Computing and Visualization in Science

, Volume 12, Issue 6, pp 267–285 | Cite as

Image segmentation using a multilayer level-set approach

  • Ginmo Chung
  • Luminita A. Vese
Regular article


We propose an efficient multilayer segmentation method based on implicit curve evolution and on variational approach. The proposed formulation uses the minimal partition problem as formulated by D. Mumford and J. Shah, and can be seen as a more efficient extension of the segmentation models previously proposed in Chan and Vese (Scale-Space Theories in Computer Vision, Lecture Notes in Computer Science, Vol. 1682, pp. 141–151, 1999, IEEE Trans Image Process 10(2):266–277, 2001), and Vese and Chan (Int J Comput Vis 50(3):271–293, 2002). The set of unknown discontinuities is represented implicitly by several nested level lines of the same function, as inspired from prior work on island dynamics for epitaxial growth (Caflisch et al. in Appl Math Lett 12(4):13, 1999; Chen et al. in J Comput Phys 167:475, 2001). We present the Euler–Lagrange equations of the proposed minimizations together with theoretical results of energy decrease, existence of minimizers and approximations. We also discuss the choice of the curve regularization and conclude with several experimental results and comparisons for piecewise-constant segmentation of gray-level and color images.


Image Segmentation Epitaxial Growth Active Contour Lagrange Equation Level Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aronsson G. (1967). Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6(6): 551–561 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Caflisch R.E., Gyure M.F., Merriman B., Osher S., Ratsch C., Vvedensky D.D. and Zinck J.J. (1999). Island dynamics and the level set method for epitaxial growth. Appl. Math. Lett. 12(4): 13 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Caselles V., Morel J.-M. and Sbert C. (1998). An axiomatic approach to image interpolation. IEEE Trans. Image Process. 7(3): 376–386 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chan T.F., Sandberg B.Y. and Vese L.A. (2000). Active contours without edges for vector-valued images. J. Vis. Commun. Image Represent. 11(2): 130–141 CrossRefGoogle Scholar
  5. 5.
    Chan, T.F., Vese, L.: An active contour model without edges. In: Scale-Space Theories in Computer Vision, Lecture Notes in Computer Science, Vol. 1682, pp. 141–151 (1999)Google Scholar
  6. 6.
    Chan T.F. and Vese L.A. (2001). Active contours without edges. IEEE Trans. Image Process. 10(2): 266–277 zbMATHCrossRefGoogle Scholar
  7. 7.
    Chen S., Kang M., Merriman B., Caflisch R.E., Ratsch C., Fedkiw R., Gyure M.F. and Osher S. (2001). Level set method for thin film epitaxial growth. J. Comput. Phys. 167: 475 zbMATHCrossRefGoogle Scholar
  8. 8.
    Chung, G., Vese, L.A.: Energy minimization based segmentation and denoising using a multilayer level set approach. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition, 5th International Workshop, EMMCVPR 2005, St Augustine, FL, USA, 9–11 November 2005. LNCS, Vol. 3757/2005, pp. 439–455 (2005)Google Scholar
  9. 9.
    Cohen L.D. (1997). Avoiding local minima for deformable curves in image analysis. In: Le Méhauté A., Rabut C., Schumaker L.L. (eds). Curves and Surfaces with Applications in CAGD, pp. 77–84Google Scholar
  10. 10.
    Cohen, L., Bardinet, E., Ayache, N.: Surface reconstruction using active contour models. In: Proceedings of SPIE 93 Conference on Geometric Methods in Computer Vision, San Diego (1993)Google Scholar
  11. 11.
    Dervieux A. and Thomasset F. (1979). A finite element method for the simulation of Rayleigh–Taylor instability. Lect. Notes Math. 771: 145–159 CrossRefMathSciNetGoogle Scholar
  12. 12.
    Dervieux A. and Thomasset F. (1980). Multifluid incompressible flows by a finite element method. Lect. Notes Phys. 141: 158–163 CrossRefGoogle Scholar
  13. 13.
    Evans L.C. and Gariepy R. (1992). Measure Theory and Fine Properties of Functions. CRC Press, London zbMATHGoogle Scholar
  14. 14.
    Gyure M.F., Ratsch C., Merriman B., Caflisch R.E., Osher S., Zinck J.J. and Vvedensky D.D. (1998). Level set methods for the simulation of epitaxial phenomena. Phys. Rev. E 58: R6927 CrossRefGoogle Scholar
  15. 15.
    Jensen R. (1993). Uniqueness of Lipschitz extensions—minimizing the sup norm of the gradient. Arch. Ration. Mech. Anal. 123(1): 51–74 zbMATHCrossRefGoogle Scholar
  16. 16.
    Kimmel R. (2003). Fast Edge Integration. In: Osher, S. and Paragios, N. (eds) Geometric Level Set Methods in Imaging, Vision and Graphics, pp 59–77. Springer, Heidelberg CrossRefGoogle Scholar
  17. 17.
    Lie J., Lysaker M. and Tai X.-C. (2006). A Binary Level Set Model and Some Applications to Mumford-Shah Image Segmentation. IEEE Trans. Image Process. 15(5): 1171–1181 CrossRefGoogle Scholar
  18. 18.
    Mumford D. and Shah J. (1989). Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42: 577–685 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Osher S. and Sethian J. (1988). Fronts propagating with curvature-dependent speed—algorithms based on Hamilton–Jacobi formulations. JCP 79(1): 12–49 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Paragios N. and Deriche R. (1999). Unifying boundary and region-based information for geodesic active tracking. Proc. Comput. Vision Pattern Recognit. 2: 23–25 Google Scholar
  21. 21.
    Paragios N. and Deriche R. (2002). Geodesic active regions: a new framework to deal with frame partition problems in computer vision. J. Vis. Commun. Image Represent. 13(1–2): 249–268 CrossRefGoogle Scholar
  22. 22.
    Paragios N. and Deriche R. (2002). Geodesic active regions and level set methods for supervised texture segmentation. IJCV 46(3): 223–247 zbMATHCrossRefGoogle Scholar
  23. 23.
    Rudin L.I., Osher S. and Fatemi E. (1992). Nonlinear Total variation based noise removal algorithms. Phys. D Nonlinear Phenomena 60(1–4): 259–268 zbMATHCrossRefGoogle Scholar
  24. 24.
    Samson, C., Blanc-Féraud, L., Aubert, G., Zérubia, J.: A Level Set Model for Image Classification. LNCS, vol. 1682, pp. 306–317Google Scholar
  25. 25.
    Samson C., Blanc-Féraud L., Aubert G. and Zérubia J. (2000). A level set model for image classification. IJCV 40(3): 187–197 zbMATHCrossRefGoogle Scholar
  26. 26.
    Tsai A., Yezzi A. and Willsky A.S. (2001). Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. Image Process. 10(8): 1169–1186 zbMATHCrossRefGoogle Scholar
  27. 27.
    Vese L.A. and Chan T.F. (2002). A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50(3): 271–293 zbMATHCrossRefGoogle Scholar
  28. 28.
    Vese L. (2003). Multiphase Object Detection and Image Segmentation. In: Osher, S. and Paragios, N. (eds) Geometric Level Set Methods in Imaging, Vision and Graphics, pp 175–194. Springer, Heidelberg CrossRefGoogle Scholar
  29. 29.
    Zhao H.-K., Chan T., Merriman B. and Osher S. (1996). Variational level set approach to multiphase motion. J. Comput. Phys. 127(1): 179–195 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

Personalised recommendations