Simplex-structured matrix factorization (SSMF) is a common task encountered in signal processing and machine learning. Minimum-volume constrained unmixing (MVCU) algorithms are among the most widely used methods to perform this task. While MVCU algorithms generally perform well in an offline setting, their direct application to online scenarios suffers from scalability limitations due to memory and computational demands. To overcome these limitations, this paper proposes an approach which can build upon any off-the-shelf MVCU algorithm to operate sequentially, i.e., to handle one observation at a time. The key idea of the proposed method consists in updating the solution of MVCU only when necessary, guided by an online check of the corresponding optimization problem constraints. It only stores and processes observations identified as informative with respect to the geometrical constraints underlying SSMF. We demonstrate the effectiveness of the approach when analyzing synthetic and real datasets, showing that it achieves estimation accuracy comparable to the offline MVCU method upon which it relies, while significantly reducing the computational cost.

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.

Image restoration aims at recovering a clean image from degraded observations. This paper presents a novel Bayesian framework for image restoration using a regularization-by-denoising (RED) prior. It introduces a probabilistic counterpart to the RED paradigm, and proposes a new Monte Carlo algorithm to efficiently sample from the resulting posterior distribution. The proposed method benefit from the recent developments of deep learning-based denoisers. Extensive numerical experiments illustrate the efficiency of the proposed method, showcasing its competitive performance against state-of-the-art methods.

Abstract 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 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.

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 black-boxes and lack 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.

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 half quadratic splitting method (HQS) and 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 and PnP-HQS) 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.

This letter proposes a fast yet efficient method to solve the hyperspectral unmixing problem in the challenging unsupervised context, i.e., when the endmember spectral signatures are unknown. First, a coarse approximation of the hyperspectral image is computed by spatially averaging neighboring pixels, which significantly reduces the amount of pixels to be handled. This reduced set of hyperspectral pixels is unmixed to derive coarse solutions of the unmixing problem, i.e., coarse estimates of the endmember signatures and the corresponding low-resolution abundance maps. Then, the plain resolution abundance maps are estimated from the corresponding hyperspectral image based on the coarse endmember signatures. A sparsity promoting prior exploiting the low resolution map complements the conventional data fitting term to promote spatial smoothness while mitigating the loss of details in the edge areas. Finally, a least square optimization problem is solved to obtain the actual endmember signatures from the hyperspectral image and the abundance maps of plain resolution estimated in the previous step. Numerical experiments show that the proposed method is fast and performs well compared to state-of-the-art approaches from the literature.

We propose a simple yet efficient sparse unmixing method for hyperspectral images. It exploits the spatial and spectral properties of hyperspectral images by designing a new regularization informed by multiscale analysis. The proposed approach consists of two steps. First, a sparse unmixing is conducted on a coarse hyperspectral image resulting from a spatial smoothing of the original data. The estimated coarse abundance map is subsequently used to design two weighting terms summarizing the spatial and spectral properties of the image. They are combined to define a sparse regularization embedded into a unmixing problem associated with the original hyperspectral image at full resolution. The performance of the proposed method is assessed with numerous experiments conducted on synthetic and real datasets. It is shown to compete favorably with state-of-the-art methods from the literature with lower computational complexity.

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. It is based on fitting the Dirichlet parameters to the 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.

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., piece-wise 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.

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.

This paper introduces two informed spatial regularizations dedicated to multiband image fusion. The fusion process combines a multispectral image with high spatial resolution and a hyperspectral image with high spectral resolution, with the aim of recovering a full resolution data-cube. In this work, we propose two spatial regularizations that exploit the spatial information of the multispectral image. A weighted Sobolev regularization identifies the sharp structures locations to locally mitigate a smoothness-promoting Sobolev regularization. A dictionary-based regularization takes advantage of spatial redundancy to recover spatial textures using a dictionary learned on the multispectral image. The proposed regularizations are evaluated on realistic simulations of James Webb Space Telescope (JWST) observations of the Orion Bar and show a better reconstruction of sharp structures compared to a non-informed regularization. Since JWST is now in orbit, we expect to use this method on real data in the near future.

This paper proposes a weighted residual nonnegative matrix factorization (NMF) with spatial regularization to unmix hyperspectral data. NMF decomposes a matrix into the product of two nonnegative matrices. However, NMF is known to be generally sensitive to noise, which makes difficult to retrieve the global minimum of the underlying objective function. To overcome this limitation, we include a residual weighting mechanism in the conventional NMF formulation. This strategy treats each row of the residual based on the weighting factor. In this manner, residuals with large values are penalized less and residuals with small values are penalized more to make NMF based unmixing problem more robust. Furthermore, we include a weight term in the form of an ℓ1 norm regularizer to provide spatial information of the abundance matrix. Experimental results are conducted to validate the effectiveness of the proposed method.

This letter proposes a simple, fast yet efficient sparse hyperspectral unmixing algorithm. The proposed method consists of three main steps. First, a coarse approximation of the hyperspectral image is built using a off-the-shelf segmentation algorithm. Then, a low-resolution approximation of the abundance map is estimated by solving a weighted ℓ1-regularized problem on this coarse approximation of the hyperspectral data. Finally, this low-resolution abundance map is subsequently used to design a sparsity-promoting penalization which acts as a spatial regularization informed by the coarse abundance map. It is incorporated into another weighted ℓ1-regularized problem whose solution is a higher resolution abundance map. The computational efficiency of the two last steps is ensured by solving the two underlying optimization problems using an alternating direction method of multipliers. Extensive experiments conducted on simulated and real data show the effectiveness of the proposed method.

Hyperspectral unmixing plays an important role in hyperspectral image processing and analysis. It aims to decompose mixed pixels into pure spectral signatures and their associated abundances. The hyperspectral image contains spatial information in neighborhood regions, and spectral signatures existing in the region also have a high correlation. However, most autoencoder (AE)-based unmixing methods are pixel-to-pixel methods and ignore these priors. It is helpful to add spectral–spatial information into unmixing methods. A recent trend to deal with this problem is to use convolutional neural networks (CNNs). Our proposed framework uses 3-D-CNN-based networks to jointly learn spectral–spatial priors. Moreover, previous AE-based unmixing methods use fixed spectral signatures for each pure material. In our work, we use a carefully designed decoder to cope with the endmember variability issue, and variational inference strategy is applied to add uncertainty property into endmembers. To avoid overfitting, we use structured sparsity regularizers to the encoder networks, and$\ell _{2,1}$-loss is added to the estimated abundances to guarantee the sparseness. Experimental results on both simulated and real data demonstrate the effectiveness of our proposed method.

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) one. 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, referred to as SNPALQ, generalizes the successive nonnegative projection algorithm (SNPA), designed for linear BSS. 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 can be used as a postprocessing step for SNPALQ. It then enables one 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 potentially easier-to-check conditions than those of SNPALQ). We show that SNPALQ with and without the BF postprocessing is relevant in realistic numerical experiments.

Archetypal scenarios for change detection generally consider two images acquired through sensors of the same modality. The resolution dissimilarity is often bypassed though a simple preprocessing, applied independently on each image to bring them to the same resolution. However, in some important situations, e.g. a natural disaster, the only images available may be those acquired through sensors of different modalities and resolutions. Therefore, it is mandatory to develop general and robust methods able to deal with this unfavorable situation. This paper proposes a coupled dictionary learning strategy to detect changes between two images with different modalities and possibly different spatial and/or spectral resolutions. The pair of observed images is modelled as a sparse linear combination of atoms belonging to a pair of coupled overcomplete dictionaries learnt from the two observed images. Codes are expected to be globally similar for areas not affected by the changes while, in some spatially sparse locations, they are expected to be different. Change detection is then envisioned as an inverse problem, namely estimation of a dual code such that the difference between the estimated codes associated with each image exhibits spatial sparsity. A comparison with state-of-the-art change detection methods evidences the proposed method superiority.

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, thus yielding an ill-conditioned problem. To enrich the model and 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 article, 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.

An important task when processing dynamic PET images is to identify the time-activity curves (TACs) of the pure tissues, along with their corresponding spatial proportions. This step, often referred to as unmixing or factor analysis, is based on a loss function which measures the discrepancy between the observed data and the model. This loss function should be chosen according to the statistical properties of the noise, which is in this case hard to characterize. Indeed, while dynamic PET images results from a decay process that can be statistically described by a Poisson distribution, acquisition and post-filtering reconstruction drastically change the nature of the noise. In the literature dedicated to factor analysis of dynamic PET images, a common and underlying assumption consists in assuming that the dynamic PET images are corrupted by an additive Gaussian or by a Poisson noise. These assumptions lead to the choice of the squared Euclidian distance and the Kullback-Leibler divergence. We propose here to consider the β-divergence, which is able to encompass a wide family of divergence measures corresponding to various noise distributions. This loss function is incorporated into three different factor models and evaluated using four sets of synthetic data.

In recent years, much research has been devoted to the restoration of Poissonian images using optimization-based methods. On the other hand, the derivation of efficient and general fully Bayesian approaches is still an active area of research and especially if standard regularization functions are used, e.g. the total variation (TV) norm. This paper proposes to use the recent split-and-augmented Gibbs sampler (SPA) to sample efficiently from an approximation of the initial target distribution when log-concave prior distributions are used. SPA embeds proximal Markov chain Monte Carlo (MCMC) algorithms to sample from possibly non-smooth log-concave full conditionals. The benefit of the proposed approach is illustrated on several experiments including different regularizers, intensity levels and with both analysis and synthesis approaches.

Markov chain Monte Carlo (MCMC) methods are an important class of computation techniques to solve Bayesian inference problems. Much recent research has been dedicated to scale these algorithms in high-dimensional settings by relying on powerful optimization tools such as gradient information or proximity operators. In a similar vein, this paper proposes a new Bayesian hierarchical model to solve large scale inference problems by taking inspiration from variable splitting methods. Similarly to the latter, the derived Gibbs sampler permits to divide the initial sampling task into simpler ones. As a result, the proposed Bayesian framework can lead to a faster sampling scheme than state-of-the-art methods by embedding them. The strength of the proposed methodology is illustrated on two often-studied image processing problems.

The spatial pixel resolution of common multispectral and hyperspectral sensors is generally not sufficient to avoid that multiple elementary materials contribute to the observed spectrum of a single pixel. To alleviate this limitation, spectral unmixing is a by‐pass procedure which consists in decomposing the observed spectra associated with these mixed pixels into a set of component spectra, or endmembers, and a set of corresponding proportions, or abundances, that represent the proportion of each endmember in these pixels. In this study, a spectral unmixing technique is proposed to handle the challenging scenario of non‐linear mixtures. This algorithm relies on a dedicated implementation of multiple‐kernel learning using self‐organising map proposed as a solver for the non‐linear unmixing problem. Based on a priori knowledge of the endmember spectra, it aims at estimating their relative abundances without specifying the non‐linear model under consideration. It is compared to state‐of‐the‐art algorithms using synthetic yet realistic and real hyperspectral images. Results obtained from experiments conducted on synthetic and real hyperspectral images assess the potential and the effectiveness of this unmixing strategy. Finally, the relevance and potential parallel implementation of the proposed method is demonstrated.

So far, the problem of unmixing large or multitemporal hyperspectral data sets 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. 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 nonconvex 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 paper can be implemented to address various matrix factorization problems.

Spectral variability is one of the major issues when conducting hyperspectral unmixing. Within a given image composed of some elementary materials (herein referred to as endmember classes), the spectral signatures 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.

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 it is 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 paper proposes to study the relevance of several divergence measures to be used within a factor analysis framework. To this end, the β-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 β, 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 β to improve the factor analysis.

Supervised classification and spectral unmixing are two methods to extract information from hyperspectral images. However, despite their complementarity, they have been scarcely considered jointly. This paper presents a new hierarchical Bayesian model to perform simultaneously both analysis in order to ensure that they benefit from each other. A linear mixture model is proposed to described the pixel measurements. Then a clustering is performed to identify groups of statistically similar abundance vectors. A Markov random field (MRF) is used as prior for the corresponding cluster labels. It promotes a spatial regularization through a Potts-Markov potential and also includes a local potential induced by the classification. Finally, the classification exploits a set of possibly corrupted labeled data provided by the end-user. Model parameters are estimated thanks to a Markov chain Monte Carlo (MCMC) algorithm. The interest of the proposed model is illustrated on synthetic and real data.

This paper deals with a fusion-based change detection (CD) framework for multi-band images with different spatial and spectral resolutions. The first step of the considered CD framework consists in fusing the two observed images. The resulting fused image is subsequently spatially or spectrally degraded to produce two pseudo-observed images, with the same resolutions as the two observed images. Finally, CD can be performed through a pixel-wise comparison of the pseudo-observed and observed images since they share the same resolutions. Obviously, fusion is a key step in this framework. Thus, this paper proposes to quantitatively and qualitatively compare state-of-the-art fusion methods, gathered into four main families, namely component substitution, multi-resolution analysis, unmixing and Bayesian, with respect to the performance of the whole CD framework evaluated on simulated and real images.

Only a few research works consider LiDAR data while conducting hyperspectral image unmixing. However, the digital surface model derived from LiDAR can provide meaningful information, in particular when spatially regularizing the inverse problem underlain by spectral unmixing. This paper proposes a general framework for spectral unmixing that incorporates LiDAR data to inform the spatial regularization applied to the abundance maps. The proposed framework is validated and compared to existing unmixing methods that incorporate spatial information derived from the hyperspectral image itself using two different simulated data and digital surface models. Results show that the spatial regularization incorporating LiDAR data significantly improves abundance estimates.

Change detection (CD) is one of the most challenging issues when analyzing remotely sensed images. Comparing several multidate images acquired through the same kind of sensor is the most common scenario. Conversely, designing robust, flexible, and scalable algorithms for CD becomes even more challenging when the images have been acquired by two different kinds of sensors. This situation arises in the case of emergency under critical constraints. This paper presents, to the best of our knowledge, the first strategy to deal with optical images characterized by dissimilar spatial and spectral resolutions. Typical considered scenarios include CD between panchromatic, multispectral, and hyperspectral images. The proposed strategy consists of a three-step procedure: 1) 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; 2) predicting two images with, respectively, the same spatial and spectral resolutions as the observed images by the degradation of the fused one; and 3) implementing a decision rule to each pair of observed and predicted images characterized by the same spatial and spectral resolutions to identify changes. To quantitatively assess the performance of the method, an experimental protocol is specifically designed, relying on synthetic yet physically plausible change rules applied to real images. The accuracy of the proposed framework is finally illustrated on real images.

Spectral unmixing (SU) methods incorporating the spatial regularizations have demonstrated increasing interest. Although spatial regularizers that 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 the 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 SU framework that incorporates external data [i.e. light detection and ranging (LiDAR) data]. The LiDAR measurements can be easily exploited to adjust the standard spatial regularizations applied to the unmixing process. The proposed framework is rigorously evaluated using two simulated data sets and a real hyperspectral image. It is compared with 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 the pixels affected by shadows.

Principal component analysis is a widely used technique to perform dimension reduction. However, selecting a finite number of significant components is essential and remains a crucial issue. Only few attempts have proposed a probabilistic approach to adaptively select this number. This paper introduces a Bayesian nonparametric model to jointly estimate the principal components and the corresponding intrinsic dimension. More precisely, the observations are projected onto a random orthogonal basis which is assigned a prior distribution defined on the Stiefel manifold. Then the factor scores take benefit of an Indian buffet process prior 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. The performances of the proposed approach are assessed thanks to experiments conducted on various examples.

This paper proposes a robust fusion-based strategy to detect changes between two multi-band optical images with different spatial and spectral resolutions, e.g., a multispectral high spatial resolution image and a hyperspectral low spatial resolution image. The dissimilarity between sensor resolutions makes the change detection problem challenging, which has been generally bypassed in the literature: most often, the two images are crudely and independently resampled in order to get the same spatial and spectral resolutions and finally, classical change detection methods are applied. However, the resampling operation tends to lose information. In this paper, we propose a method that more effectively uses the available information: the two observed images are respectively modeled as spatial and spectral degradations of two latent images characterized by the same high spatial and high spectral resolutions. Representing the same scene, these latent images are expected to be globally similar except for possible changes in sparse spatial locations. Change detection is then envisioned through the solution of an inverse problem, shown to be a specific instance of multi-band image fusion. The proposed method is applied to real images with simulated realistic changes. A comparison with state-of-the-art change detection methods evidences the proposed method superiority.

This contribution focuses, within the ℓ1-Potts model, on the automated estimation of the regularization parameter balancing the ℓ1data fidelity term and the TVℓ0penalization. Variational approaches based on total variation gained considerable interest to solve piecewise constant denoising problems thanks to their deterministic setting and low computational cost. However, the quality of the achieved solution strongly depends on the tuning of the regularization parameter. While recent works have tailored various hierarchical Bayesian procedures to additionally estimate the regularization parameter for Gaussian noise, less attention has been granted to Laplacian noise, of interested in numerous applications. This contribution promotes a fast and parameter-free denoising procedure for piecewise constant signals corrupted by Laplacian noise, that includes automated selection of the regularization parameter. It relies on the minimization of a Bayesian-driven criterion whose similarities with the ℓ1-Potts model permit to derive a computationally efficient algorithm.

This paper presents a novel nonlinear hyperspectral mixture model and its associated supervised unmixing algorithm. The model assumes a linear mixing model corrupted by an additive term which accounts for multiple scattering nonlinearities (NL). The proposed model generalizes bilinear models by taking into account higher order interaction terms. The inference of the abundances and nonlinearity coefficients of this model is formulated as a convex optimization problem suitable for fast estimation algorithms. This formulation accounts for constraints such as the sum-to-one and nonnegativity of the abundances, the non-negativity of the nonlinearity coefficients, 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 model and its unmixing algorithm 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.

A generalized Swendsen-Wang (GSW) algorithm is proposed for the joint segmentation of a set of multiple images sharing, in part, an unknown number of common classes. The class labels are a priori modeled by a combination of the hierarchical Dirichlet process (HDP) and the Potts model. The HDP allows the number of regions in each image and classes to be automatically inferred while the Potts model ensures spatially consistent segmentations. Compared to a classical Gibbs sampler, the GSW ensures a better exploration of the posterior distribution of the labels. To avoid label switching issues, the best partition is estimated using the Dahl's criterion.

The problem of band selection (BS) is of great importance to handle the curse of dimensionality for hyperspectral image (HSI) applications (e.g., classification). This letter proposes an unsupervised BS approach based on a split-and-merge concept. This new approach provides relevant spectral sub-bands by splitting the adjacent bands without violating the physical meaning of the spectral data. Next, it merges highly correlated bands and sub-bands to reduce the dimensionality of the HSI. Experiments on three public data sets and comparison with state-of-the-art approaches show the efficiency of the proposed approach.

Hyperspectral unmixing consists in determining the reference spectral signatures composing a hyperspectral image and their relative abundance fractions in each pixel. In practice, the identified signatures may be affected by a significant spectral variability resulting for instance from the temporal evolution of the imaged scene. This phenomenon can be accounted for by using a perturbed linear mixing model. This paper studies an online estimation algorithm for the parameters of this extended linear mixing model. This algorithm is of interest for the practical applications where the size of the hyper-spectral images precludes the use of batch procedures. The performance of the proposed method is evaluated on synthetic data.

This letter proposes a robust fast multiband 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 by Wei et al., the generalized Sylvester matrix equation associated with the multiband 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 their method. 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 developed by Wei et al. The proposed new algorithm is tested with different priors considered by Wei et al. Our conclusion is that the proposed fusion algorithm is more robust than the one by Wei et al. with a reduced computational cost.

This paper presents a multiband 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 nonnegativity 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 subproblems 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 compared with the state-of-the-art joint fusion and unmixing algorithms.

This paper presents a new strategy to correct the Earth data corrupted by spurious samples that are randomly included in the multiplexed data stream provided by the MADRAS instrument. The proposed strategy relies on the construction of a trellis associated with each scan of the multichannel image, modeling the possible occurrences of these erroneous data. A specific weight that promotes the smooth behavior of the signals recorded in each channel is assigned to each transition between trellis states. The joint detection and correction of the erroneous data are conducted using a dynamic programming algorithm for minimizing the overall cost function throughout the trellis. Simulation results obtained on synthetic and real MADRAS data demonstrate the effectiveness of the proposed solution.

Hyperspectral unmixing is aimed at identifying the reference spectral signatures composing a 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, the 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 with methods analyzing the images independently. However, the significant size of the 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. Finally, a comparison with independent unmixing algorithms illustrates the interest of the proposed strategy.

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 is derived and implemented efficiently for the Gaussian prior or any other regularization that can be formulated into an l2 -regularized quadratic model, i.e., an l2 - l2 optimization problem. 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 into traditional 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 existing techniques.

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This paper presents an unsupervised Bayesian algorithm for hyperspectral image unmixing accounting for endmember variability. Each image pixel is modeled by a linear combination of random endmembers to take into account endmember variability in the image. The coefficients of this linear combination (referred to as abundances) allow the proportions of each material (endmembers) to be quantified in the image pixel. An additive noise is also considered in the proposed model generalizing the normal compositional model. The proposed Bayesian algorithm exploits spatial correlations between adjacent pixels of the image and provides spectral information by achieving a spectral unmixing. It estimates both the mean and the covariance matrix of each endmember in the image. A spatial classification is also obtained based on the estimated abundances. Simulations conducted with synthetic and real data show the potential of the proposed model and the unmixing performance for the analysis of hyperspectral images.

This paper presents a variational-based approach for fusing hyperspectral and multispectral images. The fusion problem is formulated as an inverse problem whose solution is the target image assumed to live in a lower dimensional subspace. A sparse regularization term is carefully designed, relying on a decomposition of the scene on a set of dictionaries. The dictionary atoms and the supports of the corresponding active coding coefficients are learned from the observed images. Then, conditionally on these dictionaries and supports, the fusion problem is solved via alternating optimization with respect to the target image (using the alternating direction method of multipliers) and the coding coefficients. Simulation results demonstrate the efficiency of the proposed algorithm when compared with state-of-the-art fusion methods.

Texture characterization is a central element in many image processing applications. Multifractal analysis is a useful signal and image processing tool, yet, the accurate estimation of multifractal parameters for image texture remains a challenge. This is due in the main to the fact that current estimation procedures consist of performing linear regressions across frequency scales of the 2D dyadic wavelet transform, for which only a few such scales are computable for images. The strongly non-Gaussian nature of multifractal processes, combined with their complicated dependence structure, makes it difficult to develop suitable models for parameter estimation. Here, we propose a Bayesian procedure that addresses the difficulties in the estimation of the multifractality parameter. The originality of the procedure is threefold. The construction of a generic semiparametric statistical model for the logarithm of wavelet leaders; the formulation of Bayesian estimators that are associated with this model and the set of parameter values admitted by multifractal theory; the exploitation of a suitable Whittle approximation within the Bayesian model which enables the otherwise infeasible evaluation of the posterior distribution associated with the model. Performance is assessed numerically for several 2D multifractal processes, for several image sizes and a large range of process parameters. The procedure yields significant benefits over current benchmark estimators in terms of estimation performance and ability to discriminate between the two most commonly used classes of multifractal process models. The gains in performance are particularly pronounced for small image sizes, notably enabling for the first time the analysis of image patches as small as 64 × 64 pixels.

We introduce a robust mixing model to describe hyperspectral data resulting from the mixture of several pure spectral signatures. The new model extends the commonly used linear mixing model by introducing an additional term accounting for possible nonlinear effects, that are treated as sparsely distributed additive outliers. With the standard nonnegativity and sum-to-one constraints inherent to spectral unmixing, our model leads to a new form of robust nonnegative matrix factorization with a group-sparse outlier term. The factorization is posed as an optimization problem, which is addressed with a block-coordinate descent algorithm involving majorization-minimization updates. Simulation results obtained on synthetic and real data show that the proposed strategy competes with the state-of-the-art linear and nonlinear unmixing methods.

This paper presents an unsupervised Bayesian algorithm for hyperspectral image unmixing, accounting for endmember variability. The pixels are modeled by a linear combination of endmembers weighted by their corresponding abundances. However, the endmembers are assumed random to consider their variability in the image. An additive noise is also considered in the proposed model, generalizing the normal compositional model. The proposed algorithm exploits the whole image to benefit from both spectral and spatial information. It estimates both the mean and the covariance matrix of each endmember in the image. This allows the behavior of each material to be analyzed and its variability to be quantified in the scene. A spatial segmentation is also obtained based on the estimated abundances. In order to estimate the parameters associated with the proposed Bayesian model, we propose to use a Hamiltonian Monte Carlo algorithm. The performance of the resulting unmixing strategy is evaluated through simulations conducted on both synthetic and real data.

This paper proposes a fast multi-band image fusion algorithm, which combines a high-spatial low-spectral resolution image and a low-spatial high-spectral resolution image. The well admitted forward model is explored to form the likelihoods of the observations. Maximizing the likelihoods leads to solving a Sylvester equation. By exploiting the properties of the circulant and downsampling matrices associated with the fusion problem, a closed-form solution for the corresponding Sylvester equation is obtained explicitly, getting rid of any iterative update step. Coupled with the alternating direction method of multipliers and the block coordinate descent method, the proposed algorithm can be easily generalized to incorporate prior information for the fusion problem, allowing a Bayesian estimator. Simulation results show that the proposed algorithm achieves the same performance as the existing algorithms with the advantage of significantly decreasing the computational complexity of these algorithms.

This paper presents a nonlinear mixing model for joint hyperspectral image unmixing and nonlinearity detection. The proposed model assumes that the pixel reflectances are linear mixtures of endmembers, corrupted by an additional nonlinear term and an additive Gaussian noise. A Markov random field is considered for nonlinearity detection based on the spatial structure of the nonlinear terms. The observed image is segmented into regions where nonlinear terms, if present, share similar statistical properties. A Bayesian algorithm is proposed to estimate the parameters involved in the model yielding a joint nonlinear unmixing and nonlinearity detection algorithm. Simulations conducted with synthetic and real data show the accuracy of the proposed unmixing and nonlinearity detection strategy for the analysis of hyperspectral images.

Sparse signal/image recovery is a challenging topic that has captured a great interest during the last decades. To address the ill-posedness of the related inverse problem, regularization is often essential by using appropriate priors that promote the sparsity of the target signal/image. In this context, ℓ0+ ℓ1regularization has been widely investigated. In this paper, we introduce a new prior accounting simultaneously for both sparsity and smoothness of restored signals. We use a Bernoulli-generalized Gauss-Laplace distribution to perform ℓ0+ ℓ1+ ℓ2regularization in a Bayesian framework. Our results show the potential of the proposed approach especially in restoring the non-zero coefficients of the signal/image of interest.

This paper presents a Bayesian fusion technique for multi-band images. The observed images are related to the high spectral and high spatial resolution image to be recovered through physical degradations, e.g., spatial and spectral blurring and/or subsampling defined by the sensor characteristics. The fusion problem is formulated within a Bayesian estimation framework. An appropriate prior distribution related to the linear mixing model for hyperspectral images is introduced. To compute Bayesian estimators of the scene of interest from its posterior distribution, a Gibbs sampling algorithm is proposed to generate samples asymptotically distributed according to the target distribution. To efficiently sample from this high-dimensional distribution, a Hamiltonian Monte Carlo step is introduced in this Gibbs sampler. The efficiency of the proposed fusion method is evaluated with respect to several state-of-the-art fusion techniques.

The identification of texture changes is a challenging problem that can be addressed by considering local regularity fluctuations in an image. This work develops a procedure for local regularity estimation that combines a convex optimization strategy with wavelet leaders, specific wavelet coefficients recently introduced in the context of multifractal analysis. The proposed procedure is formulated as an inverse problem that combines the joint estimation of both local regularity exponent and of the optimal weights underlying regularity measurement. Numerical experiments using synthetic texture indicate that the performance of the proposed approach compares favorably against other wavelet based local regularity estimation formulations. The method is also illustrated with an example involving real-world texture.

This paper studies a new Bayesian algorithm for fusing hyperspectral and multispectral images. The observed images are related to the high spatial resolution hyperspectral image to be recovered through physical degradations, e.g., spatial and spectral blurring and/or sub-sampling defined by the sensor characteristics. In this work, we assume that the spectral response of the multispectral sensor is unknown as it may not be available in practical applications. The resulting fusion problem is formulated within a Bayesian estimation framework, which is very convenient to model the uncertainty regarding the multispectral sensor characteristics and the scene to be estimated. The high spatial resolution hyperspectral image is then inferred from its posterior distribution. More precisely, to compute the Bayesian estimators associated with this posterior, a Markov chain Monte Carlo algorithm is proposed to generate samples asymptotically distributed according to the distribution of interest. Simulation results demonstrate the efficiency of the proposed fusion method when compared with several state-of-the-art fusion techniques.

This letter considers the problem of computing the Cramer–Rao bound for the parameters of a Markov random field. Computation of the exact bound is not feasible for most fields of interest because their likelihoods are intractable and have intractable derivatives. We show here how it is possible to formulate the computation of the bound as a statistical inference problem that can be solve approximately, but with arbitrarily high accuracy, by using a Monte Carlo method. The proposed methodology is successfully applied on the Ising and the Potts models.

This paper presents a nonlinear mixing model for joint hyperspectral image unmixing and nonlinearity detection. The proposed model assumes that the pixel reflectances are linear combinations of known pure spectral components corrupted by an additional nonlinear term, affecting the end members and contaminated by an additive Gaussian noise. A Markov random field is considered for nonlinearity detection based on the spatial structure of the nonlinear terms. The observed image is segmented into regions where nonlinear terms, if present, share similar statistical properties. A Bayesian algorithm is proposed to estimate the parameters involved in the model yielding a joint nonlinear unmixing and nonlinearity detection algorithm. The performance of the proposed strategy is first evaluated on synthetic data. Simulations conducted with real data show the accuracy of the proposed unmixing and nonlinearity detection strategy for the analysis of hyperspectral images.

This paper presents a nonlinear mixing model for hyperspectral image unmixing. The proposed model assumes that the pixel reflectances are post-nonlinear functions of unknown pure spectral components contaminated by an additive white Gaussian noise. These nonlinear functions are approximated using second-order polynomials leading to a polynomial post-nonlinear mixing model. A Bayesian algorithm is proposed to estimate the parameters involved in the model yielding an unsupervised nonlinear unmixing algorithm. Due to the large number of parameters to be estimated, an efficient Hamiltonian Monte Carlo algorithm is investigated. The classical leapfrog steps of this algorithm are modified to handle the parameter constraints. The performance of the unmixing strategy, including convergence and parameter tuning, is first evaluated on synthetic data. Simulations conducted with real data finally show the accuracy of the proposed unmixing strategy for the analysis of hyperspectral images.

This paper studies a pixel by pixel nonlinearity detector for hyperspectral image analysis. The reflectances of linearly mixed pixels are assumed to be a linear combination of known pure spectral components (endmembers) contaminated by additive white Gaussian noise. Nonlinear mixing, however, is not restricted to any prescribed nonlinear mixing model. The mixing coefficients (abundances) satisfy the physically motivated sum-to-one and positivity constraints. The proposed detection strategy considers the distance between an observed pixel and the hyperplane spanned by the endmembers to decide whether that pixel satisfies the linear mixing model (null hypothesis) or results from a more general nonlinear mixture (alternative hypothesis). The distribution of this distance is derived under the two hypotheses. Closed-form expressions are then obtained for the probabilities of false alarm and detection as functions of the test threshold. The proposed detector is compared to another nonlinearity detector recently investigated in the literature through simulations using synthetic data. It is also applied to a real hyperspectral image.

Multifractal analysis has matured into a widely used signal and image processing tool. Due to the statistical nature of multifractal processes (strongly non-Gaussian and intricate dependence) the accurate estimation of multifractal parameters is very challenging in situations where the sample size is small (notably including a range of biomedical applications) and currently available estimators need to be improved. To overcome such limitations, the present contribution proposes a Bayesian estimation procedure for the multifractality (or intermittence) parameter. Its originality is threefold: First, the use of wavelet leaders, a recently introduced multiresolution quantity that has been shown to yield significant benefits for multifractal analysis; Second, the construction of a simple yet generic semi-parametric model for the marginals and covariance structure of wavelet leaders for the large class of multiplicative cascade based multifractal processes; Third, the construction of original Bayesian estimators associated with the model and the constraints imposed by multifractal theory. Performance are numerically assessed and illustrated for synthetic multifractal processes for a range of multifractal parameter values. The proposed procedure yields significantly improved estimation performance for small sample sizes.

BACKGROUND: This paper introduces a new constrained model and the corresponding algorithm, called unsupervised Bayesian linear unmixing (uBLU), to identify biological signatures from high dimensional assays like gene expression microarrays. The basis for uBLU is a Bayesian model for the data samples which are represented as an additive mixture of random positive gene signatures, called factors, with random positive mixing coefficients, called factor scores, that specify the relative contribution of each signature to a specific sample. The particularity of the proposed method is that uBLU constrains the factor loadings to be non-negative and the factor scores to be probability distributions over the factors. Furthermore, it also provides estimates of the number of factors. A Gibbs sampling strategy is adopted here to generate random samples according to the posterior distribution of the factors, factor scores, and number of factors. These samples are then used to estimate all the unknown parameters. RESULTS: Firstly, the proposed uBLU method is applied to several simulated datasets with known ground truth and compared with previous factor decomposition methods, such as principal component analysis (PCA), non negative matrix factorization (NMF), Bayesian factor regression modeling (BFRM), and the gradient-based algorithm for general matrix factorization (GB-GMF). Secondly, we illustrate the application of uBLU on a real time-evolving gene expression dataset from a recent viral challenge study in which individuals have been inoculated with influenza A/H3N2/Wisconsin. We show that the uBLU method significantly outperforms the other methods on the simulated and real data sets considered here. CONCLUSIONS: The results obtained on synthetic and real data illustrate the accuracy of the proposed uBLU method when compared to other factor decomposition methods from the literature (PCA, NMF, BFRM, and GB-GMF). The uBLU method identifies an inflammatory component closely associated with clinical symptom scores collected during the study. Using a constrained model allows recovery of all the inflammatory genes in a single factor.

This paper studies a nonlinear mixing model for hyperspectral image unmixing and nonlinearity detection. The proposed model assumes that the pixel reflectances are nonlinear functions of pure spectral components contaminated by an additive white Gaussian noise. These nonlinear functions are approximated by polynomials leading to a polynomial post-nonlinear mixing model. We have shown in a previous paper that the parameters involved in the resulting model can be estimated using least squares methods. A generalized likelihood ratio test based on the estimator of the nonlinearity parameter is proposed to decide whether a pixel of the image results from the commonly used linear mixing model or from a more general nonlinear mixing model. To compute the test statistic associated with the nonlinearity detection, we propose to approximate the variance of the estimated nonlinearity parameter by its constrained Cramér-Rao bound. The performance of the detection strategy is evaluated via simulations conducted on synthetic and real data. More precisely, synthetic data have been generated according to the standard linear mixing model and three nonlinear models from the literature. The real data investigated in this study are extracted from the Cuprite image, which shows that some minerals seem to be nonlinearly mixed in this image. Finally, it is interesting to note that the estimated abundance maps obtained with the post-nonlinear mixing model are in good agreement with results obtained in previous studies.

Linear spectral unmixing is a challenging problem in hyperspectral imaging that consists of decomposing an observed pixel into a linear combination of pure spectra (or endmembers) with their corresponding proportions (or abundances). Endmember extraction algorithms can be employed for recovering the spectral signatures while abundances are estimated using an inversion step. Recent works have shown that exploiting spatial dependencies between image pixels can improve spectral unmixing. Markov random fields (MRF) are classically used to model these spatial correlations and partition the image into multiple classes with homogeneous abundances. This paper proposes to define the MRF sites using similarity regions. These regions are built using a self-complementary area filter that stems from the morphological theory. This kind of filter divides the original image into flat zones where the underlying pixels have the same spectral values. Once the MRF has been clearly established, a hierarchical Bayesian algorithm is proposed to estimate the abundances, the class labels, the noise variance, and the corresponding hyperparameters. A hybrid Gibbs sampler is constructed to generate samples according to the corresponding posterior distribution of the unknown parameters and hyperparameters. Simulations conducted on synthetic and real AVIRIS data demonstrate the good performance of the algorithm.

This paper addresses the problem of estimating the Potts parameter β jointly with the unknown parameters of a Bayesian model within a Markov chain Monte Carlo (MCMC) algorithm. Standard MCMC methods cannot be applied to this problem because performing inference on β requires computing the intractable normalizing constant of the Potts model. In the proposed MCMC method, the estimation of β is conducted using a likelihood-free Metropolis-Hastings algorithm. Experimental results obtained for synthetic data show that estimating β jointly with the other unknown parameters leads to estimation results that are as good as those obtained with the actual value of β. On the other hand, choosing an incorrect value of β can degrade estimation performance significantly. To illustrate the interest of this method, the proposed algorithm is successfully applied to real bidimensional SAR and tridimensional ultrasound images.

This paper describes a Gaussian process based method for nonlinear hyperspectral image unmixing. The proposed model assumes a nonlinear mapping from the abundance vectors to the pixel reflectances contaminated by an additive white Gaussian noise. The parameters involved in this model satisfy physical constraints that are naturally expressed within a Bayesian framework. The proposed abundance estimation procedure is applied simultaneously to all pixels of the image by maximizing an appropriate posterior distribution which does not depend on the endmembers. After determining the abundances of all image pixels, the endmembers contained in the image are estimated by using Gaussian process regression. The performance of the resulting unsupervised unmixing strategy is evaluated through simulations conducted on synthetic data.

Subspace estimation using relatively few samples is a frequently encountered problem in numerous applications, including hyperspectral imagery the target application of this paper. We address this problem in a Bayesian framework assuming that some rough prior knowledge about the subspace is available. Our approach is based on the CS decomposition of an orthogonal matrix whose columns span the subspace of interest. This parametrization only involves mild assumptions about the distribution of the angles between the actual subspace and the prior subspace, and is intuitively appealing. We derive the posterior distribution for the matrices involved in the CS decomposition and the angles between subspaces, and we propose a Gibbs sampling scheme to compute the minimum mean-square distance estimator of the subspace of interest. The estimator accuracy is evaluated through numerical simulations and tested against real hyperspectral data.

This paper presents a nonlinear mixing model for hyperspectral image unmixing. The proposed model assumes that the pixel reflectances are nonlinear functions of pure spectral components contaminated by an additive white Gaussian noise. These nonlinear functions are approximated using polynomial functions leading to a polynomial postnonlinear mixing model. A Bayesian algorithm and optimization methods are proposed to estimate the parameters involved in the model. The performance of the unmixing strategies is evaluated by simulations conducted on synthetic and real data.

We propose a solution to the image deconvolution problem where the convolution kernel or point spread function (PSF) is assumed to be only partially known. Small perturbations generated from the model are exploited to produce a few principal components explaining the PSF uncertainty in a high-dimensional space. Unlike recent developments on blind deconvolution of natural images, we assume the image is sparse in the pixel basis, a natural sparsity arising in magnetic resonance force microscopy (MRFM). Our approach adopts a Bayesian Metropolis-within-Gibbs sampling framework. The performance of our Bayesian semi-blind algorithm for sparse images is superior to previously proposed semi-blind algorithms such as the alternating minimization algorithm and blind algorithms developed for natural images. We illustrate our myopic algorithm on real MRFM tobacco virus data.

This paper addresses the problem of jointly estimating the statistical distribution and segmenting lesions in multiple-tissue high-frequency skin ultrasound images. The distribution of multiple-tissue images is modeled as a spatially coherent finite mixture of heavy-tailed Rayleigh distributions. Spatial coherence inherent to biological tissues is modeled by enforcing local dependence between the mixture components. An original Bayesian algorithm combined with a Markov chain Monte Carlo method is then proposed to jointly estimate the mixture parameters and a label-vector associating each voxel to a tissue. More precisely, a hybrid Metropolis-within-Gibbs sampler is used to draw samples that are asymptotically distributed according to the posterior distribution of the Bayesian model. The Bayesian estimators of the model parameters are then computed from the generated samples. Simulation results are conducted on synthetic data to illustrate the performance of the proposed estimation strategy. The method is then successfully applied to the segmentation of in vivo skin tumors in high-frequency 2-D and 3-D ultrasound images.

This paper studies a hierarchical Bayesian model for nonlinear hyperspectral image unmixing. The proposed model assumes that the pixel reflectances are polynomial functions of linear mixtures of pure spectral components contaminated by an additive white Gaussian noise. The parameters involved in this model satisfy constraints that are naturally expressed within a Bayesian framework. A Gibbs sampler allows one to sample the unknown abundances and nonlinearity parameters according to the joint posterior of interest. The performance of the resulting unmixing strategy is evaluated thanks to simulations conducted on synthetic and real data.

This paper investigates a Bayesian model and a Markov chain Monte Carlo (MCMC) algorithm for gene factor analysis. Each sample in the dataset is decomposed as a linear combination of characteristic gene signatures (also referred to as factors) following a linear mixing model. To enforce the sparsity of the relative contribution (called factor score) of each gene signature to a specific sample, constrained Bernoulli-Gaussian distributions are elected as prior distributions for these factor scores. This distribution allows one to ensure non-negativity and full-additivity constraints for the scores that are interpreted as concentrations. The complexity of the resulting Bayesian estimators is alleviated by using a Gibbs sampler which generates samples distributed according to the posterior distribution of interest. These samples are then used to approximate the standard maximum a posteriori (MAP) or minimum mean square error (MMSE) estimators. The accuracy of the proposed Bayesian method is illustrated by simulations conducted on synthetic and real data.

This paper studies a variational Bayesian unmixing algorithm for hyperspectral images based on the standard linear mixing model. Each pixel of the image is modeled as a linear combination of endmembers whose corresponding fractions or abundances are estimated by a Bayesian algorithm. This approach requires to define prior distributions for the parameters of interest and the related hyperparameters. After defining appropriate priors for the abundances (uniform priors on the interval (0,1)), the joint posterior distribution of the model parameters and hyperparameters is derived. The complexity of this distribution is handled by using variational methods that allow the joint distribution of the unknown parameters and hyperparameter to be approximated. Simulation results conducted on synthetic and real data show similar performances than those obtained with a previously published unmixing algorithm based on Markov chain Monte Carlo methods, with a significantly reduced computational cost.

This paper addresses the problem of estimating the statistical distribution of multiple-tissue non-stationary ultrasound images of skin. The distribution of multiple-tissue images is modeled as a finite mixture of Heavy-Tailed Rayleigh distributions. An original Bayesian algorithm combined with a Markov chain Monte Carlo method is then derived to jointly estimate the mixture parameters and a label vector associating each voxel to a tissue. Precisely, a hybrid Metropolis-within-Gibbs sampler is proposed to draw samples that are asymptotically distributed according to the posterior distribution of the Bayesian model. These samples are then used to compute the Bayesian estimators of the model parameters. Simulation results are conducted on synthetic data to illustrate the performance of the proposed estimation strategy. The method is then successfully applied to the detection of an in-vivo skin lesion in a high frequency 3D ultrasound image.

This paper studies a linear radial basis function network (RBFN) for unmixing hyperspectral images. The proposed RBFN assumes that the observed pixel reflectances are nonlinear mixtures of known end members (extracted from a spectral library or estimated with an end member extraction algorithm), with unknown proportions (usually referred to as abundances). We propose to estimate the model abundances using a linear combination of radial basis functions whose weights are estimated using training samples. The main contribution of this paper is to study an orthogonal least squares algorithm which allows the number of RBFN centers involved in the abundance estimation to be significantly reduced. The resulting abundance estimator is combined with a fully constrained estimation procedure ensuring positivity and sum-to-one constraints for the abundances. The performance of the nonlinear unmixing strategy is evaluated with simulations conducted on synthetic and real data.

This paper studies estimation algorithms for nonlinear hyperspectral image unmixing. The proposed unmixing model assumes that the pixel reflectances are polynomial functions of linear mixtures of pure spectral components contaminated by an additive white Gaussian noise. A hierarchical Bayesian algorithm and an optimization method are proposed for solving the resulting unmixing problem. The parameters involved in the proposed model satisfy constraints that are naturally included in the estimation procedure. The performance of the unmixing strategies is evaluated thanks to simulations conducted on synthetic and real data.

Nonlinear models have recently shown interesting properties for spectral unmixing. This paper considers a generalized bilinear model recently introduced for unmixing hyperspectral images. Different algorithms are studied to estimate the parameters of this bilinear model. The positivity and sum-to-one constraints for the abundances are ensured by the proposed algorithms. The performance of the resulting unmixing strategy is evaluated via simulations conducted on synthetic and real data.

This paper describes a new algorithm for hyperspectral image unmixing. Most unmixing algorithms proposed in the literature do not take into account the possible spatial correlations between the pixels. In this paper, a Bayesian model is introduced to exploit these correlations. The image to be unmixed is assumed to be partitioned into regions (or classes) where the statistical properties of the abundance coefficients are homogeneous. A Markov random field, is then proposed to model the spatial dependencies between the pixels within any class. Conditionally upon a given class, each pixel is modeled by using the classical linear mixing model with additive white Gaussian noise. For this model, the posterior distributions of the unknown parameters and hyperparameters allow the parameters of interest to be inferred. These parameters include the abundances for each pixel, the means and variances of the abundances for each class, as well as a classification map indicating the classes of all pixels in the image. To overcome the complexity of the posterior distribution, we consider a Markov chain Monte Carlo method that generates samples asymptotically distributed according to the posterior. The generated samples are then used for parameter and hyperparameter estimation. The accuracy of the proposed algorithms is illustrated on synthetic and real data.

Nonlinear models have recently shown interesting properties for spectral unmixing. This paper studies a generalized bilinear model and a hierarchical Bayesian algorithm for unmixing hyperspectral images. The proposed model is a generalization not only of the accepted linear mixing model but also of a bilinear model that has been recently introduced in the literature. Appropriate priors are chosen for its parameters to satisfy the positivity and sum-to-one constraints for the abundances. The joint posterior distribution of the unknown parameter vector is then derived. Unfortunately, this posterior is too complex to obtain analytical expressions of the standard Bayesian estimators. As a consequence, a Metropolis-within-Gibbs algorithm is proposed, which allows samples distributed according to this posterior to be generated and to estimate the unknown model parameters. The performance of the resulting unmixing strategy is evaluated via simulations conducted on synthetic and real data.

In this paper, we address the problem of unmixing hyperspectral images in a semi-supervised framework using the normal compositional model recently introduced by Eismann and Stein. Each pixel of the image is modeled as a linear combination of random endmembers. More precisely, endmembers are modeled as Gaussian vectors whose means belong to a known spectral library. This paper proposes to estimate the number of endmembers involved in the mixture, as well as the mixture coefficients (referred to as abundances) using a trans-dimensional algorithm. Appropriate prior distributions are assigned to the abundance vector (to satisfy constraints inherent to hyperspectral imagery), the noise variance and the number of components involved in the mixture model. The computational complexity of the resulting posterior distribution is alleviated by constructing an hybrid Gibbs algorithm which generates samples distributed according to this posterior distribution. As the number of endmembers is unknown, the sampler has to jump between spaces of different dimensions. This is achieved by a reversible jump Markov chain Monte Carlo method that allows one to handle the model order selection problem. The performance of the proposed methodology is evaluated thanks to simulations conducted on synthetic data.

We consider Bayesian detection/classification of discrete random parameters that are strongly dependent locally due to some deterministic local constraint. Based on the recently introduced partially collapsed Gibbs sampler (PCGS) principle, we develop a Markov chain Monte Carlo method that tolerates and even exploits the challenging probabilistic structure imposed by deterministic local constraints. We study the application of our method to the practically relevant case of nonuniformly spaced binary pulses with a known minimum distance. Simulation results demonstrate significant performance gains of our method compared to a recently proposed PCGS that is not specifically designed for the local constraint.

Bayesian positive source separation (BPSS) is a useful unsupervised approach for hyperspectral data unmixing, where numerical nonnegativity of spectra and abundances has to be ensured, such as in remote sensing. Moreover, it is sensible to impose a sum-to-one (full additivity) constraint to the estimated source abundances in each pixel. Even though nonnegativity and full additivity are two necessary properties to get physically interpretable results, the use of BPSS algorithms has so far been limited by high computation time and large memory requirements due to the Markov chain Monte Carlo calculations. An implementation strategy that allows one to apply these algorithms on a full hyperspectral image, as it is typical in earth and planetary science, is introduced. The effects of pixel selection and the impact of such sampling on the relevance of the estimated component spectra and abundance maps, as well as on the computation times, are discussed. For that purpose, two different data sets have been used: a synthetic one and a real hyperspectral image from Mars.

This paper studies a new Bayesian unmixing algorithm for hyperspectral images. Each pixel of the image is modeled as a linear combination of so-called endmembers. These endmembers are supposed to be random in order to model uncertainties regarding their knowledge. More precisely, we model endmembers as Gaussian vectors whose means have been determined using an endmember extraction algorithm such as the famous N-finder (N-FINDR) or Vertex Component Analysis (VCA) algorithms. This paper proposes to estimate the mixture coefficients (referred to as abundances) using a Bayesian algorithm. Suitable priors are assigned to the abundances in order to satisfy positivity and additivity constraints whereas conjugate priors are chosen for the remaining parameters. A hybrid Gibbs sampler is then constructed to generate abundance and variance samples distributed according to the joint posterior of the abundances and noise variances. The performance of the proposed methodology is evaluated by comparison with other unmixing algorithms on synthetic and real images.

In this paper, we propose a Bayesian model and a Monte Carlo Markov chain (MCMC) algorithm for reconstructing images that consist of only few non-zero pixels. An appropriate distribution that promotes sparsity is proposed as prior distribution for the pixel values. The hyperparameters involved in the modeling are also assigned prior distributions, resulting in a hierarchical model. A Gibbs sampler allows us to draw samples distributed according the full posterior of interest. These samples are then used to approximate standard maximum a posteriori (MAP) estimator. By conducting some simulations, we show that the proposed estimator clearly outperforms previous estimators proposed in the literature.

This paper presents a hierarchical Bayesian model to reconstruct sparse images when the observations are obtained from linear transformations and corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is well suited to such naturally sparse image applications as it seamlessly accounts for properties such as sparsity and positivity of the image via appropriate Bayes priors. We propose a prior that is based on a weighted mixture of a positive exponential distribution and a mass at zero. The prior has hyperparameters that are tuned automatically by marginalization over the hierarchical Bayesian model. To overcome the complexity of the posterior distribution, a Gibbs sampling strategy is proposed. The Gibbs samples can be used to estimate the image to be recovered, e.g., by maximizing the estimated posterior distribution. In our fully Bayesian approach, the posteriors of all the parameters are available. Thus, our algorithm provides more information than other previously proposed sparse reconstruction methods that only give a point estimate. The performance of the proposed hierarchical Bayesian sparse reconstruction method is illustrated on synthetic data and real data collected from a tobacco virus sample using a prototype MRFM instrument.

This paper addresses the problem of unmixing hyperspectral images contamined by additive colored noise. Each pixel of the image is modeled as a linear combination of pure materials (denoted as end-members) corrupted by an additive zero mean Gaussian noise sequence with unknown covariance matrix. Appropriate priors are defined ensuring positivity and additivity constraints on the mixture coefficients (denoted as abundances). These coefficients as well as the noise covariance matrix are then estimated from their joint posterior distribution. A Gibbs sampling strategy generates abundances and noise covariance matrices distributed according to the joint posterior. These samples are then averaged for minimum mean square error estimation.

This paper addresses the problem of hyperspectral image unmixing. A new hierarchical Bayesian algorithm is proposed to estimate the coefficients of a linear mixture of spectra associated to a given pixel of the image. Appropriate priors are introduced to guaranty the positivity and additivity constraints inherent to the mixture coefficients. These coefficients referred to as abundances are then estimated from their posterior following the principles of Bayesian inference. The estimation is performed by using a Gibbs sampling strategy which generates samples distributed according the abundance posterior distribution. These samples are then averaged yielding the abundance minimum mean square error estimator.

We propose a joint segmentation algorithm for piecewise constant AR processes recorded by several independent sensors. The algorithm is based on a hierarchical Bayesian model. Appropriate priors allow to introduce correlations between the change locations of the observed signals. Numerical problems inherent to Bayesian inference are solved by a Gibbs sampling strategy. The proposed joint segmentation methodology provides interesting results compared to a signal-by-signal segmentation