Skip to main content
SpringerLink
Log in

Geodesic active contour under geometrical conditions: theory and 3D applications

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we propose a new scheme for both detection of boundaries and fitting of geometrical data based on a geometrical partial differential equation, which allows a rigorous mathematical analysis. The model is a geodesic-active-contour-based model, in which we are trying to determine a curve that best approaches the given geometrical conditions (for instance a set of points or curves to approach) while detecting the object under consideration. Formal results concerning existence, uniqueness (viscosity solution) and stability are presented as well. We give the discretization of the method using an additive operator splitting scheme which is very efficient for this kind of problem. We also give 2D and 3D numerical examples on real data sets.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alvarez, L., Lions P.L., Morel, J.M.: Image selective smoothing and edge detection by non linear diffusion. SIAM J. Numer. Anal. 29(3), 845–866 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alvarez, O., Cardaliaguet, P., Monneau, R.: Existence and uniqueness for dislocation dyna mics with nonnegative velocity. Interfaces Free Bound. 7(4), 415–434 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Apprato, D., Arcangéli, R.: Ajustement spline le long d’un ensemble de courbes (Spline fitting along a set of curves) RAIRO Modél. Math. Anal. Numér. 25(2), 193–212 (1991)

    MATH  Google Scholar 

  4. Apprato, D., Gout, C., Komatitsch, D.: Surface fitting from ship track data: application to the bathymetry of the Marianas trench. Math. Geol. 34(7), 831–843 (2002)

    Article  MATH  Google Scholar 

  5. Apprato, D., Ducassou, D., Gout, C., Laffon, E., Le Guyader, C.: Segmentation of medical image sequence under constraints: application to non-invasive assessment of pulmonary arterial hypertension. Int. J. Comput. Math. 5, 527–536 (2004)

    Google Scholar 

  6. Barles, G.: Solutions de Viscosité des Équations de Hamilton-Jacobi. Springer-Verlag, Berlin (1994)

    MATH  Google Scholar 

  7. Barles, G., Da Lio, F.: A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions. Interfaces Free Bound. 5(3), 239–274 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Caselles, V., Catté, F., Coll, T., Dibos, F.: A geometric model for active contours in image processing. Numer. Math. 66, 1–31 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caselles, V., Sbert, C.: What is the best causal scale space for three-dimensional images?. SIAM J. Appl. Math. 56(4), 1199–1246 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caselles, V., Coll, B.: Snakes in movement. SIAM J. Numer. Anal. 33(6), 2445–2456 (1997)

    Article  MathSciNet  Google Scholar 

  11. Caselles, V., Kimmel, R., Sapiro, G., Sbert, C.: Minimal surfaces: a geometric three-dimensional segmentation approach. Numer. Math. 77(4), 423–451 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–87 (1997)

    Article  MATH  Google Scholar 

  13. Chan, T.F., Vese, L.: An efficient variational multiphase motion for the Mumford–Shah segmentation mode. Sig. Sys. Comp. 1, 490–494 (2000), IEEE Asilomar Conference on Signals Systems and Computers

    Google Scholar 

  14. Chan, T.F., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    Article  MATH  Google Scholar 

  15. Chan, T.F., Shen, J., Vese, L.: Variational PDE models in image processing. Notices Am. Math. Soc. 50(1), 14–26 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Chan, T.F., Sandberg, B., Vese, L.A.: Active contours without edges for vector-valued images. J. Vis. Commun. Image Represent. 11(2), 130–141 (2000)

    Article  Google Scholar 

  17. Chen, Y.G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33(3), 749–786 (1991)

    MathSciNet  MATH  Google Scholar 

  18. Cohen, L.D.: On active contours models and balloons. CVGIP Image Underst. 53(2), 211–218 (1991)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–69 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Deckelnick, K., Elliott, C.M.: Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities. Interfaces Free Bound. 6(3), 329–349 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry Methods and Applications, Part I, The Geometry of Surfaces, Transformation Groups, and Fields. Springer-Verlag (1992)

  23. Falcone, M., Ferreti, R.: Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations. Numer. Math. 67(3), 315–344 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Freidlin, M.: Functional Integration and PDE. Princeton University Press (1985)

  25. Giga, Y., Sato, M.H.: A level set approach to semicontinuous viscosity solutions for Cauchy problems. Commun. Partial Differ. Equ. 26(5–6), 813–839 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Giga, Y., Sato, M.H.: Generalized interface evolution with boundary condition. Proc. Japan Acad. 67(A), 263–266 (1991)

    MathSciNet  MATH  Google Scholar 

  27. Gout, C.: C k surface reconstruction from surface patches. Comput. Math. Appl. 44(3–4), 389–406 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gout, C., Komatitsch, D.: Surface fitting of rapidly varying data using rank coding: application to geophysical surfaces. Math. Geol. 32(7), 873–888 (2000)

    Article  Google Scholar 

  29. Gout, C., Le Guyader, C., Vese, L.: Segmentation under geometrical conditions using geodesic active contours and interpolation using level set methods. Numer. Algorithms 39(1–3), 155–173 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gout, C., Vieira-Testé, S.: An algorithm for segmentation under interpolation conditions using deformable models. Int. J. Comput. Math. 80(1), 47–54 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gout, C., Le Guyader, C.: Segmentation of complex geophysical structures with well data. Comput. Geosci. 10(4), 361–372 (2006)

    Article  MathSciNet  Google Scholar 

  32. Ishii, H.: Viscosity solutions and their applications. Sugaku Expo. 10(2), 123–141 (1995)

    Google Scholar 

  33. Ishii, H., Sato, M.H.: Nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain. Nonlinear Anal. 57A(7–8), 1077–1098 (2004)

    Article  MathSciNet  Google Scholar 

  34. Kocan, M.: Approximation of viscosity solutions of elliptic partial differential equations on minimal grids. Numer. Math. 72(1), 73–92 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Gradient flows and geometric active contour models. In: Proceedings, Fifth International Conference on Computer Vision, pp. 810–815 (1995)

  36. Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Conformal curvature flows: from phase transitions to active vision. Arch. Ration. Mech. Anal. 134(3), 275–301 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Le Guyader, C.: Imagerie Mathématique: Segmentation sous contraintes géométriques, Théory et Applications. Ph. D. dissertation, INSA Rouen, France (2004)

  38. Le Guyader, C., Apprato, D., Gout, C.: Using a level set approach for image segmentation under interpolation conditions. Numer. Algorithms 39(1–3), 221–235 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Le Guyader, C., Vese, L.: Self-repelling snakes for topology-preserving segmentation models. Accepted for publication in IEEE Transactions on Image Processing (2008, in press)

  40. Lions, P.L., Rouy, E., Tourin, A.: Shape-from-shading, viscosity solutions and edges. Numer. Math. 64(3), 323–353 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  41. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer Verlag (2003)

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

    Article  MathSciNet  MATH  Google Scholar 

  43. Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics. Computer Vision and Material Science, Cambridge University Press, Londres, (1999)

  44. Sethian, J.A.: Evolution, implementation and application of level set and fast marching methods for advancing fronts. J. Comput. Phys. 169(2), 503–555 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tsai, Y., Giga, Y., Osher, S.: A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations. Math. Comput. 72(241), 159–181 (2003)

    MathSciNet  MATH  Google Scholar 

  46. Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50(3), 271–293 (2002)

    Article  MATH  Google Scholar 

  47. Weickert, J., Kühne, G.: Fast methods for implicit active contours models. preprint 61, Universität des Saarlandes, Saarbrücken (2002)

  48. Zhao, H.K., Osher, S., Merriman, B., Kang, M.: Implicit and non parametric shape reconstruction from unorganized data using a variational level set method. Comput. Vis. Image Underst. 80(3), 295–314 (2000)

    Article  MATH  Google Scholar 

  49. Zhao, H.K., Chan, T.F., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127, 179–195 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INSA de Rennes, Centre de Mathématiques de l’INSA, 20 Avenue des Buttes de Coësmes, CS 14315, 35043, Rennes cedex, France

    Carole Le Guyader

  2. Department of Mathematics, University of California, Los Angeles, 405, Hilgard Avenue, Los Angeles, CA, 90095-1555, USA

    Carole Le Guyader

  3. LAMAV-ISTV2, Université de Valenciennes, Le Mont Houy, 59313, Valenciennes Cedex 9, France

    Christian Gout

  4. INSA de Rouen, Laboratoire de Mathématiques de l’INSA, Place Emile Blondel, BP 08, 76 131, Mont-Saint-Aignan Cedex, France

    Christian Gout

  5. INRIA Futurs Research Center Team-Project Magique3D, Pau, France

    Christian Gout

Authors
  1. Carole Le Guyader
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Gout
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Gout.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Le Guyader, C., Gout, C. Geodesic active contour under geometrical conditions: theory and 3D applications. Numer Algor 48, 105–133 (2008). https://doi.org/10.1007/s11075-008-9174-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-008-9174-y

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation