Nicolas Papadakis

Optimal Transport with Proximal Splitting

This article reviews the use of first order convex optimization schemes to solve the discretized dynamic optimal transport problem, initially proposed by Benamou and Brenier. We develop a staggered grid discretization that is well adapted to the computation of the

Data assimilation with the weighted ensemble Kalman filter

L^2

Tracking with Occlusions via Graph Cuts

optimal transport geodesic between distributions defined on a uniform spatial grid. We show how proximal splitting schemes can be used to solve the resulting large scale convex optimization problem. A specific instantiation of this method on a centered grid corresponds to the initial algorithm developed by Benamou and Brenier. We also show how more general cost functions can be taken into account and how to extend the method to perform optimal transport on a Riemannian manifold.

Layered Estimation of Atmospheric Mesoscale Dynamics From Satellite Imagery

Abstract In this paper, two data assimilation methods based on sequential Monte Carlo sampling are studied and compared: the ensemble Kalman filter and the particle filter. Each of these techniques has its own advantages and drawbacks. In this work, we try to get the best of each method by combining them. The proposed algorithm, called the weighted ensemble Kalman filter, consists to rely on the EnsembleKalman Filter updates of samples in order to define a proposal distribution for the particle filter that depends on the history of measurement. The corresponding particle filter reveals to be efficient with a small number of samples and does not rely anymore on the Gaussian approximations of the ensemble Kalman filter. The efficiency of the new algorithm is demonstrated both in terms of accuracy and computational load. This latter aspect is of the utmost importance in meteorology or in oceanography since in these domains, data assimilation processes involve a huge number of state variables driven by highly non-linear dynamical models. Numerical experiments have been performed on different dynamical scenarios. The performances of the proposed technique have been compared to the ensemble Kalman filter procedure, which has demonstrated to provide meaningful results in geophysical sciences.

A Variational Model for Histogram Transfer of Color Images

This work presents a new method for tracking and segmenting along time-interacting objects within an image sequence. One major contribution of the paper is the formalization of the notion of visible and occluded parts. For each object, we aim at tracking these two parts. Assuming that the velocity of each object is driven by a dynamical law, predictions can be used to guide the successive estimations. Separating these predicted areas into good and bad parts with respect to the final segmentation and representing the objects with their visible and occluded parts permit handling partial and complete occlusions. To achieve this tracking, a label is assigned to each object and an energy function representing the multilabel problem is minimized via a graph cuts optimization. This energy contains terms based on image intensities which enable segmenting and regularizing the visible parts of the objects. It also includes terms dedicated to the management of the occluded and disappearing areas, which are defined on the areas of prediction of the objects. The results on several challenging sequences prove the strength of the proposed approach.

Variational Assimilation of Fluid Motion from Image Sequence

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.

Dynamically consistent optical flow estimation

In this paper, we propose a variational formulation for histogram transfer of two or more color images. We study an energy functional composed by three terms: one tends to approach the cumulative histograms of the transformed images, the other two tend to maintain the colors and geometry of the original images. By minimizing this energy, we obtain an algorithm that balances equalization and the conservation of features of the original images. As a result, they evolve while approaching an intermediate histogram between them. This intermediate histogram does not need to be specified in advance, but it is a natural result of the model. Finally, we provide experiments showing that the proposed method compares well with the state of the art.

An Optimized PatchMatch for multi-scale and multi-feature label fusion

In this paper, a variational technique derived from optimal control theory is used in order to realize a dynamically consistent motion estimation of a whole fluid image sequence. The estimation is conducted through an iterative process involving a forward integration of a given dynamical model followed by a backward integration of an adjoint evolution law. By combining physical conservation laws and image observations, a physically grounded temporal consistency is imposed, and the quality of the motion estimation is significantly improved. The method is validated on two synthetic image sequences provided by numerical simulation of fluid flows and on real world meteorological examples.

A Variational Technique for Time Consistent Tracking of Curves and Motion

In this paper, we present a framework for dynamic consistent estimation of dense motion fields over a sequence of images. The originality of the approach is to exploit recipes related to optimal control theory. This setup allows performing the estimation of an unknown state function according to a given dynamical model and to noisy and incomplete measurements. The overall process is formalized through the minimization of a global spatio-temporal cost functional w.r.t the complete sequence of motion fields. The minimization is handled considering an adjoint formulation. The resulting scheme consists in iterating a forward integration of the evolution model and a backward integration of the adjoint evolution model guided by a discrepancy measurement between the state variable and the available noisy observations. Such an approach allows us to cope with several delicate situations (such as the absence of data) which are not well managed with usual estimators.

Variational Exemplar-Based Image Colorization

Automatic segmentation methods are important tools for quantitative analysis of Magnetic Resonance Images (MRI). Recently, patch-based label fusion approaches have demonstrated state-of-the-art segmentation accuracy. In this paper, we introduce a new patch-based label fusion framework to perform segmentation of anatomical structures. The proposed approach uses an Optimized PAtchMatch Label fusion (OPAL) strategy that drastically reduces the computation time required for the search of similar patches. The reduced computation time of OPAL opens the way for new strategies and facilitates processing on large databases. In this paper, we investigate new perspectives offered by OPAL, by introducing a new multi-scale and multi-feature framework. During our validation on hippocampus segmentation we use two datasets: young adults in the ICBM cohort and elderly adults in the EADC-ADNI dataset. For both, OPAL is compared to state-of-the-art methods. Results show that OPAL obtained the highest median Dice coefficient (89.9% for ICBM and 90.1% for EADC-ADNI). Moreover, in both cases, OPAL produced a segmentation accuracy similar to inter-expert variability. On the EADC-ADNI dataset, we compare the hippocampal volumes obtained by manual and automatic segmentation. The volumes appear to be highly correlated that enables to perform more accurate separation of pathological populations.