## Section: New Results

### Fluid motion estimation

#### Stochastic uncertainty models for motion estimation

Participants : Sébastien Beyou, Etienne Mémin, Emmanuel Saunier.

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 naturally define a linear multiresolution scale-space 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 provides 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 supply a useful piece of information in the context of data assimilation. This ability has been exploited to define multiscale incremental data assimilation filtering schemes. This work has been recently published in Numerical Mathematics: Theory, Methods and Applications [14] . It is also described in Sébastien Beyou's PhD dissertation [11] . The development of an efficient GPU based version of this estimator recently started through the Inria ADT project FLUMILAB

#### 3D flows reconstruction from image data

Participants : Ioana Barbu, Kai Berger, Cédric Herzet, Etienne Mémin.

Our work focuses on the design of new tools for the estimation of 3D turbulent flow motion in the experimental setup of Tomo-PIV. 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, we investigate state-of-the-art methodologies such as “sparse representations" for the characterization of the observation and fluid motion models. On the other hand, we place the estimation problem into a probabilistic Bayesian framework and use state-of- the-art inference tools to effectively exploit the strong time-dependence on the fluid motion. In our previous work, we have focussed on the problem of reconstructing the particle positions from several two-dimensional images. Our approach was based on the exploitation of a particular family of sparse representation algorithms, leading to a good trade-off between performance and complexity. Moreover, we also tackled the problem of estimating the 3D velocity field of the fluid flow from two instances of reconstructed volumes of particles. Our approach was based on a generalization of the well-known Lucas-Kanade's motion estimator to 3D problems. A potential strength of the proposed approach is the possibility to consider a fully parallelized (and therefore very fast) hardware implementation. This year, we have focused on the design of new methodologies to jointly estimate the volume of particles and the velocity field from the received image data. Our approach is based on the minimization (with respect to both the position of the particles and the velocity field) of a cost function penalizing both the discrepancies with respect to a conservation equation and some prior estimates of particle positions. This work has led to one publication in an international conference (PIV13) [27] and one publication in a national conference (Fluvisu13) [31] .

Since October 2013, with our new postdoctoral fellow Kai Berger, we have started a new direction of research targeting the volume reconstruction problem. In particular, we address the question of devising effective reconstruction procedures taking into account the limited computational budget available in practice. Our approach is based on the design of simple thresholding operators, allowing to reduce the dimension of the initial problem and amenable to fast parallel implementations.

#### Motion estimation techniques for turbulent fluid flows

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

In this study we have devised smoothing functional adapted to the multiscale structure of homogeneous turbulent flows. These regularization constraints ensue from a classical phenomenological description of turbulence. The smoothing is in practice achieved by imposing some scale-invariance principles between histograms of motion increments computed at different scales. Relying on a Bayesian formulation, an inference technique, based on likelihood maximization and marginalization of the motion variable, has been proposed to jointly estimate the fluid motion, the regularization parameters and a proper physical models. The performance of the proposed Bayesian estimator has been assessed on several image sequences depicting synthetic and real turbulent fluid flows. The results obtained in the context of fully developed turbulence show that an improvement in terms of small-scale motion estimation can be achieved as compared to classical motion estimator. This work, performed within a collaboration with Pablo Mininni from the university of Buenos Aires, have been published in the IEEE Transactions on Pattern Analysis And Machine Learning [22] .

#### Wavelet basis for multiscale motion estimation

Participants : Patrick Héas, Cédric Herzet, Etienne Mémin.

In this study we focused on the implementation of a simple wavelet-based optical-flow motion estimator dedicated to the recovery of fluid motions. 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 divergence free wavelet basis. This study has been published in the journal of Numerical Mathematics: Theory, Methods and Applications [19] and in the international Journal of Computer Vision [23] . This work has strongly benefited from a collaboration with Souleyman Kadri-Harouna (University of La Rochelle and who was formerly on a post-doctoral position in our team). The divergence free wavelets basis proposed in [24] constitutes the building blocks on which we have elaborated our wavelet based motion estimation solutions. We have otherwise pursue our collaboration with Chico university through the post-doc of Pierre Dérian on the GPU implementation of such motion estimator for Lidar data.

#### Sparse-representation algorithms

Participant : Cédric Herzet.

The paradigm of sparse representations is a rather new concept which turns out to be central in many domains 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 plays a crucial role in the recovery of volumes of particles in the 3D Tomo-PIV problem.

Unfortunately, the standard sparse representation problem is known to be NP hard. Therefore, heuristic procedures have to be devised to try and access to the solution of this problem. Among the popular methods available in the literature, one can mention orthogonal matching pursuit, orthogonal least squares and the family of procedures based on the minimization of ${\ell}_{p}$ norms. In order to assess and improve the performance of these algorithms, theoretical works have been undertaken in order to understand under which conditions these procedures can succeed in recovering the "true" sparse vector.

This year, we have contributed to this research axis by deriving conditions of success for the algorithms mentioned above when some partial information is available about the position of the nonzero coefficients in the sparse vector. This paradigm is of interest in the Tomographic-PIV volume reconstruction problem: one can indeed expect volumes of particles at two successive instants to be pretty similar; any estimate of the position of the particles at one given instant can therefore serve as a prior estimate about their position at the next instant. The conditions of success of such procedure have been rigorously formalized in two publications in the IEEE Transactions on Information Theory [21] , [26] and one publication in an international conference (SPARS13) [28] .