**Approach one: Variational optical flow**

**Introduction**

In this project, 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. 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 (2014) and Resseguier et al. (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.