Local matching indicators for transport with concave costs

In this note, we introduce a class of indicators that enable to compute efficiently optimal transport plans associated to arbitrary distributions of $N$ demands and $N$ supplies in $\mathbf{R}$ in the case where the cost function is concave. The computational cost of these indicators is small and independent of $N$. A hierarchical use of them enables to obtain an efficient algorithm

A Wasserstein-type distance in the space of Gaussian Mixture Models

In this paper we introduce a Wasserstein-type distance on the set of Gaussian mixture models. This distance is defined by restricting the set of possible coupling measures in the optimal transport problem to Gaussian mixture models. We derive a very simple discrete formulation for this distance, which makes it suitable for high dimensional problems. We also study the corresponding multi-marginal and barycenter formulations. We show some properties of this Wasserstein-type distance, and we illustrate its practical use with some examples in image processing

A Bayesian Hyperprior Approach for Joint Image Denoising and Interpolation, with an Application to HDR Imaging

Recently, impressive denoising results have been achieved by Bayesian approaches which assume Gaussian models for the image patches. This improvement in performance can be attributed to the use of per-patch models. Unfortunately such an approach is particularly unstable for most inverse problems beyond denoising. In this work, we propose the use of a hyperprior to model image patches, in order to stabilize the estimation procedure. There are two main advantages to the proposed restoration scheme: Firstly it is adapted to diagonal degradation matrices, and in particular to missing data problems (e.g. inpainting of missing pixels or zooming). Secondly it can deal with signal dependent noise models, particularly suited to digital cameras. As such, the scheme is especially adapted to computational photography. In order to illustrate this point, we provide an application to high dynamic range imaging from a single image taken with a modified sensor, which shows the effectiveness of the proposed scheme.Comment: Some figures are reduced to comply with arxiv's size constraints. Full size images are available as HAL technical report hal-01107519v5, IEEE Transactions on Computational Imaging, 201

Stochastic Modeling and Resolution-Free Rendering of Film Grain

The realistic synthesis and rendering of film grain is a crucial goal for many amateur and professional photographers and film-makers whose artistic works require the authentic feel of analog photography. The objective of this work is to propose an algorithm that reproduces the visual aspect of film grain texture on any digital image. Previous approaches to this problem either propose unrealistic models or simply blend scanned images of film grain with the digital image, in which case the result is inevitably limited by the quality and resolution of the initial scan. In this work, we introduce a stochastic model to approximate the physical reality of film grain, and propose a resolution-free rendering algorithm to simulate realistic film grain for any digital input image. By varying the parameters of this model, we can achieve a wide range of grain types. We demonstrate this by comparing our results with film grain examples from dedicated software, and show that our rendering results closely resemble these real film emulsions. In addition to realistic grain rendering, our resolution-free algorithm allows for any desired zoom factor, even down to the scale of the microscopic grains themselves

FastDVDnet: Towards Real-Time Video Denoising Without Explicit Motion Estimation

In this paper, we propose a state-of-the-art video denoising algorithm based on a convolutional neural network architecture. Until recently, video denoising with neural networks had been a largely under explored domain, and existing methods could not compete with the performance of the best patch-based methods. The approach we introduce in this paper, called FastDVDnet, shows similar or better performance than other state-of-the-art competitors with significantly lower computing times. In contrast to other existing neural network denoisers, our algorithm exhibits several desirable properties such as fast run-times, and the ability to handle a wide range of noise levels with a single network model. The characteristics of its architecture make it possible to avoid using a costly motion compensation stage while achieving excellent performance. The combination between its denoising performance and lower computational load makes this algorithm attractive for practical denoising applications. We compare our method with different state-of-art algorithms, both visually and with respect to objective quality metrics

DVDnet: A Fast Network for Deep Video Denoising

In this paper, we propose a state-of-the-art video denoising algorithm based on a convolutional neural network architecture. Previous neural network based approaches to video denoising have been unsuccessful as their performance cannot compete with the performance of patch-based methods. However, our approach outperforms other patch-based competitors with significantly lower computing times. In contrast to other existing neural network denoisers, our algorithm exhibits several desirable properties such as a small memory footprint, and the ability to handle a wide range of noise levels with a single network model. The combination between its denoising performance and lower computational load makes this algorithm attractive for practical denoising applications. We compare our method with different state-of-art algorithms, both visually and with respect to objective quality metrics. The experiments show that our algorithm compares favorably to other state-of-art methods. Video examples, code and models are publicly available at \url{https://github.com/m-tassano/dvdnet}

Properties of Discrete Sliced Wasserstein Losses

The Sliced Wasserstein (SW) distance has become a popular alternative to the Wasserstein distance for comparing probability measures. Widespread applications include image processing, domain adaptation and generative modelling, where it is common to optimise some parameters in order to minimise SW, which serves as a loss function between discrete probability measures (since measures admitting densities are numerically unattainable). All these optimisation problems bear the same sub-problem, which is minimising the Sliced Wasserstein energy. In this paper we study the properties of $\mathcal{E}: Y \longmapsto \mathrm{SW}_2^2(\gamma_Y, \gamma_Z)$, i.e. the SW distance between two uniform discrete measures with the same amount of points as a function of the support $Y \in \mathbb{R}^{n \times d}$ of one of the measures. We investigate the regularity and optimisation properties of this energy, as well as its Monte-Carlo approximation $\mathcal{E}_p$ (estimating the expectation in SW using only $p$ samples) and show convergence results on the critical points of $\mathcal{E}_p$ to those of $\mathcal{E}$, as well as an almost-sure uniform convergence. Finally, we show that in a certain sense, Stochastic Gradient Descent methods minimising $\mathcal{E}$ and $\mathcal{E}_p$ converge towards (Clarke) critical points of these energies