Anne Cuzol

Data assimilation with the weighted ensemble Kalman filter

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 Low Dimensional Fluid Motion Estimator

In this paper we propose a new motion estimator for image sequences depicting fluid flows. The proposed estimator is based on the Helmholtz decomposition of vector fields. This decomposition consists in representing the velocity field as a sum of a divergence free component and a vorticity free component. The objective is to provide a low-dimensional parametric representation of optical flows by depicting them as deformations generated by a reduced number of vortex and source particles. Both components are approximated using a discretization of the vorticity and divergence maps through regularized Dirac measures. The resulting so called irrotational and solenoidal fields consist of linear combinations of basis functions obtained through a convolution product of the Green kernel gradient and the vorticity map or the divergence map respectively. The coefficient values and the basis function parameters are obtained by minimization of a functional relying on an integrated version of mass conservation principle of fluid mechanics. Results are provided on synthetic examples and real world sequences.

A Stochastic Filtering Technique for Fluid Flow Velocity Fields Tracking

In this paper, we present a method for the temporal tracking of fluid flow velocity fields. The technique we propose is formalized within a sequential Bayesian filtering framework. The filtering model combines an Ito diffusion process coming from a stochastic formulation of the vorticity-velocity form of the Navier-Stokes equation and discrete measurements extracted from the image sequence. In order to handle a state space of reasonable dimension, the motion field is represented as a combination of adapted basis functions, derived from a discretization of the vorticity map of the fluid flow velocity field. The resulting nonlinear filtering problem is solved with the particle filter algorithm in continuous time. An adaptive dimensional reduction method is applied to the filtering technique, relying on dynamical systems theory. The efficiency of the tracking method is demonstrated on synthetic and real-world sequences.

Vortex and source particles for fluid motion estimation

In this paper we propose a new motion estimator for image sequences depicting fluid flows. The proposed estimator is based on the Helmholtz decomposition of vector fields. This decomposition consists in representing the velocity field as a sum of a divergence free component and a curl free component. The objective is to provide a low-dimensional parametric representation of optical flows by depicting them as a flow generated by a small number of vortex and source particles. Both components are approximated using a discretization of the vorticity and divergence maps through regularized Dirac measures. The resulting so called irrotational and solenoidal fields consist then in linear combinations of basis functions obtained through a convolution product of the Green kernel gradient and the vorticity map or the divergence map respectively. The coefficient values and the basis function parameters are obtained by minimization of a functional relying on an integrated version of mass conservation principle of fluid mechanics. Results are provided on real world sequences.

Weighted ensemble transform Kalman filter for image assimilation

ABSTRACT This study proposes an extension of the Weighted Ensemble Kalman filter (WEnKF) proposed by Papadakis et al. (2010) for the assimilation of image observations. The main focus of this study is on a novel formulation of the Weighted filter with the Ensemble Transform Kalman filter (WETKF), incorporating directly as a measurement model a non-linear image reconstruction criterion. This technique has been compared to the original WEnKF on numerical and real world data of 2-D turbulence observed through the transport of a passive scalar. In particular, it has been applied for the reconstruction of oceanic surface current vorticity fields from sea surface temperature (SST) satellite data. This latter technique enables a consistent recovery along time of oceanic surface currents and vorticity maps in presence of large missing data areas and strong noise.

A novel parametric method for non-rigid image registration

This paper presents a novel non-rigid registration method. The main contribution of the method is the modeling of the vorticity (respectively divergence) of the deformation field using vortex (respectively sink and source) particles. Two parameters are associated with a particle: the vorticity (or divergence) strength and the influence domain. This leads to a very compact representation of vorticity and divergence fields. In addition, the optimal position of these particles is determined using a mean shift process. 2D experiments of this method are presented and demonstrate its ability to recover evolving phenomena (MS lesions) so as to register images from 20 patients.

A stochastic filter for fluid motion tracking

In this paper, we present a method for the tracking of fluid flows velocity fields. The technique we propose is formalized within sequential Bayesian filter framework. The filter we propose here combines an Ito diffusion process coming from a stochastic formulation of the vorticity-velocity form of Navier-Stokes equation and discrete measurements extracted from an image sequence. The resulting tracker provides robust and consistent estimations of instantaneous motion fields along the whole image sequence. In order to handle a state space of reasonable dimension for the stochastic filtering problem, we represent the motion field as a combination of adapted basis functions. The used basis functions ensue from a mollification of Biot-Savart integral and a discretization of the vorticity and divergence maps of the fluid vector field. The efficiency of the method is demonstrated on a long real world sequence showing a vortex launch at tip of airplane wing.

Conditional stochastic simulation for character animation

Realistic character animation requires elaborate rigging built on top of high quality 3D models. Sophisticated anatomically based rigs are often the choice of visual effect studios where life-like animation of CG characters is the primary objective. However, rigging a character with a muscular-skeletal system is very involving and time-consuming process, even for professionals. Although, there have been recent research efforts to automate either all or some parts of the rigging process, the complexity of anatomically based rigging nonetheless opens up new research challenges. We propose a new method to automate anatomically based rigging that transfers an existing rig of one character to another. The method is based on a data interpolation in the surface and volume domain, where various rigging elements can be transferred between different models. As it only requires a small number of corresponding input feature points, users can produce highly detailed rigs for a variety of desired character with ease. Copyright

Field of Particle Filters for Image Inpainting

We present a novel algorithm for solving the image inpainting problem based on a field of locally interacting particle filters. Image inpainting, also known as image completion, is concerned with the problem of filling image regions with new visually plausible data. In order to avoid the difficulty of solving the problem globally for the region to be inpainted, we introduce a field of local particle filters. The states of the particle filters are image patches. Global consistency is enforced by a Markov random field image model which connects neighbouring particle filters. The benefit of using locally interacting particle filters is that several competing hypotheses on inpainting solutions are kept active, allowing the method to provide globally consistent solutions on problems where other local methods may fail. We provide examples of applications of the developed method.

Monte Carlo fixed-lag smoothing in state-space models

This paper presents an algorithm for Monte Carlo fixed-lag smoothing in state-space models de-fined by a diffusion process observed through noisy discrete-time measurements. Based on a par-ticles approximation of the filtering and smoothing distributions, the method relies on a simulation technique of conditioned diffusions. The proposed sequential smoother can be applied to general 5 non linear and multidimensional models, like the ones used in environmental applications. The smoothing of a turbulent flow in a high-dimensional context is given as a practical example.