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We address the problem of vector-valued image regularization with variational methods and PDEs. From the study of existing formalisms, we propose a unifying framework based on a very local interpretation of the regularization processes. The resulting equations are then specialized into new regularization PDEs and corresponding numerical schemes that respect the local geometry of vector-valued images. They are finally applied on a wide variety of image processing problems, including color image restoration, in-painting, magnification and flow visualization.