29]. Yu, H., & Chua, C. S. (2006). GVF-based anisotropic diffusion models. IEEE Transactions on Image Processing, 15(6), 1517-1524. https://doi.org/ 10.1109/TIP.2006.871143

[30]. Zhang, L., Zhang, L., Mou, X., & Zhang, D. (2012, September). A comprehensive evaluation of full reference image quality assessment algorithms. In 2012 19th IEEE International Conference on Image Processing (pp. 1477-1480). IEEE. https://doi.org/ 10.1109/ICIP.2012.6467150

[31]. Zhong, P., & Wang, R. (2014). Jointly learning the hybrid CRF and MLR model for simultaneous denoising and classification of hyperspectral imagery. IEEE Transactions on Neural Networks and Learning Systems, 25(7), 1319-1334. https://doi.org/10.1109/TNNLS.20 13.2293061

[32]. Zhong, S., & Cherkassky, V. (2000, September). Image denoising using wavelet thresholding and model selection. In Proceedings 2000 International Conference on Image Processing (Cat. No. 00CH37101) (Vol. 3, pp. 262-265). IEEE.

Hybrid Equation Resulting from the Combination of a Second-Order Equation with a Fourth-Order Equation for the Restoration of Noisy Images

Abstract

Keywords

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