Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization

Plug-and-Play (PnP) methods solve ill-posed inverse problems through iterative proximal algorithms by replacing a proximal operator by a denoising operation. When applied with deep neural network denoisers, these methods have shown state-of-the-art visual performance for image restoration problems. However, their theoretical convergence analysis is still incomplete. Most of the existing convergence results consider nonexpansive denoisers, which is non-realistic, or limit their analysis to strongly convex data-fidelity terms in the inverse problem to solve. Recently, it was proposed to train the denoiser as a gradient descent step on a functional parameterized by a deep neural network. Using such a denoiser guarantees the convergence of the PnP version of the Half-Quadratic-Splitting (PnP-HQS) iterative algorithm. In this paper, we show that this gradient denoiser can actually correspond to the proximal operator of another scalar function. Given this new result, we exploit the convergence theory of proximal algorithms in the nonconvex setting to obtain convergence results for PnP-PGD (Proximal Gradient Descent) and PnP-ADMM (Alternating Direction Method of Multipliers). When built on top of a smooth gradient denoiser, we show that PnP-PGD and PnP-ADMM are convergent and target stationary points of an explicit functional. These convergence results are confirmed with numerical experiments on deblurring, super-resolution and inpainting.

Gradient Step Denoiser for convergent Plug-and-Play

Plug-and-Play methods constitute a class of iterative algorithms for imaging problems where regularization is performed by an off-the-shelf denoiser. Although Plug-and-Play methods can lead to tremendous visual performance for various image problems, the few existing convergence guarantees are based on unrealistic (or suboptimal) hypotheses on the denoiser, or limited to strongly convex data terms. In this work, we propose a new type of Plug-and-Play methods, based on half-quadratic splitting, for which the denoiser is realized as a gradient descent step on a functional parameterized by a deep neural network. Exploiting convergence results for proximal gradient descent algorithms in the non-convex setting, we show that the proposed Plug-and-Play algorithm is a convergent iterative scheme that targets stationary points of an explicit global functional. Besides, experiments show that it is possible to learn such a deep denoiser while not compromising the performance in comparison to other state-of-the-art deep denoisers used in Plug-and-Play schemes. We apply our proximal gradient algorithm to various ill-posed inverse problems, e.g. deblurring, super-resolution and inpainting. For all these applications, numerical results empirically confirm the convergence results. Experiments also show that this new algorithm reaches state-of-the-art performance, both quantitatively and qualitatively.

POPCORN: Progressive Pseudo-labeling with Consistency Regularization and Neighboring

Semi-supervised learning (SSL) uses unlabeled data to compensate for the scarcity of annotated images and the lack of method generalization to unseen domains, two usual problems in medical segmentation tasks. In this work, we propose POPCORN, a novel method combining consistency regularization and pseudo-labeling designed for image segmentation. The proposed framework uses high-level regularization to constrain our segmentation model to use similar latent features for images with similar segmentations. POPCORN estimates a proximity graph to select data from easiest ones to more difficult ones, in order to ensure accurate pseudo-labeling and to limit confirmation bias. Applied to multiple sclerosis lesion segmentation, our method demonstrates competitive results compared to other state-of-the-art SSL strategies.

Learning local regularization for variational image restoration

In this work, we propose a framework to learn a local regularization model for solving general image restoration problems. This regularizer is defined with a fully convolutional neural network that sees the image through a receptive field corresponding to small image patches. The regularizer is then learned as a critic between unpaired distributions of clean and degraded patches using a Wasserstein generative adversarial networks based energy. This yields a regularization function that can be incorporated in any image restoration problem. The efficiency of the framework is finally shown on denoising and deblurring applications.

On the Existence of Optimal Transport Gradient for Learning Generative Models

The use of optimal transport cost for learning generative models has become popular with Wasserstein Generative Adversarial Networks (WGAN). Training of WGAN relies on a theoretical background: the calculation of the gradient of the optimal transport cost with respect to the generative model parameters. We first demonstrate that such gradient may not be defined, which can result in numerical instabilities during gradient-based optimization. We address this issue by stating a valid differentiation theorem in the case of entropic regularized transport and specify conditions under which existence is ensured. By exploiting the discrete nature of empirical data, we formulate the gradient in a semi-discrete setting and propose an algorithm for the optimization of the generative model parameters. Finally, we illustrate numerically the advantage of the proposed framework.

GraphXCOVID: Explainable Deep Graph Diffusion Pseudo-Labelling for Identifying COVID-19 on Chest X-rays

Can one learn to diagnose COVID-19 under extreme minimal supervision? Since the outbreak of the novel COVID-19 there has been a rush for developing Artificial Intelligence techniques for expert-level disease identification on Chest X-ray data. In particular, the use of deep supervised learning has become the go-to paradigm. However, the performance of such models is heavily dependent on the availability of a large and representative labelled dataset. The creation of which is a heavily expensive and time consuming task, and especially imposes a great challenge for a novel disease. Semi-supervised learning has shown the ability to match the incredible performance of supervised models whilst requiring a small fraction of the labelled examples. This makes the semi-supervised paradigm an attractive option for identifying COVID-19. In this work, we introduce a graph based deep semi-supervised framework for classifying COVID-19 from chest X-rays. Our framework introduces an optimisation model for graph diffusion that reinforces the natural relation among the tiny labelled set and the vast unlabelled data. We then connect the diffusion prediction output as pseudo-labels that are used in an iterative scheme in a deep net. We demonstrate, through our experiments, that our model is able to outperform the current leading supervised model with a tiny fraction of the labelled examples. Finally, we provide attention maps to accommodate the radiologist's mental model, better fitting their perceptual and cognitive abilities. These visualisation aims to assist the radiologist in judging whether the diagnostic is correct or not, and in consequence to accelerate the decision.

Wasserstein Generative Models for Patch-based Texture Synthesis

In this paper, we propose a framework to train a generative model for texture image synthesis from a single example. To do so, we exploit the local representation of images via the space of patches, that is, square sub-images of fixed size (e.g. $4\times 4$). Our main contribution is to consider optimal transport to enforce the multiscale patch distribution of generated images, which leads to two different formulations. First, a pixel-based optimization method is proposed, relying on discrete optimal transport. We show that it is related to a well-known texture optimization framework based on iterated patch nearest-neighbor projections, while avoiding some of its shortcomings. Second, in a semi-discrete setting, we exploit the differential properties of Wasserstein distances to learn a fully convolutional network for texture generation. Once estimated, this network produces realistic and arbitrarily large texture samples in real time. The two formulations result in non-convex concave problems that can be optimized efficiently with convergence properties and improved stability compared to adversarial approaches, without relying on any regularization. By directly dealing with the patch distribution of synthesized images, we also overcome limitations of state-of-the art techniques, such as patch aggregation issues that usually lead to low frequency artifacts (e.g. blurring) in traditional patch-based approaches, or statistical inconsistencies (e.g. color or patterns) in learning approaches.

Multi-task deep learning for image segmentation using recursive approximation tasks

Fully supervised deep neural networks for segmentation usually require a massive amount of pixel-level labels which are manually expensive to create. In this work, we develop a multi-task learning method to relax this constraint. We regard the segmentation problem as a sequence of approximation subproblems that are recursively defined and in increasing levels of approximation accuracy. The subproblems are handled by a framework that consists of 1) a segmentation task that learns from pixel-level ground truth segmentation masks of a small fraction of the images, 2) a recursive approximation task that conducts partial object regions learning and data-driven mask evolution starting from partial masks of each object instance, and 3) other problem oriented auxiliary tasks that are trained with sparse annotations and promote the learning of dedicated features. Most training images are only labeled by (rough) partial masks, which do not contain exact object boundaries, rather than by their full segmentation masks. During the training phase, the approximation task learns the statistics of these partial masks, and the partial regions are recursively increased towards object boundaries aided by the learned information from the segmentation task in a fully data-driven fashion. The network is trained on an extremely small amount of precisely segmented images and a large set of coarse labels. Annotations can thus be obtained in a cheap way. We demonstrate the efficiency of our approach in three applications with microscopy images and ultrasound images.

Variational Osmosis for Non-linear Image Fusion

We propose a new variational model for nonlinear image fusion. Our approach incorporates the osmosis model proposed in Vogel et al. (2013) and Weickert et al. (2013) as an energy term in a variational model. The osmosis energy is known to realize visually plausible image data fusion. As a consequence, our method is invariant to multiplicative brightness changes. On the practical side, it requires minimal supervision and parameter tuning and can encode prior information on the structure of the images to be fused. We develop a primal-dual algorithm for solving this new image fusion model and we apply the resulting minimisation scheme to multi-modal image fusion for face fusion, colour transfer and some cultural heritage conservation challenges. Visual comparison to state-of-the-art proves the quality and flexibility of our method.

GraphX$^{NET}-$ Chest X-Ray Classification Under Extreme Minimal Supervision

The task of classifying X-ray data is a problem of both theoretical and clinical interest. Whilst supervised deep learning methods rely upon huge amounts of labelled data, the critical problem of achieving a good classification accuracy when an extremely small amount of labelled data is available has yet to be tackled. In this work, we introduce a novel semi-supervised framework for X-ray classification which is based on a graph-based optimisation model. To the best of our knowledge, this is the first method that exploits graph-based semi-supervised learning for X-ray data classification. Furthermore, we introduce a new multi-class classification functional with carefully selected class priors which allows for a smooth solution that strengthens the synergy between the limited number of labels and the huge amount of unlabelled data. We demonstrate, through a set of numerical and visual experiments, that our method produces highly competitive results on the ChestX-ray14 data set whilst drastically reducing the need for annotated data.