Approximate Nearest Neighbor Search under Translation Invariant Hausdorff Distance

  • Christian Knauer
  • Marc Scherfenberg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

Abstract

The Hausdorff distance is a measure for the resemblance of two geometric objects. Given a set of n point patterns and a query point pattern Q , the nearest neighbor of Q under the Hausdorff distance is the point pattern which minimizes this distance to Q . An extension of the Hausdorff distance is the translation invariant Hausdorff distance which additionally allows the translation of the point patterns in order to minimize the distance. This paper introduces the first data structure which allows to solve the nearest neighbor problem for the directed Hausdorff distance under translation in sublinear query time in a non-heuristic manner, in the sense that the quality of the results, the performance, and the space bounds are guaranteed. The data structure answers queries for both directions of the directed Hausdorff distance with a \( \sqrt{d(s-1.5)}(1+\epsilon) \)-approximation factor in \( O(\log \frac{n}{\epsilon}) \) query time for the nearest neighbor and O(k + logn) query time for the k -th nearest neighbor for any ε> 0 . (The O -notation of the latter runtime contains terms that are quadratic in ε− 1 .)

Furthermore it is shown how to find the exact nearest neighbor under the directed Hausdorff distance without transformation of the point sets within some weaker time and storage bounds.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Har-Peled, S.: A replacement for Voronoi diagrams of near linear size. Foundations of Computer Science (January 2001)Google Scholar
  2. 2.
    Kleinberg, J.M.: Two algorithms for nearest-neighbor search in high dimensions. In: STOC 1997: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pp. 599–608. ACM, New York (1997)CrossRefGoogle Scholar
  3. 3.
    Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: STOC 1984: Proceedings of the sixteenth annual ACM symposium on Theory of computing, pp. 135–143. ACM, New York (1984)CrossRefGoogle Scholar
  4. 4.
    Indyk, P.: On approximate nearest neighbors under l  ∞  norm. Journal of Computer and System Sciences (January 2001)Google Scholar
  5. 5.
    Arya, S., Mount, D., Netanyahu, N., Silverman, R., Wu, A.: An optimal algorithm for approximate nearest neighbor searching in fixed dimensions. Journal of the ACM (JACM) (January 1998)Google Scholar
  6. 6.
    Braß, P., Knauer, C.: Nearest neighbour search in Hausdorff distance pattern spaces. Technical report, Institut für Informatik, FU-Berlin (2001)Google Scholar
  7. 7.
    Farach-Colton, M., Indyk, P.: Approximate nearest neighbor algorithms for Hausdorff metrics via embeddings. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science (October 1999)Google Scholar
  8. 8.
    Vleugels, J., Veltkamp, R.C.: Efficient image retrieval through vantage objects. Pattern Recognition 35(1), 69–80 (2002)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Christian Knauer
    • 1
  • Marc Scherfenberg
    • 1
  1. 1.Institute of Computer ScienceFreie UniversitätBerlin

Personalised recommendations