## Section: New Results

Keywords : data assimilation, ill-posed problem, Tikhonov regularization, image processing.

### Data Assimilation for Image Processing

Participants : Dominique Béréziat, Isabelle Herlin.

In the image processing area, most of problematics correspond to ill-posed problems (edge detection, segmentation, optical flow estimation, deblurring, ...). A common method used to solve ill-posedness is to regularize solutions by adding smoothing constraints. For example, Tikhonov like approach proposes to minimise a functional measuring the discrepancy between model and solution and constraining the spatial variations of the solution. It is now well known [29] that the Euler-Lagrange evolution equation associated to this functional corresponds to a diffusion process.

Similar equations are found in the modelling in many physical situations (meteorology, oceanography, air quality, ...), and data assimilation techniques [31] have proven their efficiency for solving them, even when partial and/or indirect observations of state variables are available.

We propose to exploit this property in order to formulate ill-posed image processing problems in a variational data assimilation framework. The research addresses the definition of evolution models for the image information to be retrieved. In a first step this approach is studied for 3D reconstruction from a sequence of 2D observations: we try to reconstruct the evolution of a 3D object from a sequence of 2D image acquisitions. The observations consist of the 2D images, obtained by projection of the 3D space. The evolution model can be as simple as an optical-flow like constraint (transport of grey level over 3D sequence).