Optimal Lower Bound for Generalized Median Problems in Metric Space

  • Xiaoyi Jiang
  • Horst Bunke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2396)

Abstract

The computation of generalized median patterns is typically an NP-complete task. Therefore, research efforts are focused on approximate approaches. One essential aspect in this context is the assessment of the quality of the computed approximate solutions. In this paper we present a lower bound in terms of a linear program for this purpose. It is applicable to any pattern space. The only assumption we make is that the distance function used for the definition of generalized median is a metric. We will prove the optimality of the lower bound, i.e. it will be shown that no better one exists when considering all possible instances of generalized median problems. An experimental verification in the domain of strings and graphs shows the tightness, and thus the usefulness, of the proposed lower bound.

References

  1. 1.
    J. Astola, P. Haavisto, and Y. Neuvo, Vector median filters, Proceedings of the IEEE, 78(4): 678–689, 1990.CrossRefGoogle Scholar
  2. 2.
    F. Bartolini, V. Cappellini, C. Colombo, and A. Mecocci, Enhancement of local optical flow techniques, Proc. of 4th Int. Workshop on Time Varying Image Processing and Moving Object Recognition, Florence, Italy, 1993.Google Scholar
  3. 3.
    C. Gramkow, On averaging rotations, Int. Journal on Computer Vision, 42(1/2): 7–16, 2001.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. Guimond, J. Meunier, and J.-P. Thirion, Average brain models: A convergence study, Computer Vision and Image Understanding, 77(2): 192–210, 2000.CrossRefGoogle Scholar
  5. 5.
    C. de la Higuera and F. Casacuberta, Topology of strings: Median string is NP-complete, Theoretical Computer Science, 230(1–2): 39–48, 2000.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    X. Jiang, L. Schiffmann, and H. Bunke, Computation of median shapes, Proc. of 4th. Asian Conf. on Computer Vision, 300–305, Taipei, 2000.Google Scholar
  7. 7.
    X. Jiang, A. Münger, and H. Bunke, On median graphs: Properties, algorithms, and applications, IEEE Trans. on PAMI, 23(10): 1144–1151, 2001.Google Scholar
  8. 8.
    X. Jiang, H. Bunke, and J. Csirik, Median strings: A review, 2002. (submitted for publication)Google Scholar
  9. 9.
    A. Juan and E. Vidal, Fast median search in metric spaces, in A. Amin and D. Dori (eds.), Advances in Pattern Recognition, Springer-Verlag, 905–912, 1998.Google Scholar
  10. 10.
    T. Lewis, R. Owens, and A. Baddeley, Averaging feature maps, Pattern Recognition, 32(9): 1615–1630, 1999.CrossRefGoogle Scholar
  11. 11.
    D. Lopresti and J. Zhou, Using consensus sequence voting to correct OCR errors, Computer Vision and Image Understanding, 67(1): 39–47, 1997.CrossRefGoogle Scholar
  12. 12.
    L. Mico and J. Oncina, An approximate median search algorithm in non-metric spaces, Pattern Recognition Letters, 22(10): 1145–1151, 2001.MATHCrossRefGoogle Scholar
  13. 13.
    A. J. O’Toole, T. Price, T. Vetter, J. C. Barlett, and V. Blanz, 3D shape and 2D surface textures of human faces: The role of “averages” in attractiveness and age, Image and Vision Computing, 18(1): 9–19, 1999.CrossRefGoogle Scholar
  14. 14.
    C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Inc., 1982.Google Scholar
  15. 15.
    X. Pennec and N. Ayache, Uniform distribution, distance and expectation problems for geometric features processing, Journal of Mathematical Imaging and Vision, 9(1): 49–67, 1998.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    J. S. Sim and K. Park, The consensus string problem for a metric is NP-complete, Journal of Discrete Algorithms, 2(1), 2001.Google Scholar
  17. 17.
    K. Subramanyan and D. Dean, A procedure to average 3D anatomical structures, Medical Image Analysis, 4(4): 317–334, 2000.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Xiaoyi Jiang
    • 1
  • Horst Bunke
    • 2
  1. 1.Department of Electrical Engineering and Computer ScienceTechnical University of BerlinBerlinGermany
  2. 2.Department of Computer ScienceUniversity of BernBernSwitzerland

Personalised recommendations