A new PDE-based time dependent model for image restoration

Santosh Kumar; Uttam Kumar Sharma; Khursheed Alam; Girraj Kishore

doi:https://doi.org/10.14445/22315373/IJMTT-V65I11P517

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 65 | Issue 11 | Year 2019 | Article Id. IJMTT-V65I11P517 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I11P517

A new PDE-based time dependent model for image restoration

Santosh Kumar, Uttam Kumar Sharma, Khursheed Alam, Girraj Kishore

Citation :

Santosh Kumar, Uttam Kumar Sharma, Khursheed Alam, Girraj Kishore, "A new PDE-based time dependent model for image restoration," International Journal of Mathematics Trends and Technology (IJMTT), vol. 65, no. 11, pp. 168-173, 2019. Crossref, https://doi.org/10.14445/22315373/IJMTT-V65I11P517

Abstract

In this paper, we present a new time-dependent model for image restoration. This model constructed by evolving the Euler-Lagrange equations of the optimization problem. We propose to apply prior smoothness on the solution image and then denoise it by minimizing the total variation norm of the estimated solution. The main idea is to apply a priori smoothness to the solution image. 2D numerical experimental results by explicit numerical schemes are discussed.

Keywords

Total variation norm, image restoration, Lagrange’s multiplier

References

[1] L. Alvarez, P. L. Lions and J. M. Morel, Image selective smoothing and edgedetection by nonlinear diffusion II∗, SIAM J. Numer. Anal., 29(3), 1992, 845-866.
[2] F. Catte, P. L. Lions, J. M. Morel and T. Coll, Image selective smoothing and edge detection by nonliear diffusion⋆, SIAM J. Numer. Anal., 29, 1992, 182-193.
[3] T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20(6), 1999, 1964-1977.
[4] Q. Chang and I-Liang Chern, Acceleration methods for total variation-based image denoising, SIAM J. Sci. Comput., 25(3), 2003, 982-994.
[5] T. F. Chan and C. K. Wong. Total variation blind deconvolution, IEEE Transactions on Image Processing, 7, 1998, 370-375.
[6] S. Kumar, M.K. Ahmad, A time dependent model for image restoration, Journal of Signal and Information Processing, 6, 2015, 28- 38.
[7] L. Lapidus, G. F. Pinder, Numerical solution of partial differential equations inscience and engineering, SIAM Rev., 25(4), 1983, 581-582.
[8] A. Marquina, Inverse scale space methods for blind deconvolution, UCLA CAM Report, 2006, 06-36.
[9] A. Marquina and S. Osher, Explict algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM. J. Sci. Comput., 22(2), 2000, 387-405.
[10] A. Marquina, S. Osher. A new time dependent model based on level set motion for nonlinear deblurring and noise removal. In M. Nielsen, P. johansen, O. F. Olsen and J. Weickert, editors, scale-Space Theories in Computer Vision, volume 1682 of lecture notes in Computer Science, Springer Berlin, 1999, 429-434.
[11] P. Perona and J. Malik. Scale space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 1990, 629- 639.
[12] J. G. Rosen, The gradient projection method for nonlinear programming, Part II, nonlinear constraints, J. Soc. Indust. Appl. Math., 9(4), 1961, 514-532.
[13] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm, Phys. D., 60, 1992, 259-268.
[14] L. Rudin, S. Osher, Total variation based image restoration with free local constraints, Proc. IEEE Int. Conf. Imag. Proc., 1994, 31- 35.
[15] G. Strang, Accurate Partial Difference Methods II. Non Linear Problems, Numer. Math., 6, 1964, 37-46.
[16] G. Strang, On the construction and comparison of difference schemes, SIAM J. numer. Anal., 5(3), 1968, 506-517.
[17] A. P. Witkin, Scale-space filtering, Proc. IJCAI, Karlsruhe, 1983, 1019-1021.
[18] J. Weickert. Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.
[19] M. Welk, D. Theis, T. Brox, J. Weickert, PDE-based deconvolution with forward-backward diffusivities and diffusion tensors. In scale space, LNCS., Springer, Berlin2005, 585-597.
[20] C.R. Vogel and M.E. Oman, Itrative methods for total variation denoising, SIAM j. Sci. Comput., 17, 1996, 227-238.
[21] Y. L. You and M. Kaveh. Anisotropic blind image restoration. In Proc. 1996 IEEE International Conference on Image Processing, volume 2, pages 461-464, Lausanne, Switzerland, Sept. 1996.
[22] L. Rudin, S. Osher, and C. Fu, Total Variation Based Restoration of Noisy, Blurred Images, SINUM, to appear.
[23] E.R. Cole, The Removal of Unknown Imlige Blurs by Homomorphic Filters, Computer Scienc 5 Report UTEC-CSC-74-2029, (1973), University of Utah.
[24] P.-L. Lions, S. Osher, and L.I. Rudin, Denoising and Deblurring Images with Constmined Nonlinear Partial Diflerential Equations, submitted to SINUM.