Advertisement

Interpolating Sporadic Data

  • Lyle Noakes
  • Ryszard Kozera
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2351)

Abstract

We report here on the problem of estimating a smooth planar curve γ: [0, T] → ℝ2 and its derivatives from an ordered sample of interpolation points γ(t 0), γ(t 1),...,γ(t i -1),γ(t i ),...,γ(t m -1),γ(t m ), where 0 = t 0 < t 1 <... < t i - 1 < t i <...< t m - 1 < t m = T, and the t i are not known precisely for 0 < i < m. Such situtation may appear while searching for the boundaries of planar objects or tracking the mass center of a rigid body with no times available. In this paper we assume that the distribution of t i coincides with more-or-less uniform sampling. A fast algorithm, yielding quartic convergence rate based on 4-point piecewise-quadratic interpolation is analysed and tested. Our algorithm forms a substantial improvement (with respect to the speed of convergence) of piecewise 3-point quadratic Lagrange intepolation [19] and [20]. Some related work can be found in [7]. Our results may be of interest in computer vision and digital image processing [5], [8], [13], [14], [17] or [24], computer graphics [1], [4], [9], [10], [21] or [23], approximation and complexity theory [3], [6], [16], [22], [26] or [27], and digital and computational geometry [2] and [15].

Keywords

shape image analysis and features curve interpolation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barsky, B.A., DeRose, T.D.: Geometric Continuity of Parametric Curves: Three Equivalent Characterizations. IEEE. Comp. Graph. Appl. 9:6 (1989) 60–68CrossRefGoogle Scholar
  2. 2.
    Bertrand, G., Imiya, A., Klette, R. (eds): Digital and Image Geometry. Lecture Notes in Computer Science Vol. 2243, Springer-Verlag, Berlin Heidelberg New York (2001)zbMATHGoogle Scholar
  3. 3.
    Bézier, P.E.: Numerical Control: Mathematics and Applications. John Wiley, New York (1972)zbMATHGoogle Scholar
  4. 4.
    Boehm, W., Farin, G., Kahmann, J.: A Survey of Curve and Surface Methods in CAGD. Comput. Aid. Geom. Des. 1 (1988) 1–60CrossRefGoogle Scholar
  5. 5.
    Bülow, T., Klette, R.: Rubber Band Algorithm for Estimating the Length of Digitized Space-Curves. In: Sneliu, A., Villanva, V.V., Vanrell, M., Alquézar, R., Crowley. J., Shirai, Y. (eds): Proceedings of 15th International Conference on Pattern Recognition. Barcelona, Spain. IEEE, Vol. III. (2000) 551–555Google Scholar
  6. 6.
    Davis, P.J.: Interpolation and Approximation. Dover Pub. Inc., New York (1975)zbMATHGoogle Scholar
  7. 7.
    Dcabrowska, D., Kowalski, M.A.: Approximating Band-and Energy-Limited Signals in the Presence of Noise. J. Complexity 14 (1998) 557–570CrossRefMathSciNetGoogle Scholar
  8. 8.
    Dorst, L., Smeulders, A.W.M.: Discrete Straight Line Segments: Parameters, Primitives and Properties. In: Melter, R., Bhattacharya, P., Rosenfeld, A. (eds): Ser. Contemp. Maths, Vol. 119. Amer. Math. Soc. (1991) 45–62Google Scholar
  9. 9.
    Epstein, M.P.: On the Influence of Parametrization in Parametric Interpolation. SIAM. J. Numer. Anal. 13:2 (1976) 261–268zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hoschek, J.: Intrinsic Parametrization for Approximation. Comput. Aid. Geom. Des. 5 (1988) 27–31zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Klette, R.: Approximation and Representation of 3D Objects. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds): Advances in Digital and Computational Geometry. Springer, Singapore (1998) 161–194Google Scholar
  12. 12.
    Klette, R., Bülow, T.: Critical Edges in Simple Cube-Curves. In: Borgefors, G., Nyström, I., Sanniti di Baja, G. (eds): Proceedings of 9th Conference on Discrete Geometry for Computer Imagery. Uppsala, Sweden. Lecture Notes in Computer Science, Vol. 1953. Springer-Verlag, Berlin Heidelberg (2000) 467–478CrossRefGoogle Scholar
  13. 13.
    Klette, R., Kovalevsky, V., Yip, B.: On the Length Estimation of Digital Curves. In: Latecki, L.J., Melter, R.A., Mount, D.A., Wu, A.Y. (eds): Proceedings of SPIE Conference, Vision Geometry VIII, Vol. 3811. Denver, USA. The International Society for Optical Engineering (1999) 52–63Google Scholar
  14. 14.
    Klette, R., Yip, B.: The Length of Digital Curves. Machine Graphics and Vision 9 (2000) 673–703Google Scholar
  15. 15.
    Klette, R., Rosenfeld, A., Sloboda, F. (eds): Advances in Digital and Computational Geometry. Springer, Singapore (1998) 161–194zbMATHGoogle Scholar
  16. 16.
    Kvasov, B.I.: Method of Shape-Preserving Spline Approximation. World Scientific Pub. Co., Singapore, New Jersey, London, Hong Kong (2000)Google Scholar
  17. 17.
    Moran, P.A.P.: Measuring the Length of a Curve. Biometrika 53:3/4 (1966) 359–364zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Noakes, L., Kozera, R.: More-or-Less Uniform Sampling and Lengths of Curves. Quart. Appl. Maths. In pressGoogle Scholar
  19. 19.
    Noakes, L., Kozera, R., and Klette R.: Length Estimation for Curves with Different Samplings. In: Bertrand, G., Imiya, A., Klette, R. (eds): Digital and Image Geometry. Lecture Notes in Computer Science Vol. 2243, Springer-Verlag, Berlin Heidelberg New York, (2001) 339–351CrossRefGoogle Scholar
  20. 20.
    Noakes, L., Kozera, R., and Klette R.: Length Estimation for Curves with g—Uniform Sampling. In: Skarbek, W. (ed.): Proceedings of 9th International Conference on Computer Analysis of Images and Pattterns. Warsaw, Poland. Lecture Notes in Computer Science, Vol. 2124. Springer-Verlag, Berlin Heidelberg New York, (2001) 518–526CrossRefGoogle Scholar
  21. 21.
    Piegl, L., Tiller, W.: The NURBS Book. 2nd edn Springer-Verlag, Berlin Heidelberg (1997)Google Scholar
  22. 22.
    Plaskota, L.: Noisy Information and Computational Complexity. Cambridge Uni. Press, Cambridge (1996)zbMATHGoogle Scholar
  23. 23.
    Sederberg, T.W., Zhao, J., Zundel, A.K.: Approximate Parametrization of Algebraic Curves. In: Strasser, W., Seidel, H.P. (eds): Theory and Practice in Geometric Modelling. Springer-Verlag, Berlin (1989) 33–54Google Scholar
  24. 24.
    Sloboda, F., Zaťko, B., Stör, J.: On approximation of Planar One-Dimensional Continua. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds): Advances in Digital and Computational Geometry. Springer, Singapore (1998) 113–160Google Scholar
  25. 25.
    Steinhaus, H.: Praxis der Rektifikation und zur Längenbegriff. (in German) Akad. Wiss. Leipzig Ber. 82 (1930) 120–130Google Scholar
  26. 26.
    Traub, J.F., Werschulz, A.G.: Complexity and Information. Cambridge Uni. Press, Cambridge (1998)zbMATHGoogle Scholar
  27. 27.
    Werschulz, A.G., Woźniakowski, H.: What is the Complexity of Surface Integration? J. Complexity. 17 (2001) 442–466zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Lyle Noakes
    • 1
  • Ryszard Kozera
    • 2
  1. 1.Department of Mathematics & StatisticsThe University of Western AustraliaCrawleyAustralia
  2. 2.Department of Computer Science & Software EngineeringThe University of Western AustraliaCrawleyAustralia

Personalised recommendations