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A Primal-Dual Hybrid Gradient Algorithm to Solve the LLT Model for Image Denoising

Published online by Cambridge University Press:  28 May 2015

Chunxiao Liu ,
Dexing Kong  and
Shengfeng Zhu
Chunxiao Liu*
Affiliation:
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
Dexing Kong*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, China
Shengfeng Zhu*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

We propose an efficient gradient-type algorithm to solve the fourth-order LLT denoising model for both gray-scale and vector-valued images. Based on the primal-dual formulation of the original nondifferentiable model, the new algorithm updates the primal and dual variables alternately using the gradient descent/ascent flows. Numerical examples are provided to demonstrate the superiority of our algorithm.

Keywords

68U1065K10LLT modelimage denoisingprimal-dual
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 2 , May 2012 , pp. 260 - 277
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Blomgren, P. and Chan, T.F., Color TV: total variation methods for restoration of vector valued images, IEEE Trans. Image Process., 7 (1996), pp. 304–309.CrossRefGoogle Scholar
[2]Bresson, X. and Chan, T.F., Fast dual minimization of the vectorial total variation norm and applications to color image processing, Inverse Probl. Imag., 2 (2008), pp. 455–48.CrossRefGoogle Scholar
[3]Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J. and Osher, S., Fast global minimization of the active contour/snake model, J. Math. Imaging Vis., 28 (2007), pp. 151–167.CrossRefGoogle Scholar
[4]Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), pp. 89–97.Google Scholar
[5]Chambolle, A. and Lions, P, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), pp. 167–188.CrossRefGoogle Scholar
[6]Chan, T.F., Esedoglu, S. and Park, F.E., A fourth order dual method for staircase reduction in texture extraction and image restoration problems, Technical Report 05-28, UCLA CAM Reports, 2005.Google Scholar
[7]Chan, T.F., Golub, G.H. and Mulet, P, A nonlinear primal dual method for total variation based image restoration, SIAM J. Sci. Comput., 20 (1999), pp. 1964–1977.CrossRefGoogle Scholar
[8]Chan, T.F., Marquina, A. and Mulet, P., High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), pp. 503–516.CrossRefGoogle Scholar
[9]Chen, H., Song, J. and Tai, X.-C., A dual algorithm for minimization of the LLT model, Adv. Comput. Math., 31 (2009), pp. 115–130.CrossRefGoogle Scholar
[10]Goldstein, T. and Osher, S., The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), pp. 323–343.CrossRefGoogle Scholar
[11]Hajiaboli, M.R., An anisotropic fourth-order diffusion filter for image noise removal, Int. J. Comput. Vis., 92 (2011), pp. 177–191.CrossRefGoogle Scholar
[12]Li, F., Shen, C., Fan, J. and Shen, C., Image restoration combining a total variational filter and a fourth-order filter, J. Vis. Commun. Image R, 18 (2007), pp. 322–330.CrossRefGoogle Scholar
[13]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 Trans. Image Process., 12 (2003), pp. 1579–1590.CrossRefGoogle ScholarPubMed
[14]Lysaker, M. and X.-Tai, C., Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput. Vis., 66 (2006), pp. 5–18.CrossRefGoogle Scholar
[15]Rudin, L., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), pp. 259–268.CrossRefGoogle Scholar
[16]Steidl, G., A note on the dual treatment of higher-order regularization functionals, Computing, 76 (2006), pp. 135–148.CrossRefGoogle Scholar
[17]Wang, Y., Yin, W. and Zhang, Y., A fast algorithm for image deblurring with total variation regularization, Rice University CAAM Technical Report TR07-10, 2007.Google Scholar
[18]Whitaker, R.T. and S.Pizer, M., A multi-scale approach to nonuniform diffusion, Comput. Vis. Graph. Image Process.: Image Understand., 57 (1993), pp. 99–110.Google Scholar
[19]Wu, C. and Tai, X.-C., Augmented Lagrangian method, dual methods, and split bregman iteration for ROF model, SSVM 2009: pp. 502–513.Google Scholar
[20]Wu, C. and Tai, X.-C., Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), pp. 300–339.CrossRefGoogle Scholar
[21]You, Y.-L. and Kaveh, M., Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000), pp. 1723–1730.CrossRefGoogle ScholarPubMed
[22]Zhu, M. and T.Chan, F., An efficient primal-dual hybrid gradient algorithm for total variation image restoration, Technical Report 08-34, UCLA CAM Reports, 2008.Google Scholar