Section: New Results

Fluid motion estimation

Stochastic uncertainty models for motion estimation

Participants : Etienne Mémin, Manuel Saunier, Abed Malti.

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. 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.

Last year, we 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 was 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 year, we revisited the problem of volume reconstruction through the prism of some modern optimization techniques. More specifically, we focussed our attention on the family of proximal and splitting methods and showed that the standard techniques commonly adopted in the TomoPIV literature can be seen as particular cases of such methodologies. Recasting standard methodologies in a more general framework allowed us to propose extensions of the latter: i) we showed that the parcimony characterizing the sought volume can be accounted for without increasing the complexity of the algorithms (e.g., by including simple thresholding operations); ii) we emphasized that the speed of convergence of the standard reconstruction algorithms can be improved by using Nesterov's acceleration schemes; iii) we also proposed a totally novel way of reconstructing the volume by using the so-called “alternating direction of multipliers method" (ADMM) . The journal publications relative to the contributions developped this year are currently in construction.

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 access to the solution of this problem. Among the popular methods available in the literature, one can mention orthogonal matching pursuit (OMP), orthogonal least squares (OLS) 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.

Last, we 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. Another information of interest which can help the algorithms in their reconstruction process is the decay of the amplitude of the nonzero coeffcient in the sparse vector. In a TomoPIV context, this decay corresponds to the fact that not all the particles in fluid diffuse the same quantity of light (notably beacuse of illumination or radius variation). This year, we thus pursue our effort in the understanding of the success of some reconstruction algorithms when the sparse vectors obey some decay. In particular, we showed that the standard coherence-based guarantees for OMP/OLS can be relaxed by an amount which depends on the decay of the nonzero coeffcients.

Another axis of research we have dealt with is the extension of sparse methodologies to the context of nonlinear models. This type of situtation is indeed frequently encountered in fluid mechanics or geophysics where the initial/boundary conditions of a system are known to be sparse in some basis and the collected observations obey a nonlinear dynamical model (e.g., the Navier-Stokes equations). In our work, we showed that many sparse representation algorithms, designed in the linear paradigm, can be nicely extended to the nonlinear setup provided that the gradient of the functional can be evaluated efficiently. In order to do so, we suggested a methodology, well-known in the commmunity of optimal control, but surprinsingly quite uncommon in many fields of signal processing.

Our work have led to the publication of contributions in the IEEE International Conference on Speech, Acoustic and Signal Processing (ICASSP) [23] and international - Traveling Workshop on Interactions between Sparse models and Technology (iTwist) [22] ,[24]