Optimal Transport for Image Processing

Abstract : Optimal Transport is a well developed mathematical theory that defines robust metrics between probability distributions. The computation of optimal displacements between densities through the associated transport map makes this theory mainstream in several applicative fields. For image processing applications, the transport map can for instance be used to compute geodesics between images or to transfer characteristics of one image to another. In this context, it is of major interest to preserve the nature of the observed objects, in order to synthesize images that are physically and visually plausible. In this document, generalized Optimal Transport distances including relaxation and regularization are considered to improve the modeling of image processing problems. New models and algorithms within the continuous and discrete formulations of Optimal Transport are presented. With the continuous setting, the integration of physical regularization of the transport plan makes possible the interpolation of ocean images containing complex structures. In the discrete setting, the regularization of the transport plan is considered for color transfer between images. Convex and non-convex models are proposed to define automatic methods that adapt the proportion of colors required to synthesize visually plausible images. These methods are extended to the computation of barycenters to deal with the color normalization of multiple images. Finally, the entropy regularization of discrete Optimal Transport is used for image segmentation. A fast and convex model is designed to segment images, while respecting global color distribution constraints.
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Nicolas Papadakis. Optimal Transport for Image Processing. Signal and Image Processing. Université de Bordeaux, 2015. ⟨tel-01246096v8⟩

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