Abstract
The total variation semi-norm based model by Rudin-Osher-Fatemi (in Physica D 60, 259–268, 1992) has been widely used for image denoising due to its ability to preserve sharp edges. One drawback of this model is the so-called staircasing effect that is seen in restoration of smooth images. Recently several models have been proposed to overcome the problem. The mean curvature-based model by Zhu and Chan (in SIAM J. Imaging Sci. 5(1), 1–32, 2012) is one such model which is known to be effective for restoring both smooth and nonsmooth images. It is, however, extremely challenging to solve efficiently, and the existing methods are slow or become efficient only with strong assumptions on the formulation; the latter includes Brito-Chen (SIAM J. Imaging Sci. 3(3), 363–389, 2010) and Tai et al. (SIAM J. Imaging Sci. 4(1), 313–344, 2011).
Here we propose a new and general numerical algorithm for solving the mean curvature model which is based on an augmented Lagrangian formulation with a special linearised fixed point iteration and a nonlinear multigrid method. The algorithm improves on Brito-Chen (SIAM J. Imaging Sci. 3(3), 363–389, 2010) and Tai et al. (SIAM J. Imaging Sci. 4(1), 313–344, 2011). Although the idea of an augmented Lagrange method has been used in other contexts, both the treatment of the boundary conditions and the subsequent algorithms require careful analysis as standard approaches do not work well. After constructing two fixed point methods, we analyze their smoothing properties and use them for developing a converging multigrid method. Finally numerical experiments are conducted to illustrate the advantages by comparing with other related algorithms and to test the effectiveness of the proposed algorithms.
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References
Blomgren, P., Mulet, P., Chan, T.F., Wong, C.K.: Total variation image restoration: numerical methods and extensions. In: Proceedings of the 1997 IEEE International Conference on Image Processing, Santa Barbara, CA, vol. 3, pp. 384–387 (1997)
Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)
Briggs, W.: A Multigrid Tutorial. SIAM, Philadelphia (1987)
Brito-Loeza, C., Chen, K.: Multigrid algorithm for high order denoising. SIAM J. Imaging Sci. 3(3), 363–389 (2010)
Brito-Loeza, C., Chen, K.: On high-order denoising models and fast algorithms for vector-valued images. IEEE Trans. Image Process. 19(6), 1518–1527 (2010)
Chen, K.: Matrix Preconditioning Techniques and Applications. Cambridge University Press, Cambridge (2005)
Chumchobn, N., Chen, K., Brito-Loeza, C.: Fourth order variational image registration on model and its fast multigrid algorithm. SIAM Multiscale Model. Simul. 9(1), 89–128 (2011)
Chumchobn, N., Chen, K., Brito-Loeza, C.: Variational model for removal of combined additive and multiplicative noise. Int. J. Comput. Math. 90(1), 140–161 (2013)
Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Bullard, J.W., et al. (eds.) Computational and Mathematical Models of Microstructural Evolution, pp. 39–46. Materials Research Society, Warrendale (1998)
Eyre, D.J.: An unconditionally stable one-step scheme for gradient systems. See www.math.utah.edu/~eyre/research/methods/stable.ps. Unpublished article (1998)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. SIAM Multiscale Model. Simul. 7(3), 1005–1028 (2008)
Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)
Henn, S.: A multigrid method for a fourth-order diffusion equation with application to image processing. SIAM J. Sci. Comput. 27(3), 831–849 (2005)
Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)
Lou, Y., Zhang, X., Osher, S., Bertozzi, A.: Image recovery via nonlocal operators. J. Sci. Comput. 42(2), 185–197 (2010)
Lysaker, M., Osher, S., Tai, X.-C.: Noise removal using smoothed normals and surface fitting. IEEE Trans. Image Process. 13(10), 1345–1357 (2004)
Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Shyy, W., Sue, C.-S.: Development of a pressure-correction/staggered-grid based multigrid solver for incompressible recirculating flows. Comput. Fluids 22(1), 51–76 (1993)
Tai, X.-C., Hahn, J., Chung, G.J.: A fast algorithm for Euler’s elastic method using augmented Lagrangian method. SIAM J. Imaging Sci. 4(1), 313–344 (2011)
Trottenberg, U., Oosterlee, C., Schuller, A.: Multigrid. Academic Press, London (2001)
Wesseling, P.: An Introduction to Multigrid Methods. Wiley, Chichester (1992)
Wienands, R., Joppich, W.: Practical Fourier Analysis for Multigrid Methods. Chapman and Hall/CRC Press, Boca Raton (2005)
Wise, S., Kim, J., Lowengrub, J.: Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys. 226(1), 414–446 (2007)
Wu, C., 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(3), 300–339 (2010)
Yuille, A.L., Rangarajan, A.: The concave-convex procedure. Neural Comput. 15, 915–936 (2003)
Zhang, J.P., Chen, K., Yu, B.: An iterative Lagrange multiplier method for constrained total-variation-based image denoising. SIAM J. Numer. Anal. 50(3), 983–1003 (2012)
Zhu, W., Chan, T.F.: Image denoising using mean curvature. SIAM J. Imaging Sci. 5(1), 1–32 (2012)
Zhu, W., Tai, X.-C., Chan, T.F.: Augmented Lagrangian method for a mean curvature based image denoising model. UCLA CAM report 12-02 (2012)
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Communicated by Rosemary Renaut.
This work is supported by the Fundamental Research Funds for the Central Universities (lzujbky-2013-15).
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Sun, L., Chen, K. A new iterative algorithm for mean curvature-based variational image denoising. Bit Numer Math 54, 523–553 (2014). https://doi.org/10.1007/s10543-013-0448-y
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DOI: https://doi.org/10.1007/s10543-013-0448-y
Keywords
- Staircasing effect
- Denoising
- Mean curvature-based model
- Fixed point iteration method
- Nonlinear multigrid