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research-article

Spatial fractional telegraph equation for image structure preserving denoising

Authors:

Yupu YangAuthors Info & Claims

Signal Processing, Volume 107, Issue C

Pages 368 - 377

https://doi.org/10.1016/j.sigpro.2014.04.015

Published: 01 February 2015 Publication History

Abstract

In this paper, we propose a spatial fractional telegraph equation which could be applied to image denoising. The proposed equation interpolates between the second and the fourth order anisotropic diffusion equation by the use of spatial fractional derivatives. On the other hand, the telegraph equation interpolates between diffusion equation and wave equation, which leads to a mixed behavior of diffusion and wave propagation and thus it can preserve edges in the highly oscillatory regions. The existence, uniqueness and stability of the solution of our model are proved in this paper. The experimental results indicate superiority of the proposed model over the existing methods. HighlightsFractional derivatives lead to a interpolation between second and fourth order models.Telegraph equation interpolates between a diffusion equation and a wave equation.Telegraph equation is beneficial to enhance edges.We have proved that the proposed model is well-posed.The stable and convergent numerical scheme is given.

References

[1]

M. Elad, M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006) 3736-3745.

Digital Library

[2]

J. Mairal, F. Bach, J. Ponce, G. Sapiro, A. Zisserman, Non-local sparse models for image restoration, in: 2009 IEEE 12th International Conference on Computer Vision, IEEE, pp. 2272-2279.

[3]

P. Chatterjee, P. Milanfar, Clustering-based denoising with locally learned dictionaries, IEEE Trans. Image Process., 18 (2009) 1438-1451.

Digital Library

[4]

A. Buades, B. Coll, J.-M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2005) 490-530.

[5]

K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007) 2080-2095.

Digital Library

[6]

P. Chatterjee, P. Milanfar, Patch-based near-optimal image denoising, IEEE Trans. Image Process., 21 (2012) 1635-1649.

Digital Library

[7]

P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990) 629-639.

Digital Library

[8]

F. Catté, P.-L. Lions, J.-M. Morel, T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992) 182-193.

Digital Library

[9]

L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenom., 60 (1992) 259-268.

Digital Library

[10]

Y.-L. You, M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000) 1723-1730.

Digital Library

[11]

T.F. Chan, S. Esedoglu, F. Park, A fourth order dual method for staircase reduction in texture extraction and image restoration problems, in: 2010 17th IEEE International Conference on Image Processing (ICIP), IEEE, pp. 4137-4140.

[12]

T. Chan, A. Marquina, P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000) 503-516.

Digital Library

[13]

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 Process., 12 (2003) 1579-1590.

Digital Library

[14]

J. Bai, X.-C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007) 2492-2502.

Digital Library

[15]

V. Ratner, Y.Y. Zeevi, Denoising-enhancing images on elastic manifolds, IEEE Trans. Image Process., 20 (2011) 2099-2109.

Digital Library

[16]

Y. Cao, J. Yin, Q. Liu, M. Li, A class of nonlinear parabolic-hyperbolic applied to image restoration, Nonlinear Anal. Real World Appl., 11 (2010) 253-261.

[17]

W. Zeng, X. Lu, X. Tan, A class of fourth-order telegraph-diffusion for image restoration, J. Appl. Math., 2011 (2011).

[18]

E. Cuesta, M. Kirane, S.A. Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Process., 92 (2012) 553-563.

Digital Library

[19]

Wei Zhang, Jiaojie Li, Yupu Yang, A fractional diffusion-wave equation with non-local regularization for image denoising, Signal Processing, Available online 12 November 2013, ISSN 0165-1684, http://dx.doi.org/10.1016/j.sigpro.2013.10.028.

[20]

I. Podlubny, Academic Press, San Diego, California, USA, 1998.

[21]

M. Weilbeer, Efficient Numerical Methods for Fractional Differential Equations and their Analytical Background Ph.D. Thesis, Technische UniversitŁat Braunschweig, 2005.

[22]

R.A. Adams, J.J. Fournier, Sobolev Spaces, Academic Press, New York, 1975.

[23]

L. Tartar, Springer-Verlag, Berlin, Heidelberg, 2007.

[24]

M. Bergounioux, E. Trélat, A variational method using fractional order hilbert spaces for tomographic reconstruction of blurred and noised binary images, J. Funct. Anal., 259 (2010) 2296-2332.

[25]

L.C. Evans, Partial differential equations, in: Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, Rhode Island, USA, 1998.

[26]

R.A. Adams, Sobolev spaces, in: Pure and Applied Mathematics Series of Monographs and Textbooks, vol. 65, Academic Press, Inc., New York, San Francisco, London, 1975.

Cited By

Li CHe C(2023)Fractional-order diffusion coupled with integer-order diffusion for multiplicative noise removalComputers & Mathematics with Applications10.1016/j.camwa.2023.01.036136:C(34-43)Online publication date: 15-Apr-2023
https://dl.acm.org/doi/10.1016/j.camwa.2023.01.036
Rafsanjani HSedaaghi MSaryazdi S(2017)An adaptive diffusion coefficient selection for image denoisingDigital Signal Processing10.1016/j.dsp.2017.02.00464:C(71-82)Online publication date: 1-May-2017
https://dl.acm.org/doi/10.1016/j.dsp.2017.02.004
Bhrawy AZaky MMachado J(2017)Numerical Solution of the Two-Sided Space---Time Fractional Telegraph Equation Via Chebyshev Tau ApproximationJournal of Optimization Theory and Applications10.1007/s10957-016-0863-8174:1(321-341)Online publication date: 1-Jul-2017
https://dl.acm.org/doi/10.1007/s10957-016-0863-8

Spatial fractional telegraph equation for image structure preserving denoising
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
2. Theory of computation

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Published In

cover image Signal Processing

Signal Processing Volume 107, Issue C

February 2015

448 pages

ISSN:0165-1684

Issue’s Table of Contents

Copyright © Elsevier B.V.

Publisher

Elsevier North-Holland, Inc.

United States

Publication History

Published: 01 February 2015

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Cited By

Li CHe C(2023)Fractional-order diffusion coupled with integer-order diffusion for multiplicative noise removalComputers & Mathematics with Applications10.1016/j.camwa.2023.01.036136:C(34-43)Online publication date: 15-Apr-2023
https://dl.acm.org/doi/10.1016/j.camwa.2023.01.036
Rafsanjani HSedaaghi MSaryazdi S(2017)An adaptive diffusion coefficient selection for image denoisingDigital Signal Processing10.1016/j.dsp.2017.02.00464:C(71-82)Online publication date: 1-May-2017
https://dl.acm.org/doi/10.1016/j.dsp.2017.02.004
Bhrawy AZaky MMachado J(2017)Numerical Solution of the Two-Sided Space---Time Fractional Telegraph Equation Via Chebyshev Tau ApproximationJournal of Optimization Theory and Applications10.1007/s10957-016-0863-8174:1(321-341)Online publication date: 1-Jul-2017
https://dl.acm.org/doi/10.1007/s10957-016-0863-8

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