Minimizing the Weighted Directed Hausdorff Distance between Colored Point Sets under Translations and Rigid Motions

  • Christian Knauer
  • Klaus Kriegel
  • Fabian Stehn
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5598)


Matching geometric objects with respect to their Hausdorff distance is a well investigated problem in Computational Geometry with various application areas. The variant investigated in this paper is motivated by the problem of determining a matching (in this context also called registration) for neurosurgical operations. The task is, given a sequence \(\mathcal{P}\) of weighted point sets (anatomic landmarks measured from a patient), a second sequence \(\mathcal{Q}\) of corresponding point sets (defined in a 3D model of the patient) and a transformation class \(\mathcal{T}\), compute the transformations \(t\in\mathcal{T}\) that minimize the weighted directed Hausdorff distance of \(t(\mathcal{P})\) to \(\mathcal{Q}\). The weighted Hausdorff distance, as introduced in this paper, takes the weights of the point sets into account. For this application, a weight reflects the precision with which a landmark can be measured.

We present an exact solution for translations in the plane, a simple 2-approximation as well as a FPTAS for translations in arbitrary dimension and a constant factor approximation for rigid motions in the plane or in ℝ3.


Voronoi Diagram Computational Geometry Weighted Distance Iterative Close Point Rigid Motion 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alt, H., Guibas, L.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Handbook of Computational Geometry, pp. 121–153. Elsevier B.V., Amsterdam (2000)CrossRefGoogle Scholar
  2. 2.
    Hoffmann, F., Kriegel, K., Schönherr, S., Wenk, C.: A Simple and Robust Geometric Algorithm for Landmark Registration in Computer Assisted Neurosurgery. Technical Report B 99-21, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Germany (December 1999)Google Scholar
  3. 3.
    Besl, P.J., McKay, N.D.: A Method for Registration of 3-D Shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence 14(2), 239–256 (1992)CrossRefGoogle Scholar
  4. 4.
    Dimitrov, D., Knauer, C., Kriegel, K.: Registration of 3D - Patterns and Shapes with Characteristic Points. In: Proceedings of International Conference on Computer Vision Theory and Applications - VISAPP 2006, Setúbal, Portugal, pp. 393–400 (2006)Google Scholar
  5. 5.
    Dimitrov, D., Knauer, C., Kriegel, K., Stehn, F.: Approximation algorithms for a point-to-surface registration problem in medical navigation. In: Proc. Frontiers of Algorithmics Workshop, Lanzhou, China, pp. 26–37 (2007)Google Scholar
  6. 6.
    Huttenlocher, D.P., Kedem, K., Sharir, M.: The upper envelope of Voronoi surfaces and its applications. In: SCG 1991: Proceedings of the seventh annual symposium on Computational geometry, pp. 194–203. ACM, New York (1991)CrossRefGoogle Scholar
  7. 7.
    Sharir, M., Agarwal, P.K.: Davenport-Schinzel sequences and their geometric applications. Cambridge University Press, New York (1996)zbMATHGoogle Scholar
  8. 8.
    Aurenhammer, F., Edelsbrunner, H.: An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recognition 17(2), 251–257 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hershberger, J.: Finding the upper envelope of n line segments in O(n log n) time. Inf. Process. Lett. 33(4), 169–174 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rucklidge, W.: Lower Bounds for the Complexity of the Graph of the Hausdorff Distance as a Function of Transformation. Discrete & Computational Geometry 16(2), 135–153 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Christian Knauer
    • 1
  • Klaus Kriegel
    • 1
  • Fabian Stehn
    • 1
  1. 1.Institut für InformatikFreie Universität BerlinGermany

Personalised recommendations