**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 a very good performance of the proposed formulation for turbulent flow motion estimation.

#### Approach two: Deep PIV

**Introduction**

A fluid motion estimation algorithm based on deep neural networks is proposed. With the development of deep learning, it is possible to solve the problem of fluid image velocimetry by using convolutional neural network (CNN). The deep learning technology is innovatively applied to the PIV experiment. Specifically, two PIV neural networks are proposed based on FlowNetS and LiteFlowNet, respectively, which are used for optical flow estimation. The input of the networks is a particle image pair and the output is a global velocity field. In addition, a PIV data set is artificially generated for CNN training, which takes into account the physical properties and the image noise. The proposed CNN models are verified by a number of assessments and in real PIV experiments such as turbulent boundary layer. Without loss of precision, the computational efficiency is greatly improved compared with the variational optical flow method. This advantage provides possibility for real-time flow measurement and control.

**Paper**

1. S. Cai, S. Zhou, C. Xu, Q. Gao. Dense motion estimation of particle images via a convolutional neural network. Experiments in Fluids, 60: 73, 2019.

2. S. Cai, J. Liang, Q. Gao, C. Xu, R. Wei. Particle Image Velocimetry Based on a Deep Learning Motion Estimator, IEEE Transactions on Instrumentation and Measurement, PP(99):1-1, 2019.

3. S. Cai, J. Liang, S. Zhou, et al. Deep-PIV: a new framework of PIV using deep learning techniques, International Symposium on Particle Image Velocimetry. Munich, Germany, 2019.

**Patent**

1. C. Xu, S. Cai, Q. Gao, S.Zhou, One particle image velocimetry method based on convolutional neural network. Patent. Public.