Didier Auroux

Smoothing Problems in a Bayesian Framework and Their Linear Gaussian Solutions

AbstractSmoothers are increasingly used in geophysics. Several linear Gaussian algorithms exist, and the general picture may appear somewhat confusing. This paper attempts to stand back a little, in order to clarify this picture by providing a concise overview of what the different smoothers really solve, and how. The authors begin addressing this issue from a Bayesian viewpoint. The filtering problem consists in finding the probability of a system state at a given time, conditioned to some past and present observations (if the present observations are not included, it is a forecast problem). This formulation is unique: any different formulation is a smoothing problem. The two main formulations of smoothing are tackled here: the joint estimation problem (fixed lag or fixed interval), where the probability of a series of system states conditioned to observations is to be found, and the marginal estimation problem, which deals with the probability of only one system state, conditioned to past, present, and ...

A one-shot inpainting algorithm based on the topological asymptotic analysis

The aim of this article is to propose a new method for the inpainting problem. Inpainting is the problem of filling-in holes in images. We consider in this article the crack localization problem, which can be solved using the Dirichlet to Neumann approach and the topological gradient. In a similar way, we can define a Dirichlet and a Neumann inpainting problem. We then define a cost function measuring the discrepancy between the two corresponding solutions. The minimization is done using the topological asymptotic analysis, and is performed in only one iteration. The optimal solution provides the best localization of the missing edges, and it is then easy to inpaint the holes.

Image Processing by Topological Asymptotic Expansion

AbstractThe aim of this article is to recall the applications of the topological asymptotic expansion to major image processing problems. We briefly review the topological asymptotic analysis, and then present its historical application to the crack localization problem from boundary measurements. A very natural application of this technique in image processing is the inpainting problem, which can be solved by identifying the optimal localization of the missing edges. A second natural application is then the image restoration or enhancement. The identification of the main edges of the image allows us to preserve them, and smooth the image outside the edges. If the conductivity outside edges goes to infinity, the regularized image is piecewise constant and provides a natural solution to the segmentation problem. The numerical results presented for each application are very promising. Finally, we must mention that all these problems are solved with a

From restoration by topological gradient to medical image segmentation via an asymptotic expansion

\mathcal{O}(n.\log(n))

An evolution of the back and forth nudging for geophysical data assimilation: application to Burgers equation and comparisons

complexity.

Application of the Topological Gradient Method to Color Image Restoration

The aim of this article is to present an application of the topological asymptotic expansion to the medical image segmentation problem. We first recall the classical variational of the image restoration problem, and its resolution by topological asymptotic analysis in which the identification of the diffusion coefficient can be seen as an inverse conductivity problem. The conductivity is set either to a small positive coefficient (on the edge set), or to its inverse (elsewhere). In this paper a technique based on a power series expansion of the solution to the image restoration problem with respect to this small coefficient is introduced. By considering the limit when this coefficient goes to zero, we obtain a segmented image, but some numerical issues do not allow a too small coefficient. The idea is to use the series expansion to approximate the asymptotic solution with several solutions corresponding to positive (larger than a threshold) conductivity coefficients via a quadrature formula. We illustrate this approach with some numerical results on medical images.

Symmetry-Based Observers for Some Water-Tank Problems

We study in this article an improvement to the back and forth nudging (BFN) method for geophysical data assimilation. In meteorology or oceanography, the theoretical equations are usually diffusive free, but diffusion is added into the model equations in order to stabilize the numerical integrations and to take into consideration some subscale phenomena. We propose to change the sign of the diffusion in the backward nudging model, which is physically consistent and stabilizes the backward integration. We apply this method to a Burgers equation, study the convergence properties and report the results of numerical experiments. We compare the quality of the estimated initial condition with two other data assimilation techniques that are close from the algorithmic point of view: a variational method, and the quasi-inverse linear method.

Data assimilation methods for an oceanographic problem

The aim of this paper is to generalize to color images the topological gradient method for the restoration problem. First, we consider a simple extension from grey-level to color images, by considering a channel by channel method. Then, as there should be some coupling between the different channels, we propose considering a tensorial approach: the Di Zenzo gradient. We derive an asymptotic expansion of this gradient with respect to the insertion of a small crack in the image, and we propose a new algorithm based on the topological gradient for the color image restoration problem. Finally, we illustrate these two approaches with numerical results.

Generalization of the dual variational data assimilation algorithm to a nonlinear layered quasi-geostrophic ocean model

In this paper, we consider a tank containing fluid and we want to estimate the horizontal currents when the fluid surface height is measured. The fluid motion is described by shallow water equations in two horizontal dimensions. We build a simple nonlinear observer which takes advantage of the symmetries of fluid dynamics laws. As a result its structure is based on convolutions with smooth isotropic kernels, and the observer is remarkably robust to noise. We prove the convergence of the observer around a steady-state. In numerical applications local exponential convergence is expected. The observer is also applied to the problem of predicting the ocean circulation. Realistic simulations illustrate the relevance of the approach compared with some standard oceanography techniques.

Optimal parameters identification and sensitivity study for abrasive waterjet milling model

In this paper, we first focus our interest on the use of the variational adjoint method in a relatively simple ocean model in order to try to reconstruct the four-dimensional ocean system from altimetric surface observations of the ocean.