Skip to main content
SpringerLink
Log in

A Low Dimensional Fluid Motion Estimator

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

Abstract

In this paper we propose a new motion estimator for image sequences depicting fluid flows. The proposed estimator is based on the Helmholtz decomposition of vector fields. This decomposition consists in representing the velocity field as a sum of a divergence free component and a vorticity free component. The objective is to provide a low-dimensional parametric representation of optical flows by depicting them as deformations generated by a reduced number of vortex and source particles. Both components are approximated using a discretization of the vorticity and divergence maps through regularized Dirac measures. The resulting so called irrotational and solenoidal fields consist of linear combinations of basis functions obtained through a convolution product of the Green kernel gradient and the vorticity map or the divergence map respectively. The coefficient values and the basis function parameters are obtained by minimization of a functional relying on an integrated version of mass conservation principle of fluid mechanics. Results are provided on synthetic examples and real world sequences.

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

  • Alvarez, L., Weickert, J., and Sanchez, J. 2000. Reliable estimation of dense optical flow fields with large displacements. Int. J. Computer Vision, 39(1):41–56.

    Article  MATH  Google Scholar 

  • Amini, A. 1994. A scalar function formulation for optical flow. In Proc. Europ. Conf. Computer Vision, pp. 125–131.

  • Bartoli, A. and Zisserman, A. 2004. Direct estimation of non-rigid registrations. In Proceedings of the 15th British Machine Vision Conference. London, UK, vol. II, pp. 899–908.

  • Béréziat, D., Herlin, I., and Younes, L. 2000. A generalized optical flow constraint and its physical interpretation. In Proc. Conf. Comp. Vision Pattern Rec. Hilton Head Island, South Carolina, USA, vol. 2, pp. 487–492.

  • Black, M. and Anandan, P. 1996. The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields. Computer Vision and Image Understanding, 63(1):75–104.

    Google Scholar 

  • Brox, T., Bruhn, A., Papenberg, N., and Weickert, J. 2004. High accuracy optical flow estimation based on a theory for warping. In ECCV(4), Prague, Czech Republic, pp. 25–36.

  • Burt, P.J. 1984. The pyramid as a structure for efficient computation. In Multiresolution Image Processing and Analysis, A. Rozenfeld (Ed), Springer-Verlag, pp. 6–35.

  • Cheng, Y. 1995. Mean shift, mode seeking and clustering. IEEE Trans. Pattern Anal. Mach. Intell., 17(8):790–799.

    Article  Google Scholar 

  • Chorin, A.J. 1973. Numerical study of slightly viscous flow. J. Fluid Mech., 57:785–796.

    Article  MathSciNet  Google Scholar 

  • Chorin, A.J. and Marsden, J.E. 1979. A Mathematical Introduction to Fluid Mechanics. Springer-Verlag.

  • Comaniciu, D. and Meer, P. 2002. Mean shift: A robust approach toward feature space analysis. IEEE Trans. Pattern Analysis Machine Intelligence, 24(5):603–619.

    Article  Google Scholar 

  • Comaniciu, D., Ramesh, V., and Meer, P. 2000. Real-time tracking of non-rigid objects using mean shift. In Proc. Conf. Comp. Vision Pattern Rec., pp. 142–149.

  • Corpetti, T., Mémin, E., and Pérez, P. 2002. Dense estimation of fluid flows. IEEE Trans. Pattern Anal. Machine Intell., 24(3):365–380.

    Article  Google Scholar 

  • Corpetti, T., Mémin, E., and Pérez, P. 2003. Extraction of singular points from dense motion fields: An analytic approach. J. Mathematical Imaging and Vision, 19(3):175–198.

    Article  MATH  Google Scholar 

  • Cottet, G.-H. and Koumoutsakos, P. 2000. Vortex Methods: Theory and Practice. Cambridge University Press.

  • Cuzol, A., Hellier, P., and Mémin, E. 2005. A novel parametric method for non-rigid image registration. In Proc. Information Processing in Medical Imaging (IPMI’05), G. Christensen and M. Sonka (Eds), number 3565 in LNCS, Glenwood Springs, Colorado, USA, pp. 456–467.

  • Cuzol, A. and Mémin, E. 2005. A stochastic filter for fluid motion tracking. In Proc. Int. Conf. on Computer Vision (ICCV’05), Beijing, China.

  • Cuzol, A. and Mémin, E. 2005. Vortex and source particles for fluid motion estimation. In 5th Int. Conf. on Scale-Space and PDE Methods in Computer Vision (Scale-Space’05), Hofgeismar, Germany.

  • Fitzpatrick, J.M. 1985. A method for calculating velocity in time dependent images based on the continuity equation. In Proc. Conf. Comp. Vision Pattern Rec., San Francisco, USA, pp. 78–81.

  • Fitzpatrick, J.M. 1988. The existence of geometrical density-image transformations corresponding to object motion. Comput. Vision, Graphics, Image Proc., 44(2):155–174.

    Google Scholar 

  • Ford, R.M., Strickland, R., and Thomas, B. 1994. Image models for 2-d flow visualization and compression. Graph. Mod. Image Proc., 56(1):75–93.

    Google Scholar 

  • Fukanaga, K. and Hostetler, L.D. 1975. The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Trans. on Info. Theory, 21(1):32–40.

    Article  Google Scholar 

  • Gilbert, J.C. and Nocedal, J. 1992. Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optimization, 2:21–42.

    Article  MATH  MathSciNet  Google Scholar 

  • Hellier, P., Barillot, C., Mémin, E., and Pérez, P. 2001. Hierarchical estimation of a dense deformation field for 3D robust registration. IEEE Transaction on Medical Imaging, 20(5):388–402.

    Article  Google Scholar 

  • Horn, B. and Schunck, B. 1981. Determining optical flow. Artificial Intelligence, 17:185–203.

    Article  Google Scholar 

  • Kohlberger, T., Mémin, E., and Schnörr, C. 2003 Variational dense motion estimation using the Helmholtz decomposition. In Int. conf on Scale-Space theories in Computer Vision(Scale-Space’03), Isle of Skye.

  • Lamballais, E. and Silvestrini, J.H. 2002. Direct numerical simulation of interactions between a mixing layer and a wake around a cylinder. Journal of Turbulence, 3(028).

  • Larsen, R., Conradsen, K., and Ersboll, B.K. 1998. Estimation of dense image flow fields in fluids. IEEE trans. on Geoscience and Remote sensing, 36(1):256–264.

    Article  Google Scholar 

  • Leonard, A. 1980. Vortex methods for flow simulation. J. Comp. Phys., 37.

  • Lester, H. and Arridge, S. 1999. A survey of hierarchical non-linear medical image registration. Pattern Recognition, 32:129– 149.

    Article  Google Scholar 

  • Maes, F., Collignon, A., Vandermeulen, D., Marchal, G., and Suetens, P. (1997) Multimodality image registration by maximisation of mutual information. IEEE Trans. Medical Imaging, 16(2):187– 198.

    Google Scholar 

  • Maintz, J. and Viergever, M.A. 1998. A survey of medical image registration. Medical Image Analysis, 2(1):1–36.

    Article  Google Scholar 

  • Mémin, E. and Pérez, P. 1998. Dense estimation and object-based segmentation of the optical flow with robust techniques. IEEE Trans. Image Processing, 7(5):703–719.

    Google Scholar 

  • Mémin, E. and Pérez, P. 1999. Fluid motion recovery by coupling dense and parametric motion fields. In Int. Conf. on Computer, ICCV’99, pp. 620–625.

  • Nagel, H. and Enkelmann, W. 1986. An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences. IEEE Trans. Pattern Anal. Machine Intell., 8:565–593.

    Google Scholar 

  • Nomura, A., Miike, H., and Koga, K. 1991. Field theory approach for determining optical flow. Pattern Recognition Letters, 12(3):183–190.

    Article  Google Scholar 

  • Rueckert, D., Somoda, I., Hayes, C., Hill, D., Leach, M., and Hawkes, D. 1999. Nonririd registration using free-form deformations: application to breast MR images. IEEE Transactions on Medical Imaging, 18(8):712–721.

    Article  Google Scholar 

  • Schunk, B.G. 1984. The motion constraint equation for optical flow. In Proc. Int. Conf. Pattern Recognition, Montreal pp. 20–22.

  • Song, S.M. and Leahy, R.M. 1991. Computation of 3d velocity fields from 3d cine and CT images of human heart. IEEE trans. on medical imaging, 10(3):295–306.

    Article  Google Scholar 

  • Thirion, J.P. 1998. Image matching as a diffusion process: an analogy with Maxwell’s demons. Medical Image Analysis, 2(3):243– 260.

    Google Scholar 

  • Toga, A. and Thompson, P. 2001. The role of image registration in brain mapping. Image and Vision Computing, 19:3–24.

    Article  Google Scholar 

  • Weickert, J. and Schnörr, C. 2001. Variational optic-flow computation with a spatio-temporal smoothness constraint. J. Mathematical Imaging and Vision, 14(3):245–255.

    Article  MATH  Google Scholar 

  • Wildes, R., Amabile, M., Lanzillotto, A.M., and Leu, T.S. 1997. Physically based fluid flow recovery from image sequences. In Proc. Conf. Comp. Vision Pattern Rec., pp. 969–975.

  • Yuan, J., Ruhnau, P., Mémin, E., and Schnörr, C. 2005. Discrete orthogonal decomposition and variational fluid flow estimation. In 5th Int. Conf. on Scale-Space and PDE methods in Computer Vision (Scale-Space’05), Hofgeismar, Germany.

  • Zhou, L., Kambhamettu, C., and Goldgof, D. 2000. Fluid structure and motion analysis from multi-spectrum 2D cloud images sequences. In Proc. Conf. Comp. Vision Pattern Rec., Hilton Head Island, South Carolina, USA, vol. 2, pp. 744–751.

  • Zitova, B. and Flusser, J. 2003. Image registration methods: a survey. Image and Vision Computing, 21:977–1000.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IRISA, Université de Rennes 1, Campus de Beaulieu, 35 042, Rennes Cedex, France

    Anne Cuzol, Pierre Hellier & Etienne Mémin

Authors
  1. Anne Cuzol
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Hellier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Etienne Mémin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anne Cuzol.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cuzol, A., Hellier, P. & Mémin, E. A Low Dimensional Fluid Motion Estimator. Int J Comput Vis 75, 329–349 (2007). https://doi.org/10.1007/s11263-007-0037-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-007-0037-0

Keywords

Search

Navigation