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Fast Linearized Augmented Lagrangian Method for Euler's Elastica Model

Part of: Communication, information Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  20 February 2017

Jun Zhang ,
Rongliang Chen ,
Chengzhi Deng  and
Shengqian Wang
Jun Zhang*
Affiliation:
Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing, Nanchang Institute of Technology, Nanchang 330099, Jiangxi, China College of Science, Nanchang Institute of Technology, Nanchang 330099, Jiangxi, China
Rongliang Chen*
Affiliation:
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, P. R. China
Chengzhi Deng*
Affiliation:
Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing, Nanchang Institute of Technology, Nanchang 330099, Jiangxi, China
Shengqian Wang*
Affiliation:
Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing, Nanchang Institute of Technology, Nanchang 330099, Jiangxi, China
*
*Corresponding author. Email addresses:[email protected] (J. Zhang), [email protected] (R.-L. Chen), [email protected] (C.-Z. Deng), [email protected] (S.-Q. Wang)
*Corresponding author. Email addresses:[email protected] (J. Zhang), [email protected] (R.-L. Chen), [email protected] (C.-Z. Deng), [email protected] (S.-Q. Wang)
*Corresponding author. Email addresses:[email protected] (J. Zhang), [email protected] (R.-L. Chen), [email protected] (C.-Z. Deng), [email protected] (S.-Q. Wang)
*Corresponding author. Email addresses:[email protected] (J. Zhang), [email protected] (R.-L. Chen), [email protected] (C.-Z. Deng), [email protected] (S.-Q. Wang)
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Abstract

Recently, many variational models involving high order derivatives have been widely used in image processing, because they can reduce staircase effects during noise elimination. However, it is very challenging to construct efficient algorithms to obtain the minimizers of original high order functionals. In this paper, we propose a new linearized augmented Lagrangian method for Euler's elastica image denoising model. We detail the procedures of finding the saddle-points of the augmented Lagrangian functional. Instead of solving associated linear systems by FFT or linear iterative methods (e.g., the Gauss-Seidel method), we adopt a linearized strategy to get an iteration sequence so as to reduce computational cost. In addition, we give some simple complexity analysis for the proposed method. Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method, and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.

Keywords

Image denoisingEuler's elastica modellinearized augmented Lagrangian methodshrink operatorclosed form solution

MSC classification

Secondary: 65M55: Multigrid methods; domain decomposition 68U10: Image processing 94A08: Image processing (compression, reconstruction, etc.)
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 1 , February 2017 , pp. 98 - 115
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bredies, K., Kunisch, K., and Pock, T., Total generalized variation, SIAM J. Imaging Sciences, vol. 3 (2010), pp. 492–526.CrossRefGoogle Scholar
[2] Carter, J. L., Dual methods for total variation-based image restoration, Technical report, UCLA CAM Report 02-13, 2002.Google Scholar
[3] Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imaging Vis., vol. 20 (2004), pp. 89–97.Google Scholar
[4] Chambolle, A., Total variation minimization and a class of binary MRF models, EMMCVPR 2005, LNCS 3757 (2005), pp. 136–152.Google Scholar
[5] Chan, T. F., Golub, G. H., and Mulet, P., A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., vol. 20 (1999), pp. 1964–1977.Google Scholar
[6] Chan, T. F. and Shen, J., Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM Publisher, Philadelphia, 2005.Google Scholar
[7] Chen, D. Q. and Zhou, Y., Multiplicative denoising based on linearized alternating direction method using discrepancy function constraint, J. Sci. Comput., vol. 60 (2014), pp. 483–504.Google Scholar
[8] Chen, Y., Levine, S., and Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., vol. 66 (2006), pp. 1383–1406.Google Scholar
[9] Duan, Y. and Huang, W., A fixed-point augmented lagrangian method for total variation minimization problems, J. Vis. Commun. Image R., vol. 24 (2013), pp. 1168–1181.CrossRefGoogle Scholar
[10] Duan, Y., Wang, Y., and Hahn, J., A fast augmented Lagrangian method for Euler's elastica models, Numer. Math. Theor. Meth. Appl., vol. 6 (2013), pp. 47–71.Google Scholar
[11] Goldstein, T. and Osher, S., The split Bregman method for L1-regularized problems, SIAM J. Imaging Sciences, vol. 2 (2009), pp. 323–343.CrossRefGoogle Scholar
[12] Hahn, J., Tai, X. C., Borok, S., and Bruckstein, A. M., Orientation-matching minimization for image denoising and inpainting, Int. J. Comput. Vision, vol. 92 (2011), pp. 308–324.Google Scholar
[13] Hahn, J., Wu, C., and Tai, X. C., Augmented Lagrangian method for generalized TV-Stokes model, J. Sci. Comput., vol. 50 (2012), pp. 235–264.Google Scholar
[14] Hintermüller, M., Rautenberg, C. N., and Hahn, J., Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction, Inverse Probl., vol. 30 (2014), pp. 055014.Google Scholar
[15] Jeong, T., Woo, H., and Yun, S., Frame-based Poisson image restoration using a proximal linearized alternating direction method, Inverse Probl., vol. 29 (2013), pp. 075007.Google Scholar
[16] Jia, R. Q. and Zhao, H., A fast algorithm for the total variation model of image denoising, Adv. Comput. Math., vol. 33 (2010), pp. 231–241.CrossRefGoogle Scholar
[17] 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., vol. 12 (2003), pp. 1579–1590.Google Scholar
[18] Masnou, S. and Morel, J. M., Level lines based disocclusion, IEEE Int. Conf. Image Process., vol. 1998 (1998), pp. 259–263.Google Scholar
[19] Myllykoski, M., Glowinski, R., Kärkkäinen, T., and Rossi, T., A new augmented Lagrangian approach for L1-mean curvature image denoising, SIAM J. Imaging Sciences, vol. 8 (2015), pp. 95–125.Google Scholar
[20] Paragios, N., Chen, Y., and Faugeras, O., Handbook of MathematicalModels in Computer Vision, Springer, Heidelberg, 2005.Google Scholar
[21] Rudin, L. I., Osher, S., and Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, vol. 60 (1992), pp. 259–268.CrossRefGoogle Scholar
[22] Shen, J., Kang, S. H., and Chan, T. F., Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., vol. 63 (2003), pp. 564–592.Google Scholar
[23] Tai, X. C., Hahn, J., and Chung, G. J., A fast algorithm for Euler's elastica model using augmented Lagrangian method, SIAM J. Imaging Sciences, vol. 4 (2011), pp. 313–344.CrossRefGoogle Scholar
[24] Tai, X. C. and Wu, C., Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, SSVM 2009, LNCS 5567 (2009), pp. 502-513.Google Scholar
[25] Vogel, C. R. and Oman, M. E., Iterative methods for total variation denoising, SIAM J. Sci. Comput., vol. 17 (1996), pp. 227–238.Google Scholar
[26] 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 Sciences, vol. 3 (2010), pp. 300–339.Google Scholar
[27] Yang, J., Zhang, Y., and Yin, W., An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., vol. 31 (2009), pp. 2842–2865.Google Scholar
[28] You, Y. L. and Kaveh, M., Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., vol. 9 (2000), pp. 1723–1730.Google Scholar
[29] Zhu, M., Fast Numerical Algorithms for Total Variation Based Image Restoration, Ph.D. Thesis, UCLA, 2008.Google Scholar
[30] Zhu, M. and Chan, T. F., An efficient primal-dual hybrid gradient algorithm for total variation image restoration, Technical report, UCLA CAM Report 08-34, 2008.Google Scholar
[31] Zhu, M., Wright, S. J., and Chan, T. F., Duality-based algorithms for total-variation-regularized image restoration, Comput. Optim. Appl., vol. 47 (2010), pp. 377–400.Google Scholar
[32] Zhu, W. and Chan, T., Image denoising using mean curvature of image surface, SIAM J. Imaging Sciences, vol. 5 (2012), pp. 1–32.Google Scholar
[33] Zhu, W., Tai, X. C., and Chan, T., Image segmentation using Eulers elastica as the regularization, J. Sci. Comput., vol. 57 (2013), pp. 414–438.Google Scholar
[34] Zhu, W., Tai, X. C., and Chan, T., A fast algorithm for a mean curvature based image denoising model using augmented lagrangian method, Global Optimization Methods, LNCS 8293 (2014), pp. 104–118.Google Scholar