Skeletal Structures

  • Silvia Biasotti
  • Dominique Attali
  • Jean-Daniel Boissonnat
  • Herbert Edelsbrunner
  • Gershon Elber
  • Michela Mortara
  • Gabriella Sanniti di Baja
  • Michela Spagnuolo
  • Mirela Tanase
  • Remco Veltkamp
Part of the Mathematics and Visualization book series (MATHVISUAL)

Shape Descriptors are compact and expressive representations of objects suitable for solving problems like recognition, classification, or retrieval of shapes, tasks that are computationally expensive if performed on huge data sets. Skeletal structures are a particular class of shape descriptors, which attempt to quantify shapes in ways that agree with human intuition. In fact, they represent the essential structure of objects and the way basic components connect to form a whole.

In the large amount of literature devoted to a wide variety of skeletal structures, this Chapter provides a concise and non-exhaustive introduction to the subject: indeed the first structural descriptor, the medial axis, dates back to 1967, which means forty years of literature on the topic.


Voronoi Diagram Hausdorff Distance Medial Axis Voronoi Cell Skeletal Structure 
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.


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  1. 1.
    O. Aichholzer, D. Alberts, F. Aurenhammer, and B. Gartner. A novel type of skeleton for polygons. Journal of Universal Computer Science, 1:752-761, 1995.MathSciNetGoogle Scholar
  2. 2.
    N. Amenta and M. Bern. Surface reconstruction by Voronoi filtering. Discrete and Computational Geometry, 22:481-504, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    N. Amenta, S. Choi, and R.K. Kolluri. The power crust, unions of balls, and the medial axis transform. Computational Geometry: Theory and Applications, 19:127-153, 2001.zbMATHMathSciNetGoogle Scholar
  4. 4.
    C. Arcelli and G. Sanniti di Baja. Euclidean skeleton via center-of-maximal-disc extraction. Image and Vision Computing, 11:163-173, 1993.CrossRefGoogle Scholar
  5. 5.
    D. Attali. Squelettes et graphes de Voronoi 2-D et 3-D. PhD thesis, University Joseph Fourier, 1995.Google Scholar
  6. 6.
    D. Attali, J.-D. Boissonnat, and H. Edelsbrunner. Stability and computation of medial axes — A state-of-the-art report. In T. Möller, B. Hamann, and B. Russell, editors, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration. Springer-Verlag, 2005. To appear.Google Scholar
  7. 7.
    D. Attali and J.D. Boissonnat. Complexity of the Delaunay triangulation of points on polyhedral surfaces. Discrete and Computational Geometry, 30(3):437-452, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    D. Attali and J.D. Boissonnat. A linear bound on the complexity of the Delaunay triangulation of points on polyhedral surfaces. Discrete and Computational Geometry, 31:369-384, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    D. Attali, J.D. Boissonnat, and A. Lieutier. Complexity of the Delaunay triangulation of points on surfaces: the smooth case. In SCG ’03: Proc. of the 19th Annual Symposium on Computational Geometry 2003, pages 201-210. ACM Press, 2003.Google Scholar
  10. 10.
    D. Attali and A. Montanvert. Modeling noise for a better simplification of skeletons. In ICIP ’96: Proc. of the International Conference on Image Processing, volume 3, pages 13-16, 1996.CrossRefGoogle Scholar
  11. 11.
    M. Attene, S. Biasotti, and M. Spagnuolo. Shape understanding by contour-driven retiling. The Visual Computer, 19(2-3):127-138, 2003.zbMATHGoogle Scholar
  12. 12.
    F. Aurenhammer and H. Imai. Geometric relations among Voronoi diagrams. Geometria Dedicata, 27:65-75, 1988.zbMATHMathSciNetGoogle Scholar
  13. 13.
    U. Axen and H. Edelsbrunner. Auditory Morse analysis of triangulated manifolds. In Mathematical Visualization, pages 223-236. Springer-Verlag, 1998.Google Scholar
  14. 14.
    Ulrike Axen. Computing Morse functions on triangulated manifolds. In SODA ’99: Proc. of the 10th ACM-SIAM Symposium on Discrete Algoritms 1999, pages 850-851. ACM Press, 1999.Google Scholar
  15. 15.
    M. Bern, D. Eppstein, P.K. Agarwal, N. Amenta, P. Chew, T. Dey, D.P. Dobkin, H. Edelsbrunner, C. Grimm, L.J. Guibas, J. Harer, J. Hass, A. Hicks, C.K. Johnson, G. Lerman, D. Letscher, P. Plassmann, E. Sedgwick, J. Snoeyink, J. Weeks, C. Yap, and D. Zorin. Emerging challenges in computational topology. In Report from the NSF-funded Workshop on Computational Topology, 1999.Google Scholar
  16. 16.
    S. Biasotti. Computational Topology Methods for Shape Modelling Applications. PhD thesis, Universit à degli Studi di Genova, May 2004.Google Scholar
  17. 17.
    S. Biasotti. Reeb graph representation of surfaces with boundary. In SMI ’04: Proc. of Shape Modeling Applications 2004, pages 371-374, Los Alamitos, Jun 2004. IEEE Computer Society.CrossRefGoogle Scholar
  18. 18.
    S. Biasotti, B. Falcidieno, and M. Spagnuolo. Surface shape understanding based on extended Reeb graphs. In Topological Data Structures for Surfaces: An Introduction for Geographical Information Science, pages 87-103. John Wiley and Sons, 2004.Google Scholar
  19. 19.
    S. Biasotti, L. De Floriani, B. Falcidieno, and L. Papaleo. Morphological representations of scalar fields. In Shape Analysis and Structuring. Springer, 2007.Google Scholar
  20. 20.
    S. Biasotti, S. Marini, M. Mortara, and G. Patane. An overview on properties and efficacy of topological skeletons in shape modelling. In M.S. Kim, editor, SMI ’03: Proc. of Shape Modeling International 2003, pages 245-254, Los Alamitos, May 2003. IEEE Computer Society.CrossRefGoogle Scholar
  21. 21.
    H. Blum. A transformation for extracting new descriptors of shape. In Weiant Wathen Dunn, editor, Models for the Perception of Speech and Visual Form. Proc. of a Symposium, pages 362-380, Cambridge MA, Nov 1967. MIT Press.Google Scholar
  22. 22.
    J.D. Boissonnat and F. Cazals. Natural neighbor coordinates of points on a surface. Computational Geometry-Theory and Applications, 19(2-3):155-173, 2001.zbMATHMathSciNetGoogle Scholar
  23. 23.
    J.D. Boissonnat and M. Karavelas. On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of d-dimensional spheres. In SODA ’03: Proc. of the 14th ACM-SIAM Symposium on Discrete Algoritms, pages 305-312. ACM Press, 2003.Google Scholar
  24. 24.
    G. Borgefors. Distance transformations in digital images. Computer Vision, Graphics, and Image Processing, 34:344-371, 1986.CrossRefGoogle Scholar
  25. 25.
    G. Borgefors. On digital distance transform in three dimensions. Computer Vision and Image Understanding, 64(3):368-376, 1996.CrossRefGoogle Scholar
  26. 26.
    G. Borgefors, I. Nyström, G. Sanniti di Baja, and S. Svensson. Simplification of 3D skeletons using distance information. In L. J. Latecki, R. A. Melter, D. M. Mount, and A.Y. Wu, editors, Proc. of SPIE - Vision Geometry IX, volume 4117, pages 300-309, San Diego - USA, 2000.Google Scholar
  27. 27.
    J.W. Brandt. Convergence and continuity criteria for discrete approximations of the continuous planar skeletons. CVGIP: Image Understanding, 59:116-124, 1994.CrossRefGoogle Scholar
  28. 28.
    J.W. Brandt and V.R. Algazi. Continuous skeleton computation by Voronoi diagram. CVGIP: Image Understanding, 55:329-337, 1992.zbMATHCrossRefGoogle Scholar
  29. 29.
    C. Burnikel. Exact Computation of Voronoi Diagrams and Line Segment Intersections. Ph.D thesis, Universit ät des Saarlandes, March 1996.Google Scholar
  30. 30.
    C. Arcelli and G. Sanniti di Baja. A thinning algorithm based on prominence detection. Pattern Recognition, 13(3):225-235, 1981.CrossRefGoogle Scholar
  31. 31.
    C.Arcelli and G. Sanniti di Baja. Skeletons of planar patterns. In T.Y. Kong and A. Rosenfeld, editors, Topological Algorithms for Digital Image Processing, pages 99-143. North-Holland, 1996.Google Scholar
  32. 32.
    H. Carr, J. Snoeyink, and U. Axen. Computing contour trees in all dimensions. In SODA ’00: Proc. of the 11th ACM-SIAM Symposium on Discrete Algoritms 2000, pages 918-926. ACM Press, 2000.Google Scholar
  33. 33.
    The CGAL 3.1 User Manual.Google Scholar
  34. 34.
    T.M. Chan, J. Snoeyink, and C.K. Yap. Primal dividing and dual pruning: Outputsensitive construction of 4-D polytopes and 3-D Voronoi diagrams. Discrete and Computational Geometry, 18:433-454, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    F. Chazal and A. Lieutier. Stability and homotopy of a subset of the medial axis. In SMA ’04: Proc. of the 9th ACM Symposium on Solid Modeling and Applications 2004, pages 243-248. ACM Press, 2004.Google Scholar
  36. 36.
    F. Chazal and R. Soufflet. Stability and finiteness properties of medial axis and skeleton. Journal on Control Dynamics and Systems, 10:149-170, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    S.W. Cheng and A. Vigneron. Motorcycle graphs and straight skeletons. In SODA ’02: Proc. of the 13th ACM-SIAM Symposium on Discrete Algoritms 2002, pages 156-165. ACM Press, 2002.Google Scholar
  38. 38.
    S.W. Choi and H.P. Seidel. Linear one-sided stability of MAT for weakly injective 3D domain. In SMA ’02: Proc. of the 7th ACM Symposium on Solid Modeling and Applications 2002, pages 344-355. ACM Press, 2002.Google Scholar
  39. 39.
    K. Cole-McLaughlin, H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci. Loops in Reeb graphs of 2-manifolds. In SCG ’03: Proc. of the 19th Annual Symposium on Computational Geometry 2003, pages 344-350. ACM Press, 2003.Google Scholar
  40. 40.
    N. D. Cornea, D. Silver, and P. Min. Curve-skeleton applications. In Proceedings IEEE Visualization, pages 95-102, 2005.Google Scholar
  41. 41.
    T. Culver. Computing the medial axis of a polyhedron reliably and efficiently. PhD thesis, University North Carolina, Chapel Hill, North Carolina, 2000.Google Scholar
  42. 42.
    T. Culver, J. Keyser, and D. Manocha. Exact computation of the medial axis of a polyhedron. Computer Aided Geometric Design, 21(1):65-98, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    T. Dey and j. Sun. Defining and computing curve-skeletons with medial geodesic function. In Proceedings of the Symposium on Geometry Processing, pages 143-152, 2006.Google Scholar
  44. 44.
    T.K. Dey and W. Zhao. Approximating the medial axis from the Voronoi diagram with a convergence guarantee. Algorithmica, 38, 2004.Google Scholar
  45. 45.
    G. Sanniti di Baja. Well-shaped, stable and reversible skeletons from the (3,4)-distance transform. Visual Communication and Image Representation, 5:107-115, 1994.CrossRefGoogle Scholar
  46. 46.
    G. Sanniti di Baja. Representing shape by line patterns. In P. Wang and A. Rosenfeld, editors, Advances in Structural and Syntactical Pattern Recognition, volume 1121 of Lecture Notes in Computer Science, pages 230-239. Springer-Verlag, 1996.Google Scholar
  47. 47.
    G. Sanniti di Baja and I. Nyström. Skeletonization in 3D discrete binary images. In C.H. Chen and P.S.P. Wang, editors, Handbook of Pattern Recognition and Computer Vision, Chapter 2.2, pages 137-156. World Scientific, Singapore, 3rd edition, January 2005.Google Scholar
  48. 48.
    G. Sanniti di Baja and E. Thiel. Skeletonization algorithm running on path-based distance maps. Image and Vision Computing, 14:47-57, 1997.Google Scholar
  49. 49.
    G.Sanniti di Baja and S. Svensson. Surface skeletons detected on the D6 distance transform. In F.J. Ferri et al., editor, Proc. of SSSPR’2000 - Advances in Pattern Recognition, volume 1121, pages 387-396, Alicante, 2000. LNCS, Springer-Verlag.Google Scholar
  50. 50.
    S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J. Hart. Spectral surface quadrangulation. ACM Transactions on Graphics, 25(3):1057-1066, August 2006.CrossRefGoogle Scholar
  51. 51.
    D. Dutta and C. Hoffmann. On the skeleton of simple CSG objects. ASME J. of Mechanical Design, 115:87-94, 1993.CrossRefGoogle Scholar
  52. 52.
    H. Edelsbrunner, J. Harer, A. Mascarenhas, and V. Pascucci. Time-varying Reeb graphs for continuous space-time data. In Proceeding of the 20-th ACM Symposium on Computational Geometry, pages 366-372, 2004.Google Scholar
  53. 53.
    H. Edelsbrunner, J. Harer, and A. Zomorodian. Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds. Discrete and Computational Geometry, 30:87-107, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    G. Elber and M. S. Kim. Bisector curves of planar rational curves. Computer Aided Design, 30(14):1089-1096, December 1998.zbMATHCrossRefGoogle Scholar
  55. 55.
    G. Elber and M. S. Kim. The bisector surface of freeform rational space curves. ACM Trans. on Graphics, 17(1):32-50, January 1998.CrossRefGoogle Scholar
  56. 56.
    G. Elber and M. S. Kim. Computing rational bisectors. Computer Graphics and Applications, 19(6):76-81, November-December 1999.CrossRefGoogle Scholar
  57. 57.
    G. Elber and M. S. Kim. Rational bisectors of CSG primitives. In The Fifth ACM/IEEE Symposium on Solid Modeling and Applications, Ann Arbor, Michigan, pages 159-166, June 1999.Google Scholar
  58. 58.
    G. Elber and M. S. Kim. A computational model for nonrational bisector surfaces: Curve-surface and surface-surface bisectors. In Geometric Modeling and Processing 2000, Hong Kong, pages 364-372, April 2000.Google Scholar
  59. 59.
    G. Elber and M. S. Kim. Geometric constraint solver using multivariate rational spline functions. In The Sixth ACM/IEEE Symposium on Solid Modeling and Applications, Ann Arbor, Michigan, pages 1-10, June 2001.Google Scholar
  60. 60.
    D. Eppstein and J. Erickson. Raising roofs, crashing cycles, and playing pool: Applications of a data structure for finding pairwise interactions. Discrete and Computational Geometry, 22:569-592, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    R. Farouki and J. Johnstone. The bisector of a point and a plane parametric curve. Computer Aided Geometric Design, 11(2):117-151, April 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    R. Farouki and R. Ramamurthy. Specified-precision computation of curve/curve bisectors. Int. J. of Computational Geometry & Applications, 8(5-6):599-617, OctoberDecember 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    R. Farouki and R. Ramamurthy. Voronoi diagram and medial axis algorithm for planar domains with curved boundaries I. theoretical foundations. J. of Computational and Applied Mathematics, 102:119-141, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    R. Farouki and R. Ramamurthy. Voronoi diagram and medial axis algorithm for planar domains with curved boundaries II. detailed algorithm description. J. of Computational and Applied Mathematics, 102:119-141, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    S. Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2:153-174, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    P. Giblin and B.B. Kimia. A formal classification of 3D medial axis points and their local geometry. In CVPR 2000: Proc. of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition 2000, volume 1, pages 566-573, Los Alamitos, 2000. IEEE Computer Society.Google Scholar
  67. 67.
    P.J. Giblin and B.B. Kimia. On the local form and transitions of symmetry sets, medial axes, and shocks. International Journal of Computer Vision, 54:143-157, 2003.zbMATHCrossRefGoogle Scholar
  68. 68.
    S. Goswami, T. K. Dey, and C. L. Bajaj. Identifying flat and tubular regions of a shape by unstable manifolds. In SPM ’06: Proceedings of the 2006 ACM symposium on Solid and physical modeling, pages 27-37, New York, NY, USA, 2006. ACM Press.CrossRefGoogle Scholar
  69. 69.
    A. Gramain. Topologie des surfaces. Presses Universitaires de France, 1971.Google Scholar
  70. 70.
    I. Hanniel, R. Muthuganapathy, G. Elber, and M. S. Kim. Precise Voronoi cell extraction of free-form rational planar closed curves. In ACM Symposium on Solid and Physical Modeling, pages 51-59, June 2005.Google Scholar
  71. 71.
    M. Held. Vroni: An engineering approach to the reliable and efficient computation of Voronoi diagrams of points and line segments. Computational Geometry: Theory and Applications, 18:95-123, 2001.zbMATHMathSciNetGoogle Scholar
  72. 72.
    F. Hetroy and D. Attali. Topological quadrangulations of closed triangulated surfaces using the Reeb graph. Graphical Models, 65(1-3):131-148, 2003.zbMATHCrossRefGoogle Scholar
  73. 73.
    M. Hilaga, Y. Shinagawa, T. Kohmura, and T. L. Kunii. Topology matching for fully automatic similarity estimation of 3D shapes Los Angeles, CA. Computer graphics proceedings, annual conference series: SIGGRAPH conference proceedings, pages 203-212, Aug 2001.Google Scholar
  74. 74.
    D.S. Kim, Y. Cho, and D. Kim. Euclidean Voronoi diagram of 3D balls and its computation via tracing edges. In Computer Aided Design, pages 1412-1424, November 2005.Google Scholar
  75. 75.
    B. Kimia, A. Tannenbaum, and S. Zucker. Shapes, shocks, and deformations, I: The components of shape and the reaction-diffusion space. International Journal of Computer Vision, 15:189-224, 1995.CrossRefGoogle Scholar
  76. 76.
    F. Lazarus and A. Verroust. Level set diagrams of polyhedral objects. In W.F. Bronsvoort and D.C. Anderson, editors, SMA ’99: Proc. of the 5th ACM Symposium on Solid Modeling and Applications 1999, pages 130-140. ACM Press, 1999.Google Scholar
  77. 77.
    D. T. Lee. Medial axis transformation of a planar shape. IEEE Transactions on Pattern Analysis and Machine Intelligence, 4(4):363-369, 1982.zbMATHCrossRefGoogle Scholar
  78. 78.
    A. Lieutier. Any open bounded subset of Rn has the same homotopy type as its medial axis. In Proc. 8th ACM Sympos. Solid Modeling Appl., pages 65-75. ACM Press, 2003.Google Scholar
  79. 79.
    G. Matheron. Examples of topological properties of skeletons. In Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances, pages 217-238. Academic Press, 1988.Google Scholar
  80. 80.
    J. Milnor. Morse Theory. Princeton University Press, New Jersey, 1963.zbMATHGoogle Scholar
  81. 81.
    M. Mortara and G. Patané . Shape-covering for skeleton extraction. International Journal of Shape Modelling, 8:245-252, 2002.CrossRefGoogle Scholar
  82. 82.
    M. Mortara, G. Patane, M. Spagnuolo, B. Falcidieno, and J. Rossignac. Blowing bubbles for multi-scale analysis and decomposition of triangle meshes. Algorithmica, 38(1):227-248,2004.MathSciNetGoogle Scholar
  83. 83.
    R.L. Ogniewicz. Skeleton-space: A multi-scale shape description combining region and boundary information. In CVPR ’94: Proc. of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition 1994, pages 746-751, Los Alamitos, 1994. IEEE Computer Society.Google Scholar
  84. 84.
    R.L. Ogniewicz and O. Kubler. Hierarchic Voronoi skeletons. Pattern Recognition, 28:343-359, 1995.CrossRefGoogle Scholar
  85. 85.
    V. Pascucci and K. Cole-McLaughlin. Parallel computation of the topology of level sets. Algorithmica, 38:249-268, 2003.CrossRefMathSciNetGoogle Scholar
  86. 86.
    M. Peternell. Geometric properties of bisector surfaces. In Graphical Models and Image Processing, 2000.Google Scholar
  87. 87.
    M. Ramanathan and B. Gurumoorthy. Constructing medial axis transform of planar domains with curved boundaries. Computer Aided Design, 35:619-632, 2002.CrossRefGoogle Scholar
  88. 88.
    M. Ramanathan and B. Gurumoorthy. Constructing medial axis transform of extruded and revolved 3D objects with free-form boundaries. Computer-Aided Design, 37 (13):1370-1387, 2005.Google Scholar
  89. 89.
    G. Reeb. Sur les points singuliers d’une forme de Pfaff compl ètement int égrable ou d’une fonction num érique. Comptes Rendus Hebdomadaires des S éances de l’Acad émie des Sciences, 222:847-849, 1946.zbMATHMathSciNetGoogle Scholar
  90. 90.
    A. Rosenfeld. Digital geometry: Introduction and bibliography. In Advances in Digital and Computational Geometry, 1998.Google Scholar
  91. 91.
    E. Sherbrooke, N. M. Patrikalakis, and F.-E. Wolter. Differential and topological properties of medial axis transforms. Graphical Models and Image Processing, 58:574-592, 1996.CrossRefGoogle Scholar
  92. 92.
    E.C. Sherbrooke, N.M. Patrikalakis, and E. Brisson. An algorithm for the medial axis transform of 3D polyhedral solids. IEEE Trans. on Visualization and Computer Graphics, 22:44-61, 1996.CrossRefGoogle Scholar
  93. 93.
    Y. Shinagawa and T.L. Kunii. Constructing a Reeb graph automatically from cross sections. IEEE Computer Graphics and Applications, 11:44-51, 1991.CrossRefGoogle Scholar
  94. 94.
    Y. Shinagawa, T.L. Kunii, and Y.L. Kergosien. Surface coding based on Morse theory. IEEE Computer Graphics and Applications, 11:66-78, 1991.CrossRefGoogle Scholar
  95. 95.
    S. Svensson and G. Sanniti di Baja. Simplifying curve skeletons in volume images. Computer Vision and Image Understanding, 90:242-257, 2003.zbMATHCrossRefGoogle Scholar
  96. 96.
    S. Svensson, I. Nystr om, and G. Sanniti di Baja. Curve skeletonization of surfacelike objects in 3D images guided by voxel classification. Pattern Recognition Letters, 23(12):1419-1426, 2002.zbMATHCrossRefGoogle Scholar
  97. 97.
    M. Tanase and R. C. Veltkamp. A straight skeleton approximating the medial axis. Lecture Notes in Computer Science, 3221:809-821, Sep 2004.Google Scholar
  98. 98.
    M. van Kreveld, R. Oostrum, C. Bajaj, V. Pascucci, and D. Schikore. Contour trees and small seed sets for isosurface transversal. In SCG ’97: Proc. of the 13th Annual Symposium on Computational Geometry 1997, pages 212-220. ACM Press, Jun 1997.Google Scholar
  99. 99.
    M. Wan, Z. Liang, Q. Ke, L. Hong, I. Bitter, and A. Kaufman. Automatic centerline extraction for virtual colonoscopy. IEEE Trans. on Medical Imaging, 21(12):1450-1460, December 2002.Google Scholar
  100. 100.
    N. Werghi, Y. Xiao, and J. P. Siebert. A functional-based segmentation of human body scans in arbitrary postures. IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics, 36(1):153-165, 2006.CrossRefGoogle Scholar
  101. 101.
    F.E. Wolter. Cut locus & medial axis in global shape interrogation & representation. Technical Report Design Laboratory Memorandum 92-2, MIT, 1992.Google Scholar
  102. 102.
    Z. Wood, H. Hoppe, M. Desbrun, and P. Schroeder. Removing excess topology from isosurfaces. ACM Trans. on Graphics, 23:190-208, 2004.CrossRefGoogle Scholar
  103. 103.
    Z.J. Wood, M. Desbrun, P. Schroder, and D. Breen. Semi-regular mesh extraction from volumes. In VIS 2000: Proc. of IEEE Conference on Visualization 2000, pages 275-282, Los Alamitos, 2000. IEEE Computer Society.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Silvia Biasotti
    • 1
  • Dominique Attali
    • 2
  • Jean-Daniel Boissonnat
    • 3
  • Herbert Edelsbrunner
    • 4
  • Gershon Elber
    • 5
  • Michela Mortara
    • 1
  • Gabriella Sanniti di Baja
    • 6
  • Michela Spagnuolo
    • 1
  • Mirela Tanase
    • 7
  • Remco Veltkamp
    • 7
  1. 1.CNR. - Ist. di Matematica Applicata e Tecnologie InformaticheGenovaItaly
  2. 2.LIS-CNRSDomaine UniversitaireSaint Martin d'HèresFrance
  3. 3.INRIASophia-AntipolisFrance
  4. 4.Department of Computer ScienceDuke UniversityNorth CarolinaUSA
  5. 5.Technion - Israel Institute of TechnologyHaifaIsrael
  6. 6.CNR - Ist. di Cibernetica “E. Caianello”Pozzuoli, NapoliItaly
  7. 7.Universiteit Utrecht (UU)The Netherlands

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