Advertisement

Discrete elastica

  • Alfred M. Bruckstein
  • Robert J. Holt
  • Arun N. Netravali
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1176)

Abstract

This paper develops a discrete approach to the design of planar curves that minimize cost functions dependent upon their shape. The curves designed by using this approach are piecewise linear with equal length segments and obey various types of endpoint constraints.

Keywords

Penalty Function Turn Angle Subjective Contour Curve Design Good Initial Guess 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    L. Euler. Additamentum ‘De Curvis Elasticis'. In Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes. Lausanne, 1744.Google Scholar
  2. 2.
    M. Born. Stabilität der elastischen Linie in Ebene und Raum. PhD thesis, Göttingen, Göttingen, Germany, 1906.Google Scholar
  3. 3.
    A. E. H. Love. The Mathematical Theory of Elasticity. Cambridge University Press, London, 1927.Google Scholar
  4. 4.
    M. Malcolm. On the computation of nonlinear spline functions. SIAM Journal of Numerical Analysis, 14:254–282, 1977.Google Scholar
  5. 5.
    I. D. Cooper. Curve interpolation with nonlinear spiral splines. IMA Journal of Numerical Analysis, 13:327–341, 1992.Google Scholar
  6. 6.
    S. Ullman. Filling in the gaps: The shape of subjective contours and a model for their generation. Biological Cybernetics, 25:1–6, 1976.Google Scholar
  7. 7.
    W. S. Rutkowski. Shape completion. Computer Graphics and Image Processing, 9:89–101, 1979.Google Scholar
  8. 8.
    B. K. P. Horn. The curve of least energy. ACM Transactions on Mathematical Software, 9:441–460, 1983.Google Scholar
  9. 9.
    M. Kallay. Plane curves of minimal energy. ACM Transactions on Mathematical Software, 12:219–222, 1986.Google Scholar
  10. 10.
    A. M. Bruckstein and A. N. Netravali. On minimal energy trajectories. Computer Vision, Graphics, and Image Processing, 49:283–296, 1990.Google Scholar
  11. 11.
    H. P. Moreton and C. H. Séquin. Scale-invariant minimum-cost curves: Fair and robust design implements. In Proceedings of Eurographics '93, pages C-473–C-484, Barcelona, September 1993.Google Scholar
  12. 12.
    D. Mumford. Elastica and computer vision. In C. L. Bajaj, editor, Algebraic Geometry and its Applications, pages 491–506. Springer-Verlag, New York, 1994.Google Scholar
  13. 13.
    E. H. Lee and G. E. Forsythe. Variational study of nonlinear spline curves. SIAM Review, 15:120–133, 1973.Google Scholar
  14. 14.
    M. Golomb and J. Jerome. Equilibria of the curvature functional and manifolds of nonlinear interpolating spline curves. SIAM Journal of Mathematical Analysis, 13:421–458, 1982.Google Scholar
  15. 15.
    R. Bryant and P. Griffiths. Reduction for constrained variational problems and κ2/2 ds. American Journal of Mathematics, 108:525–570, 1986.Google Scholar
  16. 16.
    K. Foltinek. The hamiltonian theory of elastica. American Journal of Mathematics, 116:1479–1488, 1994.Google Scholar
  17. 17.
    V. Jurdjevic. Non-euclidean elastica. American Journal of Mathematics, 117:93–124, 1995.Google Scholar
  18. 18.
    M. E. Mortensen. Geometric Modeling. John Wiley & Sons, New York, 1985.Google Scholar
  19. 19.
    G. Farin. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide. Academic Press, Boston, 1988.Google Scholar
  20. 20.
    D. B. Parkinson and D. N. Moreton. Optimal biarc-curve fitting. Computer-Aided Design, 23:411–419, 1991.Google Scholar
  21. 21.
    C. L. Bajaj and G. Xu. NURBS approximation of surface/surface intersection curves. Advances in Computational Mathematics, 2:1–21, 1994.Google Scholar
  22. 22.
    L. E. Dubins. On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. American Journal of Mathematics, 79:497–516, 1957.Google Scholar
  23. 23.
    B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt. Maple V User's Guide. Watcom Publications Limited, Waterloo, Ontario, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Alfred M. Bruckstein
    • 1
  • Robert J. Holt
    • 1
  • Arun N. Netravali
    • 1
  1. 1.Bell LaboratoriesMurray HillUSA

Personalised recommendations