Advertisement

A Simple Algorithm for Approximate Partial Point Set Pattern Matching under Rigid Motion

  • Arijit Bishnu
  • Sandip Das
  • Subhas C. Nandy
  • Bhargab B. Bhattacharya
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5942)

Abstract

This paper deals with the problem of approximate point set pattern matching in 2D. Given a set P of n points, called sample set, and a query set Q of k points (k ≤ n), the problem is to find a match of Q with a subset of P under rigid motion (rotation and/or translation) transformation such that each point in Q lies in the ε-neighborhood of a point in P. The ε-neighborhood region of a point p i  ∈ P is an axis-parallel square having each side of length ε and p i at its centroid. We assume that the point set is well-seperated in the sense that for a given ε> 0, each pair of points p, p′ ∈ P satisfy at least one of the following two conditions (i) |x(p) − x(p′)| ≥ ε, and (ii) |y(p) − y(p′)| ≥ 3ε, and we propose an algorithm for the approximate matching that can find a match (if it exists) under rigid motion in O(n 2 k 2(klogk + logn)) time. If only translation is considered then the existence of a match can be tested in O(n k 2 logn) time. The salient feature of our algorithm for the rigid motion and translation is that it avoids the use of intersection of high degree curves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akutsu, T.: On determining the congruence of point sets in d dimensions. Computational Geometry: Theory and Applications 9, 247–256 (1998)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Alt, H., Mehlhorn, K., Wagener, H., Welzl, E.: Congruence, similarity and symmetries of geometric objects. Discrete Computational Geometry 3, 237–256 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alt, H., Guibas, L.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Handbook of Computational Geometry, pp. 121–153. Elsevier Science Publishers B.V. North-Holland, Amsterdam (1999)Google Scholar
  4. 4.
    Arkin, E.M., Kedem, K., Mitchell, J.S.B., Sprinzak, J., Werman, M.: Matching points into pairwise-disjoint noise regions: combinatorial bounds and algorithms. ORSA Journal on Computing 4, 375–386 (1992)zbMATHGoogle Scholar
  5. 5.
    Brass, P., Knauer, C.: Testing the congruence of d-dimensional point sets. Int. J. Computational Geometry and Applications 12, 115–124 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cabello, S., Giannopoulos, P., Knauer, C.: On the parameterized complexity of d-dimensional point set pattern matching. Information Processing Letters 105, 73–77 (2008)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Chew, L.P., Goodrich, M.T., Huttenlocher, D.P., Kedem, K., Kleinberg, J.M., Kravets, D.: Geometric pattern matching under euclidean motion. Computational Geometry: Theory and Applications 7, 113–124 (1997)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Efrat, A., Itai, A.: Improvements on bottleneck matching and related problems using geometry. In: Proc. 12th ACM Symposium on Computational Geometry, pp. 301–310. ACM, New York (1996)Google Scholar
  9. 9.
    Gavrilov, M., Indyk, P., Motwani, R., Venkatasubramanian, S.: Geometric pattern matching: a performance study. In: Proc. 15th ACM Symposium on Computational Geometry, pp. 79–85. ACM, New York (1999)Google Scholar
  10. 10.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, NY (1980)zbMATHGoogle Scholar
  11. 11.
    Goodrich, M.T., Mitchell, J.S.B., Orletsky, M.W.: Approximate geometric pattern matching under rigid motions. IEEE Trans. PAMI 21(4), 371–379 (1999)Google Scholar
  12. 12.
    Heffernan, P.J., Schirra, S.: Approximate decision algorithms for point set congruence. Computational Geometry: Theory and Applications 4(3), 137–156 (1994)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Imai, K., Sumino, S., Imai, H.: Minimax geometric fitting of two corresponding sets of points. In: Proc. 5th ACM Symposium on Computational Geometry, pp. 266–275. ACM, New York (1989)Google Scholar
  14. 14.
    Indyk, P., Motwani, R., Venkatasubramanian, S.: Geometric matching under noise: combinatorial bounds and algorithms. In: Proc. 10th SIAM-ACM Symposium on Discrete Algorithms, pp. 457–465. ACM-SIAM, New York (1999)Google Scholar
  15. 15.
    Irani, S., Raghavan, P.: Combinatorial and experimental results for randomized point matching algorithms. In: Proc. 12th ACM Symposium on Computational Geometry, pp. 68–77. ACM, New York (1996)Google Scholar
  16. 16.
    Maltoni, D., Maio, D., Jain, A.K., Prabhakar, S.: Handbook of Fingerprint Recognition. Springer, NY (2003)zbMATHGoogle Scholar
  17. 17.
    Mount, D.M., Netanyahu, N.S., Moigne, J.L.: Efficient algorithms for robust feature matching. Pattern Recognition 32, 17–38 (1999)CrossRefGoogle Scholar
  18. 18.
    Rezende, P.J., Lee, D.T.: Point set pattern matching in d-dimensions. Algorithmica 13, 387–404 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, NY (1995)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Arijit Bishnu
    • 1
  • Sandip Das
    • 1
  • Subhas C. Nandy
    • 1
  • Bhargab B. Bhattacharya
    • 1
  1. 1.Advanced Computing and Microelectronics UnitIndian Statistical InstituteKolkataIndia

Personalised recommendations