A Reflective Symmetry Descriptor

  • Michael Kazhdan
  • Bernard Chazelle
  • David Dobkin
  • Adam Finkelstein
  • Thomas Funkhouser
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2351)


Computing reflective symmetries of 2D and 3D shapes is a classical problem in computer vision and computational geometry. Most prior work has focused on finding the main axes of symmetry, or determining that none exists. In this paper, we introduce a new reflective symmetry descriptor that represents a measure of reflective symmetry for an arbitrary 3D voxel model for all planes through the model’s center of mass (even if they are not planes of symmetry). The main benefits of this new shape descriptor are that it is defined over a canonical parameterization (the sphere) and describes global properties of a 3D shape. Using Fourier methods, our algorithm computes the symmetry descriptor in O(N 4 log N) time for an N × N × N voxel grid, and computes a multiresolution approximation in O(N 3 log N) time. In our initial experiments, we have found the symmetry descriptor to be useful for registration, matching, and classification of shapes.


Shape Descriptor North Pole Voxel Model Symmetry Detection Voxel Grid 
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.


  1. 1.
    Mitsumoto, H., Tamura, S., Okazaki, K., Kajimi, N., Fukui, Y.: Reconstruction using mirror images based on a plane symmetry recovery method (1992)Google Scholar
  2. 2.
    Zabrodsky, H., Peleg, S., Avnir, D.: Symmetry as a continuous feature. IEEE PAMI 17 (1995) 1154–1156Google Scholar
  3. 3.
    Liu, Y., Rothfus, W., Kanade, T.: Content-based 3d neuroradiologic image retrieval: Preliminary results (1998)Google Scholar
  4. 4.
    Leou, J., Tsai, W.: Automatic rotational symmetry determination for shape analysis. Pattern Recognition 20 (1987) 571–582CrossRefGoogle Scholar
  5. 5.
    Wolfson, H., Reisfeld, D., Yeshurun, Y.: Robust facial feature detection using symmetry. Proceedings of the International Conference on Pattern Recognition (1992) 117–120Google Scholar
  6. 6.
    Atallah, M.J.: On symmetry detection. IEEE Trans. on Computers c-34 (1985) 663–666MathSciNetCrossRefGoogle Scholar
  7. 7.
    Wolter, J.D., Woo, T.C., Volz, R.A.: Optimal algorithms for symmetry detection in two and three dimensions. The Visual Computer 1 (1985) 37–48zbMATHCrossRefGoogle Scholar
  8. 8.
    Marola, G.: On the detection of the axes of symmetry of symmetric and almost symmetric planar images. IEEE PAMI 11 (1989) 104–108zbMATHGoogle Scholar
  9. 9.
    Shen, D., Ip, H., Cheung, K., Teoh, E.: Symmetry detection by generalized complex (gc) moments: A close-form solution. IEEE PAMI 21 (1999) 466–476Google Scholar
  10. 10.
    Sun, C., Si, D.: Fast reflectional symmetry detection using orientation histograms. Real-Time Imaging 5 (1999) 63–74CrossRefGoogle Scholar
  11. 11.
    O’Mara, D., Owens, R.: Measuring bilateral symmetry in digital images. IEEE-TENCON-Digital Signal Processing Applications (1996)Google Scholar
  12. 12.
    Sun, C., Sherrah, J.: 3-d symmetry detection using the extended Gaussian image. IEEE PAMI 19 (1997)Google Scholar
  13. 13.
    Kovesi, P.: Symmetry and asymmetry from local phase. Tenth Australian Joint Converence on Artificial Intelligence (1997) 2–4Google Scholar
  14. 14.
    Elad, M., Tal, A., Ar, S.: Directed search in a 3d objects database using svm (2000)Google Scholar
  15. 15.
    Osada, R., Funkhouser, T., Chazelle, B., Dobkin, D.: Matching 3d models with shape distributions. Shape Matching International (2001)Google Scholar
  16. 16.
    Knuth, D., J.H. Morris, J., Pratt, V.: Fast pattern matching in strings. SIAM Journal of Computing 6 (1977) 323–350zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Besl, P.J., Jain, R.C.: Three-dimensional object recognition. Computing Surveys 17 (1985) 75–145CrossRefGoogle Scholar
  18. 18.
    Loncaric, S.: A survey of shape analysis techniques. Pattern Recognition 31 (1998) 983–1001CrossRefGoogle Scholar
  19. 19.
    Pope, A.R.: Model-based object recognition: A survey of recent research. Technical Report TR-94-04, University of British Columbia (1994)Google Scholar
  20. 20.
    Veltkamp, R.C., Hagedoorn, M.: State-of-the-art in shape matching. Technical Report UU-CS-1999-27, Utrecht University, the Netherlands (1999)Google Scholar
  21. 21.
    Johnson, A., Hebert, M.: Efficient multiple model recognition in cluttered 3-d scenes. IEEE CVPR (1998) 671–677Google Scholar
  22. 22.
    Zhang, D., Hebert, M.: Harmonic maps and their applications in surface matching. IEEE CVPR 2 (1999)Google Scholar
  23. 23.
    Belongie, S., Malik, J.: Matching with shape contexts. IEEE Workshop on Content-based access of Image and Video-Libraries (2000)Google Scholar
  24. 24.
    Mori, G., Belongie, S., Malik, H.: Shape contexts enable efficient retrieval of similar shapes. CVPR 1 (2001) 723–730Google Scholar
  25. 25.
    B. Horn, B.: Extended gaussian images. PIEEE 72 (1984) 1656–1678Google Scholar
  26. 26.
    Fernández-Vidal, S., Bardinet, E., Malandain, G., Damas, S., de la Blanca Capilla, N.: Object representation and comparison inferred from its medial axis. ICPR 1 (2000) 712–715Google Scholar
  27. 27.
    Siddiqi, K., Shokoufandeh, A., Dickinson, S., Zucker, S.: Shock graphs and shape matching. IJCV 35 (1999) 13–32CrossRefGoogle Scholar
  28. 28.
    Bloomenthal, J., Lim, C.: Skeletal methods of shape manipulation. Shape Modeling and Applications (1999) 44–47Google Scholar
  29. 29.
    Storti, D., Turkiyyah, G., Ganter, M., Lim, C., Stal, D.: Skeleton-based modeling operations on solids. Symposium on Solid Modeling and Applications (1997) 141–154Google Scholar
  30. 30.
    Serre, J.: Linear Representations of Finite Groups. Springer-Verlag, New York (1977)zbMATHGoogle Scholar
  31. 31.
    Rubner, Y., Tomasi, C., Guibas, L.: A metric for distributions with applications to image databases. IEEE ICCV (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Michael Kazhdan
    • 1
  • Bernard Chazelle
    • 1
  • David Dobkin
    • 1
  • Adam Finkelstein
    • 1
  • Thomas Funkhouser
    • 1
  1. 1.Princeton UniversityPrincetonUSA

Personalised recommendations