Signal, Image and Video Processing

, Volume 1, Issue 4, pp 303–320 | Cite as

A discrete level set approach to image segmentation

Original Paper
First Online:
Received:
Revised:
Accepted:

Abstract

Models and algorithms in image processing are usually defined in the continuum and then applied to discrete data, that is the signal samples over a lattice. In particular, the set up in the continuum of the segmentation problem allows a fine formulation basically through either a variational approach or a moving interfaces approach. In any case, the image segmentation is obtained as the steady-state solution of a nonlinear PDE. Nevertheless the application to real data requires discretization schemes where some of the basic image geometric features have a loose meaning. In this paper, a discrete version of the level set formulation of a modified Mumford and Shah energy functional is investigated, and the optimal image segmentation is directly obtained through a nonlinear finite difference equation. The typical characteristics of a segmentation, such as its component domains area and its boundary length, are all defined in the discrete context thus obtaining a more realistic description of the available data. The existence and uniqueness of the optimal solution is proved in the class of piece wise constant functions, but with no restrictions on the nature of the segmentation boundary multiple points. The proposed algorithm compared to a standard segmentation procedure in the continuum generally provides a more accurate segmentation, with a much lower computational cost.

Keywords

Image segmentation Level set method Discrete image model Denoising Deblurring Restoration 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kass M., Witkin A. and Terzopoulos D. (1988). Snakes: active contour models. Int. J. Comput. Vis. 1: 321–332 CrossRefGoogle Scholar
  2. 2.
    Mumford D. and Shah J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42(4): 577–685 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    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 Proces. 10(8): 1169–1184 MATHCrossRefGoogle Scholar
  4. 4.
    Suri J.S., Liu K., Singh S., Laxminarayan S.N., Zeng X. and Reden L. (2002). Shape recovery algorithms using level sets in 2-D/3-D medical imagery: a state-of-the-art review. IEEE Trans. Inform. Tech. Biomed. 6(1): 8–28 CrossRefGoogle Scholar
  5. 5.
    Mansouri A.R. and Conrad J. (2003). Multiple motion segmentation with level sets. IEEE Trans. Image Proces. 12(2): 201–220 CrossRefGoogle Scholar
  6. 6.
    Kruse, F.A.: Multiresolution segmentation for improved hyperspectral mapping. In: Proc. SPIE Symposium on Defense & Security, Orlando, FL (2005)Google Scholar
  7. 7.
    Yang, T., Li, S.Z., Pan, Q., Li, J.: Real time and accurate segmentation of moving objects in dynamic scene. In: Proc. of the ACM 2-nd Intern. Workshop on Video Surveillance & Sensor networks, pp. 136–143, New York (2004)Google Scholar
  8. 8.
    Imasogie B.I. and Wendt U. (2004). Characterization of graphite particle shape in spheroidal graphite iron using a computer based image analyzer. J. Minerals Materi. Characterization Eng. 3(1): 1–12 Google Scholar
  9. 9.
    Proc. of the Seventh International Conference on Document Analysis and Recognition—ICDAR2003, http://www.cse.salford.ac.uk/prima/ICDAR2003/Google Scholar
  10. 10.
    Niethammer M., Tannenbaum A. and Angenent S. (2006). Dynamic active contours for visual tracking. IEEE Trans. Autom. Control 51(4): 562–579 CrossRefMathSciNetGoogle Scholar
  11. 11.
    Kichenassamy S., Kumar A., Olver P., Tannenbaum A. and Yezzy A. (1996). Conformal curvature flows: from phase transition to active vision. Arch. Rational Mech. Anal. 134(3): 275–301 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Caselles V. and Coll B. (1996). Snakes in movement. SIAM J. Numer. Anal. 33(6): 2445–2456 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Caselles V., Kimmel R. and Sapiro. G. (1997). Geodesic Active Contours. Int. J. Comput. Vis. 22(1): 61–79 MATHCrossRefGoogle Scholar
  14. 14.
    Osher S. and Sethian J.A. (1988). Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79: 12–49 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Caselles V., Cattè F., Coll B. and Dibos F. (1993). A geometric model for active contours in image processing. Numer. Math. 66: 1–31 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Chan T. and Vese L. (2001). Active contours without edge. IEEE Trans. Image Proces. 10(2): 266–277 MATHCrossRefGoogle Scholar
  17. 17.
    Chan T. and Vese L. (2002). A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model. Int. J. Comput. Vis. 50(3): 271–293 MATHCrossRefGoogle Scholar
  18. 18.
    Heiler M. and Schnorr C. (2005). Natural image statistics for natural image segmentation. Int. J. Comput Vis. 63(1): 5–19 CrossRefGoogle Scholar
  19. 19.
    Mansouri A.R., Mitiche A. and Vasquez C. (2006). Multiregion competition: a level set extension of region competition to multiple region image partitioning. Comput. Vis Image Understanding 101: 137–150 CrossRefGoogle Scholar
  20. 20.
    Cremers D., Sochen N. and Schnorr C. (2006). A multiphase dynamic labeling model for variational recognition-driven image segmentation. Int. J. Comput. Vis. 66(1): 67–81 CrossRefGoogle Scholar
  21. 21.
    Unal G., Yezzi A. and Krim H. (2005). Information-theoretic active polygons for unsupervised texture segmentation. Int. J. Comput. Vis. 62(3): 199–220 CrossRefGoogle Scholar
  22. 22.
    Aubert G. and Vese L. (1997). A variational method in image recovery. SIAM J. Numer. Anal. 34(5): 1948–1979 MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Shi, Y., Karl, W.C.: A fast level set method without solving PDEs. In: Proc. 2005 IEEE Int’l Conf. on Acoustics, Speech, and Signal Processing, vol. II, pp. 97–100, Philadelphia (2005)Google Scholar
  24. 24.
    Boykov Y. and Kolmogorov V. (2004). An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE Trans. PAMI, 26(9): 1124–1137 Google Scholar
  25. 25.
    Funka-Lea, G., Boykov, Y., Florin, C., Jolly, M.-P., Moreau-Gobard, R., Ramaraj, R., Rinck, D.: Automatic heart isolation for CT coronary visualization using graph-cuts. In: 3-rd International Symposium on Biomedical Imaging: from macro to nano, Arlington-Virginia, pp. 614–617Google Scholar
  26. 26.
    Gel’fand I.M. and Shilov G.E. (1964). Generalized Functions-vol 1-Properties and Operations. Academic, New York Google Scholar
  27. 27.
    Jeon M., Alexander M., Pedrycz W. and Pizzi N. (2005). Unsupervised hierarchical image segmentation with level set and additive operator splitting. Pattern Recogn. Lett. 26: 1461–1469 CrossRefGoogle Scholar
  28. 28.
    Rudin L.I., Osher S. and Fatemi E. (1992). Nonlinear total variation based noise removal algorithms. Physica D 60: 259–268 MATHCrossRefGoogle Scholar
  29. 29.
    Holliday J.D., Hu C.-Y. and Willett P. (2002). Grouping of coefficients for the calculation of inter-molecular similarity and dissimilarity using 2-D fragment bit-strings. Combin. Chem. High Throughput Screening 5: 155–166 Google Scholar

Copyright information

© Springer-Verlag London Limited 2007

Authors and Affiliations

  1. 1.Department of Computer and Systems Science “Antonio Ruberti”University “Sapienza” of RomeRomeItaly

Personalised recommendations