Skip to main content
Log in

A geometric model for active contours in image processing

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

We propose a new model for active contours based on a geometric partial differential equation. Our model is intrinsec, stable (satisfies the maximum principle) and permits a rigorous mathematical analysis. It enables us to extract smooth shapes (we cannot retrieve angles) and it can be adapted to find several contours simultaneously. Moreover, as a consequence of the stability, we can design robust algorithms which can be engineed with no parameters in applications. Numerical experiments are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alvarez, L., Lions, P.L., Morel, J.M. (1991): Image selective smoothing and edge detection by nonlinear diffusion (II). Cahier du CEREMADE no 9046, Univ. Paris IX-Dauphine, Paris

    Google Scholar 

  2. Amini, A.A., Tehrani, S., Weymouth, T.E. (1988): Using dynamic programming for minimizing the energy of active contours in the presence of hard constraints. Proc. Second ICCV. 95–99

  3. Ayache, N., Boissonat, J.D., Brunet, E., Cohen, L., Chièze, J.P., Geiger, B., Monga, O., Rocchisani, J.M., Sander, P. (1989): Building highly structured volume representations in 3d medical images. Computer Aided Radiology. Berlin

    Google Scholar 

  4. Barles, G. (1985): Remarks on a flame propagation model. Rapport INRIA,464, 1–38

    Google Scholar 

  5. Berger, M.O. (1990): Snake growing. O. Faugeras, ed., Computer Vision-ECCV90. Lect. Notes Comput. Sci.427, 570–572

    Google Scholar 

  6. Berger, M.O., Mohr, R. (1990): Towards Autonomy in Active Contour Models. Proc. 10th Int. Conf. Patt. Recogn. Atlantic City, NY, vol1, 847–851

    Google Scholar 

  7. Blake, A., Zisserman, A. (1987): Visual Reconstruction. MIT Press, Cambridge, MA

    Google Scholar 

  8. Chen, Y.-G., Giga, Y., Goto, S. (1989): Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations. Preprint Series in Math. Ser. 57. July, Hokkaido University, Sapporo, Japan

    Google Scholar 

  9. Cinquin, P. (1986): Un modèle pour la représentation d'images médicales 3d: Proceedings Euromédicine. (Sauramps Médical)86, 57–61

    Google Scholar 

  10. Cinquin, P. (1987): Application des Fonctions Spline au Traitement d'Images Numériques. Université Joseph Fourier, Grenoble

    Google Scholar 

  11. Cinquin, P., Goret, C., Marque, I., Lavallee, S. (1987): Morphoscopie et modélisation continue d'images 3d. Conférence AFCET IA & Reconnaissance des Formes. AFCET pp. 907–922, Paris

    Google Scholar 

  12. Cohen, L.D. (1991): On active Contour Models and Balloons. CVGIP: Image Understanding53, 211–218

    Google Scholar 

  13. Cohen, L.D., Cohen, I. (1990): A finite element method applied to new active contour models and 3D reconstruction from cross sections. Proc. Third ICCV, 587–591

  14. Cohen, L.D. (1989): On active Contour Models. Technical Report 1035, INRIA, Rocquencourt, Le Chesnay, France

    Google Scholar 

  15. Crandall, M.G., Lions, P.L. (1983): Viscosity Solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277, 1–42

    Google Scholar 

  16. Crandall, M.G., Evans, L.C., Lions, P.L. (1984): Some properties of Viscosity Solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.282, 487–502

    Google Scholar 

  17. Crandall, M.G., Ishii, I., Lions, P.L. (1991): User's guide to Viscosity Solutions of Second Order Partial Differential Equations. Cahier du CEREMADE no 9039. Univ. Paris IX-Dauphine, Paris

    Google Scholar 

  18. Evans, L.C., Spruck, J. (1991): Motion of level sets by mean curvature I. J. Diff. Geometry,33, 635–681

    Google Scholar 

  19. Friedman, A. (1982): Variational Principles and Free Boundary Problems. Wiley, New York

    Google Scholar 

  20. Gage, M. (1983): An isoperimetric inequality with applications to curve shortening. Duke Math. J.50, 1225–1229

    Google Scholar 

  21. Gage, M. (1984): Cuve shortening makes convex curves circular. Invent. Math.76, 357–364

    Google Scholar 

  22. Gage, M., Hamilton, R.S. (1986): The heat equation shrinking convex plane curves. J. Diff. Geom.23, 69–96

    Google Scholar 

  23. Giga, Y., Goto, S., Ishii, I., Sato, M.-H. (1990): Comparison Principle and Convexity Preserving Properties of Singular Degenerate Parabolic Equations on Unbounded Domains. Preprint Hokkaido University, 1–32, Sapporo, Japan

  24. Grayson, M.A. (1987): The heat equation shrinks embedded plane curves to round points. J. Diff. Geom.26, 285–314

    Google Scholar 

  25. Hirsch, M. (1976): Differential Topology. Springer, Berlin Heidelberg New York

    Google Scholar 

  26. Kass, M., Witkin, A., Terzopoulos, D. (1988): Snakes: active contour models. Int. J. Comput. Vision.1, 321–331

    Google Scholar 

  27. Kass, M., Witkin, A., Terzopoulos, D. (1987): Snakes: active contour models. Proc. First ICCV, 259–267

  28. Ladyzhenskaja, O.A., Solonnikov, V.A., Ural'tseva, N.N. (1968): Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, R.I.

    Google Scholar 

  29. Leroy, B. (1991): Etude de quelques propriétés des modèles de contours actifs (“snakes”). Rapport de stage de D.E.A. Univ. Paris-IX Dauphine, Septembre

  30. Lions, P.L. (1982). Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics69, Pitman, Boston

    Google Scholar 

  31. Marr, D. (1982): Vision. Freeman, San Francisco

    Google Scholar 

  32. Marr, D., Hildreth, E. (1980): A theory of edge detection. Proc. R. Soc. Lond. B207, 187–217

    Google Scholar 

  33. Osher, S., Sethian, J.A. (1988): Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Comput. J. Physics.79, 12–49

    Google Scholar 

  34. Poggio, T., Torre, V., Koch, C. (1985): Computational vision and regularization theory. Nature,317 (6035), 314–319

    Google Scholar 

  35. Terzopoulos, D. (1986): Regularization of inverse visual problems involving discontinuities. IEEE Trans. Pattern Anal. Mach. Intell.8: 413–424

    Google Scholar 

  36. Terzopoulos, D. (1988): The computation of visible surface representations. IEEE Trans. Pattern Anal. Mach. Intell.10(4), 417–438

    Google Scholar 

  37. Terzopoulos, D., Witkin, A., Kass, M. (1987): Symmetry seeking models for 3d object reconstruction. Proc. First ICCV, 269–276

  38. Terzopoulos, D., Witkin, A., Kass, M. (1988): Constraints on deformable models: recovering 3d shape and nonrigid motion. Artif. Intell.36, 91–123

    Google Scholar 

  39. Zucker, S., David, C., Dobbins, A., Iverson, L. (1988): The Organization of Curve Detection: Coarse Tangent Fields and Fine Spline Coverings. In Second International Conference on Computer Vision. pp. 568–577, Tampa Florida (USA)

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Caselles, V., Catté, F., Coll, T. et al. A geometric model for active contours in image processing. Numer. Math. 66, 1–31 (1993). https://doi.org/10.1007/BF01385685

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01385685

Mathematics Subject Classification (1991)

Navigation