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This work introduces an on-the-fly (i.e., online) linear unmixing method which is able to sequentially analyze spectral data acquired on a spectrum-by-spectrum basis. After deriving a sequential counterpart of the conventional linear mixing model, the proposed approach recasts the linear unmixing problem into a linear state-space estimation framework. Under Gaussian noise and state models, the estimation of the pure spectra can be efficiently conducted by resorting to Kalman filtering. Interestingly, it is shown that this Kalman filter can operate in a lower-dimensional subspace while ensuring the nonnegativity constraint inherent to pure spectra. This dimensionality reduction allows significantly lightening the computational burden, while leveraging recent advances related to the representation of essential spectral information. The proposed method is evaluated through extensive numerical experiments conducted on synthetic and real Raman data sets. The results show that this Kalman filter-based method offers a convenient trade-off between unmixing accuracy and computational efficiency, which is crucial for operating in an on-the-fly setting. To the best of the authors' knowledge, this is the first operational method which is able to solve the spectral unmixing problem efficiently in a dynamic fashion. It also constitutes a valuable building block for benefiting from acquisition and processing frameworks recently proposed in the microscopy literature, which are motivated by practical issues such as reducing acquisition time and avoiding potential damages being inflicted to photosensitive samples.

Radio frequency interference (RFI) have been an enduring concern in radio astronomy, particularly for the observations of pulsars which require high timing precision and data sensitivity. In most works of the literature, RFI mitigation has been formulated as a detection task that consists of localizing possible RFI in dynamic spectra. This strategy inevitably leads to a potential loss of information since parts of the signal identified as possibly RFI-corrupted are generally not considered in the subsequent data processing pipeline. Conversely, this work proposes to tackle RFI mitigation as a joint detection and restoration that allows parts of the dynamic spectrum affected by RFI to be not only identified but also recovered. The proposed supervised method relies on a deep convolutional network whose architecture inherits the performance reached by a recent yet popular image-denoising network. To train this network, a whole simulation framework is built to generate large data sets according to physics-inspired and statistical models of the pulsar signals and of the RFI. The relevance of the proposed approach is quantitatively assessed by conducting extensive experiments. In particular, the results show that the restored dynamic spectra are sufficiently reliable to estimate pulsar times-of-arrivals with an accuracy close to the one that would be obtained from RFI-free signals.

This paper introduces a Bayesian framework for image inversion by deriving a probabilistic counterpart to the regularization-by-denoising (RED) paradigm. It additionally implements a Monte Carlo algorithm specifically tailored for sampling from the resulting posterior distribution, based on an asymptotically exact data augmentation (AXDA). The proposed algorithm is an approximate instance of split Gibbs sampling (SGS) which embeds one Langevin Monte Carlo step. The proposed method is applied to common imaging tasks such as deblurring, inpainting and super-resolution, demonstrating its efficacy through extensive numerical experiments. These contributions advance Bayesian inference in imaging by leveraging data-driven regularization strategies within a probabilistic framework.

Spectral unmixing has been extensively studied with a variety of methods and used in many applications. Recently, data-driven techniques with deep learning methods have obtained great attention to spectral unmixing for its superior learning ability to automatically learn the structure information. In particular, autoencoder based architectures are elaborately designed to solve blind unmixing and model complex nonlinear mixtures. Nevertheless, these methods perform unmixing task as blackboxes and lack of interpretability. On the other hand, conventional unmixing methods carefully design the regularizer to add explicit information, in which algorithms such as plug-and-play (PnP) strategies utilize off-the-shelf denoisers to plug powerful priors. In this paper, we propose a generic unmixing framework to integrate the autoencoder network with regularization by denoising (RED), named AE-RED. More specially, we decompose the unmixing optimized problem into two subproblems. The first one is solved using deep autoencoders to implicitly regularize the estimates and model the mixture mechanism. The second one leverages the denoiser to bring in the explicit information. In this way, both the characteristics of the deep autoencoder based unmixing methods and priors provided by denoisers are merged into our well-designed framework to enhance the unmixing performance. Experiment results on both synthetic and real data sets show the superiority of our proposed framework compared with state-of-the-art unmixing approaches.

When adopting a model-based formulation, solving inverse problems encountered in multiband imaging requires to define spatial and spectral regularizations. In most of the works of the literature, spectral information is extracted from the observations directly to derive data-driven spectral priors. Conversely, the choice of the spatial regularization often boils down to the use of conventional penalizations (e.g., total variation) promoting expected features of the reconstructed image (e.g., piecewise constant). In this work, we propose a generic framework able to capitalize on an auxiliary acquisition of high spatial resolution to derive tailored data-driven spatial regularizations. This approach leverages on the ability of deep learning to extract high level features. More precisely, the regularization is conceived as a deep generative network able to encode spatial semantic features contained in this auxiliary image of high spatial resolution. To illustrate the versatility of this approach, it is instantiated to conduct two particular tasks, namely multiband image fusion and multiband image inpainting. Experimental results obtained on these two tasks demonstrate the benefit of this class of informed regularizations when compared to more conventional ones.

Normalizing flows (NF) use a continuous generator to map a simple latent (e.g. Gaussian) distribution, towards an empirical target distribution associated with a training data set. Once trained by minimizing a variational objective, the learnt map provides an approximate generative model of the target distribution. Since standard NF implement differentiable maps, they may suffer from pathological behaviors when targeting complex distributions. For instance, such problems may appear for distributions on multi-component topologies or characterized by multiple modes with high probability regions separated by very unlikely areas. A typical symptom is the explosion of the Jacobian norm of the transformation in very low probability areas. This paper proposes to overcome this issue thanks to a new Markov chain Monte Carlo algorithm to sample from the target distribution in the latent domain before transporting it back to the target domain. The approach relies on a Metropolis adjusted Langevin algorithm (MALA) whose dynamics explicitly exploits the Jacobian of the transformation. Contrary to alternative approaches, the proposed strategy preserves the tractability of the likelihood and it does not require a specific training. Notably, it can be straightforwardly used with any pre-trained NF network, regardless of the architecture. Experiments conducted on synthetic and high-dimensional real data sets illustrate the efficiency of the method.

This paper introduces a stochastic plug-and-play (PnP) sampling algorithm that leverages variable splitting to efficiently sample from a posterior distribution. The algorithm based on split Gibbs sampling (SGS) draws inspiration from the alternating direction method of multipliers (ADMM). It divides the challenging task of posterior sampling into two simpler sampling problems. The first problem depends on the likelihood function, while the second is interpreted as a Bayesian denoising problem that can be readily carried out by a deep generative model. Specifically, for an illustrative purpose, the proposed method is implemented in this paper using state-of-the-art diffusion-based generative models. Akin to its deterministic PnP-based counterparts, the proposed method exhibits the great advantage of not requiring an explicit choice of the prior distribution, which is rather encoded into a pre-trained generative model. However, unlike optimization methods (e.g., PnP-ADMM) which generally provide only point estimates, the proposed approach allows conventional Bayesian estimators to be accompanied by confidence intervals at a reasonable additional computational cost. Experiments on commonly studied image processing problems illustrate the efficiency of the proposed sampling strategy. Its performance is compared to recent state-of-the-art optimization and sampling methods.

In this paper we consider the problem of linear unmixing hidden random variables defined over the simplex with additive Gaussian noise, also known as probabilistic simplex component analysis (PRISM). Previous solutions to tackle this challenging problem were based on geometrical approaches or computationally intensive variational methods. In contrast, we propose a conventional expectation maximization (EM) algorithm which embeds importance sampling. For this purpose, the proposal distribution is chosen as a simple surrogate distribution of the target posterior that is guaranteed to lie in the simplex. This distribution is based on the Gaussian linear minimum mean squared error (LMMSE) approximation which is accurate at high signal-to-noise ratio. Numerical experiments in different settings demonstrate the advantages of this adaptive surrogate over state-of-the-art methods.

Despite their advantages, normalizing flows generally suffer from several shortcomings including their tendency to generate unrealistic data (e.g., images) and their failing to detect out-of-distribution data. One reason for these deficiencies lies in the training strategy which traditionally exploits a maximum likelihood principle only. This paper proposes a new training paradigm based on a hybrid objective function combining the maximum likelihood principle (MLE) and a sliced-Wasserstein distance. Results obtained on synthetic toy examples and real image data sets show better generative abilities in terms of both likelihood and visual aspects of the generated samples. Reciprocally, the proposed approach leads to a lower likelihood of out-of-distribution data, demonstrating a greater data fidelity of the resulting flows.

Optimal transport (OT) provides effective tools for comparing and mapping probability measures. We propose to leverage the flexibility of neural networks to learn an approximate optimal transport map. More precisely, we present a new and original method to address the problem of transporting a finite set of samples associated with a first underlying unknown distribution towards another finite set of samples drawn from another unknown distribution. We show that a particular instance of invertible neural networks, namely the normalizing flows, can be used to approximate the solution of this OT problem between a pair of empirical distributions. To this aim, we propose to relax the Monge formulation of OT by replacing the equality constraint on the push-forward measure by the minimization of the corresponding Wasserstein distance. The push-forward operator to be retrieved is then restricted to be a normalizing flow which is trained by optimizing the resulting cost function. This approach allows the transport map to be discretized as a composition of functions. Each of these functions is associated to one sub-flow of the network, whose output provides intermediate steps of the transport between the original and target measures. This discretization yields also a set of intermediate barycenters between the two measures of interest. Experiments conducted on toy examples as well as a challenging task of unsupervised translation demonstrate the interest of the proposed method. Finally, some experiments show that the proposed approach leads to a good approximation of the true OT.

In the context of Earth observation, the detection of changes is performed from multitemporal images acquired by sensors with possibly different spatial and/or spectral resolutions or even different modalities (e.g. optical, radar). Even limiting to the optical modality, this task has proved to be challenging as soon as the sensors have different spatial and/or spectral resolutions. This paper proposes a novel unsupervised change detection method dedicated to images acquired with such so-called heterogeneous optical sensors. This method capitalizes on recent advances which frame the change detection problem into a robust fusion framework. More precisely, we show that a deep adversarial network designed and trained beforehand to fuse a pair of multiband optical images can be easily complemented by a network with the same architecture to perform change detection. The resulting overall architecture itself follows an adversarial strategy where the fusion network and the additional network are interpreted as essential building blocks of a generator. A comparison with state-of-the-art change detection methods demonstrates the versatility and the effectiveness of the proposed approach.

In this work, we tackle the problem of hyperspectral (HS) unmixing by departing from the usual linear model and focusing on a Linear-Quadratic (LQ) one. The proposed algorithm, referred to as Successive Nonnegative Projection Algorithm for Linear Quadratic mixtures (SNPALQ), extends the Successive Nonnegative Projection Algorithm (SNPA), designed to address the unmixing problem under a linear model. By explicitly modeling the product terms inherent to the LQ model along the iterations of the SNPA scheme, the nonlinear contributions in the mixing are mitigated, thus improving the separation quality. The approach is shown to be relevant in a realistic numerical experiment., Comment: in Proceedings of iTWIST'20, Paper-ID: 10, Nantes, France, December, 2-4, 2020

In this work, we consider the problem of blind source separation (BSS) by departing from the usual linear model and focusing on the linear-quadratic (LQ) model. We propose two provably robust and computationally tractable algorithms to tackle this problem under separability assumptions which require the sources to appear as samples in the data set. The first algorithm generalizes the successive nonnegative projection algorithm (SNPA), designed for linear BSS, and is referred to as SNPALQ. By explicitly modeling the product terms inherent to the LQ model along the iterations of the SNPA scheme, the nonlinear contributions of the mixing are mitigated, thus improving the separation quality. SNPALQ is shown to be able to recover the ground truth factors that generated the data, even in the presence of noise. The second algorithm is a brute-force (BF) algorithm, which is used as a post-processing step for SNPALQ. It enables to discard the spurious (mixed) samples extracted by SNPALQ, thus broadening its applicability. The BF is in turn shown to be robust to noise under easier-to-check and milder conditions than SNPALQ. We show that SNPALQ with and without the BF postprocessing is relevant in realistic numerical experiments., Comment: 23 pages + 24 pages of Appendix containing the proofs

When no arterial input function is available, quantification of dynamic PET images requires a previous step devoted to the extraction of a reference time-activity curve (TAC). Factor analysis is often applied for this purpose. This paper introduces a novel approach that conducts a new kind of nonlinear factor analysis relying on a compartment model, and computes the kinetic parameters of specific binding tissues jointly. To this end, it capitalizes on data-driven parametric imaging methods to provide a physical description of the underlying PET data, directly relating the specific binding with the kinetics of the non-specific binding in the corresponding tissues. This characterization is introduced into the factor analysis formulation to yield a novel nonlinear unmixing model designed for PET image analysis. This model also explicitly introduces global kinetic parameters that allow for a direct estimation of the binding potential with respect to the free fractions in each non-specific binding tissue. The performance of the method is evaluated on synthetic and real data to demonstrate its potential interest.

Efficient sampling from a high-dimensional Gaussian distribution is an old but high-stake issue. Vanilla Cholesky samplers imply a computational cost and memory requirements which can rapidly become prohibitive in high dimension. To tackle these issues, multiple methods have been proposed from different communities ranging from iterative numerical linear algebra to Markov chain Monte Carlo (MCMC) approaches. Surprisingly, no complete review and comparison of these methods have been conducted. This paper aims at reviewing all these approaches by pointing out their differences, close relations, benefits and limitations. In addition to this state of the art, this paper proposes a unifying Gaussian simulation framework by deriving a stochastic counterpart of the celebrated proximal point algorithm in optimization. This framework offers a novel and unifying revisit of most of the existing MCMC approaches while extending them. Guidelines to choose the appropriate Gaussian simulation method for a given sampling problem in high dimension are proposed and illustrated with numerical examples., Comment: 53 pages, 11 figures

This paper discusses the reconstruction of partially sampled spectrum-images to accelerate the acquisition in scanning transmission electron microscopy (STEM). The problem of image reconstruction has been widely considered in the literature for many imaging modalities, but only a few attempts handled 3D data such as spectral images acquired by STEM electron energy loss spectroscopy (EELS). Besides, among the methods proposed in the microscopy literature, some are fast but inaccurate while others provide accurate reconstruction but at the price of a high computation burden. Thus none of the proposed reconstruction methods fulfills our expectations in terms of accuracy and computation complexity. In this paper, we propose a fast and accurate reconstruction method suited for atomic-scale EELS. This method is compared to popular solutions such as beta process factor analysis (BPFA) which is used for the first time on STEM-EELS images. Experiments based on real as synthetic data will be conducted.

This paper aims at providing a comprehensive framework to generate an astrophysical scene and to simulate realistic hyperspectral and multispectral data acquired by two JWST instruments, namely NIRCam Imager and NIRSpec IFU. We want to show that this simulation framework can be resorted to assess the benefits of fusing these images to recover an image of high spatial and spectral resolutions. To do so, we create a synthetic scene associated with a canonical infrared source, the Orion Bar. This scene combines pre-existing modelled spectra provided by the JWST Early Release Science Program 1288 and real high resolution spatial maps from the Hubble space and ALMA telescopes. We develop forward models including corresponding noises for the two JWST instruments based on their technical designs and physical features. JWST observations are then simulated by applying the forward models to the aforementioned synthetic scene. We test a dedicated fusion algorithm we developed on these simulated observations. We show the fusion process reconstructs the high spatio-spectral resolution scene with a good accuracy on most areas, and we identify some limitations of the method to be tackled in future works. The synthetic scene and observations presented in the paper are made publicly available and can be used for instance to evaluate instrument models (aboard the JWST or on the ground), pipelines, or more sophisticated algorithms dedicated to JWST data analysis. Besides, fusion methods such as the one presented in this paper are shown to be promising tools to fully exploit the unprecedented capabilities of the JWST.

Hyperspectral imaging has become a significant source of valuable data for astronomers over the past decades. Current instrumental and observing time constraints allow direct acquisition of multispectral images, with high spatial but low spectral resolution, and hyperspectral images, with low spatial but high spectral resolution. To enhance scientific interpretation of the data, we propose a data fusion method which combines the benefits of each image to recover a high spatio-spectral resolution datacube. The proposed inverse problem accounts for the specificities of astronomical instruments, such as spectrally variant blurs. We provide a fast implementation by solving the problem in the frequency domain and in a low-dimensional subspace to efficiently handle the convolution operators as well as the high dimensionality of the data. We conduct experiments on a realistic synthetic dataset of simulated observation of the upcoming James Webb Space Telescope, and we show that our fusion algorithm outperforms state-of-the-art methods commonly used in remote sensing for Earth observation.

Hyperspectral unmixing aims at identifying a set of elementary spectra and the corresponding mixture coefficients for each pixel of an image. As the elementary spectra correspond to the reflectance spectra of real materials, they are often very correlated yielding an ill-conditioned problem. To enrich the model and to reduce ambiguity due to the high correlation, it is common to introduce spatial information to complement the spectral information. The most common way to introduce spatial information is to rely on a spatial regularization of the abundance maps. In this paper, instead of considering a simple but limited regularization process, spatial information is directly incorporated through the newly proposed context of spatial unmixing. Contextual features are extracted for each pixel and this additional set of observations is decomposed according to a linear model. Finally the spatial and spectral observations are unmixed jointly through a cofactorization model. In particular, this model introduces a coupling term used to identify clusters of shared spatial and spectral signatures. An evaluation of the proposed method is conducted on synthetic and real data and shows that results are accurate and also very meaningful since they describe both spatially and spectrally the various areas of the scene.

Data augmentation, by the introduction of auxiliary variables, has become an ubiquitous technique to improve convergence properties, simplify the implementation or reduce the computational time of inference methods such as Markov chain Monte Carlo ones. Nonetheless, introducing appropriate auxiliary variables while preserving the initial target probability distribution and offering a computationally efficient inference cannot be conducted in a systematic way. To deal with such issues, this paper studies a unified framework, coined asymptotically exact data augmentation (AXDA), which encompasses both well-established and more recent approximate augmented models. In a broader perspective, this paper shows that AXDA models can benefit from interesting statistical properties and yield efficient inference algorithms. In non-asymptotic settings, the quality of the proposed approximation is assessed with several theoretical results. The latter are illustrated on standard statistical problems. Supplementary materials including computer code for this paper are available online., Comment: 63 pages

Supervised classification and representation learning are two widely used classes of methods to analyze multivariate images. Although complementary, these methods have been scarcely considered jointly in a hierarchical modeling. In this paper, a method coupling these two approaches is designed using a matrix cofactorization formulation. Each task is modeled as a factorization matrix problem and a term relating both coding matrices is then introduced to drive an appropriate coupling. The link can be interpreted as a clustering operation over a low-dimensional representation vectors. The attribution vectors of the clustering are then used as features vectors for the classification task, i.e., the coding vectors of the corresponding factorization problem. A proximal gradient descent algorithm, ensuring convergence to a critical point of the objective function, is then derived to solve the resulting non-convex non-smooth optimization problem. An evaluation of the proposed method is finally conducted both on synthetic and real data in the specific context of hyperspectral image interpretation, unifying two standard analysis techniques, namely unmixing and classification.

Factor analysis has proven to be a relevant tool for extracting tissue time-activity curves (TACs) in dynamic PET images, since it allows for an unsupervised analysis of the data. Reliable and interpretable results are possible only if considered with respect to suitable noise statistics. However, the noise in reconstructed dynamic PET images is very difficult to characterize, despite the Poissonian nature of the count-rates. Rather than explicitly modeling the noise distribution, this work proposes to study the relevance of several divergence measures to be used within a factor analysis framework. To this end, the $\beta$-divergence, widely used in other applicative domains, is considered to design the data-fitting term involved in three different factor models. The performances of the resulting algorithms are evaluated for different values of $\beta$, in a range covering Gaussian, Poissonian and Gamma-distributed noises. The results obtained on two different types of synthetic images and one real image show the interest of applying non-standard values of $\beta$ to improve factor analysis., Comment: This manuscript has been accepted for publication in IEEE Trans. Medical Imaging

Archetypal scenarios for change detection generally consider two images acquired through sensors of the same modality. However, in some specific cases such as emergency situations, the only images available may be those acquired through sensors of different modalities. This paper addresses the problem of unsupervisedly detecting changes between two observed images acquired by sensors of different modalities with possibly different resolutions. These sensor dissimilarities introduce additional issues in the context of operational change detection that are not addressed by most of the classical methods. This paper introduces a novel framework to effectively exploit the available information by modelling the two observed images as a sparse linear combination of atoms belonging to a pair of coupled overcomplete dictionaries learnt from each observed image. As they cover the same geographical location, codes are expected to be globally similar, except for possible changes in sparse spatial locations. Thus, the change detection task is envisioned through a dual code estimation which enforces spatial sparsity in the difference between the estimated codes associated with each image. This problem is formulated as an inverse problem which is iteratively solved using an efficient proximal alternating minimization algorithm accounting for nonsmooth and nonconvex functions. The proposed method is applied to real images with simulated yet realistic and real changes. A comparison with state-of-the-art change detection methods evidences the accuracy of the proposed strategy., Comment: Submitted manuscript under consideration at Computer Vision and Image Understanding

Spectral variability is one of the major issue when conducting hyperspectral unmixing. Within a given image composed of some elementary materials (herein referred to as endmember classes), the spectral signature characterizing these classes may spatially vary due to intrinsic component fluctuations or external factors (illumination). These redundant multiple endmember spectra within each class adversely affect the performance of unmixing methods. This paper proposes a mixing model that explicitly incorporates a hierarchical structure of redundant multiple spectra representing each class. The proposed method is designed to promote sparsity on the selection of both spectra and classes within each pixel. The resulting unmixing algorithm is able to adaptively recover several bundles of endmember spectra associated with each class and robustly estimate abundances. In addition, its flexibility allows a variable number of classes to be present within each pixel of the hyperspectral image to be unmixed. The proposed method is compared with other state-of-the-art unmixing methods that incorporate sparsity using both simulated and real hyperspectral data. The results show that the proposed method can successfully determine the variable number of classes present within each class and estimate the corresponding class abundances.

This paper derives two new optimization-driven Monte Carlo algorithms inspired from variable splitting and data augmentation. In particular, the formulation of one of the proposed approaches is closely related to the alternating direction method of multipliers (ADMM) main steps. The proposed framework enables to derive faster and more efficient sampling schemes than the current state-of-the-art methods and can embed the latter. By sampling efficiently the parameter to infer as well as the hyperparameters of the problem, the generated samples can be used to approximate Bayesian estimators of the parameters to infer. Additionally, the proposed approach brings confidence intervals at a low cost contrary to optimization methods. Simulations on two often-studied signal processing problems illustrate the performance of the two proposed samplers. All results are compared to those obtained by recent state-of-the-art optimization and MCMC algorithms used to solve these problems.

Unsupervised change detection techniques are generally constrained to two multi-band optical images acquired at different times through sensors sharing the same spatial and spectral resolution. This scenario is suitable for a straight comparison of homologous pixels such as pixel-wise differencing. However, in some specific cases such as emergency situations, the only available images may be those acquired through different kinds of sensors with different resolutions. Recently some change detection techniques dealing with images with different spatial and spectral resolutions, have been proposed. Nevertheless, they are focused on a specific scenario where one image has a high spatial and low spectral resolution while the other has a low spatial and high spectral resolution. This paper addresses the problem of detecting changes between any two multi-band optical images disregarding their spatial and spectral resolution disparities. We propose a method that effectively uses the available information by modeling the two observed images as spatially and spectrally degraded versions of two (unobserved) latent images characterized by the same high spatial and high spectral resolutions. Covering the same scene, the latent images are expected to be globally similar except for possible changes in spatially sparse locations. Thus, the change detection task is envisioned through a robust fusion task which enforces the differences between the estimated latent images to be spatially sparse. We show that this robust fusion can be formulated as an inverse problem which is iteratively solved using an alternate minimization strategy. The proposed framework is implemented for an exhaustive list of applicative scenarios and applied to real multi-band optical images. A comparison with state-of-the-art change detection methods evidences the accuracy of the proposed robust fusion-based strategy.

Electron microscopy has shown to be a very powerful tool to map the chemical nature of samples at various scales down to atomic resolution. However, many samples can not be analyzed with an acceptable signal-to-noise ratio because of the radiation damage induced by the electron beam. This is particularly crucial for electron energy loss spectroscopy (EELS) which acquires spectral-spatial data and requires high beam intensity. Since scanning transmission electron microscopes (STEM) are able to acquire data cubes by scanning the electron probe over the sample and recording a spectrum for each spatial position, it is possible to design the scan pattern and to sample only specific pixels. As a consequence, partial acquisition schemes are now conceivable, provided a reconstruction of the full data cube is conducted as a post-processing step. This paper proposes two reconstruction algorithms for multi-band images acquired by STEM-EELS which exploits the spectral structure and the spatial smoothness of the image. The performance of the proposed schemes is illustrated thanks to experiments conducted on a realistic phantom dataset as well as real EELS spectrum-images.

Spectral unmixing methods incorporating spatial regularizations have demonstrated increasing interest. Although spatial regularizers which promote smoothness of the abundance maps have been widely used, they may overly smooth these maps and, in particular, may not preserve edges present in the hyperspectral image. Existing unmixing methods usually ignore these edge structures or use edge information derived from the hyperspectral image itself. However, this information may be affected by large amounts of noise or variations in illumination, leading to erroneous spatial information incorporated into the unmixing procedure. This paper proposes a simple, yet powerful, spectral unmixing framework which incorporates external data (i.e. LiDAR data). The LiDAR measurements can be easily exploited to adjust standard spatial regularizations applied to the unmixing process. The proposed framework is rigorously evaluated using two simulated datasets and a real hyperspectral image. It is compared with competing methods that rely on spatial information derived from a hyperspectral image. The results show that the proposed framework can provide better abundance estimates and, more specifically, can significantly improve the abundance estimates for pixels affected by shadows.

Within a supervised classification framework, labeled data are used to learn classifier parameters. Prior to that, it is generally required to perform dimensionality reduction via feature extraction. These preprocessing steps have motivated numerous research works aiming at recovering latent variables in an unsupervised context. This paper proposes a unified framework to perform classification and low-level modeling jointly. The main objective is to use the estimated latent variables as features for classification and to incorporate simultaneously supervised information to help latent variable extraction. The proposed hierarchical Bayesian model is divided into three stages: a first low-level modeling stage to estimate latent variables, a second stage clustering these features into statistically homogeneous groups and a last classification stage exploiting the (possibly badly) labeled data. Performance of the model is assessed in the specific context of hyperspectral image interpretation, unifying two standard analysis techniques, namely unmixing and classification.

So far, the problem of unmixing large or multitemporal hyperspectral datasets has been specifically addressed in the remote sensing literature only by a few dedicated strategies. Among them, some attempts have been made within a distributed estimation framework, in particular relying on the alternating direction method of multipliers (ADMM). In this paper, we propose to study the interest of a partially asynchronous distributed unmixing procedure based on a recently proposed asynchronous algorithm. Under standard assumptions, the proposed algorithm inherits its convergence properties from recent contributions in non-convex optimization, while allowing the problem of interest to be efficiently addressed. Comparisons with a distributed synchronous counterpart of the proposed unmixing procedure allow its interest to be assessed on synthetic and real data. Besides, thanks to its genericity and flexibility, the procedure investigated in this work can be implemented to address various matrix factorization problems., Comment: 13 pages, 12 figures, accepted for publication in IEEE Trans. Geosci. Remote Sens., 2018

Principal component analysis (PCA) is very popular to perform dimension reduction. The selection of the number of significant components is essential but often based on some practical heuristics depending on the application. Only few works have proposed a probabilistic approach able to infer the number of significant components. To this purpose, this paper introduces a Bayesian nonparametric principal component analysis (BNP-PCA). The proposed model projects observations onto a random orthogonal basis which is assigned a prior distribution defined on the Stiefel manifold. The prior on factor scores involves an Indian buffet process to model the uncertainty related to the number of components. The parameters of interest as well as the nuisance parameters are finally inferred within a fully Bayesian framework via Monte Carlo sampling. A study of the (in-)consistence of the marginal maximum a posteriori estimator of the latent dimension is carried out. A new estimator of the subspace dimension is proposed. Moreover, for sake of statistical significance, a Kolmogorov-Smirnov test based on the posterior distribution of the principal components is used to refine this estimate. The behaviour of the algorithm is first studied on various synthetic examples. Finally, the proposed BNP dimension reduction approach is shown to be easily yet efficiently coupled with clustering or latent factor models within a unique framework.

To analyze dynamic positron emission tomography (PET) images, various generic multivariate data analysis techniques have been considered in the literature, such as principal component analysis (PCA), independent component analysis (ICA), factor analysis and nonnegative matrix factorization (NMF). Nevertheless, these conventional approaches neglect any possible nonlinear variations in the time activity curves describing the kinetic behavior of tissues with specific binding, which limits their ability to recover a reliable, understandable and interpretable description of the data. This paper proposes an alternative analysis paradigm that accounts for spatial fluctuations in the exchange rate of the tracer between a free compartment and a specifically bound ligand compartment. The method relies on the concept of linear unmixing, usually applied on the hyperspectral domain, which combines NMF with a sum-to-one constraint that ensures an exhaustive description of the mixtures. The spatial variability of the signature corresponding to the specific binding tissue is explicitly modeled through a perturbed component. The performance of the method is assessed on both synthetic and real data and is shown to compete favorably when compared to other conventional analysis methods. The proposed method improved both factor estimation and proportions extraction for specific binding. Modeling the variability of the specific binding factor has a strong potential impact for dynamic PET image analysis.

Hyperspectral unmixing is a blind source separation problem which consists in estimating the reference spectral signatures contained in a hyperspectral image, as well as their relative contribution to each pixel according to a given mixture model. In practice, the process is further complexified by the inherent spectral variability of the observed scene and the possible presence of outliers. More specifically, multi-temporal hyperspectral images, i.e., sequences of hyperspectral images acquired over the same area at different time instants, are likely to simultaneously exhibit moderate endmember variability and abrupt spectral changes either due to outliers or to significant time intervals between consecutive acquisitions. Unless properly accounted for, these two perturbations can significantly affect the unmixing process. In this context, we propose a new unmixing model for multitemporal hyperspectral images accounting for smooth temporal variations, construed as spectral variability, and abrupt spectral changes interpreted as outliers. The proposed hierarchical Bayesian model is inferred using a Markov chain Monte-Carlo (MCMC) method allowing the posterior of interest to be sampled and Bayesian estimators to be approximated. A comparison with unmixing techniques from the literature on synthetic and real data allows the interest of the proposed approach to be appreciated., Comment: 14 pages, 13 figures, accepted for publication in IEEE Trans. Comput. Imag., 2017

Change detection is one of the most challenging issues when analyzing remotely sensed images. Comparing several multi-date images acquired through the same kind of sensor is the most common scenario. Conversely, designing robust, flexible and scalable algorithms for change detection becomes even more challenging when the images have been acquired by two different kinds of sensors. This situation arises in case of emergency under critical constraints. This paper presents, to the best of authors' knowledge, the first strategy to deal with optical images characterized by dissimilar spatial and spectral resolutions. Typical considered scenarios include change detection between panchromatic or multispectral and hyperspectral images. The proposed strategy consists of a 3-step procedure: i) inferring a high spatial and spectral resolution image by fusion of the two observed images characterized one by a low spatial resolution and the other by a low spectral resolution, ii) predicting two images with respectively the same spatial and spectral resolutions as the observed images by degradation of the fused one and iii) implementing a decision rule to each pair of observed and predicted images characterized by the same spatial and spectral resolutions to identify changes. The performance of the proposed framework is evaluated on real images with simulated realistic changes.

Archetypal scenarios for change detection generally consider two images acquired through sensors of the same modality. However, in some specific cases such as emergency situations, the only images available may be those acquired through different kinds of sensors. More precisely, this paper addresses the problem of detecting changes between two multi-band optical images characterized by different spatial and spectral resolutions. This sensor dissimilarity introduces additional issues in the context of operational change detection. To alleviate these issues, classical change detection methods are applied after independent preprocessing steps (e.g., resampling) used to get the same spatial and spectral resolutions for the pair of observed images. Nevertheless, these preprocessing steps tend to throw away relevant information. Conversely, in this paper, we propose a method that more effectively uses the available information by modeling the two observed images as spatial and spectral versions of two (unobserved) latent images characterized by the same high spatial and high spectral resolutions. As they cover the same scene, these latent images are expected to be globally similar except for possible changes in sparse spatial locations. Thus, the change detection task is envisioned through a robust multi-band image fusion method which enforces the differences between the estimated latent images to be spatially sparse. This robust fusion problem is formulated as an inverse problem which is iteratively solved using an efficient block-coordinate descent algorithm. The proposed method is applied to real panchormatic/multispectral and hyperspectral images with simulated realistic changes. A comparison with state-of-the-art change detection methods evidences the accuracy of the proposed strategy.

Piecewise constant denoising can be solved either by deterministic optimization approaches, based on the Potts model, or by stochastic Bayesian procedures. The former lead to low computational time but require the selection of a regularization parameter, whose value significantly impacts the achieved solution, and whose automated selection remains an involved and challenging problem. Conversely, fully Bayesian formalisms encapsulate the regularization parameter selection into hierarchical models, at the price of high computational costs. This contribution proposes an operational strategy that combines hierarchical Bayesian and Potts model formulations, with the double aim of automatically tuning the regularization parameter and of maintaining computational effciency. The proposed procedure relies on formally connecting a Bayesian framework to a l2-Potts functional. Behaviors and performance for the proposed piecewise constant denoising and regularization parameter tuning techniques are studied qualitatively and assessed quantitatively, and shown to compare favorably against those of a fully Bayesian hierarchical procedure, both in accuracy and in computational load.

This paper presents two novel hyperspectral mixture models and associated unmixing algorithms. The two models assume a linear mixing model corrupted by an additive term whose expression can be adapted to account for multiple scattering nonlinearities (NL), or mismodelling effects (ME). The NL model generalizes bilinear models by taking into account higher order interaction terms. The ME model accounts for different effects such as endmember variability or the presence of outliers. The abundance and residual parameters of these models are estimated by considering a convex formulation suitable for fast estimation algorithms. This formulation accounts for constraints such as the sum-to-one and non-negativity of the abundances, the non-negativity of the nonlinearity coefficients, the spectral smoothness of the ME terms and the spatial sparseness of the residuals. The resulting convex problem is solved using the alternating direction method of multipliers (ADMM) whose convergence is ensured theoretically. The proposed mixture models and their unmixing algorithms are validated on both synthetic and real images showing competitive results regarding the quality of the inference and the computational complexity when compared to the state-of-the-art algorithms.

This paper proposes a robust fast multi-band image fusion method to merge a high-spatial low-spectral resolution image and a low-spatial high-spectral resolution image. Following the method recently developed in [1], the generalized Sylvester matrix equation associated with the multi-band image fusion problem is solved in a more robust and efficient way by exploiting the Woodbury formula, avoiding any permutation operation in the frequency domain as well as the blurring kernel invertibility assumption required in [1]. Thanks to this improvement, the proposed algorithm requires fewer computational operations and is also more robust with respect to the blurring kernel compared with the one in [1]. The proposed new algorithm is tested with different priors considered in [1]. Our conclusion is that the proposed fusion algorithm is more robust than the one in [1] with a reduced computational cost., Comment: arXiv admin note: substantial text overlap with arXiv:1502.03121

This paper presents a multi-band image fusion algorithm based on unsupervised spectral unmixing for combining a high-spatial low-spectral resolution image and a low-spatial high-spectral resolution image. The widely used linear observation model (with additive Gaussian noise) is combined with the linear spectral mixture model to form the likelihoods of the observations. The non-negativity and sum-to-one constraints resulting from the intrinsic physical properties of the abundances are introduced as prior information to regularize this ill-posed problem. The joint fusion and unmixing problem is then formulated as maximizing the joint posterior distribution with respect to the endmember signatures and abundance maps, This optimization problem is attacked with an alternating optimization strategy. The two resulting sub-problems are convex and are solved efficiently using the alternating direction method of multipliers. Experiments are conducted for both synthetic and semi-real data. Simulation results show that the proposed unmixing based fusion scheme improves both the abundance and endmember estimation comparing with the state-of-the-art joint fusion and unmixing algorithms.

Sparse representations have proven their efficiency in solving a wide class of inverse problems encountered in signal and image processing. Conversely, enforcing the information to be spread uniformly over representation coefficients exhibits relevant properties in various applications such as digital communications. Anti-sparse regularization can be naturally expressed through an $\ell_{\infty}$-norm penalty. This paper derives a probabilistic formulation of such representations. A new probability distribution, referred to as the democratic prior, is first introduced. Its main properties as well as three random variate generators for this distribution are derived. Then this probability distribution is used as a prior to promote anti-sparsity in a Gaussian linear inverse problem, yielding a fully Bayesian formulation of anti-sparse coding. Two Markov chain Monte Carlo (MCMC) algorithms are proposed to generate samples according to the posterior distribution. The first one is a standard Gibbs sampler. The second one uses Metropolis-Hastings moves that exploit the proximity mapping of the log-posterior distribution. These samples are used to approximate maximum a posteriori and minimum mean square error estimators of both parameters and hyperparameters. Simulations on synthetic data illustrate the performances of the two proposed samplers, for both complete and over-complete dictionaries. All results are compared to the recent deterministic variational FITRA algorithm.

Hyperspectral unmixing is aimed at identifying the reference spectral signatures composing an hyperspectral image and their relative abundance fractions in each pixel. In practice, the identified signatures may vary spectrally from an image to another due to varying acquisition conditions, thus inducing possibly significant estimation errors. Against this background, hyperspectral unmixing of several images acquired over the same area is of considerable interest. Indeed, such an analysis enables the endmembers of the scene to be tracked and the corresponding endmember variability to be characterized. Sequential endmember estimation from a set of hyperspectral images is expected to provide improved performance when compared to methods analyzing the images independently. However, the significant size of hyperspectral data precludes the use of batch procedures to jointly estimate the mixture parameters of a sequence of hyperspectral images. Provided that each elementary component is present in at least one image of the sequence, we propose to perform an online hyperspectral unmixing accounting for temporal endmember variability. The online hyperspectral unmixing is formulated as a two-stage stochastic program, which can be solved using a stochastic approximation. The performance of the proposed method is evaluated on synthetic and real data. A comparison with independent unmixing algorithms finally illustrates the interest of the proposed strategy., Comment: 27 pages, 11 figures, accepted in IEEE Trans. Image Process., 2016

This paper addresses the problem of single image super-resolution (SR), which consists of recovering a high resolution image from its blurred, decimated and noisy version. The existing algorithms for single image SR use different strategies to handle the decimation and blurring operators. In addition to the traditional first-order gradient methods, recent techniques investigate splitting-based methods dividing the SR problem into up-sampling and deconvolution steps that can be easily solved. Instead of following this splitting strategy, we propose to deal with the decimation and blurring operators simultaneously by taking advantage of their particular properties in the frequency domain, leading to a new fast SR approach. Specifically, an analytical solution can be obtained and implemented efficiently for the Gaussian prior or any other regularization that can be formulated into an $\ell_2$-regularized quadratic model, i.e., an $\ell_2$-$\ell_2$ optimization problem. Furthermore, the flexibility of the proposed SR scheme is shown through the use of various priors/regularizations, ranging from generic image priors to learning-based approaches. In the case of non-Gaussian priors, we show how the analytical solution derived from the Gaussian case can be embedded intotraditional splitting frameworks, allowing the computation cost of existing algorithms to be decreased significantly. Simulation results conducted on several images with different priors illustrate the effectiveness of our fast SR approach compared with the existing techniques.

This paper presents a fast spectral unmixing algorithm based on Dykstra's alternating projection. The proposed algorithm formulates the fully constrained least squares optimization problem associated with the spectral unmixing task as an unconstrained regression problem followed by a projection onto the intersection of several closed convex sets. This projection is achieved by iteratively projecting onto each of the convex sets individually, following Dyktra's scheme. The sequence thus obtained is guaranteed to converge to the sought projection. Thanks to the preliminary matrix decomposition and variable substitution, the projection is implemented intrinsically in a subspace, whose dimension is very often much lower than the number of bands. A benefit of this strategy is that the order of the computational complexity for each projection is decreased from quadratic to linear time. Numerical experiments considering diverse spectral unmixing scenarios provide evidence that the proposed algorithm competes with the state-of-the-art, namely when the number of endmembers is relatively small, a circumstance often observed in real hyperspectral applications., Comment: 11 pages

Texture segmentation constitutes a standard image processing task, crucial to many applications. The present contribution focuses on the particular subset of scale-free textures and its originality resides in the combination of three key ingredients: First, texture characterization relies on the concept of local regularity ; Second, estimation of local regularity is based on new multiscale quantities referred to as wavelet leaders ; Third, segmentation from local regularity faces a fundamental bias variance trade-off: In nature, local regularity estimation shows high variability that impairs the detection of changes, while a posteriori smoothing of regularity estimates precludes from locating correctly changes. Instead, the present contribution proposes several variational problem formulations based on total variation and proximal resolutions that effectively circumvent this trade-off. Estimation and segmentation performance for the proposed procedures are quantified and compared on synthetic as well as on real-world textures.

Pansharpening aims at fusing a panchromatic image with a multispectral one, to generate an image with the high spatial resolution of the former and the high spectral resolution of the latter. In the last decade, many algorithms have been presented in the literature for pansharpening using multispectral data. With the increasing availability of hyperspectral systems, these methods are now being adapted to hyperspectral images. In this work, we compare new pansharpening techniques designed for hyperspectral data with some of the state of the art methods for multispectral pansharpening, which have been adapted for hyperspectral data. Eleven methods from different classes (component substitution, multiresolution analysis, hybrid, Bayesian and matrix factorization) are analyzed. These methods are applied to three datasets and their effectiveness and robustness are evaluated with widely used performance indicators. In addition, all the pansharpening techniques considered in this paper have been implemented in a MATLAB toolbox that is made available to the community.