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Image Segmentation Using the Cahn–Hilliard Equation

  • Wenli Yang
  • Zhongyi HuangEmail author
  • Wei Zhu
Article
  • 38 Downloads

Abstract

In this paper, we propose a novel model for image segmentation by using the Cahn–Hilliard equation. An interesting feature of this model lies in its ability of interpolating missing contours along wide gaps in order to form meaningful object boundaries, which is often achieved by curvature dependent models in the literature. To solve the associated equation, we employ a recently developed technique, that is, the tailored-finite-point method, which helps preserve sharp jumps and thus helps locate segmentation contours more exactly. Numerical experiments are presented to demonstrate the effectiveness of the proposed model and its features. In addition, analytical results on the existence and uniqueness of the associated equation are also provided.

Keywords

Image segmentation Cahn–Hilliard equation Tailored finite point method 

Mathematics Subject Classification

Primary: 65M32 68U10 Secondary: 65K10 

Notes

References

  1. 1.
    Bae, E., Tai, X.C., Zhu, W.: Augmented lagrangian method for an Euler’s elastica based segmentation model that promotes convex contours. Inverse Probl. Imaging 11(1), 1–23 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertozzi, A., Esedo\(\bar{\rm g}\)lu, S., Gillette, A.: Analysis of a two-scale Cahn–Hilliard model for binary image inpainting. Multiscale Model. Simul. 6(3), 913–936 (2007)Google Scholar
  3. 3.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Y., Tagare, H.D., Thiruvenkadam, S.: Using prior shapes in geometric active contours in a variational framework. Int. J. Comput. Vis. 50(3), 315–328 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cherfils, L., Fakih, H., Miranville, A.: A complex version of the Cahn–Hilliard equation for grayscale image inpainting. Multiscale Model. Simul. 15(1), 575–605 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Han, H., Huang, Z., Kellogg, R.B.: A tailored finite point method for a singular perturbation problem on an unbounded domain. J. Sci. Comput. 36(2), 243–261 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huang, Z., Li, Y.: Monotone finite point method foe non-equilibrium radiation diffusion equations. BIT Numer. Math. 56(2), 659–679 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J Comput. Vis. 1(4), 321–331 (1988)CrossRefzbMATHGoogle Scholar
  10. 10.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nitzberg, M., Mumford, D., Shiota, T.: Filtering, segmentation and depth. Lecture Notes in Computer Science. Springer (1993)Google Scholar
  12. 12.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shen, J., Kang, S.H., Chan, T.F.: Euler’s elastica and curvature-based inpainting. SIAM J. Appl. Math. 63(2), 564–592 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tai, X.C., Hahn, J., Chung, G.J.: A fast algorithm for Euler’s elastica model using augmented Lagrangian method. SIAM J. Imaging Sci. 4(1), 313–344 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Taylor, J.E., Cahn, J.W.: Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Stat. Phys. 77(1), 183–197 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50(3), 271–293 (2002)CrossRefzbMATHGoogle Scholar
  18. 18.
    Yang, W., Huang, Z., Zhu, W.: An efficient tailored finite point method for Rician denoising and deblurring. Commun. Comput. Phys. 24(4), 1169–1195 (2018)Google Scholar
  19. 19.
    Zhu, W., Tai, X.C., Chan, T.: Image segmentation using Euler’s elastica as the regularization. J. Sci. Comput. 57(2), 414–438 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.Department of MathematicsUniversity of AlabamaTuscaloosaUSA

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