Skip to main content
SpringerLink
Log in

Split Bregman Method for Minimization of Fast Multiphase Image Segmentation Model for Inhomogeneous Images

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we present a fast multiphase image segmentation model in a variational level set formulation. The proposed model is mainly used for images with inhomogeneity. The newly defined energy functional combines the local intensity information, the global intensity information, and the edge information to deal with the inhomogeneity. We use a weight function varying with locations to control the force of the local and global information dynamically. The special structure of the new energy functional ensures that the split Bregman method can be used for fast minimization. We apply the split Bregman method to minimize the new energy functional and summarize important results in several theorems. Theoretical evidences for these results are given. Several numerical results are also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Goldstein, T., Bresson, X., Osher, S.: Geometric applications of the split Bregman method: segmentation and surface reconstruction. J. Sci. Comput. 45(1–3), 272–293 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Houhou, N., Thiran, J.P., Bresson, X.: Fast texture segmentation based on semi-local region descriptor and active contour. Numer. Math. Theor. Methods Appl. 2(4), 445–468 (2009)

    MATH  MathSciNet  Google Scholar 

  4. Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4, 460–489 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for L1-minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1(1), 143–168 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Wang, Y., Yin, W., Zhang, Y.: A fast algorithm for image deblurring with total variation regularization. Tech. rep., Rice University CAAM Technical Report TR07-10, Houston (2007)

  7. Boyd, S., Vandenberghe, L.: Convex Optim. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  8. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Secaucus (2006)

    MATH  Google Scholar 

  9. Yang, Y., Li, C., Kao, C., Osher, S.: Split Bregman method for minimization of region-scalable fitting energy for image segmentation. In: Proceedings of International Symposium on Visual Computing, vol. 6454 LNCS, pp. 117–128. Las Vegas (2010)

  10. Yang, Y., Wu, B.: Convex image segmentation model based on local and global intensity fitting energy and split Bregman method. J. Appl. Math. 2012, 692589 (2012)

    Google Scholar 

  11. Yang, Y., Wu, B.: A new and fast multiphase image segmentation model for color images. Math. Probl. Eng. 2012, 494761 (2012)

    Google Scholar 

  12. Yang, Y., Wu, B.: Split Bregman method for minimization of improved active contour model combining local and global information dynamically. J. Math. Anal. Appl. 389(1), 351–366 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Wu, B., Yang, Y.: Local- and global-statistics-based active contour model for image segmentation. Math. Probl. Eng. 2012, 791958 (2012)

    Google Scholar 

  14. Yang, Y., Zhao, Y., Wu, B., Wang, H.: A fast multiphase image segmentation model for gray images. Comput. Math. Appl. 67(8), 1559–1581 (2014)

    Article  MathSciNet  Google Scholar 

  15. Vandeghinste, B., Goossens, B., Beenhouwer, J.D., Pizurica, A., Philips, W., Vandenberghe, S., Staelens, S.: Split-Bregman-based sparse-view CT reconstruction. In: 11th International meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, pp. 431–434. Potsdam (2011)

  16. Feng, J., Qin, C., Jia, K., Zhu, S., Liu, K., Han, D., Yang, X., Gao, Q., Tian, J.: Total variation regularization for bioluminescence tomography with the split Bregman method. Appl. Optics 51(19), 4501–4512 (2012)

    Article  Google Scholar 

  17. Li, W., Li, Q., Gong, W., Tang, S.: Total variation blind deconvolution employing split Bregman iteration. J. Vis. Commun. Image Represent. 23(3), 409–417 (2012)

    Article  Google Scholar 

  18. Zuo, Z., Zhang, T., Lan, X., Yan, L.: An adaptive non-local total variation blind deconvolution employing split Bregman iteration. Circuits Syst. Signal Process. 32(5), 2407–2421 (2013)

    Article  MathSciNet  Google Scholar 

  19. Yang, Y., Möller, M., Osher, S.: A dual split Bregman method for fast \(l^{1}\) minimization. Math. Comp. 82, 2061–2085 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang, L., Li, C., Sun, Q., Xia, D., Kao, C.: Active contours driven by local and global intensity fitting energy with application to brain MR image segmentation. J. Comput. Med. Imaging Graphics 33(7), 520–531 (2009)

    Article  Google Scholar 

  22. Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imag. Vis. 28, 151–167 (2007)

    Article  MathSciNet  Google Scholar 

  23. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)

    Article  MATH  Google Scholar 

  24. Yang, Y., Wu, B.: Fast multiphase image segmentation model for images with inhomogeneity. J. Electron. Imaging 21(1), 013015-1-14 (2012).

  25. Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    Article  MATH  Google Scholar 

  26. 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)

    Article  MATH  Google Scholar 

  27. Li, C., Kao, C., Gore, J., Ding, Z.: Implicit active contours driven by local binary fitting energy. In: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1–7. IEEE Computer Society, Washington, DC (2007)

  28. Li, C., Kao, C., Gore, J.C., Ding, Z.: Minimization of region-scalable fitting energy for image segmentation. IEEE Trans. Image Process. 17(10), 1940–1949 (2008)

    Article  MathSciNet  Google Scholar 

  29. Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7(3), 200–217 (1967)

    Article  Google Scholar 

  30. Cai, J.F., Osher, S., Shen, Z.: Linearized Bregman iterations for compressed sensing. Math. Comput. 78(267), 1515–1536 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Osher, S., Mao, Y., Dong, B., Yin, W.: Fast linearized Bregman iteration for compressive sensing and sparse denoising. Commun. Math. Sci. 8(1), 93–111 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  32. Xu, J., Osher, S.: Iterative regularization and nonlinear inverse scale space applied to wavelet-based denoising. IEEE Trans. Image Process. 16(2), 534–544 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 61301208), China Postdoctoral Science Foundation (No. 2013M531018), Natural Science Foundation Project of Guangdong (No. S2013040016230) and Shenzhen Fundamental Research Plan (Nos. JC201005260116A and JCYJ20120613144110654).

Author information

Authors and Affiliations

  1. Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, China

    Yunyun Yang & Yi Zhao

  2. Department of Mathematics, Harbin Institute of Technology, Harbin, China

    Boying Wu

Authors
  1. Yunyun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Boying Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunyun Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, Y., Zhao, Y. & Wu, B. Split Bregman Method for Minimization of Fast Multiphase Image Segmentation Model for Inhomogeneous Images. J Optim Theory Appl 166, 285–305 (2015). https://doi.org/10.1007/s10957-014-0597-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0597-4

Keywords

Mathematics Subject Classification

Search

Navigation