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A statistical framework in variational methods of image and video processing problems with high dimensions

Abstract : This thesis addresses variational formulation of image and video processing problems. This formulation expresses the solution through a minimization of an energy. These energies can be defined as deterministic or stochastic. It is known that the first approach corresponds to the parametric class of the second one. The second class is then chosen as it is more general when relaxing parametric assumptions. In return, the energy must be expressed as a function of the data considered as random variables. These probabilities are classically estimated with fixed-sized kernels on marginal distributions of the data, assuming the different channels are independent. These methods have two limitations, the inhomogeneous and scarce repartition of the data in the space. These difficulties, as well as the independence assumption are enhanced when the data of the image are high dimensional (color channels, or other channels describing local patterns of natural images). At the foundation of the statistics, the k-th nearest neighbor method can solve these difficulties by locally adapting to the repartition of the data and treating the channels jointly. We propose a general statistical framework based on k-th nearest neighbors and information theory. This new framework is dedicated to variational problems as it efficiently estimates, high dimensional energies, derivates of these energies and local probabilities. This framework is applied to three problems of image and video processing: optical flow, object tracking and segmentation. This framework circumvents the problem of dimensionality and allows us to introduce new measures and probabilities more adapted to natural images. Some results obtained have been applied in an industrial context.
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Sylvain Boltz. A statistical framework in variational methods of image and video processing problems with high dimensions. Human-Computer Interaction [cs.HC]. Université Nice Sophia Antipolis, 2008. English. ⟨tel-00507488⟩

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