Hostname: page-component-cb9f654ff-p5m67 Total loading time: 0 Render date: 2025-08-18T13:45:06.181Z Has data issue: false hasContentIssue false
  • English
  • Français

Fast Algorithms for the Anisotropic LLT Model in Image Denoising

Published online by Cambridge University Press:  28 May 2015

Zhi-Feng Pang ,
Li-Lian Wang  and
Yu-Fei Yang
Zhi-Feng Pang*
Affiliation:
College of Mathematics and Information Science, Henan University, Kaifeng, 475004, China Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
Li-Lian Wang*
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
Yu-Fei Yang*
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China
*
Corresponding author. Email: zhifengpang@163.com
Corresponding author. Email: lilian@ntu.edu.sg
Corresponding author. Email: yfyang@hnu.edu.cn
Get access

Abstract

In this paper, we propose a new projection method for solving a general minimization problems with two L1-regularization terms for image denoising. It is related to the split Bregman method, but it avoids solving PDEs in the iteration. We employ the fast iterative shrinkage-thresholding algorithm (FISTA) to speed up the proposed method to a convergence rate O(k−2). We also show the convergence of the algorithms. Finally, we apply the methods to the anisotropic Lysaker, Lundervold and Tai (LLT) model and demonstrate their efficiency.

Keywords

65M1078A48Image denoisinganisotropic LLT modelDouglas-Rachford splitting methodsplit Bregman methodprojection methodfast projection method

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 3 , August 2011 , pp. 264 - 283
Copyright
Copyright © Global-Science Press 2011

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]Ambrosio, L., Fusco, N., Pallara, D.. Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, 2000.Google Scholar
[2]Aubert, G. and Kornprobst, P.. Mathematical Problems in Image Processing: Partial Diffrential Equations and the Calculus of Variations. Springer-Verlag, New York, 2002.CrossRefGoogle Scholar
[3]Aujol, J.. Some first-order algorithms for total variation based image restoration. Journal of Mathematical Imaging and Vision, 34(3)(2009), 307–327.Google Scholar
[4]Beck, A. and Teboulle, M.. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1)(2009), 183–202.Google Scholar
[5]Blomgern, P., Chan, T., Mulet, P., and Wong, C.. Total variation image restoration: Numerical methods and extensions. In: Proceedings, IEEE International Conference on Image Processing, III(1997), 384–387.CrossRefGoogle Scholar
[6]Brune, C., Sawatzky, A., and Burger, M.. Primal and dual Bregman methods with application to optical nanoscopy. International Journal of Computer Vision, 92(2)(2011), 211–229.Google Scholar
[7]Burger, M., Frick, K., Osher, S., and Scherzer, O.. Inverse total variation flow. Multiscale Modeling Simulation, 6(2)(2007), 366–395.Google Scholar
[8]Cai, J. F., Osher, S., and Shen, Z.. Split Bregman methods and frame based image restoration. Multiscale Modeling Simulation, 8(2)(2009), 337–369.CrossRefGoogle Scholar
[9]Chambolle, A.. Total variation minimization and a class of binary MRF models. In EMMCVPR 05, volume 3757 of Lecture Notes in Computer Sciences, (2005), 136–152.Google Scholar
[10]Chambolle, A.. An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20(1-2)(2004), 89–97.Google Scholar
[11]Chambolle, A. and Lions, P. L.. Image recovery via total variation minimization and related problems. Numerische Maththematik, 76(1997), 167–188.Google Scholar
[12]Chan, T., Golub, G., and Mulet, P.. A nonlinear primal-dual method for total variation-based image restoration. Journal on Scientific Computing, 20(1999), 1964–1977.Google Scholar
[13]Chan, T., Marquina, A., and Mulet, P.. High-order total variation-based image restoration. SIAM Journal on Scientific Computing, 22(2)(2000), 503–516.Google Scholar
[14]Chan, T. and Shen, J.. Image Processing and Analysis-Variational, PDE, wavelet, and stochastic methods. SIAM Publisher, Philadelphia, 2005.Google Scholar
[15]Chen, H. Z., Song, J. P., and Tai, X. C.. A dual algorithm for minimization of the LLT model. Advances in Computational Mathematics, 31(1–3)(2009), 115–130.Google Scholar
[16]Chen, K. and Tai, X. C.. A nonlinear multigrid method for total variation minimization from image restoration. Journal of Scientific Computing, 33(2)(2007), 115–138.Google Scholar
[17]Combettes, P. L.. Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization, 53(5-6)(2204), 475–504.Google Scholar
[18]Darbon, J. and Sigelle, M.. A fast and exact algorithm for total variation minimization. The second ICPRIA, 3522(2005), 351–359.Google Scholar
[19]Douglas, J. and Rachford, H.. On the numerical solution of heat conduction problems in two and three space variables. Transactions of the American Mathematical Society, 82(2)(1956), 421–439.Google Scholar
[20]Esser, E., Zhang, X., and Chan, T.. A general framework for a class of first order primal-dual algorithms for TV minimization. SIAM Journal on Imaging Sciences, 3(4)(2010), 1015–1046.CrossRefGoogle Scholar
[21]Goldstein, T., Bresson, X., and Osher, S.. Geometric applications of the split Bregman method: segmentation and surface reconstruction. Journal of Scientific Computing, 45(1-3)(2010), 272–293.Google Scholar
[22]Goldstein, T. and Osher, S.. The split Bregman for L 1 regularized problems. SIAM Journal on Imaging Sciences, 2(2)(2009), 323–343.Google Scholar
[23]He, L., Marquina, A., and Osher, S.. Blind deconvolution using TV regularization and Bregman iteration. International Journal of Imaging Systems and Technology, 15(1)(2005), 74–83.Google Scholar
[24]Jia, R. Q. and Zhao, H.. A fast algorithm for the total variation model of image denoising. Advances in Computational Mathematics, 33(2)(2010), 231–241.Google Scholar
[25]Jia, R. Q., Zhao, H., and Zhao, W.. Relaxation methods for image denoising based on difference schemes. Multiscale Modeling and Simulation, 9(1)(2011), 355–372.CrossRefGoogle Scholar
[26]Li, Y. and Santosa, F.. A computational algorithm for minimizing total variation in image restoration. IEEE Transactions on Image Processing, 5(6)(1996), 987–995.Google Scholar
[27]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 Transactions on Image Processing, 12(12)(2003), 1579–1590.CrossRefGoogle ScholarPubMed
[28]Marquina, A.1 and Osher, S.. Image super-resolution by TV-regularization and Bregman iteration. Journal of Scientific Computing, 37(3)(2008), 367–382.Google Scholar
[29]Micchelli, C., Shen, L.-X, and Xu, Y.-S. Proximity algorithms for image models: denoising. Inverse Problems, 27(4)(2011). (Doi: 10.1088/0266-5611.27/4/045009).CrossRefGoogle Scholar
[30]Ng, M. K., Qi, L., Yang, Y. F., and Huang, Y. M.. On semismooth Newton's methods for total variation minimization. Journal of Mathematical Imaging and Vision, 27(3)(2007), 265–276.Google Scholar
[31]Osher, S., Burger, M., Goldfarb, D., Xu, J., and Yin, W.. An iterative regularization method for total variation-based image restoration. Multiscale Modeling and Simulation, 4(2)(2005), 460–489.Google Scholar
[32]Pang, Z. F. and Yang, Y. F.. Semismooth Newton's methods for minimization of the LLT model. Inverse problems and Imaging, 3(4)(2009), 677–691.Google Scholar
[33]Paragios, N., Chen, Y., and Faugeras, O.. Handbook of Mathematical Models in Computer Vision. Springer, Heidelberg, 2005.Google Scholar
[34]Plonka, G. and Ma, J.. Curvelet-wavelet regularized split Bregman iteration for compressed sensing. International Journal of Wavelets. Multiresolution and Information Processing, (2011), Preprint.Google Scholar
[35]Rudin, L., Osher, S., and Fatemi, E.. Nonlinear total variation based noise removal algorithms. Physica D, 60(1992), 259–268.CrossRefGoogle Scholar
[36]Setzer, S.. Splitting methods in image processing. PhD Dissertation, (2009).Google Scholar
[37]Scherzer, O.. Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing, 60(1998), 1–27.Google Scholar
[38]Steidl, G.. A note on the dual treatment of higher order regularization functionals. Computing, 76(1-2)(2006), 135–148.Google Scholar
[39]Steidl, G. and Teuber, T.. Removing multiplicative noise by Douglas-Rachford splitting methods. Journal of Mathematical Imaging and Vision, 36(2)(2010), 168–184.Google Scholar
[40]Tai, X. C. and Wu, C.. Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. The 2nd international conference, SSVM, LNCS 5567(2009), 502–513.Google Scholar
[41]Vogel, C. R. and Oman, M.. Iterative methods for total variation denoising. SIAM Journal on Scientific Computing, 17(1)(1996), 227–238.Google Scholar
[42]Wang, Y., Yang, J., Yin, W., and Zhang, Y.. A new alternating minimization algorithm for total variation image reconstruction. SIAM Journal on Imaging Sciences, 1(3)(2008), 248–272.CrossRefGoogle Scholar
[43]Nocedal, J. and Wright, S.. Numerical Optimization. Springer, 1999.CrossRefGoogle Scholar
[44]Wu, C. and Tai, X.-C.. Augmented Lagrangian method, dual methods and split-Bregman iterations for ROF, vectorial TV and higher order models. SIAM Journal on Imaging Sciences, 3(3)(2010) 300–339.CrossRefGoogle Scholar
[45]Yin, W., Osher, S., Goldfarb, D. and Darbon, J.. Bregman iterative algorithms for ℓ 1 minimization with applications to compressed sensing. SIAM Journal on Imaging Sciences, 1(1)(2008), 143–168.CrossRefGoogle Scholar
[46]You, Y. L. and Kaveh, M.. Fourth-order partial differential equation for noise removal. IEEE Transactions on Image Processing, 9(10)(2000), 1723–1730.CrossRefGoogle ScholarPubMed