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Transport optimal pour l'assimilation de données images

Abstract : Forecasting of a physical system is computed by the help of a mathematical model. This model needs to be initialized by the state of the system at initial time. But this state is not directly measurable and data assimilation techniques are generally used to estimate it. They combine all sources of information such as observations (that may be sparse in time and space and potentially include errors), previous forecasts, the model equations and error statistics. The main idea of data assimilation techniques is to find an initial state accounting for the different sources of informations. Such techniques are widely used in meteorology, where data and particularly images are more and more numerous due to the increasing number of satellites and other sources of measurements. This, coupled with developments of meteorological models, have led to an ever-increasing quality of the forecast.Spatial consistency is one specificity of images. For example, human eyes are able to notice structures in an image. However, classical methods of data assimilation do not handle such structures because they take only into account the values of each pixel separately. In some cases it leads to a bad initial condition. To tackle this problem, we proposed to change the representation of an image: images are considered here as elements of the Wasserstein space endowed with the Wasserstein distance coming from the optimal transport theory. In this space, what matters is the positions of the different structures.This thesis presents a data assimilation technique based on this Wasserstein distance. This technique and its numerical procedure are first described, then experiments are carried out and results shown. In particularly, it appears that this technique was able to give an analysis of corrected position.
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Nelson Feyeux. Transport optimal pour l'assimilation de données images. Analyse numérique [math.NA]. Université Grenoble Alpes, 2016. Français. ⟨NNT : 2016GREAM076⟩. ⟨tel-01480695v2⟩

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