Trainable fourth-order partial differential equations for image noise removal

Document Type : Research Article

Authors

• N. Khoeiniha
• S.M. Hosseini
• R. Davoudi

Department of Applied athematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran.

10.22067/ijnao.2021.67760.1002

Abstract

Image processing by partial differential equations (PDEs) has been an active topic in the area of image denoising, which is an important task in computer vision. In PDE-based methods for unprocessed image process ing, the original image is considered as the initial value for the PDE and the solution of the equation is the outcome of the model. Despite the advan tages of using PDEs in image processing, designing and modeling different equations for various types of applications have always been a challenging and interesting problem. In this article, we aim to tackle this problem by introducing a fourth-order equation with flexible and trainable coefficients, and with the help of an optimal control problem, the coefficients are determined; therefore the proposed model adapts itself to each particular application. At the final stage, the image enhancement is performed on the noisy test image and the performance of our proposed method is compared to other PDE-based models.

Keywords

• Partial Differential Equations
• Image Processing
• Image Denoising
• Optimal Control

20.1001.1.24236977.2021.11.2.1.0

Main Subjects

• Numerical Analysis

---

References

 1. Awate, S.P., and Whitaker, R.T. Unsupervised, information-theoretic, adaptive image filtering for image restoration. IEEE Transactions on pattern analysis and machine intelligence 28, 3 (2006), 364–376.
2. Barash, D. Fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation. IEEE Transactions on Pattern Analysis and Machine Intelligence 24, 6 (2002), 844–847.
3. Besser, H. Visual access to visual images: the UC berkeley image database project. Library Trends. 38, 1990.
4. Buades, A., Coll, B., and Morel, J.-M. A review of image denoising algorithms, with a new one.Multiscale Model. Simul. 4 (2005), 490–530.
5. Chen, Q., Montesinos, P., Sun, Q.S., and Xia, D.S. Ramp preserving Perona–Malik model. Signal Processing, 90 (6) (2010), 1963–1975.
6. Dabov, K., Foi, A., Katkovnik, V., and Egiazarian, K. Image denoising with block-matching and 3d filtering. In Image Processing: Algorithms  and Systems, Neural Networks, and Machine Learning (2006), vol. 6064, International Society for Optics and Photonics, p. 606414.
7. Danielyan, A., Katkovnik, V., and Egiazarian, K. BM3D frames and variational image deblurring. IEEE Trans. Image Process. 21(4) (2012), 1715–1728.
8. Dautov, Ç.P., and Özerdem, M.S. Wavelet transform and signal denoising using wavelet method. 26th Signal Processing and Communications
Applications Conference (SIU), Izmir, Turkey, 2018, pp. 1-4.
9. Didas, S., Weickert, J., and Burgeth, B. Properties of higher order nonlinear diffusion filtering. J. Math. Imaging Vision 35 (3) (2009), 208–226.
10. Elad, M. On the origin of the bilateral filter and ways to improve it.IEEE Trans. Image Process. 11 (10) (2002), 1141–1151.
11. Gabor, D. Information theory in electron microscopy. Lab Invest. 14 (1965), 801–807.
12. Greer, J.B., and Bertozzi, A.L. Traveling wave solutions of fourth order PDEs for image processing. SIAM J. Math. Anal. 36 (1) (2004), 38–68.
13. Hajiaboli, M.R. An anisotropic fourth-order diffusion filter for imagenoise removal. Int. J. Comput. Vis. 92 (2) (2011), 177–191.
14. Hinze, M., Pinnau, R., Ulbrich, M., and Ulbrich, S. Optimization with PDE constraints, Mathematical Modelling: Theory and Applications. 23. Springer, New York, 2009.
15. Jain, A.K. Partial differential equations and finite-difference methods in image processing, part 1: Image representation. J. Optim. Theory Appl. 23 (1) (1977), 65–91.
16. Kichenassamy, S. The Perona-Malik paradox. SIAM J. Appl. Math. 57(5) (1997), 1328–1342.
17. Koenderink, J.J. The structure of images. Biol. Cybernet. 50 (5) (1984), 363–370.
18. Li, M. An improved non-local filter for image denoising. 1–4.
19. Lin, Z., Zhang, W., and Tang, X. Learning partial differential equations for computer vision. Peking Univ., Chin. Univ. of Hong Kong 2008.
20. Lions, J.L. Optimal control of systems governed by partial differential equations Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin 1971. 

21. Liu, Q., Yao, Z., and Ke, Y. Entropy solutions for a fourth-order nonlin ear degenerate problem for noise removal. Nonlinear Anal. 67 (6) (2007), 1908–1918.
22. Liu, R., Lin, Z., Zhang, W., and Su, Z. Learning PDEs for image restoration via optimal control. Daniilidis K., Maragos P., Paragios N. (eds) Computer Vision – ECCV 2010. ECCV 2010. 115–128, Lecture Notes in Computer Science, vol 6311. Springer, Berlin, Heidelberg.
23. Liu, R., Zhong, G., Cao, J., Lin, Z., Shan, S., and Luo, Z. Learning to diffuse: A new perspective to design pdes for visual analysis. IEEE transactions on pattern analysis and machine intelligence 38 (12) (2016), 2457–2471.
24. Lysaker, M., Lundervold, A., and Tai, X.-C. 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) (2003), 1579–1590.
25. Malfait, M., and Roose, D. Wavelet-based image denoising using a Markov random field a priori model. IEEE Transactions on image processing 6 (4) (1997), 549–565.
26. Milanfar, P. A tour of modern image filtering: New insights and methods, both practical and theoretical. IEEE signal processing magazine 30 (1) (2012), 106–128.
27. Mo, H., and Li, H. Image differential invariants. arXiv:1911.05327 (2019).

28. Own, C., Tsai, H., Yu, P., and Lee, Y. Adaptive type-2 fuzzy median filter design for removal of impulse noise. NSIP 2005. Abstracts. IEEE Eurasip Nonlinear Signal and Image Processing, 2005., Sapporo, Japan, 2005, Imaging Sci. J. 54 (1) (2006), 3–18.
29. Perona, P., and Malik, J. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on pattern analysis and machine intelli gence 12 (7) (1990), 629–639.
30. Philip, P. Optimal control of partial differential equations. Lecture Notes, Ludwig-Maximilians-Universität, Germany (2009).
31. Rudin, L.I., Osher, S., and Fatemi, E. Nonlinear total variation based noise removal algorithms. Experimental mathematics: computational is sues in nonlinear science (Los Alamos, NM, 1991). Phys. D 60 (1-4) (1992), 259–268.
32. Shewchuk, J.R. An introduction to the conjugate gradient method with out the agonizing pain. School of computer science. Carnegie Mellon University, Pittsburgh, PA 15213 (1994), 10.

33. Siddig, A., Guo, Z., Zhou, Z., and Wu, B. An image denoising model based on a fourth-order nonlinear partial differential equation. Comput. Math. Appl. 76 (5) (2018), 1056–1074.
34. Stoer, J., and Bulirsch, R. Introduction to numerical analysis, vol. 12. Springer Science & Business Media, 2013.
35. Tomasi, C., and Manduchi, R. Bilateral filtering for gray and color images. Sixth International Conference on Computer Vision (IEEE Cat.
No.98CH36271), Bombay, India, 1998, 839–846.
36. Tröltzsch, F. Optimal control of partial differential equations: theory, methods, and applications, Translated from the 2005 German original by Jürgen Sprekels. Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010.
37. Wang, Z., Bovik, A.C., Sheikh, H.R., and Simoncelli, E.P. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing 13 (4) (2004), 600–612.
38. Wang, Y., Chen, W., Zhou, S., Yu, T., and Zhang, Y. Mtv: modified total variation model for image noise removal. IEE Electronics Letters, 47 (10) (2011), 592–594.
39. Wang, Y., Guo, J., Chen, W., and Zhang, W. Image denoising using modified Perona–Malik model based on directional Laplacian. Signal Processing 93 (9) (2013), 2548–2558.
40. Weber, A.G. The USC-SIPI image database version 5. USC-SIPI Report 315, 1 (1997).
41. Witkin, A.P. Scale-space filtering. Readings in Computer Vision (1987) 329–332.
42. You, Y.-L., and Kaveh, M. Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing 9 (10) (2000), 1723–1730.
43. You, Y.-L., Xu, W., Tannenbaum, A., and Kaveh, M. Behavioral analysis of anisotropic diffusion in image processing. IEEE Transactions on Image Processing 5 (11) (1996), 1539–1553.
44. Zeng, W., and Lu, X. A robust variational approach to super- resolution with nonlocal tv regularisation term. Imaging Sci. J. 61 (2) (2013), 268–278.
45. Zeng, W., Lu, X., and Tan, X. A local structural adaptive partial differential equation for image denoising. Multimed. Tools Appl. 74 (3) (2015),743–757.
 46. Zhang, M., and Desrosiers, C. Image denoising based on sparse repre sentation and gradient histogram. IET Image Processing 11 (1) (2016), 54–63.
47. Zhang, X., and Ye, W. An adaptive fourth-order partial differential equation for image denoising. Comput. Math. Appl. 74 (10) (2017), 2529–2545.