Piecewise Constant Level Set Method for 3D Image Segmentation

  • Are Losnegård
  • Oddvar Christiansen
  • Xue-Cheng Tai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4485)


Level set methods have been proven to be efficient tools for tracing interface problems. Recently, some variants of the Osher- Sethian level set methods, which are called the Piecewise Constant Level Set Methods (PCLSM), have been proposed for some interface problems. The methods need to minimize a smooth cost functional under some special constraints. In this paper a PCLSM for 3D image segmentation is tested. The algorithm uses the gradient descent method combined with a Quasi-Newton method to minimize an augmented Lagrangian functional. Experiments for medical image segmentation are shown on synthetic three dimensional MR brain images. The efficiency of the algorithm and the quality of the obtained images are demonstrated.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Osher, S., Fedkiw, R.: An overview and some recent results. J. Comput. Phys 169(2), 463–502 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Tai, X.C., Chan, T.F.: A survey on multiple level set methods with applications for identifying piecewise constant functions. Int. J. Numer. Anal. Model 1(1), 25–47 (2004)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Osher, S., Burger, M.: A survey on level set methods for inverse problems and optimal design. Cam-report-04-02, UCLA, Applied Mathematics (2004)Google Scholar
  4. 4.
    Vese, L.A., Chan, T.F.: A new multiphase level set framework for image segmentation via the Mumford and Shah model. International Journal of Computer Vision 50, 271–293 (2002)zbMATHCrossRefGoogle Scholar
  5. 5.
    Chan, T.F., Vese, L.A.: Image segmentation using level sets and the piecewise constant mumford-shah model. Technical report, CAM Report 00-14, UCLA, Math. Depart (April 2000, revised December 2000)Google Scholar
  6. 6.
    Osher, S., Sethian, J.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1) (1988)Google Scholar
  7. 7.
    Chan, T., Vese, L.: Active contours without edges. In: IEEE Image Proc., vol. 10, pp. 266–277 (2001)Google Scholar
  8. 8.
    Lie, J., Lysaker, M., Tai, X.C.: A variant of the level set method and applications to image segmentation. Math. Comp. 75(255) (2006)Google Scholar
  9. 9.
    Tai, X.-c., Yao, C.-h.: Image segmentation by piecewise constant Mumford-Shah model without estimating the constants. J. Comput. Math. 24(3), 435–443 (2006)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Christiansen, O., Tai, X.-C.: Fast implementation of piecewise constant level set methods. In: Image processing based on partial differential equations, pp. 253–272. Springer, Heidelberg (2006)Google Scholar
  11. 11.
    Zhao, H.-K., et al.: A variational level set approach to multiphase motion. J. Comput. Phys. 127(1), 179–195 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Vese, L., Chan, T.: A new multiphase level set framework for image segmentation via the Mumford and Shah model. International Journal of Computer Vision 50, 271–293 (2002)zbMATHCrossRefGoogle Scholar
  13. 13.
    Chung, J.T., Vese, A.: Energy minimization based segmentation and denoising using a multilayer level set approach. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Song, B., Chan, T.F.: Fast algorithm for level set segmentation. UCLA CAM report 02-68 (2002)Google Scholar
  15. 15.
    Gibou, F., Fedkiw, R.: A fast hybrid k-means level set algorithm for segmentation. In: 4th Annual Hawaii International Conference on Statistics and Mathematics, pp. 281–291 (2005)Google Scholar
  16. 16.
    Esedoḡlu, S., Tsai, Y.H.R.: Threshold dynamics for the piecewise constant Mumford-Shah functional. J. Comput. Phys. 211(1), 367–384 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Shi, Y., Karl, W.C.: A fast level set method without solving pdes. In: ICASSP’05 (2005)Google Scholar
  18. 18.
    Goldenberg, R., et al.: Cortex Segmentation: A Fast Variational Geometric Approach. IEEE Transactions on medical imaging 21(2) (2002)Google Scholar
  19. 19.
    Holtzman-Gazit, M., et al.: Segmentation of Thin Structures in Volumetric Medical Images. IEEE Transactions on image processing 15(2) (2006)Google Scholar
  20. 20.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math, 42 (1989)Google Scholar
  21. 21.
    McConnell Brain Imaging Centre, Montréal Neurological Institute, M.U.: Brainweb (12.6.2006),

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Are Losnegård
    • 1
  • Oddvar Christiansen
    • 1
  • Xue-Cheng Tai
    • 1
  1. 1.Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008 BergenNorway

Personalised recommendations