Image Restoration Based on Partial Differential Equations (PDEs)

Abstract:

Image restoration plays an important role in both the quantitative analysis and qualitative analysis of image. It directly affects the further works of analysis and processing. At present, a large number of image restoration methods are recorded in the literatures. And image restoration method based on partial differential equations(PDEs) is one of the main tools in this area. Although these methods often seem powerless for the images with complex features, image restoration method based on PDEs still has its advantages cannot be replaced. In this paper, we make a summary and appraisal on image restoration methods based on PDEs on basis of the analysis for image characteristics and predict the development trend of image restoration methods based on PDEs.

• Periodical: Advanced Materials Research (Volume 647)
• Pages: 912-917
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.647.912
• Online since: January 2013
• Authors: Xin Jiang, Ren Jie Zhang
• Keywords: De-Noising, Image Restoration, Partial Differential Equation (PDE)

References

[1] P. perona, J. Malik. Scale Space and Edge Detection Using Anisotropic Diffusion. Proc. IEEE Comp. Soc. Workshop on Computer Vision, IEEE Computer Society Press, Washington, 16-22, (1987).

[2] Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms[ J ]. Physica D, 1992, 60: 259- 268.

DOI: 10.1016/0167-2789(92)90242-f

[3] 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[J]. IEEE Transactions on Image Processing, 2003, 12( 12) : 1579- 1590.

DOI: 10.1109/tip.2003.819229

[4] Frederic Guichard and Jean-Michel Morel. Partial Differential Equations and Image Iterative Filtering. Oct., (1995).

[5] L. Alvarez, F. Guichard, P-L. Lions, J. -M. Axioms and fundamental equations in image processing. Morel. Arch. Rational Mech. Anal. Vo1. 123, 199-257, (1993).

DOI: 10.1007/bf00375127

[6] A. Witkin. Scale-space Filtering. Int. Joint Conof. On AL, Karlsruhe, 1983: 1019-1022.

[7] P. Charbonnier, L. Blanc-Feraud, G. Aubert, M. barland. Two deterministic half-quadratic regularization algorithms for computed image. Proc. IEEE Int. Conf. Image Processing, Vo1. 2, IEEE Computer Society Press, Los Alamitos, 168-172, (1994).

DOI: 10.1109/icip.1994.413553

[8] B. Fischl E.L. Schwartz. Adaptive Nonlinear Filtering for Nonlinear Diffusion Approximation in Image Processing. Proc. 13th Int. Conf. Pattern Recognition, Vol. D, 276-280, (1996).

DOI: 10.1109/icpr.1996.547430

[9] G. -H. Cottet, L. Germain. Image Processing through Reaction Combined with nonlinear Diffusion. Math. Comp., Vo1. 61，659-673, (1993).

DOI: 10.1090/s0025-5718-1993-1195422-2

[10] G. -H. Cottet, M. EL Ayyadi. Nonlinear PDE Operators with Memory terms for Image Processing. Proc. IEEE Int. Conf. Image Processing, Vol. l，481-483, (1996).

DOI: 10.1109/icip.1996.559538

[11] F Catte, P L Lions, J M Morel et al. Image selective smoothing and edge detection by nonlinear diffusion[J]. SIAM J Num Anal, 1992; 29(1) : 182～193.

DOI: 10.1137/0729012

[12] Guy Gilboa, Yehoshua Y. Zeevi, Nir A. Sochen. Resolution Enhancement by Forward and Backward Nonlinear Diffusion Processes.

[13] Guy Gilboa, Nir A. Sochen Y. Zeevi. Forward and Backward Diffusion Processes for Adaptive Image Enhancement and Denosing. IEEE Transactions on Image Processing. 2002 Vol. l 1(7), 689703.

DOI: 10.1109/tip.2002.800883

[14] Guy Gilboa, Yehoshua Y. Zeevi, Nir A. Sochen. Image Enhancement Segmentation and Denoising by Time Dependent Nonlinear Diffusion Processes.

DOI: 10.1109/icip.2001.958069

[15] 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[J]. IEEE Transactions on Image Processing, 2003, 12( 12) : 1579- 1590.

DOI: 10.1109/tip.2003.819229

[16] You Y L, Kaveh M. Fourth - order partial differential equation for noise removal[J]. IEEE Transactions on Image Processing, 2000, 9( 10) : 1723- 1730.

DOI: 10.1109/83.869184

[17] Lysaker M, Osher S, Tai X C. Noise removal using smoothed normal and surface fitting[J]. IEEE Transactions on Image Processing, 2004, 13( 10) : 1345- 1357.

DOI: 10.1109/tip.2004.834662

[18] Osher S, Sole A, Vese L. Image decomposition and restoration using total variation minimization and the H-1 rm[J]. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 2003, 1( 3) : 349-370.

DOI: 10.1137/s1540345902416247

[19] Greer J B, Bertozzi A L. H1 solutions of a class of fourth order nonlinear equations for image processing[C]/Chepyzhov V, Efendiev M, Miranville A, et al. Discrete and Continuous Dynamical Systems 2004: Special Issue in Honor of Mark Vishik, 2004, 1- 2( 10) : 349-366.

DOI: 10.3934/dcds.2004.10.349

[20] Greer J B, Bertozzi A L. Traveling wave solutions of fourth order PDE for image processing[R]. Tech Report( 03- 25) , UCLA, Applied mathematics, (2003).

[21] Bertozzi A L, Greer J B. Low curvature image simplifiers: global regularity of smooth solutions and laplacian limiting schemes[R]. Tech report( 03- 26) , UCLA, Applied Mathematics, (2003).

DOI: 10.1002/cpa.20019

[22] Yu-Li You, Wenguan Xu, Allen Tannenbaum and Mostafa Kaveh, Behavioral analysis of anisotropic diffusion in image processing, IEEE Trans. Image Processing, vol. 5, No. 11, 1996, pp.1539-1553.

DOI: 10.1109/83.541424

[23] Yu-Li You, M. Kaveh, Fourth order partial differential equations for noise removal, IEEE Trans. Image Processing, vol. 9, No. 10, 2000, pp.1723-1730.

DOI: 10.1109/83.869184

[24] T. Chan, A. Marquina, and P. Mulet. High-order total variation based image restoration. SIAM J. Sci. Comp., 22(2), 2000, pp.503-516.

DOI: 10.1137/s1064827598344169

[25] P. V. Blomgren and T. F. Chan. Modular Solvers for Constrained Image Restoration Problems. Technical report, UCLA Dept. of lhlath., CAM 97-52. (1997).