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Fast Linearized Augmented Lagrangian Method for Euler's Elastica Model

  • Jun Zhang (a1) (a2), Rongliang Chen (a3), Chengzhi Deng (a1) and Shengqian Wang (a1)

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.

Corresponding author
*Corresponding author. Email addresses: (J. Zhang), (R.-L. Chen), (C.-Z. Deng), (S.-Q. Wang)
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[1] K. Bredies , K. Kunisch , and T. Pock , Total generalized variation, SIAM J. Imaging Sciences, vol. 3 (2010), pp. 492526.

[5] T. F. Chan , G. H. Golub , and P. Mulet , A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., vol. 20 (1999), pp. 19641977.

[6] T. F. Chan and J. Shen , Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM Publisher, Philadelphia, 2005.

[7] D. Q. Chen and Y. Zhou , Multiplicative denoising based on linearized alternating direction method using discrepancy function constraint, J. Sci. Comput., vol. 60 (2014), pp. 483504.

[8] Y. Chen , S. Levine , and M. Rao , Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., vol. 66 (2006), pp. 13831406.

[9] Y. Duan and W. Huang , A fixed-point augmented lagrangian method for total variation minimization problems, J. Vis. Commun. Image R., vol. 24 (2013), pp. 11681181.

[11] T. Goldstein and S. Osher , The split Bregman method for L1-regularized problems, SIAM J. Imaging Sciences, vol. 2 (2009), pp. 323343.

[12] J. Hahn , X. C. Tai , S. Borok , and A. M. Bruckstein , Orientation-matching minimization for image denoising and inpainting, Int. J. Comput. Vision, vol. 92 (2011), pp. 308324.

[13] J. Hahn , C. Wu , and X. C. Tai , Augmented Lagrangian method for generalized TV-Stokes model, J. Sci. Comput., vol. 50 (2012), pp. 235264.

[14] M. Hintermüller , C. N. Rautenberg , and J. Hahn , Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction, Inverse Probl., vol. 30 (2014), pp. 055014.

[15] T. Jeong , H. Woo , and S. Yun , Frame-based Poisson image restoration using a proximal linearized alternating direction method, Inverse Probl., vol. 29 (2013), pp. 075007.

[16] R. Q. Jia and H. Zhao , A fast algorithm for the total variation model of image denoising, Adv. Comput. Math., vol. 33 (2010), pp. 231241.

[17] M. Lysaker , A. Lundervold , and 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., vol. 12 (2003), pp. 15791590.

[19] M. Myllykoski , R. Glowinski , T. Kärkkäinen , and T. Rossi , A new augmented Lagrangian approach for L1-mean curvature image denoising, SIAM J. Imaging Sciences, vol. 8 (2015), pp. 95125.

[21] L. I. Rudin , S. Osher , and E. Fatemi , Nonlinear total variation based noise removal algorithms, Physica D, vol. 60 (1992), pp. 259268.

[22] J. Shen , S. H. Kang , and T. F. Chan , Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., vol. 63 (2003), pp. 564592.

[23] X. C. Tai , J. Hahn , and G. J. Chung , A fast algorithm for Euler's elastica model using augmented Lagrangian method, SIAM J. Imaging Sciences, vol. 4 (2011), pp. 313344.

[25] C. R. Vogel and M. E. Oman , Iterative methods for total variation denoising, SIAM J. Sci. Comput., vol. 17 (1996), pp. 227238.

[26] C. Wu and X. C. Tai , 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. 300339.

[27] J. Yang , Y. Zhang , and W. Yin , An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., vol. 31 (2009), pp. 28422865.

[28] Y. L. You and M. Kaveh , Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., vol. 9 (2000), pp. 17231730.

[31] M. Zhu , S. J. Wright , and T. F. Chan , Duality-based algorithms for total-variation-regularized image restoration, Comput. Optim. Appl., vol. 47 (2010), pp. 377400.

[32] W. Zhu and T. Chan , Image denoising using mean curvature of image surface, SIAM J. Imaging Sciences, vol. 5 (2012), pp. 132.

[33] W. Zhu , X. C. Tai , and T. Chan , Image segmentation using Eulers elastica as the regularization, J. Sci. Comput., vol. 57 (2013), pp. 414438.

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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
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