Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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COMPUTATION OF THE HAUSDORFF DISTANCE BETWEEN TWO ELLIPSES

  • Kim, Ik-Sung (Division of Data Information, Korea Maritime and Ocean University)
  • Received : 2016.08.23
  • Accepted : 2016.12.05
  • Published : 2016.12.25
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Abstract

We are interested in the problem of finding the Hausdorff distance between two objects in ${\mathbb{R}}^2$, or in ${\mathbb{R}}^3$. In this paper, we develop an algorithm for computing the Hausdorff distance between two ellipses in ${\mathbb{R}}^3$. Our algorithm is mainly based on computing the distance between a point $u{\in}{\mathbb{R}}^3$ and a standard ellipse $E_s$, equipped with a pruning technique. This algorithm requires O(log M) operations, compared with O(M) operations for a direct method, to achieve a comparable accuracy. We give an example,and observe that the computational cost needed by our algorithm is only O(log M).

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References

  1. Helmut Alt, Peter Brab, Michael Godau, Christian Knauer and Carola Wenk, Computing the Hausdorff distance of geometric patterns and shapes, in B. Aronov, S. Basu, J. Pach, M. Sharir (Eds.), Discrete and Computational Geometry: the Goodman-Pollack Festschrift, Springer, (2003).
  2. E. Belogay, C. Cabrelli, U. Molter and R. Shankwiler, Calculating the Hausdorff distance between curves, Information Processing Letters, 64, (1997), 17-22. https://doi.org/10.1016/S0020-0190(97)00140-3
  3. Mark Bouts, Comparing images using the Hausdorff distance, unpublished manuscript, (2006).
  4. I. S. Kim, An algorithm for finding the distance between two ellipses, Commun. Korean Math. Soc. 21(3),(2006) , 559-567. https://doi.org/10.4134/CKMS.2006.21.3.559
  5. C. A. Neff, Finding the distance between two circles in three dimensional space, IBM J. Res. Develop., 34, (1990),770-775. https://doi.org/10.1147/rd.345.0770
  6. G. Rote,Computing the minimum Hausdorff distance between two point sets on a line under translation, Information Processing Letters,38, (1991), 123-127. https://doi.org/10.1016/0020-0190(91)90233-8
  7. Ludmilla Scharf, Computing the Hausdorff distance between sets of curves, PhD Thesis, Freie Universitat, Berlin, (2003).
  8. Richard P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, (1973).