Weighted Minimal Hypersurfaces and Their Applications in Computer Vision

  • Bastian Goldlücke
  • Marcus Magnor
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3022)


Many interesting problems in computer vision can be formulated as a minimization problem for an energy functional. If this functional is given as an integral of a scalar-valued weight function over an unknown hypersurface, then the minimal surface we are looking for can be determined as a solution of the functional’s Euler-Lagrange equation. This paper deals with a general class of weight functions that may depend on the surface point and normal. By making use of a mathematical tool called the method of the moving frame, we are able to derive the Euler-Lagrange equation in arbitrary-dimensional space and without the need for any surface parameterization. Our work generalizes existing proofs, and we demonstrate that it yields the correct evolution equations for a variety of previous computer vision techniques which can be expressed in terms of our theoretical framework. In addition, problems involving minimal hypersurfaces in dimensions higher than three, which were previously impossible to solve in practice, can now be introduced and handled by generalized versions of existing algorithms. As one example, we sketch a novel idea how to reconstruct temporally coherent geometry from multiple video streams.


Minimal Surface Surface Point Active Contour Model Minimal Hypersurface Visual Hull 
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.


  1. 1.
    Chen, Y., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow. Journal of Differential Geometry 33, 749–786 (1991)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Siddiqi, K., Lauziere, Y.B., Tannenbaum, A., Zucker, S.W.: Area and length minimizing flows for shape segmentation. IEEE Transactions on Image Processing 3, 433–443 (1998)CrossRefGoogle Scholar
  3. 3.
    Clelland, J.: MSRI Workshop on Lie groups and the method of moving frames. Lecture Notes. Department of Mathematics, University of Colorado (1999), http://spot.Colorado.EDU/~jnc/MSRI.html
  4. 4.
    Sharpe, R.: Differential Geometry. Graduate Texts in Mathematics. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  5. 5.
    Osher, S., Sethian, J.: Fronts propagating with curvature dependent speed: Algorithms based on the Hamilton-Jacobi formulation. Journal of Computational Physics 79, 12–49 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chop, D.: Computing minimal surfaces via level set curvature flow. Journal of Computational Physics 106, 77–91 (1993)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods, 2nd edn. Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  8. 8.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: Proc. International Conference on Computer Vision, pp. 694–699 (1995)Google Scholar
  9. 9.
    Caselles, V., Kimmel, R., Sapiro, G., Sbert, C.: Three dimensional object modeling via minimal surfaces. In: Buxton, B.F., Cipolla, R. (eds.) ECCV 1996. LNCS, vol. 1065, pp. 97–106. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  10. 10.
    Caselles, V., Kimmel, R., Sapiro, G., Sbert, C.: Minimal surfaces based object segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 19, 394–398 (1997)CrossRefGoogle Scholar
  11. 11.
    Zhao, H., Osher, S., Fedkiw, R.: Fast surface reconstruction using the level set method. In: 1st IEEE Workshop on Variational and Level Set Methods, 8th ICCV, vol. 80, pp. 194–202 (2001)Google Scholar
  12. 12.
    Paragios, N., Deriche, R.: Geodesic active contours and level sets for the detection and tracking of moving objects. IEEE Transactions on Pattern Analysis and Machine Intelligence 22, 266–280 (2000)CrossRefGoogle Scholar
  13. 13.
    Faugeras, O., Keriven, R.: Variational principles, surface evolution, PDE’s, level set methods and the stereo problem. IEEE Transactions on Image Processing 3, 336–344 (1998)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Jin, H., Soatto, S., Yezzi, A.: Multi-view stereo beyond Lambert. In: IEEE Conference on Computer Vision and Pattern Recognition, Madison, Wisconsin, USA, vol. I, pp. 171–178 (2003)Google Scholar
  15. 15.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. International Journal of Computer Vision 1, 321–331 (1988)CrossRefGoogle Scholar
  16. 16.
    Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A., Yezzi, A.: Gradient flows and geometric active contour models. In: ICCV, pp. 810–815 (1995)Google Scholar
  17. 17.
    Zhao, H., Osher, S., Merriman, B., Kang, M.: Implicit and non-parametric shape reconstruction from unorganized points using variational level set method. In: Computer Vision and Image Understanding, pp. 295–319 (2000)Google Scholar
  18. 18.
    Kutukalos, K.N., Seitz, S.M.: A theory of shape by space carving. International Journal of Computer Vision 38, 197–216 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Bastian Goldlücke
    • 1
  • Marcus Magnor
    • 1
  1. 1.Graphics – Optics – Vision, Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations