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Approximate Geometric Ellipsoid Fitting: A CG-Approach

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Recent Advances in Optimization and its Applications in Engineering

Summary

The problem of geometric ellipsoid fitting is considered. In connection with a conjugate gradient procedure a suitable approximation for the Euclidean distance of a point to an ellipsoid is used to calculate the fitting parameters. The approach we follow here ensures optimization over the set of all ellipsoids with codimension one rather than allowing for different conics as well. The distance function is analyzed in some detail and a numerical example supports our theoretical considerations.

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References

  1. F. L. Bookstein. Fitting conic sections to scattered data. Computer Graphics and Image Processing, 9:56–71, 1981.

    Article  Google Scholar 

  2. P. Ciarlini, M. Cox, F. Pavese, and D. Richter, editors. Advanced mathematical tools in metrology II. Proceedings of the 2nd international workshop, Oxford, UK, September 27–30, 1995.Singapore: World Scientific, 1996.

    MATH  Google Scholar 

  3. F. Deutsch. Best approximation in inner product spaces. NY Springer, 2001.

    Google Scholar 

  4. A. Fitzgibbon, M. Pilu, and R. B. Fisher. Direct least square fitting of ellipses. IEEE Trans. Pattern Anal. Mach. Intell., 21(5):476–480, 1999.

    Article  Google Scholar 

  5. A. Forbes. Generalised regression problems in metrology. Numer. Algorithms, 5(1-4):523–533, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  6. W. Gander, G. Golub, and R. Strebel. Least-squares fitting of circles and ellipses. BIT, 34(4):558–578, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. Golub and C. F. Van Loan. Matrix computations. 3rd ed. Baltimore: The Johns Hopkins Univ. Press, 1996.

    MATH  Google Scholar 

  8. K. Kanatani. Statistical optimization for geometric computation: theory and practice. Amsterdam: North-Holland, 1996.

    MATH  Google Scholar 

  9. K. Kanatani. Statistical optimization for geometric fitting: theoretical accuracy bound and high order error analysis. Int. J. of Comp. Vision, 80:167–188, 2008.

    Article  Google Scholar 

  10. I. Markovsky, A. Kukush, and S. Van Huffel. Consistent least squares fitting of ellipsoids. Numer. Math., 98(1):177–194, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  11. M.Baeg, H. Hashimoto, F. Harashima, and J. Moore. Pose estimation of quadratic surface using surface fitting technique. In Proc. of the International Conference on Intelligent Robots and Systems, vol. 3, pages 204–209, 1995.

    Google Scholar 

  12. J. Nocedal and S. J.Wright. Numerical optimization.New York: Springer, 1999.

    MATH  Google Scholar 

  13. E. Rimon and S. P. Boyd. Obstacle collision detection using best ellipsoid fit. J. Intell. Robotics Syst., 18(2):105–126, 1997.

    Article  MATH  Google Scholar 

  14. P. D. Sampson. Fitting conic sections to ’very scattered’ data: An iterative refinement of the Bookstein algorithm. Computer Graphics and Image Processing, 18(1):97–108, 1982.

    Article  Google Scholar 

  15. S. Shklyar, A. Kukush, I. Markovsky, and S. van Huffel. On the conic section fitting problem. J. Multivariate Anal., 98(3):588–624, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Stoer. On the relation between quadratic termination and convergence properties of minimization algorithms. Numer. Math., 28:343–366, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Strebel, D. Sourlier, and W. Gander. A comparison of orthogonal least squares fitting in coordinate metrology. Van Huffel, Sabine (ed.), Recent advances in total least squares techniques and errors-in-variables modeling. Philadelphia: SIAM. 249-258, 1997.

    Google Scholar 

  18. P. Thomas, R. Binzel, M. Gaffey, B. Zellner, A. Storrs, and E. Wells. Vesta: Spin pole, size, and shape from Hubble Space Telescope images. Icarus, 128, 1997.

    Google Scholar 

  19. J. Varah. Least squares data fitting with implicit functions. BIT, 36(4):842–854, 1996.

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work has been supported in parts by CoTeSys - Cognition for Technical Systems, Cluster of Excellence, funded by Deutsche Forschungsgemeinschaft.

Author information

Authors and Affiliations

  1. Institute of Data Processing, Technische Universität München, München, Germany

    Martin Kleinsteuber

  2. Department of Mathematics, University of Würzburg, Würzburg, Germany

    Knut Hüper

Authors
  1. Martin Kleinsteuber
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  2. Knut Hüper
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Corresponding author

Correspondence to Martin Kleinsteuber .

Editor information

Editors and Affiliations

  1. Dept. Electrical Engineering (ESAT), Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, Leuven, 3001, Belgium

    Moritz Diehl

  2. CORE-UCL, Voie du Roman Pays 34, Louvain-la-Neuve, 1348, Belgium

    Francois Glineur

  3. , Department Computerwetenschappen, Katholieke Universiteit Leuven, Leuven, 3001, Belgium

    Elias Jarlebring

  4. Dept. Computer Science, Katholieke Universiteit Leuven, Celesijnenlaan 200a, Leuven, 3001, Belgium

    Wim Michiels

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Kleinsteuber, M., Hüper, K. (2010). Approximate Geometric Ellipsoid Fitting: A CG-Approach. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds) Recent Advances in Optimization and its Applications in Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12598-0_7

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  • DOI: https://doi.org/10.1007/978-3-642-12598-0_7

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-12597-3

  • Online ISBN: 978-3-642-12598-0

  • eBook Packages: EngineeringEngineering (R0)

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