Abstract
In this paper, we introduce a variational framework derived from data assimilation principles in order to realize a temporal Bayesian smoothing of fluid flow velocity fields. The velocity measurements are supplied by an optical flow estimator. These noisy measurement are smoothed according to the vorticity-velocity formulation of Navier-Stokes equation. Following optimal control recipes, the associated minimization is conducted through an iterative process involving a forward integration of our dynamical model followed by a backward integration of an adjoint evolution law. Both evolution laws are implemented with second order non-oscillatory scheme. The approach is here validated on a synthetic sequence of turbulent 2D flow provided by Direct Numerical Simulation (DNS) and on a real world meteorological satellite image sequence depicting the evolution of a cyclone.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bennet, A.F.: Inverse Methods in Physical Oceanography. Cambridge University Press, Cambridge (1992)
Carlier, J., Wieneke, B.: Report on production and diffusion of fluid mechanics images and data. Technical report, Fluid Project deliverable 1.2 (2005)
Corpetti, T., Mémin, E., Pérez, P.: Dense estimation of fluid flows. IEEE Trans. Pattern Anal. Machine Intell. 24(3), 365–380 (2002)
Corpetti, T., Mémin, E., Pérez, P.: Extraction of singular points from dense motion fields: an analytic approach. J. Math. Imaging and Vision 19(3), 175–198 (2003)
Cuzol, A., Mémin, E.: A stochastic filter for fluid motion tracking. In: Int. Conf. on Computer Vision (ICCV’05), Beijing, China (October 2005)
Le Dimet, F.-X., Talagrand, O.: Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus 38A, 97–110 (1986)
Fitzpatrick, J.M.: 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 (1985)
Ford, R.M., Strickland, R., Thomas, B.: Image models for 2-d flow visualization and compression. Graph. Mod. Image Proc. 56(1), 75–93 (1994)
Giering, R., Kaminski, T.: Recipes for adjoint code construction. ACM Trans. Math. Softw. 24(4), 437–474 (1998)
Giles, M.: On the use of runge-kutta time-marching and multigrid for the solution of steady adjoint equations. Technical Report 00/10, Oxford University Computing Laboratory (2000)
Harten, A., et al.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. of Comput. Phys. 71(2), 231–303 (1987)
Kohlberger, T., Mémin, E., Schnörr, C.: Variational dense motion estimation using the helmholtz decomposition. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 432–448. Springer, Heidelberg (2003)
Kurganov, A., Levy, D.: A third-order semidiscrete central scheme for conservation laws and convection-diffusion equations. SIAM J. Sci. Comput. 22(4), 1461–1488 (2000)
Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convetion-diffusion equations. J. Comput. Phys. 160(1), 241–282 (2000)
Larsen, R., Conradsen, K., Ersboll, B.K.: Estimation of dense image flow fields in fluids. IEEE trans. on Geosc. and Remote sensing 36(1), 256–264 (1998)
Levy, D., Puppo, G., Russo, G.: A third order central weno scheme for 2d conservation laws. Appl. Num. Math.: Trans. of IMACS 33(1–4), 415–421 (2000)
Levy, D., Tadmor, E.: Non-oscillatory central schemes for the incompressible 2-d euler equations. Math. Res. Let 4, 321–340 (1997)
Mémin, E., Pérez, P.: Fluid motion recovery by coupling dense and parametric motion fields. In: Int. Conf. on Computer, ICCV’99, pp. 620–625 (1999)
Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. of Comput. Phys. 87(2), 408–463 (1990)
Oksendal, B.: Stochastic differential equations. Springer, Heidelberg (1998)
Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Heidelberg (1998)
Talagrand, O.: Variational assimilation. Adjoint equations. Kluwer Academic Publishers, Dordrecht (2002)
Talagrand, O., Courtier, P.: Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory. J. of Roy. Meteo. soc. 113, 1311–1328 (1987)
Schnörr, C., et al.: Discrete Orthogonal Decomposition and Variational Fluid Flow Estimation. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 267–278. Springer, Heidelberg (2005)
Yuan, J., Schnörr, C., Mémin, E.: Discrete orthogonal decomposition and variational fluid flow estimation. Accepted for publication in J. of Math. Imaging and Vision (2006)
Zhou, L., Kambhamettu, C., Goldgof, D.: Fluid structure and motion analysis from multi-spectrum 2D cloud images sequences. In: Proc. Conf. Comp. Vision Pattern Rec., vol. 2, Hilton Head Island, USA, pp. 744–751 (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Papadakis, N., Mémin, É. (2007). A Variational Framework for Spatio-temporal Smoothing of Fluid Motions. In: Sgallari, F., Murli, A., Paragios, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer Science, vol 4485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72823-8_52
Download citation
DOI: https://doi.org/10.1007/978-3-540-72823-8_52
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72822-1
Online ISBN: 978-3-540-72823-8
eBook Packages: Computer ScienceComputer Science (R0)