Sharpness in Trajectory Estimation by Piecewise-quadratics(-cubics) and Cumulative Chords

  • Ryszard Kozera
  • Mateusz Śmietanka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7594)


In this paper we verify sharpness of the theoretical results concerning the asymptotic orders of trajectory approximation of the unknown parametric curve γ in arbitrary Euclidean space. The pertinent interpolation schemes (based on piecewise-quadratics and piecewise-cubics) are here considered for the so-called reduced data. The latter forms an ordered collection of points without provision of the associated interpolation knots. To complement such data i.e. to determine the missing knots, cumulative chord parameterization is invoked. Sharpness of cubic and quartic orders of convergence are demonstrated for piecewise-quadratics and piecewise-cubics, respectively. This topic has its ramification in computer vision (e.g. image segmentation), computer graphics (e.g. trajectory modeling) or in engineering (e.g. robotics).


Convergence Rate Interpolation Scheme Length Estimation Fast Convergence Rate Asymptotic Order 
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  1. 1.
    de Boor, C.: A Practical Guide to Splines. Springer, Berlin (2001)zbMATHGoogle Scholar
  2. 2.
    Ferziger, J.H.: Numerical Methods for Engineering Application. John Wiley & Sons, New York (1998)zbMATHGoogle Scholar
  3. 3.
    Floater, M.S.: Chordal cubic spline interpolation is fourth order accurate. IMA Journal of Numerical Analysis 26, 25–33 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kozera, R., Noakes, L.: C 1 interpolation with cumulative chord cubics. Fundamenta Informaticae 31(3-4), 285–301 (2004)MathSciNetGoogle Scholar
  5. 5.
    Kozera, R.: Curve modeling via interpolation based on multidimensional reduced data. Studia Informatica 25(4B(61)), 1–140 (2004)Google Scholar
  6. 6.
    Janik, M., Kozera, R., Kozioł, P.: Reduced data for curve modeling - applications in graphics. Computer Vision and Physics (submitted)Google Scholar
  7. 7.
    Noakes, L., Kozera, R.: Cumulative chords and piecewise-quadratics and piecewise-cubics. In: Klette, R., Kozera, R., Noakes, L., Weickert, J. (eds.) Geometric Properties of Incomplete Data. Computational Imaging and Vision, The Netherlands, vol. 31, pp. 59–75 (2006)Google Scholar
  8. 8.
    Noakes, L., Kozera, R., Klette, R.: Length Estimation for Curves with Different Samplings. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 339–351. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Kvasov, B.: Methods of Shape-Preserving Spline Approximation. World Scientific, Singapore (2000)Google Scholar
  10. 10.
    Piegl, L., Tiller, W.: The NURBS Book. Springer, Berlin (1995)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ryszard Kozera
    • 1
  • Mateusz Śmietanka
    • 1
  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

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