Section: New Results

Minimax filtering

In minimax filtering for state estimation, the initial state error, the model error and the observational errors are classically supposed to belong to one joint ellipsoid. In this case, it is only assumed that the errors, stochastic or deterministic, are bounded. For each assimilation experiment, the filter computes an ellipsoid where one will find at least all states compatible with observations and errors description. The state estimate is taken as the center of the ellipsoid. No assumption on the actual distribution of the errors in needed and the state estimate minimizes the worst-case error, which makes the filter robust.

A posteriori minimax motion estimation

Participants : Sergiy Zhuk [IBM Research, Ireland] , Isabelle Herlin, Olexander Nakonechnyi [Taras Shevchenko National University of Kyiv] , Jason Frank [CWI, the Netherlands] .

Data assimilation algorithms based on the 4D-Var formulation look for the so-called conditional mode estimate. The latter maximizes the conditional probability density function, provided the initial condition, model error and observation noise are realizations of independent Gaussian random variables. However this Gaussian assumption is often not satisfied for geophysical flows. Moreover, the estimation error of the conditional mode estimate is not a first-hand result of these methods. The issues above can be addressed by means of the Minimax State Estimation (MSE) approach. It allows to filter out any random (with bounded correlation operator) or deterministic (with bounded energy) noise and assess the worst-case estimation error.

The iterative MSE algorithm was developed for the problem of optical flow estimation from a sequence of 2D images. The main idea of the algorithm is to use the "bi-linear" structure of the Navier-Stokes equations and optical flow constraint in order to iteratively estimate the velocity. The algorithm consists of the following parts:

1) we construct pseudo-observations $\widehat{I}$ as the estimate of the image brightness function $I(x,y,t)$ solving the optical flow constraint such that $\widehat{I}$ fits (in the sense of least-squares) the observed sequence of images. To do so, we set the velocity field in the optical flow constraint to be the current minimax estimate of the velocity field $𝐰$, obtained at the previous iteration of the algorithm, and construct the minimax estimate $\widehat{I}$ of the solution of the resulting linear advection equation using the observed image sequence as discrete measurements of the brightness function;

2) we plug the estimate of the image gradient, obtained out of pseudo-observations $\widehat{I}$ in 1), into the optical flow constraint and the current minimax estimate $𝐰$ of the velocity field into the non linear part of Navier-Stokes equations so that we end up with a system of linear PDEs, which represents an extended state equation: it contains a linear parabolic equation for the velocity field and linear advection equation for the image brightness function; we construct the minimax estimate of the velocity field from the extended state equation using again the observed image sequence as discrete measurements of the brightness function;

3) we use the minimax estimate of the velocity field obtained in 2) in order to start 1) again.

Point 1) has been implemented and tested. As Point 2) is currently under development, it is replaced by one of our motion estimation method in order to be plugged in Point 3).