Reduced modeling in high-dimensional reproducing kernel Hilbert spaces offers the opportunity to approximate efficiently non-linear dynamics. In this work, we devise an algorithm based on low rank constraint optimization and kernel-based computation that generalizes a recent approach called "kernel-based dynamic mode decomposition". This new algorithm is characterized by a gain in approximation accuracy, as evidenced by numerical simulations, and in computational complexity.

Dynamic Mode Decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of non-linear systems from experimental datasets. Recently, several attempts have extended DMD to the context of low-rank approximations. This extension is of particular interest for reduced-order modeling in various applicative domains, e.g., for climate prediction, to study molecular dynamics or microelectromechanical devices. This low-rank extension takes the form of a non-convex optimization problem. To the best of our knowledge, only sub-optimal algorithms have been proposed in the literature to compute the solution of this problem. In this paper, we prove that there exists a closed-form optimal solution to this problem and design an effective algorithm to compute it based on Singular Value Decomposition (SVD). A toy-example illustrates the gain in performance of the proposed algorithm compared to state-of-the-art techniques.

The objective of this paper is to investigate how noisy and incomplete observations can be integrated in the process of building a reduced-order model. This problematic arises in many scientific domains where there exists a need for accurate low-order descriptions of highly-complex phenomena, which can not be directly and/or deterministically observed. Within this context, the paper proposes a probabilistic framework for the construction of "POD-Galerkin" reduced-order models. Assuming a hidden Markov chain, the inference integrates the uncertainty of the hidden states relying on their posterior distribution. Simulations show the benefits obtained by exploiting the proposed framework.

In this work, we propose a novel procedure for video superresolution, that is, the recovery of a sequence of high-resolution images from its low-resolution counterpart. Our approach is based on a “sequential” model (i.e., each high-resolution frame is supposed to be a displaced version of the preceding one) and considers the use of sparsity-enforcing priors. Both the recovery of the high-resolution images and the motion fields relating them is tackled. This leads to a large-dimensional, nonconvex and nonsmooth problem. We propose an algorithmic framework to address the latter. Our approach relies on fast gradient evaluation methods and modern optimization techniques for nondifferentiable/nonconvex problems. Unlike some other previous works, we show that there exists a provably convergent method with a complexity linear in the problem dimensions. We assess the proposed optimization method on several video benchmarks and emphasize its good performance with respect to the state of the art.

Following recent contributions in non-linear sparse representations, this work focuses on a particular non-linear model, defined as the nested composition of functions. Recalling that most linear sparse representation algorithms can be straightforwardly extended to non-linear models, we emphasize that their performance highly relies on an efficient computation of the gradient of the objective function. In the particular case of interest, we propose to resort to a well-known technique from the theory of optimal control to evaluate the gradient. This computation is then implemented into the “ℓ1-reweighted” procedure proposed by Candès et al., leading to a non-linear extension of it.

This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergence-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.

Based on physical laws describing the multiscale structure of turbulent flows, this paper 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 hyperparameters, and to select the most likely physical prior among a set of models. Hyperparameter and model inference are conducted by posterior 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 exceed the state-of-the-art results in fluid flow estimation.

Selecting optimal models and hyperparameters is crucial for accurate optical-flow estimation. This paper provides a solution to the problem in a generic Bayesian framework. The method is based on a conditional model linking the image intensity function, the unknown velocity field, hyperparameters, and the prior and likelihood motion models. Inference is performed on each of the three levels of this so-defined hierarchical model by maximization of marginalized a posteriori probability distribution functions. In particular, the first level is used to achieve motion estimation in a classical a posteriori scheme. By marginalizing out the motion variable, the second level enables to infer regularization coefficients and hyperparameters of non-Gaussian M-estimators commonly used in robust statistics. The last level of the hierarchy is used for selection of the likelihood and prior motion models conditioned to the image data. The method is evaluated on image sequences of fluid flows and from the "Middlebury" database. Experiments prove that applying the proposed inference strategy yields better results than manually tuning smoothing parameters or discontinuity preserving cost functions of the state-of-the-art methods.

Based on scaling laws describing the statistical structure of turbulent motion across scales, we propose a multiscale and non-parametric regularizer for optic-flow estimation. Regularization is achieved by constraining motion increments to behave through scales as the most likely self-similar process given some image data. In a first level of inference, the hard constrained minimization problem is optimally solved by taking advantage of lagrangian duality. It results in a collection of first-order regularizers acting at different scales. This estimation is non-parametric since the optimal regularization parameters at the different scales are obtained by solving the dual problem. In a second level of inference, the most likely self-similar model given the data is optimally selected by maximization of Bayesian evidence. The motion estimator accuracy is first evaluated on a synthetic image sequence of simulated bi-dimensional turbulence and then on a real meteorological image sequence. Results obtained with the proposed physical based approach exceeds the best state of the art results. Furthermore, selecting from images the most evident multiscale motion model enables the recovery of physical quantities, which are of major interest for turbulence characterization.

In this paper, Bayesian inference is used to select the most evident Gibbs prior model for motion estimation given some image sequence. The proposed method supplements the maximuma posteriorimotion estimation scheme proposed in He¿as et al. (2008). Indeed, in this recent work, the authors have introduced a family of multiscale spatial priors in order to cure the ill-posed inverse motion estimation problem. We propose here a second level of inference where the most likely prior model is optimally chosen given the data by maximization of Bayesian evidence. Model selection and motion estimation are assessed on Meteorological Second Generation (MSG) image sequences. Selecting from images the most evident multiscale model enables the recovery of physical quantities which are of major interest for atmospheric turbulence characterization.

The complexity of dynamical laws governing 3D atmospheric flows associated with incomplete and noisy observations make the recovery of atmospheric dynamics from satellite images sequences very difficult. In this paper, we face the challenging problem of estimating physical sound and time-consistent horizontal motion fields at various atmospheric depths for a whole image sequence. Based on a vertical decomposition of the atmosphere, we propose a dynamically consistent atmospheric motion estimator relying on a multi-layer dynamical model. This estimator is based on a weak constraint variational data assimilation scheme and is applied on noisy and incomplete pressure difference observations derived from satellite images. The dynamical model consists in a simplified vorticity-divergence form of a multi-layer shallow-water model. Average horizontal motion fields are estimated for each layer. The performance of the proposed technique is assessed on real world meteorological satellite image sequences.

In this paper, we address the problem of estimating 3-D motions of a stratified atmosphere from satellite image sequences. The analysis of 3-D atmospheric fluid flows associated with incomplete observation of atmospheric layers due to the sparsity of cloud systems is very difficult. This makes the estimation of dense atmospheric motion field from satellite image sequences very difficult. The recovery of the vertical component of fluid motion from a monocular sequence of image observations is a very challenging problem for which no solution exists in the literature. Based on a physically sound vertical decomposition of the atmosphere into cloud layers of different altitudes, we propose here a dense motion estimator dedicated to the extraction of 3-D wind fields characterizing the dynamics of a layered atmosphere. Wind estimation is performed over the complete 3-D space, using a multilayer model describing a stack of dynamic horizontal layers of evolving thickness, interacting at their boundaries via vertical winds. The efficiency of our approach is demonstrated on synthetic and real sequences.

In this paper, we address the problem of estimating mesoscale dynamics of atmospheric layers from satellite image sequences. Due to the great deal of spatial and temporal distortions of cloud patterns and because of the sparse 3-D nature of cloud observations, standard dense-motion field-estimation techniques used in computer vision are not well adapted to satellite images. Relying on a physically sound vertical decomposition of the atmosphere into layers, we propose a dense-motion estimator dedicated to the extraction of multilayer horizontal wind fields. This estimator is expressed as the minimization of a global function including data and spatio-temporal smoothness terms. A robust data term relying on the integrated-continuity equation mass-conservation model is proposed to fit sparse-transmittance observations related to each layer. A novel spatio-temporal smoother derived from large eddy prediction of a shallow-water momentum-conservation model is used to build constraints for large-scale temporal coherence. These constraints are combined in a global smoothing framework with a robust second-order smoother, preserving divergent and vorticity structures of the flow. For optimization, a two-stage motion estimation scheme is proposed to overcome multiresolution limitations when capturing the dynamics of mesoscale structures. This alternative approach relies on the combination of correlation and optical-flow observations in a variational context. An exhaustive evaluation of the novel method is first performed on a scalar image sequence generated by direct numerical simulation of a turbulent 2-D flow. By qualitative comparisons, the method is then assessed on a METEOSAT image sequence.

During the last decades, satellites have acquired incessantly high-resolution images of many Earth observation sites. New products have arisen from this intensive acquisition process: high-resolution satellite image time-series (SITS). They represent a large data volume with a rich information content and may open a broad range of new applications. This paper presents an information mining concept which enables a user to learn and retrieve spatio-temporal structures in SITS. The concept is based on a hierarchical Bayesian modeling of SITS information content which enables us to link the interest of a user to specific spatio-temporal structures. The hierarchy is composed of two inference steps: an unsupervised modeling of dynamic clusters resulting in a graph of trajectories, and an interactive learning procedure based on graphs which leads to the semantic labeling of spatio-temporal structures. Experiments performed on a SPOT image time-series demonstrate the concept capabilities.

A visual information mining concept is proposed for spatio-temporal patterns discovery in remotely sensed image time-series (ITS). An information theory framework is adopted to first model information content. It results in the inference of a relevant directed graph characterizing ITS. Then the user conjecture is modeled via visual information representations: similarity measures between sub-graphs, which represents spatio-temporal events are derived and included in an interactive learning and probabilistic retrieval procedure of user-specific event-types. The present concept for ITS mining is demonstrated on multitemporal SPOT data.

In this paper, a dynamic scene understanding concept is proposed and applied on multispectral image time series. Information mining enables the explorations and discovery of spatio temporal patterns localized in given spatio temporal windows. With this in mind, a hierarchical information representation is developed. It comprises different levels in which the data is modeled so that the relevant information is transmitted through the architecture, according to a query and to several assumptions made on the models employed. There are mainly four components: feature extraction, reduction of dimensionality, clustering and interactive exploration.