Cédric Richard

Xiuheng Wang, Jie Chen, Cédric Richard

Deconvolution is a widely used strategy to mitigate the blurring and noisy degradation of hyperspectral images~(HSI) generated by the acquisition devices. This issue is usually addressed by solving an ill-posed inverse problem. While investigating proper image priors can enhance the deconvolution performance, it is not trivial to handcraft a powerful regularizer and to set the regularization parameters. To address these issues, in this paper we introduce a tuning-free Plug-and-Play (PnP) algorithm for HSI deconvolution. Specifically, we use the alternating direction method of multipliers (ADMM) to decompose the optimization problem into two iterative sub-problems. A flexible blind 3D denoising network (B3DDN) is designed to learn deep priors and to solve the denoising sub-problem with different noise levels. A measure of 3D residual whiteness is then investigated to adjust the penalty parameters when solving the quadratic sub-problems, as well as a stopping criterion. Experimental results on both simulated and real-world data with ground-truth demonstrate the superiority of the proposed method.

Xiuheng Wang, Ricardo Borsoi, Arnaud Breloy et al.

Non-parametric change-point detection in streaming time series data is a long-standing challenge in signal processing. Recent advancements in statistics and machine learning have increasingly addressed this problem for data residing on Riemannian manifolds. One prominent strategy involves monitoring abrupt changes in the center of mass of the time series. Implemented in a streaming fashion, this strategy, however, requires careful step size tuning when computing the updates of the center of mass. In this paper, we propose to leverage robust centroid on manifolds from M-estimation theory to address this issue. Our proposal consists of comparing two centroid estimates: the classical Karcher mean (sensitive to change) versus one defined from Huber's function (robust to change). This comparison leads to the definition of a test statistic whose performance is less sensitive to the underlying estimation method. We propose a stochastic Riemannian optimization algorithm to estimate both robust centroids efficiently. Experiments conducted on both simulated and real-world data across two representative manifolds demonstrate the superior performance of our proposed method.

ChenTong Wang, Jincheng Gao, Fei Zhu et al.

Transformers have shown significant success in hyperspectral unmixing (HU). However, challenges remain. While multi-scale and long-range spatial correlations are essential in unmixing tasks, current Transformer-based unmixing networks, built on Vision Transformer (ViT) or Swin-Transformer, struggle to capture them effectively. Additionally, current Transformer-based unmixing networks rely on the linear mixing model, which lacks the flexibility to accommodate scenarios where nonlinear effects are significant. To address these limitations, we propose a multi-scale Dilated Transformer-based unmixing network for nonlinear HU (DTU-Net). The encoder employs two branches. The first one performs multi-scale spatial feature extraction using Multi-Scale Dilated Attention (MSDA) in the Dilated Transformer, which varies dilation rates across attention heads to capture long-range and multi-scale spatial correlations. The second one performs spectral feature extraction utilizing 3D-CNNs with channel attention. The outputs from both branches are then fused to integrate multi-scale spatial and spectral information, which is subsequently transformed to estimate the abundances. The decoder is designed to accommodate both linear and nonlinear mixing scenarios. Its interpretability is enhanced by explicitly modeling the relationships between endmembers, abundances, and nonlinear coefficients in accordance with the polynomial post-nonlinear mixing model (PPNMM). Experiments on synthetic and real datasets validate the effectiveness of the proposed DTU-Net compared to PPNMM-derived methods and several advanced unmixing networks.

Xiuheng Wang, Jie Chen, Cédric Richard

To overcome inherent hardware limitations of hyperspectral imaging systems with respect to their spatial resolution, fusion-based hyperspectral image (HSI) super-resolution is attracting increasing attention. This technique aims to fuse a low-resolution (LR) HSI and a conventional high-resolution (HR) RGB image in order to obtain an HR HSI. Recently, deep learning architectures have been used to address the HSI super-resolution problem and have achieved remarkable performance. However, they ignore the degradation model even though this model has a clear physical interpretation and may contribute to improve the performance. We address this problem by proposing a method that, on the one hand, makes use of the linear degradation model in the data-fidelity term of the objective function and, on the other hand, utilizes the output of a convolutional neural network for designing a deep prior regularizer in spectral and spatial gradient domains. Experiments show the performance improvement achieved with this strategy.

Roula Nassif, Stefan Vlaski, Cedric Richard et al.

The problem of learning simultaneously several related tasks has received considerable attention in several domains, especially in machine learning with the so-called multitask learning problem or learning to learn problem [1], [2]. Multitask learning is an approach to inductive transfer learning (using what is learned for one problem to assist in another problem) and helps improve generalization performance relative to learning each task separately by using the domain information contained in the training signals of related tasks as an inductive bias. Several strategies have been derived within this community under the assumption that all data are available beforehand at a fusion center. However, recent years have witnessed an increasing ability to collect data in a distributed and streaming manner. This requires the design of new strategies for learning jointly multiple tasks from streaming data over distributed (or networked) systems. This article provides an overview of multitask strategies for learning and adaptation over networks. The working hypothesis for these strategies is that agents are allowed to cooperate with each other in order to learn distinct, though related tasks. The article shows how cooperation steers the network limiting point and how different cooperation rules allow to promote different task relatedness models. It also explains how and when cooperation over multitask networks outperforms non-cooperative strategies.

Ricardo Augusto Borsoi, Tales Imbiriba, Pau Closas et al.

The recent evolution of hyperspectral imaging technology and the proliferation of new emerging applications presses for the processing of multiple temporal hyperspectral images. In this work, we propose a novel spectral unmixing (SU) strategy using physically motivated parametric endmember representations to account for temporal spectral variability. By representing the multitemporal mixing process using a state-space formulation, we are able to exploit the Bayesian filtering machinery to estimate the endmember variability coefficients. Moreover, by assuming that the temporal variability of the abundances is small over short intervals, an efficient implementation of the expectation maximization (EM) algorithm is employed to estimate the abundances and the other model parameters. Simulation results indicate that the proposed strategy outperforms state-of-the-art multitemporal SU algorithms.

Ricardo Augusto Borsoi, Tales Imbiriba, José Carlos Moreira Bermudez et al.

Multiple Endmember Spectral Mixture Analysis (MESMA) is one of the leading approaches to perform spectral unmixing (SU) considering variability of the endmembers (EMs). It represents each EM in the image using libraries of spectral signatures acquired a priori. However, existing spectral libraries are often small and unable to properly capture the variability of each EM in practical scenes, which compromises the performance of MESMA. In this paper, we propose a library augmentation strategy to increase the diversity of existing spectral libraries, thus improving their ability to represent the materials in real images. First, we leverage the power of deep generative models to learn the statistical distribution of the EMs based on the spectral signatures available in the existing libraries. Afterwards, new samples can be drawn from the learned EM distributions and used to augment the spectral libraries, improving the overall quality of the SU process. Experimental results using synthetic and real data attest the superior performance of the proposed method even under library mismatch conditions.

Ricardo Augusto Borsoi, Tales Imbiriba, José Carlos Moreira Bermudez et al.

Introducing spatial prior information in hyperspectral imaging (HSI) analysis has led to an overall improvement of the performance of many HSI methods applied for denoising, classification, and unmixing. Extending such methodologies to nonlinear settings is not always straightforward, specially for unmixing problems where the consideration of spatial relationships between neighboring pixels might comprise intricate interactions between their fractional abundances and nonlinear contributions. In this paper, we consider a multiscale regularization strategy for nonlinear spectral unmixing with kernels. The proposed methodology splits the unmixing problem into two sub-problems at two different spatial scales: a coarse scale containing low-dimensional structures, and the original fine scale. The coarse spatial domain is defined using superpixels that result from a multiscale transformation. Spectral unmixing is then formulated as the solution of quadratically constrained optimization problems, which are solved efficiently by exploring their strong duality and a reformulation of their dual cost functions in the form of root-finding problems. Furthermore, we employ a theory-based statistical framework to devise a consistent strategy to estimate all required parameters, including both the regularization parameters of the algorithm and the number of superpixels of the transformation, resulting in a truly blind (from the parameters setting perspective) unmixing method. Experimental results attest the superior performance of the proposed method when comparing with other, state-of-the-art, related strategies.

Ricardo Augusto Borsoi, Tales Imbiriba, José Carlos Moreira Bermudez et al.

Sparse hyperspectral unmixing from large spectral libraries has been considered to circumvent limitations of endmember extraction algorithms in many applications. This strategy often leads to ill-posed inverse problems, which can benefit from spatial regularization strategies. While existing spatial regularization methods improve the problem conditioning and promote piecewise smooth solutions, they lead to large nonsmooth optimization problems. Thus, efficiently introducing spatial context in the unmixing problem remains a challenge, and a necessity for many real world applications. In this paper, a novel multiscale spatial regularization approach for sparse unmixing is proposed. The method uses a signal-adaptive spatial multiscale decomposition based on superpixels to decompose the unmixing problem into two simpler problems, one in the approximation domain and another in the original domain. Simulation results using both synthetic and real data indicate that the proposed method can outperform state-of-the-art Total Variation-based algorithms with a computation time comparable to that of their unregularized counterparts.

Ibrahim El Khalil Harrane, Rémi Flamary, Cédric Richard

The rise of digital and mobile communications has recently made the world more connected and networked, resulting in an unprecedented volume of data flowing between sources, data centers, or processes. While these data may be processed in a centralized manner, it is often more suitable to consider distributed strategies such as diffusion as they are scalable and can handle large amounts of data by distributing tasks over networked agents. Although it is relatively simple to implement diffusion strategies over a cluster, it appears to be challenging to deploy them in an ad-hoc network with limited energy budget for communication. In this paper, we introduce a diffusion LMS strategy that significantly reduces communication costs without compromising the performance. Then, we analyze the proposed algorithm in the mean and mean-square sense. Next, we conduct numerical experiments to confirm the theoretical findings. Finally, we perform large scale simulations to test the algorithm efficiency in a scenario where energy is limited.

Alec Koppel, Santiago Paternain, Cedric Richard et al.

We consider multi-agent stochastic optimization problems over reproducing kernel Hilbert spaces (RKHS). In this setting, a network of interconnected agents aims to learn decision functions, i.e., nonlinear statistical models, that are optimal in terms of a global convex functional that aggregates data across the network, with only access to locally and sequentially observed samples. We propose solving this problem by allowing each agent to learn a local regression function while enforcing consensus constraints. We use a penalized variant of functional stochastic gradient descent operating simultaneously with low-dimensional subspace projections. These subspaces are constructed greedily by applying orthogonal matching pursuit to the sequence of kernel dictionaries and weights. By tuning the projection-induced bias, we propose an algorithm that allows for each individual agent to learn, based upon its locally observed data stream and message passing with its neighbors only, a regression function that is close to the globally optimal regression function. That is, we establish that with constant step-size selections agents' functions converge to a neighborhood of the globally optimal one while satisfying the consensus constraints as the penalty parameter is increased. Moreover, the complexity of the learned regression functions is guaranteed to remain finite. On both multi-class kernel logistic regression and multi-class kernel support vector classification with data generated from class-dependent Gaussian mixture models, we observe stable function estimation and state of the art performance for distributed online multi-class classification. Experiments on the Brodatz textures further substantiate the empirical validity of this approach.

Simone Scardapane, Jie Chen, Cédric Richard

In this chapter, we analyze nonlinear filtering problems in distributed environments, e.g., sensor networks or peer-to-peer protocols. In these scenarios, the agents in the environment receive measurements in a streaming fashion, and they are required to estimate a common (nonlinear) model by alternating local computations and communications with their neighbors. We focus on the important distinction between single-task problems, where the underlying model is common to all agents, and multitask problems, where each agent might converge to a different model due to, e.g., spatial dependencies or other factors. Currently, most of the literature on distributed learning in the nonlinear case has focused on the single-task case, which may be a strong limitation in real-world scenarios. After introducing the problem and reviewing the existing approaches, we describe a simple kernel-based algorithm tailored for the multitask case. We evaluate the proposal on a simulated benchmark task, and we conclude by detailing currently open problems and lines of research.

Roula Nassif, Cédric Richard, André Ferrari et al.

The multitask diffusion LMS is an efficient strategy to simultaneously infer, in a collaborative manner, multiple parameter vectors. Existing works on multitask problems assume that all agents respond to data synchronously. In several applications, agents may not be able to act synchronously because networks can be subject to several sources of uncertainties such as changing topology, random link failures, or agents turning on and off for energy conservation. In this work, we describe a model for the solution of multitask problems over asynchronous networks and carry out a detailed mean and mean-square error analysis. Results show that sufficiently small step-sizes can still ensure both stability and performance. Simulations and illustrative examples are provided to verify the theoretical findings. The framework is applied to a particular application involving spectral sensing.

T. Imbiriba, J. C. M. Bermudez, C. Richard et al.

The profusion of spectral bands generated by the acquisition process of hyperspectral images generally leads to high computational costs. Such difficulties arise in particular with nonlinear unmixing methods, which are naturally more complex than linear ones. This complexity, associated with the high redundancy of information within the complete set of bands, make the search of band selection algorithms relevant. With this work, we propose a band selection strategy in reproducing kernel Hilbert spaces that allows to drastically reduce the processing time required by nonlinear unmixing techniques. Simulation results show a complexity reduction of two orders of magnitude without compromising unmixing performance.

Rita Ammanouil, André Ferrari, Cédric Richard

This paper introduces a graph Laplacian regularization in the hyperspectral unmixing formulation. The proposed regularization relies upon the construction of a graph representation of the hyperspectral image. Each node in the graph represents a pixel's spectrum, and edges connect spectrally and spatially similar pixels. The proposed graph framework promotes smoothness in the estimated abundance maps and collaborative estimation between homogeneous areas of the image. The resulting convex optimization problem is solved using the Alternating Direction Method of Multipliers (ADMM). A special attention is given to the computational complexity of the algorithm, and Graph-cut methods are proposed in order to reduce the computational burden. Finally, simulations conducted on synthetic data illustrate the effectiveness of the graph Laplacian regularization with respect to other classical regularizations for hyperspectral unmixing.

Masa-aki Takizawa, Masahiro Yukawa, Cedric Richard

This paper presents a stochastic behavior analysis of a kernel-based stochastic restricted-gradient descent method. The restricted gradient gives a steepest ascent direction within the so-called dictionary subspace. The analysis provides the transient and steady state performance in the mean squared error criterion. It also includes stability conditions in the mean and mean-square sense. The present study is based on the analysis of the kernel normalized least mean square (KNLMS) algorithm initially proposed by Chen et al. Simulation results validate the analysis.

Jie Chen, Wei Gao, Cédric Richard et al.

The kernel least-mean-square (KLMS) algorithm is an appealing tool for online identification of nonlinear systems due to its simplicity and robustness. In addition to choosing a reproducing kernel and setting filter parameters, designing a KLMS adaptive filter requires to select a so-called dictionary in order to get a finite-order model. This dictionary has a significant impact on performance, and requires careful consideration. Theoretical analysis of KLMS as a function of dictionary setting has rarely, if ever, been addressed in the literature. In an analysis previously published by the authors, the dictionary elements were assumed to be governed by the same probability density function of the input data. In this paper, we modify this study by considering the dictionary as part of the filter parameters to be set. This theoretical analysis paves the way for future investigations on KLMS dictionary design.

Jie Chen, Cédric Richard, Alfred O. Hero

Incorporating spatial information into hyperspectral unmixing procedures has been shown to have positive effects, due to the inherent spatial-spectral duality in hyperspectral scenes. Current research works that consider spatial information are mainly focused on the linear mixing model. In this paper, we investigate a variational approach to incorporating spatial correlation into a nonlinear unmixing procedure. A nonlinear algorithm operating in reproducing kernel Hilbert spaces, associated with an $\ell_1$ local variation norm as the spatial regularizer, is derived. Experimental results, with both synthetic and real data, illustrate the effectiveness of the proposed scheme.

Wei Gao, Jie Chen, Cédric Richard et al.

Adaptive filtering algorithms operating in reproducing kernel Hilbert spaces have demonstrated superiority over their linear counterpart for nonlinear system identification. Unfortunately, an undesirable characteristic of these methods is that the order of the filters grows linearly with the number of input data. This dramatically increases the computational burden and memory requirement. A variety of strategies based on dictionary learning have been proposed to overcome this severe drawback. Few, if any, of these works analyze the problem of updating the dictionary in a time-varying environment. In this paper, we present an analytical study of the convergence behavior of the Gaussian least-mean-square algorithm in the case where the statistics of the dictionary elements only partially match the statistics of the input data. This allows us to emphasize the need for updating the dictionary in an online way, by discarding the obsolete elements and adding appropriate ones. We introduce a kernel least-mean-square algorithm with L1-norm regularization to automatically perform this task. The stability in the mean of this method is analyzed, and its performance is tested with experiments.