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The Persistent Morse Complex Segmentation of a 3-Manifold

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Modelling the Physiological Human (3DPH 2009)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5903))

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Abstract

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.

This research was partially supported by Geomagic, Inc., and by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011-05-1-0057.

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References

  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. Beucher, S.: Watersheds of functions and picture segmentation. In: Proc. IEEE Intl. Conf. Acoustic, Speech, Signal Process, pp. 1928–1931 (1982)

    Google Scholar 

  3. Beucher, S.: The watershed transformation applied to image segmentation. Scanning Microscopy Suppl. 6, 299–314 (1992)

    Google Scholar 

  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)

    Article  Google Scholar 

  5. Edelsbrunner, H.: Geometry and Topology for Mesh Generation. Cambridge Univ. Press, England (2001)

    Book  MATH  Google Scholar 

  6. Edelsbrunner, H.: Surface tiling with differential topology. In: Proc. 3rd Eurographics Sympos. Geom. Process., pp. 9–11 (2005)

    Google Scholar 

  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. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)

    MathSciNet  MATH  Google Scholar 

  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. Haralick, R.M., Shapiro, L.G.: Image segmentation techniques. Comput. Vis. Graph. Im. Proc. 29, 100–132 (1985)

    Article  Google Scholar 

  11. Haykin, S.: Neural Networks: a Comprehensive Foundation. MacMillan College, New York (1994)

    MATH  Google Scholar 

  12. Horowitz, S.L., Pavlidis, T.: Picture segmentation by a tree traversal algorithm. J. Assoc. Comput. Mach. 23, 368 (1976)

    MATH  Google Scholar 

  13. Maintz, J.B.A., Viergever, M.A.: A survey of medical image registration. Med. Im. Anal. 2, 1–36 (1998)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  15. Matsumoto, Y.: An Introduction to Morse Theory. Translated from Japanese by K. Hudson and M. Saito, Amer. Math. Soc. (2002)

    Google Scholar 

  16. McInerney, T., Terzopoulos, D.: Deformable models in medical image analysis: a survey. Med. Im. Anal. 1, 91–108 (1996)

    Article  Google Scholar 

  17. Milnor, J.: Morse Theory. Princeton Univ. Press, New Jersey (1963)

    MATH  Google Scholar 

  18. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Redwood City (1984)

    MATH  Google Scholar 

  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. Roerdink, J., Meijster, A.: The watershed transform: definitions, algorithms, and parallelization strategies. Fundamenta Informaticae 41, 187–228 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Sahoo, P.K., Soltani, S., Wong, A.K.C.: A survey of thresholding techniques. Comput. Vision Graphics Im. Process. 41, 233–260 (1988)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  24. Thom, R.: Sur une partition en cellules associée à une fonction sur une variété. Comptes Rendus l’Acad. Sci. 228, 973–975 (1949)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  26. Yang, G.Z., Burger, P., Firmin, D.N., Underwood, S.R.: Structure adaptive anisotropic image filtering. Image Vision Comput. 14, 135–145 (1996)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. IST Austria (Institute of Science and Technology Austria),  

    Herbert Edelsbrunner

  2. Dept. Comput. Sci., Duke Univ., Durham, North Carolina, USA

    Herbert Edelsbrunner

  3. Geomagic, Research Triangle Park, North Carolina, USA

    Herbert Edelsbrunner

  4. Dept. Math., Duke Univ., Durham, North Carolina, USA

    John Harer

  5. Program Comput. Bio. Bioinf., Duke Univ., Durham, North Carolina, USA

    John Harer

Authors
  1. Herbert Edelsbrunner
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  2. John Harer
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Editor information

Editors and Affiliations

  1. MIRALab/C.U.I, University of Geneva, Battelle, 7 route de Drize, 1227, Carouge / Geneva, Switzerland

    Nadia Magnenat-Thalmann

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Edelsbrunner, H., Harer, J. (2009). The Persistent Morse Complex Segmentation of a 3-Manifold. In: Magnenat-Thalmann, N. (eds) Modelling the Physiological Human. 3DPH 2009. Lecture Notes in Computer Science, vol 5903. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10470-1_4

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  • DOI: https://doi.org/10.1007/978-3-642-10470-1_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10468-8

  • Online ISBN: 978-3-642-10470-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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