Efficient Approximate 3-Dimensional Point Set Matching Using Root-Mean-Square Deviation Score

  • Yoichi Sasaki
  • Tetsuo Shibuya
  • Kimihito Ito
  • Hiroki Arimura
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9371)


In this paper, we study approximate point subset match (APSM) problem with minimum RMSD score under translation, rotation, and one-to-one correspondence in d-dimension. Since this problem seems computationally much harder than the previously studied APSM problems with translation only or distance evaluation only, we focus on speed-up of exhaustive search algorithms that can find all approximate matches. First, we present an efficient branch-and-bound algorithm using a novel lower bound function of the minimum RMSD score. Next, we present another algorithm that runs fast with high probability when a set of parameters are fixed. Experimental results on real 3-D molecular data sets showed that our branch-and-bound algorithm achieved significant speed-up over the naive algorithm still keeping the advantage of generating all answers.


3D point set matching RMSD Geometric transformation One-to-one correspondence Branch and bound Probabilistic analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akutsu, T.: On determining the congruence of point sets in d dimensions. Computational Geometry 9(4), 247–256 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alt, H., Guibas, L.: Discrete geometric shapes: Matching, interpolation, and approximation, pp. 121–153. Elsevier Science Publishers B.V. North-Holland (1999)Google Scholar
  3. 3.
    Alt, H., Mehlhorn, K., Wagener, H., Welzl, E.: Congruence, similarity and symmetries of geometric objects. Discret. Comput. Geom. 3, 237–256 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arimura, H., Uno, T., Shimozono, S.: Time and space efficient discovery of maximal geometric graphs. In: Corruble, V., Takeda, M., Suzuki, E. (eds.) DS 2007. LNCS (LNAI), vol. 4755, pp. 42–55. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  5. 5.
    Carpentier, M., Brouillet, S., Pothier, J.: Yakusa: a fast structural database scanning method. Proteins 61(1), 137–151 (2005)CrossRefGoogle Scholar
  6. 6.
    Cho, M., Mount, D.M.: Improved approximation bounds for planar point pattern matching. Algorithmica 50(2), 175–207 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications. Springer-Verlag (2000)Google Scholar
  8. 8.
    de Rezende, P.J., Lee, D.: Point set pattern matching in \(d\)-dimensions. Algorithmica 13(4), 387–404 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer (1999)Google Scholar
  10. 10.
    Goodrich, M.T., Mitchell, J.S., Orletsky, M.W.: Approximate geometric pattern matching under rigid motions. IEEE Trans. PAMI 21(4), 371–379 (1999)CrossRefGoogle Scholar
  11. 11.
    Kabsch, W.: A solution for the best rotation to relate two sets of vectors. Acta Crystallographica A32(5), 922–923 (1976)CrossRefGoogle Scholar
  12. 12.
    Mäkinen, V., Ukkonen, E.: Point pattern matching. In: Kao, M. (ed.) Encyclopedia of Algorithms, pp. 657–660. Springer (2008)Google Scholar
  13. 13.
    Nowozin, S., Tsuda, K.: Frequent subgraph retrieval in geometric graph databases. In: 8th IEEE Int’l Conf. on Data Mining, pp. 953–958 (2008)Google Scholar
  14. 14.
    Pinsky, M., Karlin, S.: An introduction to stochastic modeling. Academic Press (2010)Google Scholar
  15. 15.
    Schwartz, J.T., Sharir, M.: Identification of partially obscured objects in two and three dimensions by matching noisy characteristic curves. The Int’l J. of Robotics Res. 6(2), 29–44 (1987)CrossRefGoogle Scholar
  16. 16.
    Shibuya, T.: Geometric suffix tree: Indexing protein 3-d structures. Journal of the ACM 57(3), 15 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tam, G.K., et al.: Registration of 3d point clouds and meshes: a survey from rigid to nonrigid. IEEE Trans. Vis. Comput. Graphics 19(7), 1199–1217 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yoichi Sasaki
    • 1
  • Tetsuo Shibuya
    • 2
  • Kimihito Ito
    • 3
  • Hiroki Arimura
    • 1
  1. 1.IST, Hokkaido UniversitySapporoJapan
  2. 2.University of TokyoTokyoJapan
  3. 3.CZC, Hokkaido UniversitySapporoJapan

Personalised recommendations