We have addressed the problem of capturing a light field using a single traditional camera, by solving the inverse problem of dense light field reconstruction from a focal stack containing only very few images captured at different focus distances. An end-to-end joint optimization framework has been designed, where a novel unrolled optimization method is jointly optimized with a view synthesis deep neural network. The proposed unrolled optimization method constructs Fourier Disparity Layers (FDL), a compact representation of light fields which samples Lambertian non-occluded scenes in the depth dimension and from which all the light field viewpoints can be computed 24. Solving the optimization problem in the FDL domain allows us to derive a closed-form expression of the data-fit term of the inverse problem. Furthermore, unrolling the FDL optimization allows us to learn a prior directly in the FDL domain 8. In order to widen the FDL representation to more complex scenes, a Deep Convolutional Neural Network (DCNN) is trained to synthesize novel views from the optimized FDL. We show that this joint optimization framework reduces occlusion issues of the FDL model (see Figure 1 below), and outperforms recent state-of-the-art methods for light field reconstruction from focal stack measurements.

Conventional microscopy systems have limited depth of field, which often necessitates depth scanning techniques hindered by light scattering. Various techniques have been developed to address this challenge, but they have limited extended depth of field (EDOF) capabilities. Extended Depth of Field (EDOF) lenses can solve this problem, however there is a trade-off between the spatial resolution and the depth of field of the captured images.

To overcome this challenge, in collaboration with the university of Tampere (Prof. Humeyra Caglayan, Erdem Sahin) in the context of the Plenoptima Marie Curie project, we have adressed this problem of EDOF while preserving a high resolution by designing a pipeline for end-to-end learning and optimization of an optical layer together with a deep learning based image reconstruction method. In our design, instead of phase masks, we consider meta-surface materials. Meta-surfaces are periodic sub-wavelength structures that can change the amplitude, phase, and polarization of the incident wave. The meta-surface parameters are end-to-end optimized to have the highest depth of field with good spatial resolution and high reconstruction quality 7. Our end-to-end optimization framework allows building a computational EDOF microscope that combines a 4f microscopy optical setup incorporating a learned optics at the Fourier plane and a post-processing deblurring neural network (see the illustration of the principle in the Figure 2 below). Utilizing the end-to-end differentiable model, we have introduced a systematic design methodology for computational EDOF microscopy based on the specific visualization requirements of the sample under examination. We have shown that the metasurface optics provides key advantages for extreme EDOF imaging conditions, where the extended DOF range is well beyond what is demonstrated in state of the art, achieving superior EDOF performance.

Neural Radiance Fields (NeRF) enable novel view synthesis of 3D scenes when trained with a set of 2D images. One of the key components of NeRF is the input encoding, i.e. mapping the coordinates to higher dimensions to learn high-frequency details, which has been proven to increase the quality. Among various input mappings, hash encoding is gaining increasing attention for its efficiency. However, its performance, when using sparse inputs, is limited. To address this limitation, we have proposed a new input encoding scheme that improves hash-based NeRF for sparse inputs, i.e. few and distant cameras, specifically for 360◦ view synthesis 21. We have combined frequency encoding and hash encoding and shown that this combination can increase dramatically the quality of hash-based NeRF for sparse inputs. Additionally, we have explored scene geometry by estimating vanishing points in omnidirectional images (ODI) of indoor and city scenes in order to align frequency encoding with scene structures (see Figure 3 below). We have shown that our vanishing point-aided scene alignment further improves deterministic and non-deterministic encodings on image regression (view reconstruction and rendering) and view synthesis tasks, where sharper textures and more accurate geometry of scene structures can be reconstructed.

Today’s satellite images have a high resolution, thanks to high-performance sensors. This increase in the volume of data to be transmitted to Earth requires efficient compression methods. Designing efficient compression algorithms for satellite images must take into account several constraints. First, (i) it must be well adapted to the raw data format. In particular, we have considered the cameras for the Lion satellite constellation with ultra-high spatial resolution at the price of a lower spectral resolution. More precisely, the three spectral bands (RGB) are acquired with a single sensor with an in-built colour filter array. Second, (ii) it should be adapted to the image statistics. Indeed, satellite mages contain very high-frequency details with small objects spread over very few pixels only. Finally, (iii) the compression must be quasi-lossless to allow accurate on-ground interpretation. In addition, the images are also affected by noise during the acquisition process, which hinders the compression process.

While existing compression solutions mostly assume that the captured raw data has been demosaicked prior to compression, we have designed an end-to-end trainable neural network for joint decoding, demosaicking and denoising satellite images on the ground at the decoder side, while encoding the raw data. We have first introduced a training loss combining a perceptual loss with the classical mean square error, which has been shown to better preserve the high-frequency details present in satellite images. We then developed a multi-loss balancing strategy which significantly improves the performance of the proposed joint demosaicking-compression solution 16. We also proposed a novel guidance branch which is used during training, to enforce intermediate features of the demosaicked and decoded images to be close to those that would be obtained with ground truth full colour (non-mosaicked) images during decoding. We have shown that our method achieves better rate-distortion trade-offs compared with current satellite baselines while reducing the overall complexity 42.

We also addressed the limitations of learned image compression neural networks, which have difficulties adapting to certain satellite image characteristics, especially high frequencies that disappear at a high bit-rate in the blur generated in the reconstruction. To answer this problem we described in 15 a joint end-to-end trainable neural network. It is separated into a general compression network and a smaller specialized network. We trained this specialized network to compress the residual part of the image to best preserve the high-frequency details present in the satellite images. The proposed model achieves higher rate-distortion performance than current lossy image compression standards and also manages to retrieve details previously poorly reconstructed.

Modeling 3D scenes from image data to render novel photorealistic views has been a central topic in the field of computer graphics and vision. The most recent advances have come from the use of neural volumetric representations, such as the volumetric Neural Radiance Fields (NeRF). Although the implicit representation of NERF is able to encapsulate the scene information, its rendering is slow. PlenOctree (Plenoptic Octrees) models have been proposed in order to cope with the rendering speed issue.

A scene is first encoded in a NeRF-like network through training. However, instead of predicting the RGB color of each pixel in each view directly, the network predicts spherical harmonic coefficients. The spherical harmonics are sherical basis functions which allow the encoding of the appearances and preserve view-dependent effects such as specularities. The color is then calculated by summing the weighted spherical harmonic bases evaluated at the corresponding ray direction. The spherical harmonics enable the representation to model view-dependent appearance. To build the PlenOctree model, the NERF-SH model is densely sampled. This allows extracting the octree parameters (the density σ, and the Spherical Harmonics (SH) coefficients for each octant) and thus generate the octree-based radiance field. Then, this octree model is fine-tuned and compressed, e.g. by controlling the visibility thtreshold and the number of spherical harmonics. However, decreasing the number of harmonics affects the capability to represent view-dependent effects, such as specularity and non- Lambertian surfaces, and this proves to be the biggest bottleneck to reduce the bit rate. Plenoctree models have a large size, which is an issue for storage and transmission.

We have addressed this weakness by proposing an efficient compression scheme for the plenoctree model which allows maintaining high rendering quality and speed. The proposed approach both improves the training of the plenoctree model with geometry and sparsity contrained regularization, reduces the number of occupied voxels by controlling the visibility threshold, and then efficiently encodes the spherical harmonics. Results over a set of test camera poses show that we can reduce about eight times the bit rates of the encoded models and still obtain a higher quality of the synthesized images when compared to the original PlenOctree models or, alternatively, a reduction of about 50 times while presenting minimal degradation for novel view synthesis 20.

Neural implicit representations have appeared as promising techniques for representing a variety of signals, among which light fields, offering several advantages over traditional grid-based representations, such as independence to the signal resolution. While it has been shown to be a good representation for a given type of signal, i.e. for a particular light field, we realized that exploiting the features shared between different scenes remains an understudied problem. We have therefore made one step towards this end by developing a method for sharing the representation between thousands of light fields, splitting the representation between a part that is shared between all light fields and a part which varies individually from one light field to another 25.

Assuming that the neural representations of the different scenes lie on a manifold, the algorithm learns a factorisation (in the same vein as a singular value decomposition) of the weights matrices forming the layers of a deep neural network, to learn the basis defining the common manifold. For each layer, a base of matrices $U$ and $V$ is thus learned that serves as a representation shared between all light fields (a.k.a. scene) of the dataset, together with, for each scene in the dataset, a set of coefficients $\sigma$ with respect to this base, which acts as individual representations. The matrices formed by taking the linear combinations of the base matrices with the coefficients corresponding to a given scene serve as the weight matrices of the network. In addition to the set of coefficients, we also learn an individual bias vector for each scene. The architecture is therefore composed of two parts: 1)- a representation provider, which takes the index i of a scene, and outputs a weight matrix and bias vector for each layer. 2)- The synthesis network which, using the weights and biases, computes the values of the pixels by querying the network on the coordinates of all pixels. The model's parameters U, V and (the relevant column of) $\sigma$ are then updated by gradient backprogagation using a MSE minimization objective.

We have shown that this joint representation possesses good interpolation properties, and allows for a more light-weight storage of a whole database of light fields, exhibiting a ten-fold reduction in the size of the representation when compared to using a separate representation for each light field. Based on these results, we have then focused on the problem of generalizability of this learned joint representation to new scenes that have not been seen during training.

Graph Neural Networks (GNN) have become the de-facto deep architectures for most learning problems on data structured as graphs, such as graphs and nodes classification or regression, learning-based graph sampling or graph compression, and so on. Nevertheless, GNN act on different irregular domains represented by one or several graphs that may vary between the training phase and the testing phase, which severely constrain the manner in which they can be built compared to traditional Convolutional Neural Network, as they must respect rules of permutation invariance and equivariance. Moreover, in the absence of traditional convolution, computational complexity quickly becomes an issue, as modern graphs can comprise several millions of nodes. As a result, GNNs encounter many unique challenges limiting their performance, and a better understanding of their theoretical and empirical properties is required to adress them.

We have produced a new analysis of the properties of GNNs on large graphs by leveraging random graphs models, which are traditionally used to represent graphs sampled from manifolds, or real-world large graphs such as relational databases. In 22, we completely characterize the function space that such GNN can produce in the infinite-node limit, which emphasize GNNs' strength but also limitations. We outline the role of so-called graph Positional Encodings, a recent methodology inspired by Transformer models. In 35, we examine the effects of a crucial part in every GNN architecture on this infinite-node limit, the so-called aggregation function. We outline how different functions used in practice yield extremely different rates of convergence to the limit. Finally, in 36, we characterize the generalization properties of simple estimators related to the message-passing process at the core of GNNs on random graphs, a first step toward a more complete characterization.

Recent optimization methods for inverse problems have been introduced with the goal of combining the advantages of well understood iterative optimization techniques with those of learnable complex image priors. A first category of methods, referred to as ”Plug-and-play” methods, has been introduced where a learned network-based prior is plugged in an iterative optimization algorithm. These learnable priors can take several forms, the most common ones being: a projection operator on a learned image subspace, a proximal operator of a regularizer or a denoiser. Plug-and-Play optimization methods have in particular been introduced for solving inverse problems in imaging by plugging a denoiser into the ADMM (Alternating Direction Multiplier Method) optimization algorithm. The denoiser accounts for the regularization and therefore implicitly determines the prior knowledge on the data, hence replacing typical handcrafted priors.

In the context of the AI chair DeepCIM, we have shown that it is possible to train a network directly modeling the gradient of a MAP regularizer while jointly training the corresponding MAP denoiser. We have used this network in gradient-based optimization methods and obtain better results comparing to other generic Plug-and-Play approaches. We have also shown that the regularizer can be used as a pre-trained network for unrolled gradient descent. Lastly, we have shown that the resulting denoiser allows for a better convergence of the Plug-and-Play ADMM.

Although the design of learned denoisers has garnered significant research attention in the context of traditional 2D images, omnidirectional image denoising has received relatively limited attention in the literature. Furthermore, extending processing models and tools designed for 2D images to the sphere presents many challenges due to the inherent distortions and non-uniform pixel distributions associated with spherical representations and their underlying projections. We have addressed the problem of omnidirectional image denoising and studied the advantage of denoising the spherical image directly rather than its mapping 49. We have introduced a novel network called SphereDRUNet to denoise spherical images using deep learning tools on a spherical sampling. We have shown that denoising directly the sphere using our network gives better performance, compared to denoising the projected equirectangular images with a similarly learned model.

One advantage of plug-and-play methods with their learned priors is their genericity in the sense that they can be used for any inverse problem, and do not need to be re-trained for each new problem, in contrast with deep models learned as a regression function for a specific task. However, priors learned independently of the targeted problem may not yield the best solution. Unrolling a fixed number of iterations of optimization algorithms is another way of coupling optimization and deep learning techniques. The learnable network is trained end to- end within the iterative algorithm so that performing a fixed number of iterations yields optimized results for a given inverse problem. Several optimization algorithms (Iterative Shrinkage Thresholding Algorithm (ISTA), Half S Quadratic Splitting (HQS), and Alternating Direction Method of Multipliers (ADMM)) have been unrolled in the literature, where a learned regularization network is used at each iteration of the optimization algorithm. While usual iterative methods iterate until idempotence, i.e. until the difference between the input and the output is sufficiently small, the number of iterations in unrolled optimization methods is set to a small value, to cope with memory issues.

Another approach to cope with the memory issue of unrolled optimization methods is the Deep equilibrium (DEQ) model which extends unrolled methods to have a theoretically infinite amount of iterations by leveraging fixed-point properties. This allows for simpler back-propagation, which can even be done in a Jacobian-free manner. The resulting back-propagation has a memory-footprint independent of the number of iterations used. Deep equilibrium models, indeed avoid large memory consumption by back-propagating through the architecture’s fixed point. This results in an architecture that is intended to converge and allows for the prior to be more general. However, unrolled methods train a network end-to-end for a specific degradation and necessitate retraining for each specific degradation.

We have leveraged deep equilibrium models to train a plug-and-play style prior that can be used to solve inverse problems for a range of tasks 32. Deep equilibrium models (DEQs), iterate an unrolled method until convergence and thereby enable end-to-end training on the reconstruction error with simplified back-propagation. We have investigated to what extent a solution for several inverse problems can be found by employing a multi-task DEQ. This MT-DEQ is used to train a prior on the actual estimation error, in contrast to a theoretical noise model used for plug-and-play methods. This has the advantage that the resulting prior is trained for a range of degradations beyond pure Gaussian denoising.

The Retinex theory is particularly well suited to the restoration of nighttime images, as it can be used to decompose an image into the illumination of the scene and its reflectance. It is however not well suited for outdoor scenes where we can have colored light. In addition, it does not take into account the non linearities present in the image acquisition pipeline. We have therefore revisited the retinex model to better take into account the physics of light, i.e. by including a colored illumination, and by taking into account some non linearities of the acquisition pipeline.

Based on this new decomposition model, we have proposed a deep learning-based architecture inspired by the style transfer methods, with two generative adversarial networks (GAN) trained in an unsupervised manner 11. This deep neural network is composed of two branches, one for each of the components (illumination and reflectance) and includes instance normalization, as done in style transfer, to enforce a better separation between the style (illumination) and the content (reflectance), and thus better handle the component separation problem. The network is trained with additional loss terms corresponding to physical priors, the reflectance being the shared information between the night and daylight image distributions.

Zero-error source coding is at the core of practical compression algorithms. Here, zero error refers to the case where the data is decompressed without any error for any block-length. This must be differentiated from the vanishing error setup, where the probability of error of the decompressed data decreases to zero as the length of the processed data tends to infinity. In point-to point communication (one sender/one receiver), zero-error and vanishing-error information theoretic compression bounds coincides. This is however not the case in networks, with possibly many senders and/or receivers. For instance, in the source coding problem with a helper at the decoder, the compression bounds under both hypotheses do no coincide. More precisely, there exist examples, where both bounds differ. But more importantly, the zero-error compression characterization for the general case is still an open problem. This is due to the fact that the compression bounds are given by the so-called complementary graph entropy, which requires to compute the coloring of an infinite product of graphs. Therefore, this complementary graph entropy has no single-letter formula, except for some particular cases such as: perfect graphs, the pentagon graph, and their disjoint union. In 18, we derived a structural result that equates the complementary graph entropies of AND product and disjoint unions (up to a multiplicative constant), which has two consequences. First, we prove that the cases where the complementary graph entropies of these two compositions (AND product and disjoint union) can be linearized coincide. Second, we determine the complementary graph entropies in cases where it was unknown: AND products of perfect graphs; product of perfect graph and the pentagon graph, which are not perfect in general.

Due to the ever-growing amount of visual data produced and stored, there are now evidences that these data will not only be viewed by humans but also processed by machines. In this context, practical implementations aim to provide better compression efficiency when the primary consumers are machines that perform certain computer vision tasks. One difficulty arises from the fact that these tasks are not known upon compression. This problem has been studied in Information theory and is called the coding for computing problem. However, no single letter formula exists for this problem, due to the fact that the optimal coding performance depends on an infinite product of characteristic graphs, that represents the joint distribution of the data and the tasks. In 19, we introduced a generalization of this problem, where the encoder has some partial information about the request, for instance the type of the request that will be made at the decoder. We proposed a condition, which allows to have a single letter characterization of the optimal compression performance.

Learning on entropy coded data has many benefits. First, it avoids decoding, but also it allows one to process compact data. However, this type of learning has been overlooked due to the chaos introduced by entropy coding functions.

In 29, we proposed an empirical study to see whether learning with convolutional neural networks (CNNs) on entropy coded data is possible. First, we define spatial and semantic closeness, two key properties that we experimentally show to be necessary to guarantee the efficiency of the convolution. Then, we show that these properties are not satisfied by the data processed by an entropy coder. Despite this, our experimental results show that learning in such difficult conditions is still possible, and that the performances are far from a random guess. These results have been obtained thanks to the construction of CNN architectures designed for 1D data (one based on VGG, the other on ResNet). Finally, we propose some experiments that explain why CNN are still performing reasonably well on entropy coded data.

In 30, we introduced a new metric that measures the chaos in the data representation. This measure is easy to compute as it depends on the encoded data only. Moreover, for a family of entropy coders, this measure allows one to predict the accuracy of the learning algorithm that process entropy coded data.

An image is traditionally compressed with the aim of minimizing the error made in the reconstruction. The Mean Squared Error (MSE) naturally comes as a simple and efficient criterion to evaluate the fidelity of the output in terms of distortion. This metric remains widely used to evaluate methods compressing images ranging from high to low bitrates. But what happens when compression is pushed to the extreme ($\sim 0.02$ bpp)? While excessive compression of sensitive data such as movies or vacations photos is not desirable, a drastic reduction of storage could be welcomed for the so-called "cold data". This massive amount of data that is stored but almost never accessed is estimated to represent $60\%$ of today's storage while projected to become $80\%$ by 2025. In that case, compression at extremely low bitrates could be an alternative to deleting potentially useful data or keeping a huge amount of data that might not be used. However, when targeting such bitrates, the relevance of the MSE as an evaluation criterion drops. Indeed, there exists a tradeoff between the perceived quality of the output and the fidelity in terms of pixels. This is called the perception-distortion tradeoff. Moreover, it is shown that decreasing the bitrate exacerbates the opposing goals of the two metrics. Optimization of the MSE then leads to the apparition of numerous compression artifacts, which makes the use of such bitrates really unattractive.

In that context, we proposed in 17 a framework for image compression in which the fidelity criterion is replaced by a semantic and quality preservation objective. Encoding the image thus becomes a simple extraction of semantic, enabling to reach drastic compression ratios. The decoding side is handled by a generative model relying on the diffusion process for the reconstruction of images. We first propose to describe the semantic using low resolution segmentation maps as guide. We further improve the generation, introducing color maps guidance without retraining the generative decoder. We show that it is possible to produce images of high visual quality with preserved semantic at extremely low bitrates when compared with classical codecs (see Figure below).

Image sampling based on user perception has recently increased interest given the unstoppable increase of stored datasets. For example, let's think of the large-scale image collections associated with Instagram and other social media accounts, online retailers, or your own personal photo Gallery saved on the cloud. Storing such large-scale datasets has become a big burden, from an economical as well as sustainability perspective. Traditional image data sampling methods can be understood as a way to alleviate such burden, by retaining only pictures from a dataset with the ideal goal of providing a summary (in the sense of a brief description) of the database. The challenging aspect is however to understand which is the best set of images to select (not to delete), especially given the user's preferences. This is highly challenging as the user will perceive the deleted images as lost information.

In 14, we proposed an end-to-end user-centric sampling method aimed at selecting the images from an image collection that are able to maximize the information perceived by a given user. As main contributions, we first introduce novel metrics that assess the amount of perceived information retained by the user when experiencing a set of images. Given the actual information present in a set of images, which is the volume spanned by the set in the corresponding latent space, we show how to take into account the user's preferences in such a volume calculation to build a user-centric metric for the perceived information. Finally, we propose a sampling strategy seeking the minimum set of images that maximize the information perceived by a given user. Experiments using the coco dataset show the ability of the proposed approach to accurately integrate user preference while keeping a reasonable diversity in the sampled image set.

In 28, we proposed an algorithm to extract, from a given rectangular matrix, a submatrix with maximum volume, whose number of extracted columns is greater than the initial matrix rank. This problem arises in compression and summarization of databases, recommender systems, learning, numerical analysis or applied linear algebra. We use a continuous relaxation of the maximum volume matrix extraction problem, which admits a simple and closed form solution: the nonzero singular values of the extracted matrix must be equal. The proposed algorithm extracts matrices with singular values, which are close to be equal. It is inspired by a water-filling technique, traditionally dedicated to equalization strategies in communication channels. Simulations show that the proposed algorithm performs better than sampling methods based on determinantal point processes (DPPs) and achieves similar performance as the best known algorithm, but with a lower complexity.

In 2018, online video streaming constituted $60\%$ of the global data flow, and, by itself, generated $1\%$ of the global emissions (as much as a country like Spain emits). And this quantity explodes: video traffic is estimated to grow by 79$\%$ between 2021 and 2027. At the same time, the sixth report of the Intergovernmental Panel on Climate Change (IPCC) states that if we want to keep the global warming under 1.5°C (Paris agreement), one should target a global emissions decrease of $50\%$ when compared to those of 2019. This corresponds to a decrease of $7.6\%$ per year. They also state that this is not the path that is currently taken. Hence, every part of our society must urgently aim sobriety. This is for example the case of online video streaming. So the question is simple: what are the solutions that should be envisaged to halve by 2030 the GHG emissions due to video streaming ?

As these emissions are correlated (not necessarily proportional) to the video data flow, one could rely on the progress of video compression algorithms to decrease the size of every individual video. For example, the last H.266/VVC standard reaches $50\%$ of gain when compared with the previous codec H.265/HEVC published 7 years before. Even though similar gains can be expected for the future codec, this cannot, by itself, enables a drastic emissions reduction, because of at least three reasons. First, the size reduction is achieved on individual videos and not on the global flow. In other words, video compression does not fight against the number of videos that is created or consumed. Yet, this amount of video streamed will explode in the coming years. Second, compression gains are usually not exploited to decrease the required bandwidth but rather to increase the video resolution or to create new video usages (IoT, Virtual Reality, etc.). Last, compression gains are achieved only at the expense of huge algorithm complexity. Encoding a video becomes more and more complex and thus requires, for each codec, more energy Another cause of video streaming gas emissions is precisely the complexity of the operations all along the transmission chain (including the video coding algorithms as mentioned just above). Intensive research efforts have been made to decrease this complexity and thus the consumed energy.

Even though these research results are crucial and make the energy expenditure more optimized, we demonstrate in 27 that this is insufficient to achieve the goal set by the IPCC. We build a simple model derived from the Kaya identity. This model enables to compare the order of magnitude of the energy reduction on one side and the digital affluence on the other side. Results show that even drastic energy reduction cannot cope with the forecasted video data flow explosion. Said differently, to achieve Paris agreement objectives (half emissions in 2030 when compared to 2019), the academic, industrial and politic actors must also consider laws and regulation to limit the video data consumption.