Abstract
Optical flow is a classical problem in computer vision, but the concepts must be adapted for applications to other fields such as fluid mechanics and dynamical systems. Our approaches are based on an inverse problem formalism, considering imposed scientific priors in the form of a cost function that rewards an assumed infinitesimal generator commensurate with assumed physics of the observed density evolution. This leads to a practical and principled approach to analyze an observed dynamical system. Additionally we present here for the first time a new multi-frame version of the functional coupling of multiple images. Following the calculus of variations, this yields a coupled set of Euler–Lagrange PDEs which serve as an assimilation method that inputs video frames as driving terms. The solution of the PDE which follows is the vector field, as designed. Data from an oceanographic system will be highlighted. It is also shown here how these flow fields can be used to analyze mixing and mass transport in the fluid system being imaged.
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References
Basnayake, R., Luttman, A., Bollt, E.: A lagged diffusivity method for computing total variation regularized fluid flow. Contemp. Math. 586, 59–66 (2013)
Bollt, E., Luttman, A., Kramer, S., Basnayake, R.: Measurable dynamics analysis of transport in the Gulf of Mexico during the oil spill. Int. J. Bifurcat. Chaos 22(03), 1230012 (2012)
Brox, T., Bruhn, A., Papenberg, N., Weickert, J.: High accuracy optical flow estimation based on a theory for warping. In: Pajdla, T., Matas, J. (eds.) Proceedings of 8th European Conference on Computer Vision, vol. 4, pp. 25–36 (2004)
Corpetti, T., Mémin, E., Pérez, P.: Adaptation of standard optic flow methods to fluid motion. In: 9th International Symposium on Flow Visualization, pp. 1–10 (2000)
Froyland, G., Santitissadeekorn, N., Monahan, A.: Transport in time-dependent dynamical systems: finite-time coherent sets. Preprint arXiv:1008.1613 (2010)
Gelfand, I., Fomin, S.: Calculus of Variations. Dover Publications, Mineola (2000)
Haller, G.: Lagrangian coherent structures from approximate velocity data. Phys. Fluid. 14, 1851 (2002)
Haller, G., Poje, A.: Finite time transport in aperiodic flows. Phys. D Nonlinear Phenom. 119(3–4), 352–380 (1998)
Hansen, P.C.: The L-curve and its use in the numerical treatment of inverse problems. In: Johnston, P. (ed.) Computational Inverse Problems in Electrocardiology. Advances in Computational Bioengineering, pp. 119–142. WIT Press, Southampton (2000)
Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artif. Intell. 17, 185–203 (1981)
Kabanikhin, S.I.: Inverse and Ill-Posed Problems: Theory and Applications, vol. 55. De Gruyter, Germany (2011)
Krawczyk-StańDo, D., Rudnicki, M.: Regularization parameter selection in discrete ill-posed problems: the use of the U-curve. Int. J. Appl. Math. Comput. Sci. 17(2), 157–164 (2007)
Krawczyk-Stańdo, D., Rudnicki, M.: The use of L-curve and U-curve in inverse electromagnetic modelling. Intell. Comput. Tech. Appl. Electromagn. 119, 73–82 (2008)
Luttman, A., Bollt, E., Basnayake, R., Kramer, S.: A stream function approach to optical flow with applications to fluid transport dynamics. Proc. Appl. Math. Mech. 11(1), 855–856 (2011)
McCane, B., Novins, K., Crannitch, D., Galvin, B.: On benchmarking optical flow. Comput. Vis. Image Understand. 84, 126–143 (2001). doi:10.1006/cviu.2001.0930
Office of Satellite Operation. http://www.oso.noaa.gov/goes/ (2011)
Schmidt, M., Fung, G., Rosales, R.: Fast optimization methods for l1 regularization: a comparative study and two new approaches. Machine Learning: ECML 2007, pp. 286–297 (2007)
Shadden, S., Lekien, F., Marsden, J.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys. D Nonlinear Phenom. 212(3), 271–304 (2005)
Vogel, C.R.: Computational Methods for Inverse Problems. Frontiers in Mathematics. SIAM, Philadelphia (2002)
Weickert, J., Bruhn, A., Papenberg, N., Brox, T.: Variational optic flow computation: from continuous models to algorithms. In: Alvarez, L. (ed.) International Workshop on Computer Vision and International Workshop on Computer Vision and Image Analysis, IWCVIAÃ’03, Las Palmas de Gran Canaria (2003)
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Basnayake, R., Bollt, E.M. (2014). A Multi-time Step Method to Compute Optical Flow with Scientific Priors for Observations of a Fluidic System. In: Bahsoun, W., Bose, C., Froyland, G. (eds) Ergodic Theory, Open Dynamics, and Coherent Structures. Springer Proceedings in Mathematics & Statistics, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0419-8_4
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DOI: https://doi.org/10.1007/978-1-4939-0419-8_4
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