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Séminaire Images Optimisation et Probabilités
Deux exposés
Laura Girometti et Léo Portales
( Université de Bologne et de Toulouse )Salle de conférénces.
le 21 mars 2024 à 11:00
Title Léo : Convergence of the iterates of Lloyd's algorithm
Summary : The finding of a discrete measure that approaches a target density, called quantization, is an important aspect of machine learning and is usually done using Lloyd’s algorithm; a continuous counterpart to the K-means algorithm. We have studied two variants of this algorithm: one where we specify the former measure to be uniform (uniform quantization) and one where the weights associated to each point is adjusted to fit the target density (optimal quantization). In either case it is not yet known in the literature whether the iterates of these algorithms converge simply. We proved so with the assumption that the target density is analytic and restricted to a semi algebraic compact and convex set. We do so using tools from o-minimal geometry as well as the Kurdyka-Lojasiewicz inequality. We also proved along the way the definability in an o-minimal structure of functions of the form Y := (y1, ..., yN ) → D(\mu,1/N sum_{i=1}^N \delta_{y_i}) for the following divergences D: the general Wp Wasserstein distance, the max-sliced Wasserstein distance and the entropic regularized Wasserstein distance.
Title Laura : Quaternary Image Decomposition
Summary : Decomposing an image into meaningful components is a challenging inverse problem in image processing and has been widely applied to cartooning, texture removal, denoising, soft shadow/spotlight removal, detail enhancement etc. All the valuable contributions to this problem rely on a variational-based formulation where the intrinsic difficulty comes from the numerical intractability of the considered norms, from the tuning of the numerous model parameters, and, overall, from the complexity of extracting noise from a textured image, given the strong similarity between these two components. In this talk, I will present a two-stage variational model for the additive decomposition of images into piecewise constant, smooth, textured and white noise components, focusing on the regularization parameter selection and presenting numerical results of decomposition of textured images corrupted by several kinds of additive white noises.
Summary : The finding of a discrete measure that approaches a target density, called quantization, is an important aspect of machine learning and is usually done using Lloyd’s algorithm; a continuous counterpart to the K-means algorithm. We have studied two variants of this algorithm: one where we specify the former measure to be uniform (uniform quantization) and one where the weights associated to each point is adjusted to fit the target density (optimal quantization). In either case it is not yet known in the literature whether the iterates of these algorithms converge simply. We proved so with the assumption that the target density is analytic and restricted to a semi algebraic compact and convex set. We do so using tools from o-minimal geometry as well as the Kurdyka-Lojasiewicz inequality. We also proved along the way the definability in an o-minimal structure of functions of the form Y := (y1, ..., yN ) → D(\mu,1/N sum_{i=1}^N \delta_{y_i}) for the following divergences D: the general Wp Wasserstein distance, the max-sliced Wasserstein distance and the entropic regularized Wasserstein distance.
Title Laura : Quaternary Image Decomposition
Summary : Decomposing an image into meaningful components is a challenging inverse problem in image processing and has been widely applied to cartooning, texture removal, denoising, soft shadow/spotlight removal, detail enhancement etc. All the valuable contributions to this problem rely on a variational-based formulation where the intrinsic difficulty comes from the numerical intractability of the considered norms, from the tuning of the numerous model parameters, and, overall, from the complexity of extracting noise from a textured image, given the strong similarity between these two components. In this talk, I will present a two-stage variational model for the additive decomposition of images into piecewise constant, smooth, textured and white noise components, focusing on the regularization parameter selection and presenting numerical results of decomposition of textured images corrupted by several kinds of additive white noises.
