A Revisit to Least Squares Orthogonal Distance Fitting of Parametric Curves and Surfaces

  • Yang Liu
  • Wenping Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4975)

Abstract

Fitting of data points by parametric curves and surfaces is demanded in many scientific fields. In this paper we review and analyze existing least squares orthogonal distance fitting techniques in a general numerical optimization framework. Two new geometric variant methods (Open image in new window and Open image in new window) are proposed. The geometric meanings of existing and modified optimization methods are also revealed.

Keywords

orthogonal distance fitting parametric curve and surface fitting nonlinear least squares numerical optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Heidelberg (2006)MATHGoogle Scholar
  2. 2.
    Hoschek, J.: Intrinsic parameterization for approximation. Computer Aided Geometric Design 5(1), 27–31 (1988)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Hoschek and D. Lasser. Fundamentals of Computer Aided Geometric Design. Peters, A. K. (1993)Google Scholar
  4. 4.
    Weiss, V., Andor, L., Renner, G., Várady, T.: Advanced surface fitting techniques. Computer Aided Geometric Design 19(1), 19–42 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bjorck, A.: Numerical Methods for Least Squares Problems. In: Mathematics Society for Industrial and Applied Mathematics, Philadelphia (1996)Google Scholar
  6. 6.
    Helfrich, H.P., Zwick, D.: A trust region algorithm for parametric curve and surface fitting. Journal of Computational and Applied Mathematics 73(1-2), 119–134 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Speer, T., Kuppe, M., Hoschek, J.: Global reparameterization for curve approximation. Computer Aided Geometric Design 15(9), 869–877 (1998)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Golub, G., Pereyra, V.: Separable nonlinear least squares: the variable projection method and its applications. Inverse Problems 19, 1–26 (2003)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Borges, C.F., Pastva, T.: Total least squares fitting of Bézier and B-spline curves to ordered data. Computer Aided Geometric Design 19(4), 275–289 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Atieg, A., Watson, G.A.: A class of methods for fitting a curve or surface to data by minimizing the sum of squares of orthogonal distances. Journal of Computational and Applied Mathematics 158, 227–296 (2003)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Ahn, S.J.: Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space. LNCS, vol. 3151. Springer, Heidelberg (2004)MATHGoogle Scholar
  12. 12.
    Wang, W., Pottmann, H., Liu, Y.: Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Transactions on Graphics 25, 214–238 (2006)CrossRefGoogle Scholar
  13. 13.
    Liu, Y., Pottmann, H., Wang, W.: Constrained 3D shape reconstruction using a combination of surface fitting and registration. Computer Aided Design 38(6), 572–583 (2006)CrossRefGoogle Scholar
  14. 14.
    Wallner, J.: Gliding spline motions and applications. Computer Aided Geometric Design 21(1), 3–21 (2004)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Piegl, L.A., Tiller, W.: The NURBS Book, 2nd edn. Springer, Heidelberg (1996)Google Scholar
  16. 16.
    Saux, E., Daniel, M.: An improved hoschek intrinsic parametrization. Computer Aided Geometric Design 20(8-9), 513–521 (2003)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Yang Liu
    • 1
  • Wenping Wang
    • 1
  1. 1.Dept. of Computer ScienceThe University of Hong KongHong Kong SARP.R. China

Personalised recommendations