Measuring Resemblance of Complex Patterns

  • Michiel Hagedoorn
  • Remco Veltkamp
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1568)

Abstract

On a collection of subsets of a space, fundamentally different metrics may be defined. In pattern matching, it is often required that a metric is invariant for a given transformation group. In addition, a pattern metric should be robust for defects in patterns caused by discretisation and unreliable feature detection. Furthermore, a pattern metric should have sufficient discriminative power. We formalise these properties by presenting five axioms. Finding invariant metrics without requiring such axioms is a trivial problem. Using our axioms, we analyse various pattern metrics, including the Hausdorff distance and the symmetric difference. Finally, we present the reflection metric. This metric is defined on finite unions of n — 1-dimensional hyper-surfaces in ℝn. The reflection metric is affine invariant and satisfies our axioms.

Keywords

Topological Space Open Neighbourhood Pattern Match Computational Geometry Invariance Group 
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.

References

  1. 1.
    O. Aichholzer, H. Alt, and G. Rote. Matching shapes with a reference point. In Int. J. of Computational Geometry and Applications, volume 7, pages 349–363, August 1997.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    H. Alt, U. Fuchs, G. Rote, and G. Weber. Matching convex shapes with respect to the symmetric difference. In Algorithms ESA’ 96, Proc. 4th Annual European Symp. on Algorithms, Barcelona, Spain, September’ 96, pages 320–333. LNCS1136, Springer, 1996.Google Scholar
  3. 3.
    H. Alt and M. Godeau. Computing the Fréchet distance between two polygonal curves. Int. J. of Computational Geometry & Applications, pages 75–91, 1995.Google Scholar
  4. 4.
    L.P. Chew, M.T. Goodrich, D.P. Huttenlocher, K. Kedem, Jon M. Kleinberg, and Dina Kravets. Geometric pattern matching under Euclidean motion. In Fifth Canadian Conference on Computational Geometry, pages 151–156, 1993.Google Scholar
  5. 5.
    S.D. Cohen and L.J. Guibas. Partial matching of planar polylines under similarity transformations. In Eight Annual ACM-SIAM Symp. on Discrete Algorithms, pages 777–786, January 1997.Google Scholar
  6. 6.
    E.T. Copson. Metric spaces. Cambridge University Press, 1968.Google Scholar
  7. 7.
    M. Hagedoorn and R.C. Veltkamp. A general method for partial point set matching. In Jean-Daniel Boissonnat, editor, Proc. 13th Annual ACM Symp. Computational Geometry, pages 406–408. ACM Press, 1997.Google Scholar
  8. 8.
    M. Hagedoorn and R.C. Veltkamp. Reliable and e_cient pattern matching using an affine invariant metric. Technical Report UU-CS-1997-33, Utrecht University, 1997. revision accepted for publication in IJCV.Google Scholar
  9. 9.
    M. Hagedoorn and R.C. Veltkamp. New visibility partitions with applications in affine pattern matching. Manuscript, 1998.Google Scholar
  10. 10.
    D.P. Huttenlocher and K. Kedem. Computing the minimum Hausdorff distance for point sets under translation. In Proc. 6th Annual ACM Symp. Computational Geometry, pages 340–349, 1990.Google Scholar
  11. 11.
    D.P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete and Computational Geometry, 9:267–291, 1993.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D.P. Huttenlocher and W.J. Rucklidge. A multi-resolution technique for comparing images using the Hausdorff distance. Technical Report 92-1321, Cornell University, 1992.Google Scholar
  13. 13.
    W. Rucklidge. Efficient Visual Recognition Using the Hausdorff Distance. Lecture Notes in Computer Science. Springer-Verlag, 1996.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Michiel Hagedoorn
    • 1
  • Remco Veltkamp
    • 1
  1. 1.Department of Computer ScienceUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations