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A Fast Augmented Lagrangian Method for Euler’s Elastica Models

Published online by Cambridge University Press:  28 May 2015

Yuping Duan ,
Yu Wang  and
Jooyoung Hahn
Yuping Duan*
Affiliation:
Institute for Infocomm Research, Singapore
Yu Wang*
Affiliation:
Computer Science Department, Technion, Haifa 32000, Israel
Jooyoung Hahn*
Affiliation:
Institute for Mathematics and Scientific Computing, University of Graz, Austria
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

In this paper, a fast algorithm for Euler’s elastica functional is proposed, in which the Euler’s elastica functional is reformulated as a constrained minimization problem. Combining the augmented Lagrangian method and operator splitting techniques, the resulting saddle-point problem is solved by a serial of subproblems. To tackle the nonlinear constraints arising in the model, a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution. We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic, real-world and medical images for image denoising, image inpainting and image zooming problems.

Keywords

68U1065N2174S20Euler’s elasticaaugmented Lagrangian methodimage denoisingimage inpaintingimage zooming
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 1 , February 2013 , pp. 47 - 71
Copyright
Copyright © Global Science Press Limited 2013

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References

[1] Ambrosio, L. and Masnou, S., A direct variational approach to a problem arising in image reconstruction, Interfaces and Free Boundaries, 5(1) (2003), pp. 63–82.Google Scholar
[2] Ambrosio, L. and Masnou, S., On a variational problem arising in image reconstruction, Free Boundary Problems, 147(1) (2005), pp. 17–26.Google Scholar
[3] Bae, E., Shi, J. and Tai, X. C., Graph cuts for curvature based image denoising, IEEE Trans. Image Proc., 20(5) (2011), pp. 1199–1210.Google Scholar
[4] Ballester, C., Bertalmio, M., Caselles, V., Sapiro, G. and Verdera, J., Filling-in by joint interpolation of vector fields and gray levels, IEEE Trans. Image Proc., 10(8) (2002), pp. 1200–1211.Google Scholar
[5] Ballester, C., Caselles, V. and Verdera, J., Disocclusion by joint interpolation of vector fields and gray levels, Multiscale Model. Sim., 2(1) (2004), pp. 80–123.Google Scholar
[6] Bresson, X. and Chan, T. F., Fast dual minimization of the vectorial total variation norm and applications to color image processing, Inverse Problems Imaging, 2(4) (2008), pp. 455–484.CrossRefGoogle Scholar
[7] Brito-Loeza, C. and Chen, K., Fast numerical algorithms for eulers elastica inpainting model, Int. J. Modern Math., 5 (2010), pp. 157–182.Google Scholar
[8] Brito, C.-LOEZA and Chen, K., Multigrid algorithm for high order denoising, SIAM J. Imaging Sci., 3(3) (2010), pp. 363–389.Google Scholar
[9] Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20(1) (2004), pp. 89–97.Google Scholar
[10] Chambolle, A. and Lions, P. L., Image recovery via total variation minimization and related problems, Numer. Math., 76(2) (1997), pp. 167–188.CrossRefGoogle Scholar
[11] Chan, T., Golub, G. and Mulet, P., A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20(6) (1999), pp. 1964–1977.CrossRefGoogle Scholar
[12] Chan, T., Marquina, A. and Mulet, P., High-order total variation-based image restoration, SIAM J. Sci. Comput., 22(2) (2000), pp. 503–516.Google Scholar
[13] Chan, T. F., Kang, S. H. and Shen, J., Total variation denoising and enhancement of color images based on the CB and HSV color models, J. Visual Commun. Image Representation, 12(4) (2001), pp. 422–435.Google Scholar
[14] Chan, T. F., Kang, S. H. and Shen, J., Euler’s elastica and curvature-based inpainting, SIAM J. Appl. Math., 63(2) (2002), pp. 564–592.Google Scholar
[15] Droske, M. and Bertozzi, A., Higher-order feature-preserving geometric regularization, SIAM J. Imaging Sci., 3(1) (2010), pp. 21–51.Google Scholar
[16] Esedoglu, S. and March, R., Segmentation with depth but without detecting junctions, J. Math. Imaging Vision, 18(1) (2003), pp. 7–15.Google Scholar
[17] Esedoglu, S., Ruuth, S. and Tsai, R., Threshold dynamics for shape reconstruction and disocclusion, in IEEE Int. Conference Image Processing, 2 (2005), pp. 502–505.Google Scholar
[18] Gao, R., Song, J. P. and Tai, X. C., Image zooming algorithm based on partial differential equations technique, Int. J. Numer. Anal. Modeling, 6(2) (2009), pp. 284–292.Google Scholar
[19] Hahn, J., Chung, G. J., Wang, Y. and Tai, X. C., Fast algorithms for p-elastica energy with the application to image inpainting and curve reconstruction, in SSVM’11: Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision, pp. 169–182. Springer, 2011.Google Scholar
[20] Hahn, J., Wu, C. and Tai, X. C., Augmented lagrangian method for generalized TV-stokes model, J. Sci. Computing, 50(2) (2012), pp. 235–264.Google Scholar
[21] Komodakis, N. and Paragios, N., Beyond pairwise energies: efficient optimization for higherorder MRFs, in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 2985–2992, 2009.Google Scholar
[22] Lai, R. and Chan, T. F., A framework for intrinsic image processing on surfaces, Comput. Vision Image Understanding, 115(12) (2011), pp. 1647–1661.Google Scholar
[23] Lehmann, T. M., Gonner, C. and Spitzer, K., Survey: interpolation methods in medical image processing, IEEE Trans. Medical Imaging, 18(11) (1999), pp. 1049–1075.Google Scholar
[24] 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 Processing, 12(12) (2003), pp. 1579–1590.Google Scholar
[25] Maeland, E., On the comparison of interpolation methods, IEEE Trans. Medical Imaging, 7(3) (1988), pp. 213–217.CrossRefGoogle ScholarPubMed
[26] Masnou, S. and Morel, J. M., Level lines based disocclusion, in IEEE Int. Conference Image Processing, pp. 259–263, 1998.Google Scholar
[27] Masnou, S. and Morel, J. M., Level lines based disocclusion, in Int. Conference Image Processing, pp. 259–263, 1998.Google Scholar
[28] Mumford, D. and Center for Intelligent Control Systems (US), Elastica and computer vision. Center for Intelligent Control Systems, Massachusetts Institute of Technology, 1991.Google Scholar
[29] Nitzberg, M., Mumford, D. and Shiota, T., Filtering, segmentation and depth, Lecture Notes Computer Sci., 1993.Google Scholar
[30] Rockafellar, R. T., Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Operations Research, 1(2) (1976), pp. 97–116.Google Scholar
[31] Rudin, L. I., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D Nonlinear Phenomena, 60(1-4) (1992), pp. 259–268.Google Scholar
[32] Schönlieb, C. and Bertozzi, A., Unconditionally stable schemes for higher order inpainting, Commun. Math. Sci., 9(2) (2011), pp. 413–457.Google Scholar
[33] Tai, X. C., Hahn, J. and Chung, G. J., A fast algorithm for Euler’s elastica model using augmented lagrangian method, SIAM J. Imaging Sci., 4(1) (2011), pp. 313–344.Google Scholar
[34] Tai, X. C. and Wu, C., Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, in SSVM’09: Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision, pages 502–513. Springer, 2009.CrossRefGoogle Scholar
[35] Unser, M., Aldroubi, A. and Eden, M., Enlargement or reduction of digital images with minimum loss of information, IEEE Trans. Image Processing, 4(3) (1995), pp. 247–258.CrossRefGoogle ScholarPubMed
[36] 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(3) (2010), pp. 300–339.Google Scholar
[37] Wu, C., Zhang, J., Duan, Y. and Tai, X. C., Augmented lagrangian method for total variation based image restoration and segmentation over triangulated surfaces, J. Sci. Computing, 50(1) (2012), pp. 145–166.Google Scholar
[38] Wu, C., Zhang, J. and Tai, X. C., Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems Imaging, 5(1) (2011), pp. 237–261.Google Scholar