Regularized Optimal Transport and the Rot Mover's Distance

This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. We develop two generic schemes that we respectively call the alternate scaling algorithm and the non-negative alternate scaling algorithm, to compute efficiently the regularized optimal plans depending on whether the domain of the regularizer lies within the non-negative orthant or not. These schemes are based on Dykstra's algorithm with alternate Bregman projections, and further exploit the Newton-Raphson method when applied to separable divergences. We enhance the separable case with a sparse extension to deal with high data dimensions. We also instantiate our proposed framework and discuss the inherent specificities for well-known regularizers and statistical divergences in the machine learning and information geometry communities. Finally, we demonstrate the merits of our methods with experiments using synthetic data to illustrate the effect of different regularizers and penalties on the solutions, as well as real-world data for a pattern recognition application to audio scene classification

Multi-physics Optimal Transportation and Image Interpolation

International audienceOptimal transportation theory is a powerful tool to deal with image interpolation. This was first investigated by Benamou and Brenier \cite{BB00} where an algorithm based on the minimization of a kinetic energy under a conservation of mass constraint was devised. By structure, this algorithm does not preserve image regions along the optimal interpolation path, and it is actually not very difficult to exhibit test cases where the algorithm produces a path of images where high density regions split at the beginning before merging back at its end. However, in some applications to image interpolation this behaviour is not physically realistic. Hence, this paper aims at studying how some physics can be added to the optimal transportation theory, how to construct algorithms to compute solutions to the corresponding optimization problems and how to apply the proposed methods to image interpolation

Accounting for correlated observation errors in image data assimilation

International audienceSatellites images can provide a lot of information on the earth system evolution. Although those sequences are frequently used, the importance of spatial error correlation are rarely taken into account in practice. This results in discarding a huge part of the information content of satellite image sequences. In this paper, we investigate a method based on wavelet or curvelet transforms to represent (at an affordable cost) some of the observation error correlation in a data assimilation context. We address the topic of monitoring the initial state of a system through the variational assimilation of images corrupted by a spatially correlated noise. The feasibility and the reliability of the approach is demonstrated in an academic context.Les images satellites sont une source importante d'information sur l'évolution du système terre. Bien que ces séquences d'images soient de plus en plus utilisées, l'importance des corrélations spatiales entre les erreurs présentes en leur sein est rarement prise en compte en pratique. Cela conduit à une sous utilisation de l'information contenue dans ces données. Dans cet article, une nouvelle manière (peu coûteuse) d'intégrer cette information dans le cadre de l'assimilation de données est proposée. Le problème de l'utilisation d'images corrompues par un bruit fortement corrélé en espace afin de contrôler l'état initial du système est abordé. La faisabilité et la pertinence de l'approche proposée est démontrée dans le cadre d'une configuration académique

Inverse problem regularization with hierarchical variational autoencoders

In this paper, we propose to regularize ill-posed inverse problems using a deep hierarchical variational autoencoder (HVAE) as an image prior. The proposed method synthesizes the advantages of i) denoiser-based Plug \& Play approaches and ii) generative model based approaches to inverse problems. First, we exploit VAE properties to design an efficient algorithm that benefits from convergence guarantees of Plug-and-Play (PnP) methods. Second, our approach is not restricted to specialized datasets and the proposed PnP-HVAE model is able to solve image restoration problems on natural images of any size. Our experiments show that the proposed PnP-HVAE method is competitive with both SOTA denoiser-based PnP approaches, and other SOTA restoration methods based on generative models