## Section: New Results

### Fluid motion estimation

#### Stochastic uncertainty models for motion estimation

Participants : Thomas Corpetti, Etienne Mémin.

In this study we have proposed a stochastic formulation of the brightness consistency used principally in motion estimation problems. In this formalization the image luminance is modeled as a continuous function transported by a flow known only up to some uncertainties.
Stochastic calculus then enables to built conservation principles which take into account the motion uncertainties. These uncertainties defined either from isotropic or anisotropic models can be estimated jointly to the motion estimates. Such a formulation besides providing estimates of the velocity field and of its associated uncertainties allows us to define a natural linear scale space multiresolution framework.
The corresponding estimator, implemented within a local least squares approach, has shown to improve significantly the results of the corresponding deterministic estimator (Lucas and Kanade estimator). This fast local motion estimator has been shown to provide results that are of the same order of accuracy than state-of-the-art dense fluid flow motion estimator for particle images. The uncertainties estimated provide a useful piece of information in the context of data assimilation. This ability has been exploited to define multiscale data assimilation filtering schemes. These works have been recently published in *IEEE trans. on Image Processing* and in Numerical Mathematics: Theory, Methods and Applications [16] , [18] . We intend to pursue this formalization to define dense motion estimators that allow us handling, in the same way, luminance conservation under motion uncertainty principles. An efficient GP-GPU implementation of the local estimator is also targeted.

#### 3D flows reconstruction from image data

Participants : Ioana Barbu, Cédric Herzet, Etienne Mémin.

Our work focuses on the design of new tools for the problem of 3D reconstruction of a turbulent flow motion. This task includes both the study of physically-sound models on the observations and the fluid motion, and the design of low-complexity and accurate estimation algorithms. On the one hand, state-of-the-art methodologies such as "sparse representations" will be investigated for the characterization of the observation and fluid motion models. Sparse representations are well-suited to the representation of signals with very few coefficients and offer therefore advantages in terms of computational and storage complexity. On the other hand, the estimation problem will be placed into a probabilistic Bayesian framework. This will allow the use of state-of-the-art inference tools to effectively exploit the strong time-dependence of the fluid motion. In particular, we will investigate the use of "ensemble Kalman" filter to devise low-complexity sequential estimation algorithms.

At the beginning of Ioana Barbu's PhD, we concentrated our efforts on the problem of reconstructing the particle positions from several two-dimensional images. Our approach is based on the exploitation of a particular family of sparse representation algorithms, namely the so-called "pursuit algorithms". Indeed, the pursuit procedures generally allow a good trade-off between performance and complexity. Hence, we have performed a thorough study comparing the reconstruction performance and the complexity of different state-of-the-art algorithms to that achieved with pursuit algorithms. This work has led to the publication of two conference papers in experimental fluid mechanics.

This year, our work has focused on: i) the estimation of the 3D velocity field of the fluid flow from the reconstructed volumes of particles; ii) the design of new methodologies allowing to jointly estimate the volume of particles and the velocity field from the received image data. More particularly, we have implemented a motion estimator generalizing the local Lucas-Kanade's procedure to a 3D problem. A potential strength of the proposed approach is the possibility to consider a fully parallel (and therefore very fast) implementation. On the other hand, we have started investigating the problem of jointly estimating the volumes of particles and the velocity field. Our approach is based on the combination of sparse representation algorithms and "Lucas-Kanade"-like motion estimation methods. We are about testing the proposed approach on experimental data in order to assess its performance in practical scenarios of fluid mechanics. We also intend to collaborate with the group of Fulvio Scarano at TU Delft to assess and compare our method on experimental 3D data.

#### Motion estimation techniques for turbulent fluid flows

Participants : Patrick Héas, Dominique Heitz, Cédric Herzet, Etienne Mémin.

Based on physical laws describing the multi-scale structure of turbulent flows, this study concerns the proposition of smoothing functional for the estimation of homogeneous turbulent flow velocity fields from an image sequence. This smoothing is achieved by imposing some scale invariance property between histograms of motion increments computed at different scales. By reformulating this problem from a Bayesian perspective, an algorithm is proposed to jointly estimate motion, regularization hyper-parameters, and to select the most likely physical prior among a set of models. Hyper-parameter and model inference is conducted by likelihood maximization, obtained by marginalizing out non-Gaussian motion variables. The Bayesian estimator is assessed on several image sequences depicting synthetic and real turbulent fluid flows. Results obtained with the proposed approach in the context of fully developped turbulence improve significantly the results of state of the art fluid flow dedicated motion estimators. This series of works, which have been done in close collaboration with P. Minnini (University of Buenos Aires), have been published in several journals [21] , [22] , [23] .

#### Wavelet basis for multi-scale motion estimation

Participants : Pierre Dérian, Patrick Héas, Cédric Herzet, Souleymane Kadri Harouna, Etienne Mémin.

This work describes the implementation of a simple wavelet-based optical-flow motion estimator dedicated to the recovery of fluid motion. The wavelet representation of the unknown velocity

field is considered. This scale-space representation, associated to a simple gradient-based optimization algorithm, sets up a natural multiscale/multigrid optimization framework for the optical flow estimation that can be combined to more traditional incremental multiresolution approaches. Moreover, a very simple closure mechanism, approximating locally the solution by high-order polynomials, is provided by truncating the wavelet basis at intermediate scales. This offers a very interesting alternative to traditional Particle Image Velocimetry techniques. As another alternative to this medium-scale estimator, we explored strategies to define estimation at finer scales. These strategies rely on the encoding of high-order smoothing functional on appropriate wavelet basis. Divergence-free bi-othogonal wavelet bases enable to further nicely enforce volume preserving motion field. Numerical results on several examples have demonstrated the relevance of the method for divergence free-2D flows. These studies have been published in the journal of Numerical Mathematics: Theory, Methods and Applications [19] and in the journal of Computer Vision [24] . The extension to 3D flows would be an interesting perspective.

#### Wavelet-based divergence-free fBm prior: application to turbulent flow estimation

Participant : Patrick Héas.

This work is concerned with the estimation of turbulent flows from the observation of an image sequence. From a Bayesian perspective, we propose to study divergence-free isotropic fractional Brownian motion (fBm) as a prior model for instantaneous turbulent velocity fields. These priors are self-similar stochastic processes, which characterize accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Although, these models belong to a well-identified family of rotation invariant regularizers, there is a lack of effective algorithms in the literature to deal in practice with their fractional nature. To respond to this problem, we propose to decompose fBms on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters for divergence-free isotropic fBms, which are correlated both in space and scale. The second alternative is to use a divergence-free wavelet basis, which will take implicitly into account the divergence-free constraint arising form the physics.

#### Sparse-representation algorithms

Participant : Cédric Herzet.

The paradigm of sparse representations is a rather new concept which appears to be central in many field of signal processing. In particular, in the field of fluid motion estimation, sparse representation appears to be potentially useful at several levels: i) it provides a relevant model for the characterization of the velocity field in some scenarios; ii) it turns out to be central in the recovering of volumes of particles in the 3D Tomo-PIV problem. In these contexts, the dimensionality of the problem can be very large and the use of sparse-representation algorithms allowing for a good trade-off between complexity and effectiveness is needed.

This year, we have therefore pursued our study of efficient sparse decomposition algorithms. In particular, we have extended our work addressing the problem of finding good sparse representations into a probabilistic framework. First, we have proposed a new family of pursuit algorithms able to take into account any type of dependence (e.g. spatial or temporal) between the atoms of the sparse decomposition. This work has led to the publication of a paper in the proceedings of the international conference IEEE ICASSP 2012.

Exploiting further this probabilistic framework, we have then considered the design of structured soft pursuit algorithms. In particular, instead of making hard decisions on the support of the sparse representation and the amplitude of the non-zero coefficients, our soft procedures iteratively update probability on the latter values. The proposed algorithms are designed within the framework of the mean-field approximations and resort to the so-called variational Bayes EM algorithm to implement an efficient minimization of a Kullback-Leibler criterion. On the other hand, the proposed methodologies can handle "structured" sparse representations, that is, sparse decompositions where some dependence exists between the non-zero elements of the support. The prior model on the support of the sparse decomposition is based on a Boltzmann machine which encompasses as particular cases many type of dependence (Markov chain, Ising model, tree-like structure, etc). This work has been published in the journal IEEE Trans. on Signal Processing in 2012.