International Journal of Computer Vision

, Volume 50, Issue 3, pp 295–313 | Cite as

Diffusion Snakes: Introducing Statistical Shape Knowledge into the Mumford-Shah Functional

  • Daniel Cremers
  • Florian Tischhäuser
  • Joachim Weickert
  • Christoph Schnörr


We present a modification of the Mumford-Shah functional and its cartoon limit which facilitates the incorporation of a statistical prior on the shape of the segmenting contour. By minimizing a single energy functional, we obtain a segmentation process which maximizes both the grey value homogeneity in the separated regions and the similarity of the contour with respect to a set of training shapes. We propose a closed-form, parameter-free solution for incorporating invariance with respect to similarity transformations in the variational framework. We show segmentation results on artificial and real-world images with and without prior shape information. In the cases of noise, occlusion or strongly cluttered background the shape prior significantly improves segmentation. Finally we compare our results to those obtained by a level set implementation of geodesic active contours.

image segmentation shape recognition statistical learning variational methods diffusion snake geodesic active contours 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alcouffe, R.E., Brandt, A., Dendy, Jr.,J.E., and Painter, J.W. 1981. The multi-grid method for the diffusion equation with strongly discontinuous coefficients. SIAM J.Sci.Stat.Comp., 2(4):430–454.Google Scholar
  2. Blake, A. and Isard, M. 1998. Active Contours, Springer: London.Google Scholar
  3. Briggs, W.L., Henson, V.E., and McCormick, S.F. 2000. A Multigrid Tutorial, 2nd edn. SIAM: Philadelphia.Google Scholar
  4. Caselles, V., Kimmel, R., and Sapiro, G. 1995. Geodesic active contours. In Proc.IEEE Internat.Conf.on Comp.Vis., Boston, USA, pp. 694–699.Google Scholar
  5. Chan, T. and Vese, L. 2001. A level set algorithm for minimizing the Mumford-Shah functional in image processing. In IEEE Workshop on Variational and Level Set Methods, Vancouver, CA, pp. 161–168.Google Scholar
  6. Chen, Y., Thiruvenkadam, S., Tagare, H., Huang, F., Wilson, D., and Geiser, E. 2001. On the incorporation of shape priors into geometric active contours. In IEEE Workshop on Variational and Level Set Methods, Vancouver, CA, pp. 145–152.Google Scholar
  7. Cootes, T.F., Taylor, C.J., Cooper, D.M., and Graham, J. 1995. Active shape models—their training and application. Comp.Vision Image Underst., 61(1):38–59.Google Scholar
  8. Cremers, D., Kohlberger, T., and Schnörr, C. 2002. Nonlinear shape statistics in Mumford-Shah based segmentation. In Proc.of the Europ.Conf.on Comp.Vis., Copenhagen, A. Heyden et al. (Eds.), vol. 2351 of LNCS. Springer: Berlin, pp. 93–108.Google Scholar
  9. Cremers, D. and Schnärr, C. 2002. Motion competition: Variational integration of motion segmentation and shape regularization. In Pattern Recognition, L. van Gool (Ed.), Zürich, LNCS, Springer: Berlin.Google Scholar
  10. Cremers, D., Schnörr, C., and Weickert, J. 2001. Diffusion snakes: Combining statistical shape knowledge and image information in a variational framework. In IEEE First Workshop on Variational and Level Set Methods, Vancouver, pp. 137–144.Google Scholar
  11. Cremers, D., Schnörr, C., Weickert, J., and Schellewald, C. 2000. Diffusion snakes using statistical shape knowledge. In Alge-braic Frames for the Perception-Action Cycle, G. Sommer and Y.Y. Zeevi (Eds.), vol. 1888 of LNCS, pp. 164–174. Springer: Berlin.Google Scholar
  12. Dendy, J.E. 1982. Black box multigrid. J.Comp.Phys., 48:366–386.Google Scholar
  13. Dryden, I.L. and Mardia, K.V. 1998. Statistical Shape Analysis. Wiley: Chichester.Google Scholar
  14. Farin, G. 1997. Curves and Surfaces for Computer–Aided Geometric Design. Academic Press: San Diego, CA.Google Scholar
  15. Goodall, C. 1991. Procrustes methods in the statistical analysis of shape. J.Roy.Statist.Soc., Ser.B., 53(2):285–339.Google Scholar
  16. Grenander, U., Chow, Y., and Keenan, D.M. 1991. Hands: A Pattern Theoretic Study of Biological Shapes. Springer: New York.Google Scholar
  17. Kass, M., Witkin, A., and Terzopoulos, D. 1988. Snakes: Active contour models. Int.J.of Comp.Vis., 1(4):321–331.Google Scholar
  18. Kervrann, C. 1995. Modèles statistiques pour la segmentation et le suivi de structures déformables bidimensionnelles dans une séquence d'images. Ph.D. Thesis, Université de Rennes I, France.Google Scholar
  19. Kervrann, C. and Heitz, F. 1999. Statistical deformable model-based segmentation of image motion. IEEE Trans.on Image Processing, 8:583–588.Google Scholar
  20. Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A., and Yezzi, A.J. 1995. Gradient flows and geometric active contour models. In Proc.IEEE Internat.Conf.on Comp.Vis., Boston, USA, pp. 810–815.Google Scholar
  21. Leventon, M.E., Grimson, W.E.L., and Faugeras, O. 2000. Statistical shape influence in geodesic active contours. In Proc.Conf.Computer Vis.and Pattern Recog., Hilton Head Island, SC, vol. 1, pp. 316–323. June 13–15.Google Scholar
  22. Mantegazza, C. 1993. Su Alcune Definizioni Deboli di Curvatura per Insiemi Non Orientati. Ph.D. Thesis, Dept. of Mathematics, SNS Pisa, Italy.Google Scholar
  23. Moghaddam, B. and Pentland, A. 1995. Probabilistic visual learning for object detection. In Proc.IEEE Internat.Conf.on Comp.Vis., pp. 786–793.Google Scholar
  24. Morel, J.-M. and Solimini, S. 1988. Segmentation of images by variational methods: A constructive approach. Revista Matematica de la Universidad Complutense de Madrid, 1(1–3):169–182.Google Scholar
  25. Mumford, D. and Shah, J. 1989. Optimal approximations by piecewise smooth functions and associated variational problems. Comm.Pure Appl.Math., 42:577–685.Google Scholar
  26. Paragios, N. and Deriche, R. 2000. Coupled geodesic active regions for image segmentation: a level set approach. In ECCV, D. Vernon (Ed.), vol. 1843 of LNCS, Springer: Berlin, pp. 224–240.Google Scholar
  27. Roweis., S. EM algorithms for PCA and SPCA. 1998. In Advances in Neural Information Processing Systems 10, M. Jordan, M. Kearns, and S. Solla (Eds.), MIT Press: Cambridge, MA, pp. 626–632.Google Scholar
  28. Terzopoulos, D. 1983. Multilevel computational processes for visual surface reconstruction. Comp.Vis., Graph., and Imag.Proc., 24:52–96.Google Scholar
  29. Tipping, M.E. and Bishop, C.M. 1997. Probabilistic principal component analysis. Neural Computing Research Group, Aston University, UK, Technical Report Woe-19.Google Scholar
  30. Tischhäuser, F. 2001. Development of a multigrid algorithm for diffusion snakes. Diploma thesis Department of Mathematics and Computer Science, University of Mannheim, Mannheim, Germany (in German).Google Scholar
  31. Wang, Y. and Staib, L.H. 1998. Boundary finding with correspondence using statistical shape models. In Proc.Conf.Computer Vis.and Pattern Recog., Santa Barbara, CA, pp. 338–345.Google Scholar
  32. Weickert, J. 2001. Applications of nonlinear diffusion filtering in image processing and computer vision. Acta Mathematica Universitatis Comenianae, LXX(1):33–50.Google Scholar
  33. Werman, M. and Weinshall, D. 1995. Similarity and affine invariant distances between 2d point sets. IEEE Trans.on Patt.Anal.and Mach.Intell., 17(8):810–814.Google Scholar
  34. Wesseling, P. 1992. An Introduction to Multigrid Methods. John Wiley: Chichester.Google Scholar
  35. Yezzi, A., Soatto, S., Tsai, A., and Willsky, A. 2002. The Mumford-Shah functional: From segmentation to stereo. Mathematics and Multimedia.Google Scholar
  36. de Zeeuw, P.M. 1990. Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J.Comp.Appl.Math., 33:1–27.Google Scholar
  37. Zhu, S.C. and Mumford, D. 1997. Prior learning and Gibbs reaction–diffusion. IEEE Trans.on Patt.Anal.and Mach.Intell., 19(11):1236–1250.Google Scholar
  38. Zhu, S.C. and Yuille, A. 1996. Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Trans.on Patt.Anal.and Mach.Intell., 18(9):884–900.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Daniel Cremers
    • 1
  • Florian Tischhäuser
    • 1
  • Joachim Weickert
    • 1
  • Christoph Schnörr
    • 1
  1. 1.Computer Vision, Graphics, and Pattern Recognition Group, Department of Mathematics and Computer ScienceUniversity of Mannheim, D-68131MannheimGermany

Personalised recommendations