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Augmented Lagrangian Methods for p-Harmonic Flows with the Generalized Penalization Terms and Application to Image Processing

Published online by Cambridge University Press:  28 May 2015

Huibin Chang  and
Xue-Cheng Tai
Huibin Chang*
Affiliation:
School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387; Department of Mathematics, East China Normal University, Shanghai 200241, China
Xue-Cheng Tai*
Affiliation:
Department of Mathematics, University of Bergen, Johaness Brunsgate 12, Bergen 5007, Norway
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

In this paper, we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in [Kang & March, IEEE T. Image Process., 16 (2007), 2251-2261]. We use fast algorithms to solve the subproblems, such as the dual projection methods, primal-dual methods and augmented Lagrangian methods. With a special penalization term, some special algorithms are presented. Numerical experiments are given to demonstrate the performance of the proposed methods. We successfully show that our algorithms are effective and efficient due to two reasons: the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately (even 2 inner iterations of the subproblem are enough). It is also observed that better PSNR values are produced using the new algorithms.

Keywords

68U1065N2174S20p-harmonic flowsdenoisinggeneralized penalization termssaddle-point problemimage processingaugmented Lagrangian methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 1 , February 2013 , pp. 1 - 20
Copyright
Copyright © Global Science Press Limited 2013

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References

[1] Bae, E. and Tai, X.-C., Graph cut optimization for the piecewise constant level set method applied to multiphase image segmentation, Scale Space and Variational Methods in Computer Vision, SSVM 2009, pp. 1–13.Google Scholar
[2] Bresson, X. and Chan, T. F., Fast dual minimization of the vectorial total variation norm and applications to color image processing, Inverse Probl. Image., 2(4) (2008), pp. 455–484.Google Scholar
[3] Barrett, J. W., Bartels, S., Feng, X. and Prohl, A., A convergent and constraint-preserving finite element methodfor the p-harmonic flow into spheres, SIAM J. Numer. Anal., 45(3) (2007), pp. 905–927.Google Scholar
[4] Bartels, S., Stability and convergence of finite-element approximation schemes for harmonic maps, SIAM J. Numer. Anal., 43(1) (2005), pp. 220–238.Google Scholar
[5] Bartels, S., Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces, Math. Comput., 79(271) (2010), pp. 1263–1301.Google Scholar
[6] Bartels, S. and Prohl, A., Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44(4) (2006), pp. 1405–1419.Google Scholar
[7] Bartels, S. and Prohl, A., Convergence of an implicit, constraint preserving finite element discretization of p-harmonic heat flow into spheres, Numer. Math., 109(4) (2008), pp. 489–507.Google Scholar
[8] Bethuel, F., Brezis, H. and F. HÉLEIN, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1(2) (1993), pp. 123–148.Google Scholar
[9] Bethuel, F., Brezis, H. and F. HÉLEIN, Singular limit for the minimization of Ginzburg-Landau functionals, C. R. Acad. Sci. Paris Sér. I Math., 314(12) (1992), pp. 891–895.Google Scholar
[10] Cecil, T., Osher, S. and Vese, L., Numerical methods for minimization problems constrained to S1 and S2 , J. Comput. Phys., 198(2) (2004), pp. 567–579.Google Scholar
[11] Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20(1-2) (2004), pp. 89–97.Google Scholar
[12] Chambolle, A. and Pock, T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40(1) (2010), pp. 120–145.Google Scholar
[13] Chan, T. F., Golub, G. H. and Mulet, P., A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20(6) (1999), pp. 1964–1977.Google Scholar
[14] Cohen, R., Hardt, R., Kinderlehrer, D., Lin, S. Y. and Luskin, M., Minimum energy configurations for liquid crystals: Computational results, in Theory and Applications of Liquid Crystals, IMA Math. Appl., 5 (1987), pp. 99–121.Google Scholar
[15] Cohen, R., Lin, S. Y. and Luskin, M., Relaxation and gradient methods for molecular orientation in liquid crystals, Comput. Phys. Commun., 53(1-3) (1989), pp. 455–465.Google Scholar
[16] Glowinski, R. and Tallec, P. L., Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989.Google Scholar
[17] Goldfarb, D., Wen, Z. W. and Yin, W. T., A curvilinear search method for p-harmonic flows on spheres, SIAM J. Imaging Sci., 2(1) (2009), pp. 84–109.Google Scholar
[18] Goldstein, T. and Osher, S., The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2(2) (2009), pp. 323–343.Google Scholar
[19] Hu, Q., Tai, X.-C. and Winther, R., A saddle point approach to the computation of harmonic maps, SIAM J. Numer. Anal., 47(2) (2009), pp. 1500–1523.Google Scholar
[20] Ha Kang, Sung and March, Riccardo, Variational models for image colorization via chromaticity and brightness decomposition, IEEE T. Image Process., 16(9) (2007), pp. 2251–2261.Google Scholar
[21] Kwatra, V., Schodl, A., Essa, I., Turk, G. and Bobick, A., Graphcut textures: image and video synthesis using graph cuts, ACM Trans. Graphics, 22(3) (2003), pp. 277–286.Google Scholar
[22] Lysaker, M., Osher, S. and X.-Tai, C., Noise removal using smoothed normals and surface fitting, IEEE T. Image Process., 13(10) (2004), pp. 1345–1357.Google Scholar
[23] Marquina, A. and Osher, S., Image super-resolution by TV-regularization and Bregman iteration, J. Sci. Comput., 37(3) (2008), pp. 367–382.Google Scholar
[24] Misawa, M., Approximation of p-harmonic maps by the penalized equation, Nonlinear Anal., 47(2) (2001), pp. 1069–1080.Google Scholar
[25] Rother, C., Kolmogorov, V. and Blake, A., “GrabCut”: interactive foreground extraction using iterated graph cuts, ACM Trans. Graphics, 23(3) (2004), pp. 309–314.Google Scholar
[26] Rudin, L. I., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259–268.Google Scholar
[27] Setzer, S., Splitting Bregman algorithm, Douglas-Rachford splitting and frame shrinkage, in SSVM 09: Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision, Springer-Verlag, Berlin, Heidelberg, pp. 464–476, 2009.Google Scholar
[28] Vese, L. A. and Osher, S. J., Numerical methods for p-harmonic flows and applications to image processing, SIAM J. Numer. Anal., 40(6) (2002), pp. 2085–2104.Google Scholar
[29] Wen, Z. and Yin, W., A feasible method for optimization with orthogonality constraints, CAM Report 10-77, UCLA, Los Angeles, CA, 2010.Google Scholar
[30] Wu, C. L. and X.-Tai, C., Augmented Lagrangian method, Dual methods and Split-Bregman Iterations for ROF, vectorial TV and higher order models, SIAM J. Imaging Sci., 3(3) (2010), pp. 300–339.Google Scholar
[31] X.-Tai, C., Hahn, J. Y. and Chung, G. J., A fast algorithm for Euler’s elastica model using augmented lagrangian, SIAM J. Imaging Sci., 4(1) (2011), pp. 313–344.Google Scholar
[32] Yin, W., Osher, S., Goldfarb, D. and Darbon, F., Bregman iterative algorithms for l1-minimization with applications to compressed sensing, SIAM J. Imaging Sci., 1(1) (2008), pp. 143–168.Google Scholar
[33] Zhang, X., Burger, M. and Osher, S., A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration, UCLA CAM Report 09-99, UCLA, Los Angeles, CA, 2009.Google Scholar
[34] Zhu, M. and Chan, T., An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration, UCLA CAM Report 08-34, UCLA, Los Angeles, CA, 2008.Google Scholar