Abstract
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.
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Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artif. Intell. 17(1–3), 185–203 (1981)
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)
Ruhnau, P., Schnörr, C.: Optical Stokes flow: an imaging based control approach. Exp. Fluids 42, 61–78 (2007)
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)
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics. Springer, Berlin (1986)
Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112, American Mathematical Society, USA (2010)
Liu, T., Shen, L.: Fluid flow and optical flow. J. Fluid Mech 614, 253–291 (2008)
Heitz, D., Mémin, E., Schnörr, C.: Variational fluid flow measurement from image sequences: synopsis and perspectives. Exp. Fluids 48, 369–393 (2010)
Cayula, J.-F., Cornillon, P.: Cloud detection from a sequence of SST images. Remote Sens. Environ. 55, 80–88 (1996)
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)
Fogel, S.V.: The estimation of velocity vector-fields from time varying image sequences. CVGIP: Image Underst. 53, 253–287 (1991)
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)
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)
Logg, A., Mardal, K.-A., Wells, G.N.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Netherlands. ISBN 978-1-4020-8839-1 (2009)
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)
Aubert, G., Deriche, R., Kornprobst, P.: Computing optical flow via variational techniques. SIAM J. Math. Anal. 60(1), 156–182 (1999)
Nagel, H.-H.: Displacement vectors derived from second-order intensity variations in image sequences. CGIP 21, 85–117 (1983)
Nagel, H.-H.: On the estimation of optical flow: relations between different approaches and some new results. AI 33, 299–324 (1987)
Nagel, H.-H.: On a constraint equation for the estimation of displacement rates in image sequences. IEEE Trans. PAMI 11, 13–30 (1989)
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)
Papadakis, N., Mémin, E.: Variational assimilation of fluid motion from image sequence. SIAM J. Imaging Sci. 1(4), 343–363 (2008)
Mukawa, N.: Estimation of shape, reflection coefficients and illuminant direction from image sequences. In: ICCV90, pp. 507–512 (1990)
Corpetti, T., Mémin, E., Pérez, P.: Dense estimation of fluid flows. IEEE Trans. Pattern Anal. Mach. Intell. 24(3), 365380 (2002)
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)
Arnaud, E., Mémin, E., Sosa, R., Artana, G.: A fluid motion estimator for schlieren image velocimetry. In: ECCV06, I, pp. 198–210 (2006)
Wayne Roberts, A., Varberg, Dale E.: Convex Functions. Academic Press, New York (1973)
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)
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)
Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)
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)
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)
Blanc, V.L.: \(L^1\)-stability of periodic stationary solutions of scalar convection-diffusion equations. J. Differ. Equ. 247, 1746–1761 (2009)
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)
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)
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Roy, S., Chandrashekar, P. & Murthy, A.S.V. A variational approach to optical flow estimation of unsteady incompressible flows. Int J Adv Eng Sci Appl Math 7, 149–167 (2015). https://doi.org/10.1007/s12572-015-0147-9
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DOI: https://doi.org/10.1007/s12572-015-0147-9
Keywords
- Euler
- Navier–Stokes
- Vorticity
- Streamfunction
- Discontinuous Galerkin
- Optimization
- Linearization
- Helmholtz decomposition
- Incompressible