The Persistent Morse Complex Segmentation of a 3-Manifold

  • Herbert Edelsbrunner
  • John Harer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5903)


We describe an algorithm for segmenting three-dimensional medical imaging data modeled as a continuous function on a 3-manifold. It is related to watershed algorithms developed in image processing but is closer to its mathematical roots, which are Morse theory and homological algebra. It allows for the implicit treatment of an underlying mesh, thus combining the structural integrity of its mathematical foundations with the computational efficiency of image processing.


Span Tree Betti Number Homotopy Type Morse Function Integral 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.
    Attali, D., Cohen-Steiner, D., Edelsbrunner, H.: Extraction and simplification of iso-surfaces in tandem. In: Proc. 3rd Eurographics Sympos. Geom. Process., pp. 139–148 (2005)Google Scholar
  2. 2.
    Beucher, S.: Watersheds of functions and picture segmentation. In: Proc. IEEE Intl. Conf. Acoustic, Speech, Signal Process, pp. 1928–1931 (1982)Google Scholar
  3. 3.
    Beucher, S.: The watershed transformation applied to image segmentation. Scanning Microscopy Suppl. 6, 299–314 (1992)Google Scholar
  4. 4.
    Clarke, L.P., Velthuizen, R.P., Camacho, M.A., Heine, J.J., Vaidyanathan, M., Hall, L.O., Thatcher, R.W., Silbiger, M.L.: MRI segmentation: methods and applications. Magn. Reson. Imag. 13, 343–368 (1995)CrossRefGoogle Scholar
  5. 5.
    Edelsbrunner, H.: Geometry and Topology for Mesh Generation. Cambridge Univ. Press, England (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    Edelsbrunner, H.: Surface tiling with differential topology. In: Proc. 3rd Eurographics Sympos. Geom. Process., pp. 9–11 (2005)Google Scholar
  7. 7.
    Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Morse-Smale complexes for piecewise linear 3-manifolds. In: Proc. 19th Ann. Sympos. Comput. Geom., pp. 361–370 (2003)Google Scholar
  8. 8.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Friedlander, F., Meyer, F.: A sequential algorithm for detecting watersheds on a gray level image. Acta Stereol. 6, 663–668 (1987)Google Scholar
  10. 10.
    Haralick, R.M., Shapiro, L.G.: Image segmentation techniques. Comput. Vis. Graph. Im. Proc. 29, 100–132 (1985)CrossRefGoogle Scholar
  11. 11.
    Haykin, S.: Neural Networks: a Comprehensive Foundation. MacMillan College, New York (1994)zbMATHGoogle Scholar
  12. 12.
    Horowitz, S.L., Pavlidis, T.: Picture segmentation by a tree traversal algorithm. J. Assoc. Comput. Mach. 23, 368 (1976)zbMATHGoogle Scholar
  13. 13.
    Maintz, J.B.A., Viergever, M.A.: A survey of medical image registration. Med. Im. Anal. 2, 1–36 (1998)CrossRefGoogle Scholar
  14. 14.
    Manousakas, I.N., Undrill, P.E., Cameron, G.G., Redpath, T.W.: Split-and-merge segmentation of magnetic resonance medical images: performance evaluation and extension to three dimensions. Comput. Biomed. Res. 31, 393–412 (1998)CrossRefGoogle Scholar
  15. 15.
    Matsumoto, Y.: An Introduction to Morse Theory. Translated from Japanese by K. Hudson and M. Saito, Amer. Math. Soc. (2002)Google Scholar
  16. 16.
    McInerney, T., Terzopoulos, D.: Deformable models in medical image analysis: a survey. Med. Im. Anal. 1, 91–108 (1996)CrossRefGoogle Scholar
  17. 17.
    Milnor, J.: Morse Theory. Princeton Univ. Press, New Jersey (1963)zbMATHGoogle Scholar
  18. 18.
    Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Redwood City (1984)zbMATHGoogle Scholar
  19. 19.
    Pham, D.L., Xu, C., Prince, J.L.: Current methods in medical image segmentation. Annu. Rev. Biomed. Engin., 315–337 (2000)Google Scholar
  20. 20.
    Roerdink, J., Meijster, A.: The watershed transform: definitions, algorithms, and parallelization strategies. Fundamenta Informaticae 41, 187–228 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Sahoo, P.K., Soltani, S., Wong, A.K.C.: A survey of thresholding techniques. Comput. Vision Graphics Im. Process. 41, 233–260 (1988)CrossRefGoogle Scholar
  22. 22.
    Sijbers, J., Scheunders, P., Verhoye, M., Van der Linden, A., van Dyck, D., Raman, E.: Watershed-based segmentation of 3D MR data for volume quantization. Magn. Reson. Imag. 15, 679–688 (1997)CrossRefGoogle Scholar
  23. 23.
    Suri, J.S., Liu, K., Singh, S., Laxminarayan, S.N., Zeng, X., Reden, L.: Shape recovery algorithms using level sets in 2-D/3-D medical imagery: a state-of-the-art review. IEEE Trans. Inform. Techn. Biomed. 6, 8–28 (2002)CrossRefGoogle Scholar
  24. 24.
    Thom, R.: Sur une partition en cellules associée à une fonction sur une variété. Comptes Rendus l’Acad. Sci. 228, 973–975 (1949)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Vincent, L., Soille, P.: Watersheds of digital spaces: an efficient algorithm based on immersion simulations. IEEE Trans. Pattern Anal. Mach. Intell. 13, 583–593 (1991)CrossRefGoogle Scholar
  26. 26.
    Yang, G.Z., Burger, P., Firmin, D.N., Underwood, S.R.: Structure adaptive anisotropic image filtering. Image Vision Comput. 14, 135–145 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Herbert Edelsbrunner
    • 1
    • 2
    • 3
  • John Harer
    • 4
    • 5
  1. 1.IST Austria (Institute of Science and Technology Austria) 
  2. 2.Dept. Comput. Sci.Duke Univ.DurhamUSA
  3. 3.Geomagic, Research Triangle ParkUSA
  4. 4.Dept. Math., Duke Univ.DurhamUSA
  5. 5.Program Comput. Bio. Bioinf.Duke Univ.DurhamUSA

Personalised recommendations