Hostname: page-component-74656f9884-c52kr Total loading time: 0 Render date: 2026-11-09T09:20:25.618Z Has data issue: false hasContentIssue false
  • English
  • Français

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:xxliul98431@126.com
Corresponding author.Email address:kong@cms.zju.edu.cn
Corresponding author.Email address:shengfengzhu@zju.edu.cn
Get access

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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