Assimilation of Images: A variational framework
Résumé
At the present time, the Earth is observed by dozens of satellites giving a permanent information on the evolution of the atmosphere and of the ocean. This information is partly used by transforming radiances into state variables of the models then performing a regular method of Data Assimilation. But the dynamics of these images also contains an important information, such as the evolution of fronts or vortices, which is not taken into account. The first problem is to isolate the information in the images. Visually the information is borne by the discontinuities: a front is identified by a discontinuity in the images, the information has to be extracted from the evolution of these discontinuities. We can't expect getting information from a "flat" image or from an image with a weak gradient. We will present how to use this information. We will distinguish two classes of methods:
- Indirect Assimilation of Images. These methods are based on conservation of luminance between successive images. Conservation of luminance is an advection equation containing the velocity. Retrieving velocity can be considered as an inverse problem but it is ill posed because the velocity orthogonal to the gradient of luminance can't be identified. To alleviate this difficulty a classical approach is to introduce a so-called regularization term in the cost-function, we will present a new method based on regularization by a generalized diffusion equation. After being retrieved velocities are plugged in a regular variational data assimilation scheme.
- Direct Assimilation of Images. In this case we define an extended state variable of the model: the image. The question is to give a mathematical definition to images. The first idea is to consider the image as a set of pixel but we will obtain a very large problem (e.g. a Meteosat image has 25 millions of pixels) if we consider the evolution of an episode. Another method is to project the images in a base of appropriate functions we will demonstrate that curvelets have the properties of conserving the shape of discontinuities and requiring few functions to discretise an image. We will discuss the problem of thresholding the development and adding a regularization to avoid Gibbs phenomena.