Matching Point Sets with Respect to the Earth Mover’s Distance

  • Sergio Cabello
  • Panos Giannopoulos
  • Christian Knauer
  • Günter Rote
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)

Abstract

The Earth Mover’s Distance (EMD) between two weighted point sets (point distributions) is a distance measure commonly used in computer vision for color-based image retrieval and shape matching. It measures the minimum amount of work needed to transform one set into the other one by weight transportation.

We study the following shape matching problem: Given two weighted point sets A and B in the plane, compute a rigid motion of A that minimizes its Earth Mover’s Distance to B. No algorithm is known that computes an exact solution to this problem. We present simple FPTAS and polynomial-time (2 + ε)-approximation algorithms for the minimum Euclidean EMD between A and B under translations and rigid motions.

References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Alt, H., Guibas, L.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Comp. Geom., pp. 121–153. Elsevier Science Publishers B.V, North-Holland (1999)Google Scholar
  3. 3.
    Atkinson, D.S., Vaidya, P.M.: Using geometry to solve the transportation problem in the plane. Algorithmica 13, 442–461 (1995)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Bose, P., Maheshwari, A., Morin, P.: Fast approximations for sums of distances clustering and the Fermat-Weber problem. Comp. Geom. Theory & Appl. 24, 135–146 (2003)MathSciNetMATHGoogle Scholar
  5. 5.
    Callahan, P.B., Kosaraju, S.R.: Faster algorithms for some geometric graph problems in higher dimensions. In: Proc. of the 4th ACM-SIAM SODA, pp. 291–300 (1993)Google Scholar
  6. 6.
    Chandrasekaran, R., Tamir, A.: Algebraic optimization: The Fermat-Weber location problem. Math. Programming 46(2), 219–224 (1990)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Cohen, S., Guibas, L.: The Earth Mover’s Distance under transformation sets. In: Proc. of the 7th IEEE ICCV, pp. 173–187 (1999)Google Scholar
  8. 8.
    Giannopoulos, P., Veltkamp, R.C.: A pseudo-metric for weighted point sets. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 715–731. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Grauman, K., Darell, T.: Fast contour matching using approximate Earth Mover’s Distance. In: Proc. of the IEEE CVPR, pp. 220–227 (2004)Google Scholar
  10. 10.
    Indyk, P., Thaper, N.: Fast image retrieval via embeddings. In: 3rd Int. Workshop on Statistical and Computational Theories of Vision (2003)Google Scholar
  11. 11.
    Klein, O., Veltkamp, R.C.: Approximation algorithms for the Earth Mover’s Distance under transformations using reference points. Technical Report UU-CS-2005-003, IICS, Utrecht University, The Netherlands (2005)Google Scholar
  12. 12.
    Lv, Q., Charikar, M., Li, K.: Image similarity search with compact data structures. In: Proc. of the 13th ACM CIKM, pp. 208–217 (2004)Google Scholar
  13. 13.
    Mumford, D.: Mathematical theories of shape: Do they model perception? In: SPIE Geometric Methods in Comp. Vision, vol. 1570, pp. 2–10 (1991)Google Scholar
  14. 14.
    Orlin, J.B.: A faster strongly polynomial minimum cost flow algorithm. Operations Research 41(2), 338–350 (1993)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Rubner, Y., Tomasi, C., Guibas, L.J.: The Earth Mover’s Distance as a metric for image retrieval. Int. Journal of Computer Vision 40(2), 99–121 (2000)CrossRefMATHGoogle Scholar
  16. 16.
    Typke, R., Giannopoulos, P., Veltkamp, R.C., Wiering, F., van Oostrum, R.: Using transportation distances for measuring melodic similarity. In: Proc of 4th Int. Symp. on Music Inf. Retrieval (ISMIR), pp. 107–114 (2003)Google Scholar
  17. 17.
    Varadarajan, K.R., Agarwal, P.K.: Approximation algorithms for bipartite and non-bipartite matching in the plane. In: Proc. of the 10th ACM-SIAM SODA 1999, pp. 805–814 (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Sergio Cabello
    • 1
  • Panos Giannopoulos
    • 2
  • Christian Knauer
    • 2
  • Günter Rote
    • 2
  1. 1.Department of MathematicsIMFMLjubljanaSlovenia
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany

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