Skip to main content
SpringerLink
Log in

Kernel Density Estimation and Intrinsic Alignment for Shape Priors in Level Set Segmentation

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

In this paper, we make two contributions to the field of level set based image segmentation. Firstly, we propose shape dissimilarity measures on the space of level set functions which are analytically invariant under the action of certain transformation groups. The invariance is obtained by an intrinsic registration of the evolving level set function. In contrast to existing approaches to invariance in the level set framework, this closed-form solution removes the need to iteratively optimize explicit pose parameters. The resulting shape gradient is more accurate in that it takes into account the effect of boundary variation on the object’s pose.

Secondly, based on these invariant shape dissimilarity measures, we propose a statistical shape prior which allows to accurately encode multiple fairly distinct training shapes. This prior constitutes an extension of kernel density estimators to the level set domain. In contrast to the commonly employed Gaussian distribution, such nonparametric density estimators are suited to model aribtrary distributions.

We demonstrate the advantages of this multi-modal shape prior applied to the segmentation and tracking of a partially occluded walking person in a video sequence, and on the segmentation of the left ventricle in cardiac ultrasound images. We give quantitative results on segmentation accuracy and on the dependency of segmentation results on the number of training shapes.

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

  • Basri, R., Costa, L., Geiger, D., and Jacobs D. 1998. Determining the similarity of deformable shapes. Vision Research, 38:2365–2385.

    Article  Google Scholar 

  • Bresson, X., Vandergheynst, P., and Thirau, J.P. 2003. A priori information in image segmentation: Energy functional based on shape statistical model and image information. In Proc. IEEE Int. Conf. Image Processing, pp. 425–428.

  • Brox, T. and Weickert, J. 2004. A TV flow based local scale measure for texture discrimination. In (eds.), T.Pajdla and V.Hlavac European Conf. on Computer Vision, volume 3022 of Lect. Not. Comp. Sci., Prague, Springer.

    Google Scholar 

  • Caselles, V., Kimmel, R., and Sapiro, G. 1995. Geodesic active contours. In Proc. IEEE Intl. Conf. on Comp. Vis., pages 694–699, Boston, USA pp. 694–699..

  • Chan, T. and Zhu, W. 2003. Level set based shape prior segmentation. Technical Report 03-66, Computational Applied Mathematics, UCLA, Los Angeles.

    Google Scholar 

  • Chan, T.F. and Vese, L.A. 2001. Active contours without edges. IEEE Trans. Image Processing, 10(2):266–277.

    Article  MATH  Google Scholar 

  • Charpiat, G., Faugeras, O., and Keriven, R. 2005. Approximations of shape metrics and application to shape warping and empirical shape statistics. Journal of Foundations Of Computational Mathematics, 5(1):1–58.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, Y., Tagare, H., Thiruvenkadam, S., Huang, F., Wilson, D., Gopinath, K.S., Briggs, R.W., and Geiser, E. 2002. Using shape priors in geometric active contours in a variational framework. Int.J.ofComputer Vision, 50(3):315–328.

    Article  MATH  Google Scholar 

  • Chow, Y.S., Geman, S., and Wu, L.D. 1983. Consistent cross-validated density estimation. Annals of Statistics, 11:25–38.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Cootes T.F. and Taylor C.J. 1999. A mixture model for representing shape variation. Image and Vision Computing, 17(8):567–574.

    Article  Google Scholar 

  • Cremers, D. 2006. Dynamical statistical shape priors for level set based tracking. IEEE Trans. on Patt. Anal. and Mach. Intell.. To appear.

  • Cremers, D., Kohlberger, T., and Schnörr, C. 2003. Shape statistics in kernel space for variational image segmentation. Pattern Recognition, 36(9):1929–1943.

    Article  MATH  Google Scholar 

  • Cremers, D., Osher, S., and Soatto, S. 2004. Kernel density estimation and intrinsic alignment for knowledge-driven segmentation: Teaching level sets to walk. In Pattern Recogn., volume 3175 of Lect. Not. Comp. Sci., pp. 36–44. Springer, C.E. Rasmussen (ed.).

  • Cremers, D. and Soatto, S. 2003. A pseudo-distance for shape priors in level set segmentation. In, IEEE 2nd Int. Workshop on Variational, Geometric and Level Set Methods, N.Paragios (eds.), Nice, pp. 169–176.

  • Cremers, D. and Soatto, S. May 2005. Motion Competition: A variational framework for piecewise parametric motion segmentation. Int.J.ofComputer Vision, 62(3):249–265.

    Article  Google Scholar 

  • Cremers, D., Sochen, N., and Schnörr, C. 2003. Towards recognition-based variational segmentation using shape priors and dynamic labeling. In, Int. Conf. on Scale Space Theories in Computer Vision, volume 2695 of Lect. Not. Comp. Sci., L.Griffith, editor, pp. 388–400, Isle of Skye, Springer.

  • Cremers, D., Sochen, N., and Schnörr, C. 2006. A multiphase dynamic labeling model for variational recognition-driven image segmentation. Int.J.ofComputer Vision, 66(1):67–81.

    Article  Google Scholar 

  • Cremers, D., Tischhäuser, F., Weickert, J., and Schnörr, C. 2002. Diffusion Snakes: Introducing statistical shape knowledge into the Mumford—Shah functional. Int.J.ofComputer Vision, 50(3):295–313.

    Article  MATH  Google Scholar 

  • Deheuvels, P. 1977. Estimation non paramétrique de la densité par histogrammes généralisés. Revue de Statistique Appliquée, 25:5–42.

    MathSciNet  Google Scholar 

  • Dervieux, A. and Thomasset, F. 1979. A finite element method for the simulation of Raleigh-Taylor instability. Springer Lect. Notes in Math., 771:145–158.

    Article  MathSciNet  Google Scholar 

  • Devroye, L. and Györfi, L. 1985. Nonparametric Density Estimation. The L1 View. John Wiley, New York.

    Google Scholar 

  • Dryden, I.L. and Mardia, K.V. 1998. Statistical Shape Analysis. Wiley, Chichester.

    MATH  Google Scholar 

  • Duin, R.P.W. 1976. On the choice of smoothing parameters for Parzen estimators of probability density functions. IEEE Trans. on Computers, 25:1175–1179.

    MATH  MathSciNet  Google Scholar 

  • Fréchet, M. 1961. Les courbes aléatoires. Bull. Inst. Internat. Stat., 38:499–504.

    MATH  Google Scholar 

  • Gdalyahu, Y. and Weinshall, D. 1999. Flexible syntactic matching of curves and its application to automatic hierarchical classication of silhouettes. IEEE Trans. on Patt. Anal. and Mach. Intell., 21(12):1312–1328.

    Article  Google Scholar 

  • Grenander, U. 1976. Lectures in Pattern Theory. Springer, Berlin.

    MATH  Google Scholar 

  • Grenander, U., Chow, Y., and Keenan, D.M. 1991. Hands: A Pattern Theoretic Study of Biological Shapes. Springer, New York.

    Google Scholar 

  • Heiler, M. and Schnörr, C. 2003. Natural image statistics for natural image segmentation. In IEEE Int. Conf. on Computer Vision, pp. 1259–1266.

  • Kendall, D.G. 1977. The diffusion of shape. Advances in Applied Probability, 9:428–430.

    Article  Google Scholar 

  • Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A., and Yezzi, A.J. 1995. Gradient flows and geometric active contour models. In IEEE Int. Conf. on Computer Vision, pp. 810–815.

  • Klassen, E., Srivastava, A., Mio, W., and Joshi, S.H. 2004. Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. on Patt. Anal. and Mach. Intell., 26(3):372–383.

    Article  Google Scholar 

  • Leventon, M., Grimson, W., and Faugeras, O. 2000. Statistical shape influence in geodesic active contours. In CVPR, Hilton Head Island, SC.1:316–323.

    Google Scholar 

  • Malladi, R., Sethian, J.A., and Vemuri, B.C. 1995. Shape modeling with front propagation: A level set approach. IEEE Trans. on Patt. Anal. and Mach. Intell., 17(2):158–175.

    Article  Google Scholar 

  • Matheron, G. 1975. Random Sets and Integral Geometry. Wiley & Sons.

  • Mio, W., Srivastava, A., and Liu, X. 2004. Learning and Bayesian shape extraction for object recognition. In European Conf. on Computer Vision, volume 3024 of Lect. Not. Comp. Sci., Prague, Springer, pp. 62–73.

    Google Scholar 

  • Moelich, M. and Chan, T. 2003. Tracking objects with the Chan-Vese algorithm. Technical Report 03-14, Computational Applied Mathematics, UCLA, Los Angeles.

    Google Scholar 

  • Mumford, D., and Shah, J. 1989. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42:577–685.

    MathSciNet  MATH  Google Scholar 

  • Osher, S.J. and Fedkiw, R.P. 2002. Level Set Methods and Dynamic Implicit Surfaces. Springer, New York.

    Google Scholar 

  • Osher, S.J. and Paragios, N. 2003. Geometric Level Set Methods in Imaging, Vision and Graphics. Springer, Telos.

    MATH  Google Scholar 

  • Osher, S.J. and Sethian, J.A. 1988. Fronts propagation with curvature dependent speed: Algorithms based on Hamilton–Jacobi formulations. J. of Comp. Phys., 79:12–49.

    Article  MathSciNet  MATH  Google Scholar 

  • Paragios, N. and Deriche, R. 2002. Geodesic active regions and level set methods for supervised texture segmentation. Int.J.ofComputer Vision, 46(3):223–247.

    Article  MATH  Google Scholar 

  • Parzen, E. On the estimation of a probability density function and the mode. Annals of Mathematical Statistics, 33:1065–1076.

  • Pons, J.-P., Hermosillo, G., Keriven, R., and Faugeras, O. 2003. How to deal with point correspondences and tangential velocities in the level set framework. In IEEE Int. Conf. on Computer Vision, pages 894–899.

  • Rasmussen, C.-E. and Williams, C.K.I. 2006. Gaussian Processes for Machine Learning, MIT Press.

  • Rathi, Y., Vaswani, N., Tannenbaum, A., and Yezzi, A. 2005. Particle filtering for geometric active contours and application to tracking deforming objects. In IEEE Int. Conf. on Computer Vision and Pattern Recognition, volume2, pp.2–9.

    Google Scholar 

  • Riklin-Raviv, T., Kiryati, N. and Sochen, N. 2004. Unlevel sets: Geometry and prior-based segmentation. In T.Pajdla and V.Hlavac, editors, European Conf. on Computer Vision, volume 3024 of Lect. Not. Comp. Sci., pp.50–61, Prague. Springer.

    Google Scholar 

  • Rosenblatt, F. 1956. Remarks on some nonparametric estimates of a density function. Annals of Mathematical Statistics, 27:832–837.

    MATH  MathSciNet  Google Scholar 

  • Rousson, M., Brox, T., and Deriche, R. 2003. Active unsupervised texture segmentation on a diffusion based feature space. In Proc.IEEE Conf.on Comp.Vision Patt.Recog., pp. 699–704, Madison, WI.

  • Rousson, M. and Cremers, D. 2005. Efficient kernel density estimation of shape and intensity priors for level set segmentation. In Intl.Conf.on Medical Image Computing and Comp.Ass.Intervention (MICCAI), 1:757–764.

    Article  Google Scholar 

  • Rousson, M. and Paragios, N. 2002. Shape priors for level set representations. In A. Heyden etal., editors, Europ. Conf. on Comp. Vis., volume 2351 of Lect. Not. Comp. Sci., pages 78–92. Springer.

  • Rousson, M., Paragios, N., and Deriche, R. 2004. Implicit active shape models for 3d segmentation in MRI imaging. In Intl.Conf.on Medical Image Computing and Comp.Ass.Intervention (MICCAI), volume 2217 of Lect. Not. Comp. Sci., Springer, pp. 209–216

  • Silverman, B.W. 1978. Choosing the window width when estimating a density. Biometrika, 65:1–11.

    Article  MATH  MathSciNet  Google Scholar 

  • Silverman, B.W. 1992. Density estimation for statistics and data analysis. Chapman and Hall, London.

    Google Scholar 

  • Sussman, M., Smereka P., and Osher S.J. 1994. A level set approach for computing solutions to incompressible twophase flow. J. of Comp. Phys., 94:146–159.

    Article  Google Scholar 

  • Trouvé, A. 1998. Diffeomorphisms, groups and pattern matching in image analysis. Int.J.ofComputer Vision, 28(3):213–21.

    Article  Google Scholar 

  • Tsai, A., Yezzi, A., Wells, W., Tempany, C., Tucker, D., Fan, A., Grimson, E., and Willsky A. 2001. Model–based curve evolution technique for image segmentation. In Comp.Vision Patt.Recog., Kauai, Hawaii, pp. 463–468.

  • Tsai, A., Yezzi, A.J., and Willsky, A.S. 2001. Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. on Image Processing, 10(8):1169–1186.

    Article  MATH  Google Scholar 

  • Wagner, T.J. 1975. Nonparametric estimates of probability densities. IEEE Trans. on Inform. Theory, 21:438–440.

    Article  MATH  Google Scholar 

  • Yezzi, A. and Soatto, S. 2003. Deformotion: Deforming motions and shape averages. Int.J.ofComputer Vision, 53(2):153–167.

    Article  Google Scholar 

  • Younes, L. 1998. Computable elastic distances between shapes. SIAM J. Appl. Math., 58(2):565–586.

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao, H.-K., Chan, T., Merriman, B. and Osher, S. 1996. A variational level set approach to multiphase motion. J. of Comp. Phys., 127:179–195.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Bonn, Germany

    Daniel Cremers

  2. Department of Mathematics, University of California, Los Angeles

    Stanley J. Osher

  3. Department of Computer Science, University of California, Los Angeles

    Stefano Soatto

Authors
  1. Daniel Cremers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stanley J. Osher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefano Soatto
    View author publications

    You can also search for this author in PubMed Google Scholar

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cremers, D., Osher, S.J. & Soatto, S. Kernel Density Estimation and Intrinsic Alignment for Shape Priors in Level Set Segmentation. Int J Comput Vision 69, 335–351 (2006). https://doi.org/10.1007/s11263-006-7533-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-006-7533-5

Keywords

Search

Navigation