A variational approach to optical flow estimation of unsteady incompressible flows

  • Souvik Roy
  • Praveen Chandrashekar
  • A. S. Vasudeva Murthy


We consider optical flow estimation of flows with vorticity governed by 2D incompressible Euler and Navier–Stokes equations . A vorticity-streamfunction formulation and optimization techniques are used. We use Helmholtz decomposition of the velocity field and prove existence of an unique velocity and vorticity field for the linearized vorticity equations. Discontinuous galerkin finite elements are used to solve the vorticity equation for Euler’s flow to efficiently track discontinuous vortices. Finally we test our method with two vortex flows governed by Euler and Navier–Stokes equations at high Reynolds number which support our theoretical results.


Euler Navier–Stokes Vorticity Streamfunction Discontinuous Galerkin Optimization Linearization Helmholtz decomposition Incompressible 

Mathematics Subject Classification



  1. 1.
    Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artif. Intell. 17(1–3), 185–203 (1981)CrossRefGoogle Scholar
  2. 2.
    Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. In: Dilcher, K., Taylor, K. (eds.) CMS Books in Mathematics. Springer, New York. ISBN: 978-1-4419-9466-0 (2011)Google Scholar
  3. 3.
    Ruhnau, P., Schnörr, C.: Optical Stokes flow: an imaging based control approach. Exp. Fluids 42, 61–78 (2007)CrossRefGoogle Scholar
  4. 4.
    Yuan, J., Ruhnau, P., Mémin, E., Schnörr, C.: Discrete orthogonal decomposition and variational fluid flow estimation. In: Scale-Space 2005, volume 3459 of Lecture Notes Computer Science, pp. 267–278. Springer (2005)Google Scholar
  5. 5.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics. Springer, Berlin (1986)Google Scholar
  6. 6.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112, American Mathematical Society, USA (2010)Google Scholar
  7. 7.
    Liu, T., Shen, L.: Fluid flow and optical flow. J. Fluid Mech 614, 253–291 (2008)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Heitz, D., Mémin, E., Schnörr, C.: Variational fluid flow measurement from image sequences: synopsis and perspectives. Exp. Fluids 48, 369–393 (2010)CrossRefGoogle Scholar
  9. 9.
    Cayula, J.-F., Cornillon, P.: Cloud detection from a sequence of SST images. Remote Sens. Environ. 55, 80–88 (1996)CrossRefGoogle Scholar
  10. 10.
    Leese, J.A., Novak, C.S., Taylor, V.R.: The determination of cloud pattern motions from geosynchronous satellite image data. Pattern Recognit. 2, 279–292 (1970)CrossRefGoogle Scholar
  11. 11.
    Fogel, S.V.: The estimation of velocity vector-fields from time varying image sequences. CVGIP: Image Underst. 53, 253–287 (1991)MATHCrossRefGoogle Scholar
  12. 12.
    Wu, Q.X.: A correlation-relaxation labeling framework for computing optical flow—template matching from a new perspective. IEEE Trans. Pattern Anal. Mach. Intell. 17, 843–853 (1995)CrossRefGoogle Scholar
  13. 13.
    Parikh, J.A., DaPonte, J.S., Vitale, J.N., Tselioudis, G.: An evolutionary system for recognition and tracking of synoptic scale storm systems. Pattern Recognit. Lett. 20, 1389–1396 (1999)CrossRefGoogle Scholar
  14. 14.
    Logg, A., Mardal, K.-A., Wells, G.N.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)MATHCrossRefGoogle Scholar
  15. 15.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Netherlands. ISBN 978-1-4020-8839-1 (2009)Google Scholar
  16. 16.
    Aubert, G., Kornprobst, P.: A mathematical study of the relaxed optical flow problem in the space. SIAM J. Math. Anal. 30(6), 1282–1308 (1999)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Aubert, G., Deriche, R., Kornprobst, P.: Computing optical flow via variational techniques. SIAM J. Math. Anal. 60(1), 156–182 (1999)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Nagel, H.-H.: Displacement vectors derived from second-order intensity variations in image sequences. CGIP 21, 85–117 (1983)Google Scholar
  19. 19.
    Nagel, H.-H.: On the estimation of optical flow: relations between different approaches and some new results. AI 33, 299–324 (1987)Google Scholar
  20. 20.
    Nagel, H.-H.: On a constraint equation for the estimation of displacement rates in image sequences. IEEE Trans. PAMI 11, 13–30 (1989)MATHCrossRefGoogle Scholar
  21. 21.
    Nagel, H.-H., Enkelmann, W.: An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences. IEEE Trans. PAMI 8, 565–593 (1986)CrossRefGoogle Scholar
  22. 22.
    Papadakis, N., Mémin, E.: Variational assimilation of fluid motion from image sequence. SIAM J. Imaging Sci. 1(4), 343–363 (2008)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Mukawa, N.: Estimation of shape, reflection coefficients and illuminant direction from image sequences. In: ICCV90, pp. 507–512 (1990)Google Scholar
  24. 24.
    Corpetti, T., Mémin, E., Pérez, P.: Dense estimation of fluid flows. IEEE Trans. Pattern Anal. Mach. Intell. 24(3), 365380 (2002)CrossRefGoogle Scholar
  25. 25.
    Nakajima, Y., Inomata, H., Nogawa, H., Sato, Y., Tamura, S., Okazaki, K., Torii, S.: Physics-based flow estimation of fluids. Pattern Recognit. 36(5), 1203–1212 (2003)CrossRefGoogle Scholar
  26. 26.
    Arnaud, E., Mémin, E., Sosa, R., Artana, G.: A fluid motion estimator for schlieren image velocimetry. In: ECCV06, I, pp. 198–210 (2006)Google Scholar
  27. 27.
    Wayne Roberts, A., Varberg, Dale E.: Convex Functions. Academic Press, New York (1973)MATHGoogle Scholar
  28. 28.
    Haussecker, H.W., Fleet, D.J.: Computing optical flow with physical models of brightness variation. IEEE Trans. Pattern Anal. Mach. Intell. 23(6), 661–673 (2001)CrossRefGoogle Scholar
  29. 29.
    Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin methods. Theory, computation and applications. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2000)Google Scholar
  32. 32.
    Arlotti, L., Banasiak, J., Lods, B.: A new approach to transport equations associated to a regular field: trace results and well-posedness. Mediterr. J. Math. 6, 367–402 (2009)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Blanc, V.L.: \(L^1\)-stability of periodic stationary solutions of scalar convection-diffusion equations. J. Differ. Equ. 247, 1746–1761 (2009)MATHCrossRefGoogle Scholar
  34. 34.
    Roy, S.: Reconstruction of a class of fluid flows by variational methods and inversion of integral transforms in tomography, Ph.D. dissertation, Tata Institute of Fundamental Research, CAM, Bangalore. https://www.dropbox.com/s/cbxblk3dg7kwxp5/Souvik_phd_thesis_compressed.pdf?dl=0 (2015)
  35. 35.
    Roy, S., Chandrashekar, P., Vasudeva Murthy, A.S.: A variational approach to optical flow estimation of incompressible fluid flow. J. Comput. Vis. Sci. (2015, submitted) Google Scholar

Copyright information

© Indian Institute of Technology Madras 2015

Authors and Affiliations

  • Souvik Roy
    • 1
  • Praveen Chandrashekar
    • 1
  • A. S. Vasudeva Murthy
    • 1
  1. 1.TIFR Center for Applicable MathematicsBangaloreIndia

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