## Section: New Results

### Fluid motion estimation

#### Multiscale PIV method based on turbulent kinetic energy decay

Participants : Patrick Héas, Dominique Heitz, Etienne Mémin.

We have proposed a new multiscale PIV method based on turbulent kinetic energy decay. The technique is based on scaling power laws describing the statistical structure of turbulence. A spatial regularization constraints the solution to behave through scales as a self similar process via second-order structure function and a given power law. The real parameters of the power-law, corresponding to the distribution of the turbulent kinetic energy decay, have been estimated from a simple hot-wire measurement. The method has been assessed in a turbulent wake flow and grid turbulence through comparisons with HWA measurements and other PIV approaches. Results have indicated that the present method is superior because it accounts for the whole dynamic range involved in the flows.

#### Stochastic uncertainty models for motion estimation

Participants : Thomas Corpetti, Etienne Mémin.

In this work 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 enables to built then 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 in the same order of accuracy than state-of-the-art dense fluid flow motion estimator for particle images. This work has been published in a conference proceeding and has been accepted for publication in the journal *Ieee trans. on image processing* [23] , [16] . We intend to pursue this formalization to define dense motion estimators that allows handling in the same way luminance conservation under motion uncertainty principles.

#### 3D flows reconstruction from image data

Participants : Ioana Barbu, Dominique Heitz, Cédric Herzet, Etienne Mémin.

Our work focusses 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.

This year, we have more particularly focussed 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 [21] , [34] .

#### 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 article proposes a regularizer for fluid motion estimation from an image sequence. Regularization 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 work has been published in several conferences and in the journal Tellus, Serie A [18] .

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

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

This work aims at exploring wavelet representations for fluid motion estimation from consecutive images. This scale-space representation, associated to a simple gradient-based optimization algorithm, sets up a natural multi-resolution framework for the optical flow estimation well suited to medium range velocity magnitude. Moreover, a very simple closure mechanism, approaching locally the solution by high-order polynomials, is provided by truncating the wavelet basis at fine scales. Well-known turbulence regularities and multifractal behaviors on the reconstructed motion field can also be imposed on the wavelet coefficients. Accuracy and efficiency of the proposed method has been evaluated on scalar and particles image sequences of turbulent fluid flows. Particularly good results have been observed for particle image velocimetry. This offers a very interresting alternative to traditional PIV techniques. This work has been published in computer vision or turbulence conferences [26] , [25] .

#### Divergence-free wavelet basis and high-order regularization

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

Expanding on a wavelet basis the solution of an inverse problem provides several advantages. Wavelet bases yield a natural multiresolution analysis that may alleviate the use of Gauss Newton strategy for medium range motion amplitude. The continuous representation of the solution with wavelets enables analytical calculation of regularization integrals over the spatial domain. By choosing differentiable wavelets, high-order derivative regularizers can be designed, either taking advantage of the wavelet differentiation properties or via the basis's mass and stiffness matrices. Moreover, differential constraints on vector solutions, such as the divergence-free volume preserving constraint, can be handled with biorthogonal wavelet bases. Numerical results on synthetic and real images of incompressible turbulence show that divergence-free wavelets and high-order regularizers are particularly relevant in the context of incompressible fluid flows. This work has been partly published in a conference proceeding [25] .

#### Divergence-free wavelet basis and high-order regularization

Participants : Pierre Dérian, Patrick Héas, Souleymane Kadri Harouna.

This works presents a method for regularization of inverse problems. The vectorial bi-dimensional unknown is assumed to be the realization of an isotropic divergence-free fractional Brownian Motion (fBm). The method is based on fractional Laplacian and divergence-free wavelet bases. The main advantage of these bases is to enable an easy formalization in a Bayesian framework of fBm priors, by simply sampling wavelet coefficients according to Gaussian white noise. Fractional Laplacians and the divergence-free projector can naturally be implemented in the Fourier domain. An interesting alternative is to remain in the spatial domain. This is achieved by the analytical computation of the connection coefficients of divergence-free fractional Laplacian wavelets, which enables to easily rotate this simple prior in any sufficiently “regular” wavelet basis. Taking advantage of the tensorial structure of a separable fractional wavelet basis approximation, isotropic regularization is then computed in the spatial domain by low-dimensional matrix products. The method is successfully applied to fractal image restoration and turbulent optic-flow estimation.

#### Bayesian inference of hyper-parameters and models in motion estimation

Participants : Patrick Héas, Cédric Herzet, Etienne Mémin.

Bayes rule provides a nice framework for motion estimation from image sequences. We rely on a hierarchical modeling linking the image intensity function variable, the motion field variable, hyper-parameters composed of the likelihood and prior model inverse variances and of robust parameters, and finally the observation and prior model. The variable dependence can thus be expressed as a 4-level hierarchy. Applying the Bayes rule on this hierarchy, we obtain three levels of inference, which enable us to obtain, by marginalizing out intermediate variables, a direct dependence of the variable of interest to the image intensity function. Thus, the estimates of regularization parameters, of robust parameters associated to semi-quadratic norms of a family of M-estimators, and of observation and prior models are inferred in a maximum likelihood sense while maximizing jointly the motion field a posteriori probability. The quality of the method is demonstrated on synthetic and real two-dimensional turbulent flows and on several computer vision scenes of the "Middlebury" data-base. This work has been accepted for publication in IEEE transaction on Image Processing (IP) [17] .

#### Method to quantify the uncertainty of motion measurement

Participants : Patrick Héas, Dominique Heitz, Cédric Herzet.

Measurement uncertainty is a general concept associated with any measurement that can be used to quantify the confidence of the estimation. The `Guide to the expression of uncertainty in measurement' (GUM) provides a framework to account for all uncertainties and then to propagate them. However, in particle image velocimetry (PIV), measurement uncertainty estimation is a tricky task since it has to be done through the propagation of distributions via Monte Carlo simulations for each velocity components and all pixel location in the image. Considering a standard Bayesian formulation of the optical flow problem together with a Gaussian assumption the uncertainty associated to the estimated velocity field has been provided and described. First PIV measurement uncertainty estimations and discussions have been recently published in a conference issue [36] .

#### Sparse-representation algorithms

Participant : Cédric Herzet.

We have pursued the study of efficient sparse decomposition algorithms. In particular, we have addressed the problem of finding good sparse representations into a probabilistic framework [24] . First, we have showed that one of the standard formulations - the Lagrangian formulation - of this problem can be interpreted as a limit case of a maximum a posteriori (MAP) problem involving Bernoulli-Gaussian variables. Then, we have proposed different tractable implementations of this MAP problem and explained some well-known pursuit algorithms (MP, OMP, StOMP, CoSaMP and SP) as particular cases of the proposed algorithms. Experimentations led on synthetic data show a good general behavior of the proposed methods.

Exploiting further this probabilistic framework, we have then considered the design of *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.