Advertisement

Computing and Visualization in Science

, Volume 6, Issue 4, pp 211–225 | Cite as

Computational and qualitative aspects of evolution of curves driven by curvature and external force

  • Karol Mikula
  • Daniel Ševčovič
Regular article

Abstract

We propose a direct method for solving the evolution of plane curves satisfying the geometric equation v=β(x,k,ν) where v is the normal velocity, k and ν are the curvature and tangential angle of a plane curve Γ⊂R2 at a point x∈Γ. We derive and analyze the governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves. The governing equations include a nontrivial tangential velocity functional yielding uniform redistribution of grid points along the evolving family of curves preventing thus numerically computed solutions from forming various instabilities. We also propose a full space-time discretization of the governing system of equations and study its experimental order of convergence. Several computational examples of evolution of plane curves driven by curvature and external force as well as the geodesic curvatures driven evolution of curves on various complex surfaces are presented in this paper.

Keywords

Grid Point Tangential Velocity Normal Velocity Qualitative Aspect Closed Geodesic 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Diff. Geom. 23, 175–196 (1986) MathSciNetGoogle Scholar
  2. 2.
    Angenent, S.B.: Parabolic equations for curves on surfaces I: Curves with p–integrable curvature. Annals of Mathematics 132, 451–483 (1990) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Angenent, S.B.: Nonlinear analytic semiflows. Proc. R. Soc. Edinb., Sect. A 115, 91–107, (1990) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Angenent, S.B., Gurtin, M.E.: Multiphase thermomechanics with an interfacial structure 2. Evolution of an isothermal interface. Arch. Rat. Mech. Anal. 108, 323–391 (1989) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Angenent, S.B., Gurtin, M.E.: General contact angle conditions with and without kinetics. Quarterly of Appl. Math. 54(3), 557–569 (1996) MathSciNetGoogle Scholar
  6. 6.
    Beneš, M.: Mathematical and computational aspects of solidification of pure crystallic materials. Acta Math. Univ. Comenianae 70, 123–151 (2001) Google Scholar
  7. 7.
    Caginalp, G.: The dynamics of a conserved phase field system: Stefan–like, Hele–Shaw, and Cahn–Hilliard models as asymptotic limits. IMA J. Appl. Math. 44, 77–94 (1990) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Caselles, V., Catté, F., Coll, T, Dibos, F.: A geometric model for active contours in image processing. Numerische Matematik 66, 1–31 (1993) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. International Journal of Computer Vision 22, 61–79 (1997) CrossRefGoogle Scholar
  10. 10.
    Caselles, V., Kimmel, R., Sapiro, G., Sbert, C.: Minimal surfaces: a geometric three dimensional segmentation approach. Numerische Matematik 77, 423–451 (1997) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Deckelnick, K.: Weak solutions of the curve shortening flow Calc. Var. Partial Differ. Equ. 5, 489–510 (1997) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dziuk, G.: Convergence of a semi discrete scheme for the curve shortening flow. Math. Models Methods Appl. Sci. 4, 589–606 (1994) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dziuk, G.: Discrete anisotropic curve shortening flow. SIAM J. Numer. Anal. 36, 1808–1830 (1999) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Diff. Geom. 23, 69–96 (1986) MathSciNetGoogle Scholar
  15. 15.
    Grayson, M.: The heat equation shrinks embedded plane curves to round points. J. Diff. Geom. 26, 285–314 (1987) MathSciNetGoogle Scholar
  16. 16.
    Hou, T.Y., Lowengrub, J., Shelley, M.: Removing the stiffness from interfacial flows and surface tension. J. Comput. Phys. 114, 312–338 (1994) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hou, T.Y., Klapper, I., Si, H.: Removing the stiffness of curvature in computing 3-d filaments. J. Comput. Phys. 143, 628–664 (1998) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kačur, J., Mikula, K.: Solution of nonlinear diffusion appearing in image smoothing and edge detection. Applied Numerical Mathematics 17, 47–59 (1995) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kass, M., Witkin, A., Terzopulos, D.: Snakes: active contour models. International Journal of Computer Vision 1, 321–331 (1987) CrossRefGoogle Scholar
  20. 20.
    Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Gradient flows and geometric active contours models, in Proceedings International Conference on Computer Vision’95, Boston, 1995 Google Scholar
  21. 21.
    Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Conformal curvature flows: from phase transitions to active vision. Arch. Rational Mech. Anal. 134, 275–301 (1996) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kimura, M.: Numerical analysis for moving boundary problems using the boundary tracking method. Japan J. Indust. Appl. Math. 14, 373–398 (1997) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Malladi, R., Sethian, J., Vemuri, B.: Shape modeling with front propagation: a level set approach. IEEE Trans. Pattern Anal. Machine Intelligence 17, 158–174 (1995)CrossRefGoogle Scholar
  24. 24.
    Mikula, K., Kačur, J.: Evolution of convex plane curves describing anisotropic motions of phase interfaces. SIAM J. Sci. Comput. 17, 1302–1327 (1996) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mikula, K. Solution of nonlinear curvature driven evolution of plane convex curves. Appl. Numer. Math. 21, 1–14 (1997) Google Scholar
  26. 26.
    Mikula, K., Ševčovič, D.: Solution of nonlinearly curvature driven evolution of plane curves. Appl. Numer. Math. 31, 191–207 (1999) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mikula, K., Ševčovič, D. Evolution of plane curves driven by a nonlinear function of curvature and anisotropy. SIAM J. Appl. Math. 61, 1473–1501 (2001) Google Scholar
  28. 28.
    Mikula, K., Ševčovič, D.: A direct method for solving an anisotropic mean curvature flow of plane curves with an external force. submitted Google Scholar
  29. 29.
    Mikula, K., Ševčovič, D.: Evolution of curves on a surface driven by the geodesic curvature and external force. submitted Google Scholar
  30. 30.
    Nochetto, R., Paolini, M., Verdi, C.: Sharp error analysis for curvature dependent evolving fronts. Math. Models Methods Appl. Sci. 3, 711–723 (1993) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. Proc. IEEE Computer Society Workshop on Computer Vision 1987 Google Scholar
  32. 32.
    Sapiro, G., Tannenbaum, A.: On affine plane curve evolution. J. Funct. Anal. 119, 79–120 (1994) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sarti, A., Mikula, K., Sgallari, F.: Nonlinear multiscale analysis of three-dimensional echocardiographic sequences. IEEE Trans. on Medical Imaging 18, 453–466 (1999) CrossRefGoogle Scholar
  34. 34.
    Schmidt, A.: Computation of three dimensional dendrites with finite elements. J. Comput. Phys. 125, 293–312 (1996) CrossRefGoogle Scholar
  35. 35.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science. New York: Cambridge University Press 1999Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TechnologyBratislavaSlovak Republic
  2. 2.Institute of Applied Mathematics, Faculty of Mathematics, Physics and InformaticsComenius UniversityBratislavaSlovak Republic

Personalised recommendations