Experiments in Fluids

, 59:8 | Cite as

Motion estimation under location uncertainty for turbulent fluid flows

  • Shengze Cai
  • Etienne Mémin
  • Pierre Dérian
  • Chao XuEmail author
Research Article


In this paper, we propose a novel optical flow formulation for estimating two-dimensional velocity fields from an image sequence depicting the evolution of a passive scalar transported by a fluid flow. This motion estimator relies on a stochastic representation of the flow allowing to incorporate naturally a notion of uncertainty in the flow measurement. In this context, the Eulerian fluid flow velocity field is decomposed into two components: a large-scale motion field and a small-scale uncertainty component. We define the small-scale component as a random field. Subsequently, the data term of the optical flow formulation is based on a stochastic transport equation, derived from the formalism under location uncertainty proposed in Mémin (Geophys Astrophys Fluid Dyn 108(2):119–146, 2014) and Resseguier et al. (Geophys Astrophys Fluid Dyn 111(3):149–176, 2017a). In addition, a specific regularization term built from the assumption of constant kinetic energy involves the very same diffusion tensor as the one appearing in the data transport term. Opposite to the classical motion estimators, this enables us to devise an optical flow method dedicated to fluid flows in which the regularization parameter has now a clear physical interpretation and can be easily estimated. Experimental evaluations are presented on both synthetic and real world image sequences. Results and comparisons indicate very good performance of the proposed formulation for turbulent flow motion estimation.


  1. Atcheson B, Heidrich W, Ihrke I (2009) An evaluation of optical flow algorithms for background oriented schlieren imaging. Exp Fluids 46(3):467–476CrossRefGoogle Scholar
  2. Batchelor G (1959) Small-scale variation of convected quantities like temperature in turbulent fluid. J Fluid Mech 5(01):113–133MathSciNetCrossRefzbMATHGoogle Scholar
  3. Becker F, Wieneke B, Petra S, Schroder A, Schnorr C (2012) Variational adaptive correlation method for flow estimation. IEEE Trans Image Process 21(6):3053–3065MathSciNetCrossRefzbMATHGoogle Scholar
  4. Beyou S, Cuzol A, Gorthi S, Mémin E (2013) Weighted ensemble transform kalman filter for image assimilation. Tellus A Dyn Meteorol Oceanogr 65:18803CrossRefGoogle Scholar
  5. Black M, Anandan P (1996) The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields. Comput Vis Image Underst 63(1):75–104CrossRefGoogle Scholar
  6. Boffetta G, Ecke R (2012) Two-dimensional turbulence. Annu Rev Fluid Mech 44:427–451MathSciNetCrossRefzbMATHGoogle Scholar
  7. Brox T, Bruhn A, Papenberg N, Weickert J (2004) High accuracy optical flow estimation based on a theory for warping. In: Proceedings of the European Conference on Computer Vision, pp 25–36Google Scholar
  8. Bruhn A, Weickert J, Schnörr C (2005) Lucas/Kanade meets Horn/Schunck: Combining local and global optic flow methods. Int J Comput Vision 61(3):211–231CrossRefGoogle Scholar
  9. Carlier J (2005) Second set of fluid mechanics image sequences. European Project Fluid Image Analysis and Description (FLUID).
  10. Cassisa C, Simoens S, Prinet V, Shao L (2011) Subgrid scale formulation of optical flow for the study of turbulent flow. Exp Fluids 51(6):1739–1754CrossRefGoogle Scholar
  11. Chen X, Zillé P, Shao L, Corpetti T (2015) Optical flow for incompressible turbulence motion estimation. Exp Fluids 56(1):1–14CrossRefGoogle Scholar
  12. Corpetti T, Heitz D, Arroyo G, Mémin E, Santa-Cruz A (2006) Fluid experimental flow estimation based on an optical-flow scheme. Exp Fluids 40(1):80–97CrossRefGoogle Scholar
  13. Corpetti T, Mémin E, Pérez P (2002) Dense estimation of fluid flows. IEEE Trans Pattern Anal Mach Intell 24(3):365–380CrossRefzbMATHGoogle Scholar
  14. Crisan D, Flandoli F, Holm D (2017) Solution properties of a 3D stochastic Euler fluid equation. arXiv:1704.06989 (preprint)
  15. Cuzol A, Mémin E (2009) A stochastic filtering technique for fluid flow velocity fields tracking. IEEE Trans Pattern Anal Mach Intell 31(7):1278–1293CrossRefGoogle Scholar
  16. Dérian P, Héas P, Herzet C, Mémin E (2013) Wavelets and optical flow motion estimation. Numer Math Theory Methods Appl 6(1):116–137MathSciNetzbMATHGoogle Scholar
  17. Deusch S, Merava H, Dracos T, Rys P (2000) Measurement of velocity and velocity derivatives based on pattern tracking in 3D LIF images. Exp Fluids 29(4):388–401CrossRefGoogle Scholar
  18. Héas P, Mémin E, Heitz D, Mininni P (2012) Power laws and inverse motion modeling: application to turbulence measurements from satellite images. Tellus A Dyn Meteorol Oceanogr 64:10962CrossRefGoogle Scholar
  19. Heitz D, Héas P, Mémin E, Carlier J (2008) Dynamic consistent correlation-variational approach for robust optical flow estimation. Exp Fluids 45(4):595–608CrossRefGoogle Scholar
  20. Heitz D, Mémin E, Schnörr C (2010) Variational fluid flow measurements from image sequences: synopsis and perspectives. Exp Fluids 48(3):369–393CrossRefGoogle Scholar
  21. Holm D (2015) Variational principles for stochastic fluid dynamics. In: Proc. R. Soc. A, vol 471. The Royal Society, London, pp 20140963Google Scholar
  22. Horn B, Schunck B (1981) Determining optical flow. Artif Intell 17(1–3):185–203CrossRefGoogle Scholar
  23. Jullien M, Castiglione P, Tabeling P (2000) Experimental observation of Batchelor dispersion of passive tracers. Phys Rev Lett 85(17):3636CrossRefGoogle Scholar
  24. Kadri-Harouna S, Dérian P, Héas P, Mémin E (2013) Divergence-free wavelets and high order regularization. Int J Comput Vision 103(1):80–99MathSciNetCrossRefzbMATHGoogle Scholar
  25. Kadri-Harouna S, Mémin E (2017) Stochastic representation of the reynolds transport theorem: revisiting large-scale modeling. Comput Fluids 156:456–469MathSciNetCrossRefGoogle Scholar
  26. Lilly K (1966) On the application of the eddy viscosity concept in the inertial subrange of turbulence. Technical Report 123, NCARGoogle Scholar
  27. Liu T (2013) Extraction of skin-friction fields from surface flow visualizations as an inverse problem. Meas Sci Technol 24(12):124004CrossRefGoogle Scholar
  28. Liu T, Merat A, Makhmalbaf H, Fajardo C, Merati P (2015) Comparison between optical flow and cross-correlation methods for extraction of velocity fields from particle images. Exp Fluids 56(8):166CrossRefGoogle Scholar
  29. Liu T, Shen L (2008) Fluid flow and optical flow. J Fluid Mech 614:253–291MathSciNetCrossRefzbMATHGoogle Scholar
  30. Mémin E (2014) Fluid flow dynamics under location uncertainty. Geophys Astrophys Fluid Dyn 108(2):119–146MathSciNetCrossRefGoogle Scholar
  31. Mémin E, Pérez P (1998) Dense estimation and object-based segmentation of the optical flow with robust techniques. IEEE Trans Image Process 7(5):703–719CrossRefGoogle Scholar
  32. Papadakis N, Mémin E (2008) Variational assimilation of fluid motion from image sequence. SIAM J Imaging Sci 1(4):343–363MathSciNetCrossRefzbMATHGoogle Scholar
  33. Papadakis N, Mémin E, Cuzol A, Gengembre N (2010) Data assimilation with the weighted ensemble Kalman filter. Tellus A 62(5):673–697CrossRefGoogle Scholar
  34. Quénot G, Pakleza J, Kowalewski T (1998) Particle image velocimetry with optical flow. Exp Fluids 25(3):177–189CrossRefGoogle Scholar
  35. Raffel M, Willert C, Wereley S, Kompenhans J (2007) Particle image velocimetry: a practical guide. Springer, New YorkGoogle Scholar
  36. Resseguier V, Mémin E, Chapron B (2017a) Geophysical flows under location uncertainty, part i: random transport and general models. Geophys Astrophys Fluid Dyn 111(3):149–176MathSciNetCrossRefGoogle Scholar
  37. Resseguier V, Mémin E, Chapron B (2017b) Geophysical flows under location uncertainty, part ii: quasigeostrophic models and efficient ensemble spreading. Geophys Astrophys Fluid Dyn 111(3):177–208MathSciNetCrossRefGoogle Scholar
  38. Resseguier V, Mémin E, Chapron B (2017c) Geophysical flows under location uncertainty, Part III: SQG and frontal dynamics under strong turbulence. Geophys Astrophys Fluid Dyn 111(3):209–227MathSciNetCrossRefGoogle Scholar
  39. Resseguier V, Mémin E, Heitz D, Chapron B (2017d) Stochastic modelling and diffusion modes for POD models and small-scale flow analysis. J Fluid Mech 828:888–917CrossRefGoogle Scholar
  40. Ruhnau P, Kohlberger T, Schnörr C, Nobach H (2005) Variational optical flow estimation for particle image velocimetry. Exp Fluids 38(1):21–32CrossRefGoogle Scholar
  41. Stapf J, Garbe C (2014) A learning-based approach for highly accurate measurements of turbulent fluid flows. Exp Fluids 55(8):1799CrossRefGoogle Scholar
  42. Su LK, Dahm WJ (1996) Scalar imaging velocimetry measurements of the velocity gradient tensor field in turbulent flows. I. Assessment of errors. Phys Fluids 8(7):1869–1882CrossRefGoogle Scholar
  43. Sun D, Roth S, Black M (2010) Secrets of optical flow estimation and their principles. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, pp 2432–2439Google Scholar
  44. Tokumaru PT, Dimotakis PE (1995) Image correlation velocimetry. Exp Fluids 19(1):1–15CrossRefGoogle Scholar
  45. Yuan J, Schnörr C, Mémin E (2007) Discrete orthogonal decomposition and variational fluid flow estimation. J Math Imaging Vision 28(1):67–80MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.State Key Laboratory of Industrial Control Technology and The Institute of Cyber-Systems and ControlZhejiang UniversityHangzhouChina
  2. 2.National Institute for Research in Computer Science and Control (INRIA)RennesFrance

Personalised recommendations