Combinatorial Properties of Scale Space Singular Points

  • Atsushi Imiya
  • Tomoya Sakai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4040)


Singular points in the linear scale space provide fundamental features for the extraction of dominant parts of an image. Employing the geometrical configuration of singular points, it is possible to construct a tree in scale space. This tree expresses a hierarchical structure of dominant parts. In this paper, we clarify the graphical grammar for the construction of this tree in the linear scale space and morphological scale space. Furthermore, we show a combinatorial structure of singular points in the linear scale space and morphological scale space using conformal mapping from Euclidean space to the spherical surface.


Singular Point Hessian Matrix Scale Space Combinatorial Property Voronoi Tessellation 
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  1. 1.
    Bookstein, F.L.: The line-skeleton. CVGIP 11, 1233–1237 (1979)Google Scholar
  2. 2.
    Rosenfeld, A.: Axial representations of shapes. CVGIP 33, 156–173 (1986)Google Scholar
  3. 3.
    Serra, J.: Mathematical Morphology. Academic Press, London (1982)MATHGoogle Scholar
  4. 4.
    Iijima, T.: Pattern Recognition, Corona-sha, Tokyo (1974) (in Japanese)Google Scholar
  5. 5.
    Witkin, A.P.: Scale space filtering. In: Pros. of 8th IJCAI, pp. 1019–1022 (1993)Google Scholar
  6. 6.
    Lindeberg, T.: Scale-Space Theory in Computer Vision. Kluwer, Boston (1994)Google Scholar
  7. 7.
    ter Haar Romeny, B.M.: Front-End Vision and Multi-Scale Image Analysis Multi-scale Computer Vision Theory and Applications, written in Mathematica. Springer, Heidelberg (2003)Google Scholar
  8. 8.
    Lindeberg, T.: Feature detection with automatic selection. International Journal of Computer Vision 30, 79–116 (1998)CrossRefGoogle Scholar
  9. 9.
    Otsu, N.: Mathematical Studies on Feature Extraction in Pattern Recognition. Researches of The Electrotechnical Laboratory 818 (1981) (in Japanese)Google Scholar
  10. 10.
    Weicker, J.: Anisotropic Diffusion in Image Processing. Teubner (1998)Google Scholar
  11. 11.
    Imiya, A., Sugiura, T., Sakai, T., Kato, Y.: Temporal structure tree in digital linear scale space. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 356–371. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Zhao, N.-Y., Iijima, T.: Theory on the method of determination of view-point and field of vision during observation and measurement of figure IECE Japan. Trans. D. J68-D, 508–514 (1985) (in Japanese)Google Scholar
  13. 13.
    Zhao, N.-Y., Iijima, T.: A theory of feature extraction by the tree of stable view-points. IECE Japan, Trans. D. J68-D, 1125–1135 (1985) (in Japanese)Google Scholar
  14. 14.
    Zhao, N.-Y.: A Study of Feature Extraction by the Tree of Stable View-Points, Dissertation to Doctor of Engineering, Tokyo Institute of Technology (1985) (in Japanese)Google Scholar
  15. 15.
    Pelillo, M., Siddiqi, K., Zucker, S.W.: Matching hierarchical structures using Association graphs. IEEE Trans. PAMI 21, 1105–1120 (1999)Google Scholar
  16. 16.
    Yuille, A.L., Poggio, T.: Scale space theory for zero crossings. IEEE PAMI 8, 15–25 (1986)MATHGoogle Scholar
  17. 17.
    Kuijper, A., Florack, L.M.J., Viergever, M.A.: Scale Space Hierarchy. Journal of Mathematical Imaging and Vision 18, 169–189 (2003)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kuijper, A., Florack, L.M.J.: The hierarchical structure of images. IEEE, Trans. Image Processing 12, 1067–1079 (2003)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Imiya, A., Katsuta, R.: Extraction of a structure feature from three-dimensional objects by scale-space analysis. In: ter Haar Romeny, B.M., Florack, L.M.J., Viergever, M.A. (eds.) Scale-Space 1997. LNCS, vol. 1252, pp. 353–356. Springer, Heidelberg (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Atsushi Imiya
    • 1
  • Tomoya Sakai
    • 1
  1. 1.Institute of Media and Information TechnologyChiba UniversityChibaJapan

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