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Efficient Dual Algorithms for Image Segmentation Using TV-Allen-Cahn Type Models

Published online by Cambridge University Press:  20 August 2015

Li-Lian Wang  and
Ying Gu
Li-Lian Wang*
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
Ying Gu
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
*
*Corresponding author.Email:lilian@ntu.edu.sg
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Abstract

Variational image segmentation based on the Mumford and Shah model [31], together with implementation by the piecewise constant level-set method (PCLSM) [26], leads to fully nonlinear Total Variation (TV)-Allen-Cahn equations. The commonly-used numerical approaches usually suffer from the difficulties not only with the non-differentiability of the TV-term, but also with directly evolving the discontinuous piecewise constant-structured solutions. In this paper, we propose efficient dual algorithms to overcome these drawbacks. The use of a splitting-penalty method results in TV-Allen-Cahn type models associated with different “double-well” potentials, which allow for the implementation of the dual algorithm of Chambolle [8]. Moreover, we present a new dual algorithm based on an edge-featured penalty of the dual variable, which only requires to solve a vectorial Allen-Cahn type equation with linear ∇(div)-diffusion rather than fully nonlinear diffusion in the Chambolle’s approach. Consequently, more efficient numerical algorithms such as time-splitting method and Fast Fourier Transform (FFT) can be implemented. Various numerical tests show that two dual algorithms are much faster and more stable than the primal gradient descent approach, and the new dual algorithm is at least as efficient as the Chambolle’s algorithm but is more accurate. We demonstrate that the new method also provides a viable alternative for image restoration.

Keywords

65N2265N5574S2049J40Variational modelsimage segmentationTV-Allen-Cahn type equationsdual algorithmssplitting-penalty methodsFFT
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 4 , April 2011 , pp. 859 - 877
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Allen, S. M. and Cahn, J. W., A microscope theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall., 27(6) (1979), 1085–1095.Google Scholar
[2]Aubert, G. and Kornprobst, P., Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Springer-Verlag, New York Inc, 2006.Google Scholar
[3]Badshah, N. and Chen, K., Multigrid method for the Chan-Vese model in variational segmentation, Commun. Comput. Phys., 4(2) (2008), 294–316.Google Scholar
[4]Badshah, N. and Chen, K., Image selective segmentation under geometrical constraints using an active contour approach, Commun. Comput. Phys., 7(4) (2010), 759–778.Google Scholar
[5]Bae, E., Yuan, J. and Tai, X. C., Global minimization for continuous multiphase partitioning problems using a dual approach, UCLA, CAM-Report (09-75), 2009.Google Scholar
[6]Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J. P., and Osher, S., Fast global minimization of the active contour/snake model, J. Math. Imaging. Vis., 28(2) (2007), 151–167.CrossRefGoogle Scholar
[7]Caselles, V., Kimmel, R., and Sapiro, G., Geodesic active contours, Int. J. Comput. Vis., 22(1) (1997), 61–79.Google Scholar
[8]Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imaging. Vis., 20(1) (2004), 89–97.Google Scholar
[9]Chambolle, A., Total variation minimization and a class of binary MRF models, In Energy Minimization Methods in Computer Vision and Pattern Recognition, pages 136–152, Springer, 2005.Google Scholar
[10]Chambolle, A. and Lions, P. L., Image recovery via total variation minimization and related problems, Numer. Math., 76(2) (1997), 167–188.CrossRefGoogle Scholar
[11]Chan, T. F., Chen, K., and Tai, X. C., Nonlinear multilevel schemes for solving the total variation image minimization problem, In Image Processing Based on Partial Differential Equations, Series of Math Vision, pages 265–288, Springer, Berlin, 2007.Google Scholar
[12]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), 1964–1977.CrossRefGoogle Scholar
[13]Chan, T. F. and Vese, L. A., Active contours without edges, IEEE Trans. Image. Proc., 10(2) (2001), 266–277.CrossRefGoogle ScholarPubMed
[14]Christiansen, O. and Tai, X. C., Fast implementation of piecewise constant level set methods, In Image Processing Based on Partial Differential Equations, Series of Math Vision, pages 289-308, Springer, Berlin, 2007.Google Scholar
[15]Courant, R., Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., 49 (1943), 1–23.CrossRefGoogle Scholar
[16]Destuynder, P., Jaoua, M., and Sellami, H., A dual algorithm for denoising and preserving edges in image processing, J. Inverse. Ill-Posed. Probl., 15(2) (2007), 149–165.Google Scholar
[17]Feng, W. M., Yu, P., Hu, S. Y., Liu, Z. K., Du, Q., and Chen, L. Q., Spectral implementation of an adaptive moving mesh method for phase-field equations, J. Comput. Phys., 220(1) (2006), 498–510.CrossRefGoogle Scholar
[18]Gurholt, T. P. and Tai, X. C., 3D multiphase piecewise constant level set method based on graph-cut minimization, Numer. Math. Theor. Meth. Appl., 2 (2009), 403–420.Google Scholar
[19]Huang, Y., Ng, M. K., and Wen, Y. W., A fast total variation minimization method for image restoration, Multi. Model. Sim., 7 (2008), 774–795.Google Scholar
[20]Kass, M., Witkin, A., and Terzopoulos, D., Snakes: active contour models, Int. J. Comput. Vis., 1(4) (1988), 321–331.Google Scholar
[21]Li, C., Xu, C., Gui, C., and Fox, M., Level set evolution without re-initialization: a new varia-tional formulation, Proc. CVPR. IEEE., Citeseer, 2005.Google Scholar
[22]Li, F. and Ng, M. K., Kernel density estimation based multiphase fuzzy region competition method for texture image segmentation, Commun. Comput. Phys., 8(3) (2010), 623–641.Google Scholar
[23]Li, F., Ng, M. K., Zeng, T., and Shen, C., A multiphase image segmentaion method based on fuzzy region competition, SIAM J. Imaging Sciences, 3(3) 2010, 277–299.CrossRefGoogle Scholar
[24]Li, H. W. and Tai, X. C., Piecewise constant level set method for multiphase motion, Int. J. Numer. Anal. Model., 4(2) (2007), 291–305.Google Scholar
[25]Lie, J., Lysaker, M., and Tai, X. C., A binary level set model and some applications to Mumford-Shah image segmentation, IEEE Trans. Image. Proc., 15(5) (2006), 1171–1181.CrossRefGoogle ScholarPubMed
[26]Lie, J., Lysaker, M., and Tai, X. C., A variant of the level set method and applications to image segmentation, Math. Comput., 75(255) (2006), 1155–1174.Google Scholar
[27]Lu, T., Neittaanmaki, P., and Tai, X. C., A parallel splitting up method and its application to Navier-Stokes equations, Appl. Math. Lett., 4(2) (1991), 25–29.CrossRefGoogle Scholar
[28]Lu, T., Neittaanmaki, P., and Tai, X. C., A parallel splitting up method for partial diferen-tial equations and its application to Navier-Stokes equations, RAIRO Math. Model. Numer. Anal., 26(6) (1992), 673–708.Google Scholar
[29]Luo, Z., Tong, L. Y., J. Z Luo, Wei, P., and Wang, M. Y., Design of piezoelectric actuators using a multiphase level set method of piecewise constants, J. Comput. Phys., 228 (2009), 2643–2659.Google Scholar
[30]Marchuk, G. I., Splitting and alternating direction methods, Handb. Numer. Anal. I., pages 197–462, North-Holland, Amsterdam, 1990.Google Scholar
[31]Mumford, D. and Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure. Appl. Math., 42(5) (1989), 577–685.Google Scholar
[32]Osher, S. and Fedkiw, R. P., Level Set Methods and Dynamic Implicit Surfaces, Springer Verlag, 2002.Google Scholar
[33]Osher, S. and Sethian, J. A., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79(1) (1988), 12–49.CrossRefGoogle Scholar
[34]Rong, Z. J., Wang, L. L., and Tai, X. C., Adaptive wavelet collocation methods for image seg-menation using TV-Allen-Cahn models, UCLA CAM-Report, 09-91, 2009.Google Scholar
[35]Rudin, L., Osher, S., and Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D., 60(1-4) (1992), 259–268.CrossRefGoogle Scholar
[36]Shen, J. and Yang, X. F., An efficient moving mesh spectral method for the phase-field model of two-phase flows, J. Comput. Phys., 228(8) (2009), 2978–2992.Google Scholar
[37]Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506–517.Google Scholar
[38]Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations, Society for Industrial Mathematics, 2004.Google Scholar
[39]Tai, X. C. and Chan, T. F., A survey on multiple level set methods with applications for identifying piecewise constant functions, Int. J. Numer. Anal. Model., 1(1) (2004), 25–47.Google Scholar
[40]Tai, X. C., Christiansen, O., Lin, P., and Skjælaaen, I., Image segmentation using some piecewise constant level set methods with MBO type of projection, Int. J. Comput. Vis., 73(1) (2007), 61–76.Google Scholar
[41]Tai, X. C. and Wu, C., Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, In Scale Space and Variational Methods in Computer Vision-LNCS 5567, pages 502–514, Springer, Berlin, 2009.Google Scholar
[42]Trefethen, L. N., Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.Google Scholar
[43]Wang, Y., Yang, J., Yin, W., and Zhang, Y., A new alternating minimization algorithm for total variation image reconstruction,SIAM J. Imaging. Sci., 1(3) (2008), 248–272.Google Scholar
[44]Weickert, J., Romeny, B., and Viergever, M. A., Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image. Proc., 7(3) (1998), 398–410.Google Scholar