Recovering radionuclide contributions from γ -ray spectra is a central task in nuclear signal processing, which goal is to estimate both the underlying source spectra and their proportions in a measured signal. This source separation problem relies on prior spectral knowledge to guide the estimation, but it remains challenging due to noise and spectral variabilities due to not well-known measurement conditions, caused by γ -photon interactions in the source surroundings or differing source geometries. The problem is semi-blind in nature: the radionuclide sources are known, but their exact spectral responses are unknown. Existing unmixing algorithms either rely on iterative solvers which are slow, or use “black-box” neural networks overlooking the underlying physical structure of the problem. In this work, we propose GLUPSS, a Generative Learned Unrolling algorithm for Poisson Source Separation. Our approach leverages the statistical nature of the data by incorporating a Poisson log-likelihood loss and models the spectral variability using a 1D manifold learned through a generative Interpolating AutoEncoder (IAE). Furthermore, building on algorithm unrolling, we propose a neural network which mimics a proximal optimization scheme but drastically reduces the number of iterations, making GLUPSS usable for real-life, time-sensitive applications. Experimental results on realistic synthetic data show that GLUPSS achieves an estimation accuracy within 1% (in terms of Absolute Relative Error) of the state-of-the-art SEMSUN iterative methods while offering a 60 × speedup compared to this method 1 .

The problem of sparse Blind Source Separation (BSS) has been extensively studied when the noise is additive and Gaussian. This is however not the case when the measurements follow Poisson or shot noise statistics, which is customary with counting-based measurements. To that purpose, we introduce a novel sparse BSS algorithm coined pGMCA (poisson-Generalized Morphological Component Analysis) that specifically tackles the blind separation of sparse sources from measurements following Poisson statistics. The proposed algorithm builds upon Nesterov's smoothing technique to define a smooth approximation of sparse BSS, with a data fidelity term derived from the Poisson likelihood. This allows to design a block coordinate descent-based minimization procedure with a simple choice of the regularization parameter. Numerical experiments have been carried out that illustrate the robustness of the proposed method with respect to Poisson noise. The pGMCA algorithm has been further evaluated in a realistic astrophysical X-ray imaging setting.

Blind source separation (BSS) methods are well suited for the analysis of multichannel data. In many applications, the observations are corrupted by an additional structured noise, which hinders most of the standard BSS techniques. In this article, we propose a novel robust BSS approach able to jointly unmix the sources and separate the source contribution from the structured noise or outliers. It first builds on a new sparse component modeling that allows combining both the spectral and morphological/spatial diversity of the sources and the outliers. We introduce the tr-rGMCA algorithm (robust generalized morphological component analysis in transform domains) to tackle the underlying robust BSS problem. Numerical experiments highlight the robustness and precision of the proposed method in a wide variety of settings, including the full-rank regime.

Blind source separation (BSS) is a challenging matrix factorization problem that plays a central role in multichannel imaging science. In a large number of applications, such as astrophysics, current unmixing methods are limited since real-world mixtures are generally affected by extra instrumental effects like blurring. Therefore, BSS has to be solved jointly with a deconvolution problem, which requires tackling a new inverse problem: deconvolution BSS (DBSS). In this article, we introduce an innovative DBSS approach, called DecGMCA (deconvolved generalized morphological component analysis), based on sparse signal modeling and an efficient alternative projected least-squares algorithm. Numerical results demonstrate that the DecGMCA algorithm performs very well on simulations. It further highlights the importance of jointly solving BSS and deconvolution instead of considering these two problems independently. Furthermore, the performance of the proposed DecGMCA algorithm is demonstrated on simulated radio-interferometric data.

Blind source separation is a widely used technique to analyze multichannel data. In many real-world applications, its results can be significantly hampered by the presence of unknown outliers. In this paper, a novel algorithm coined rGMCA (robust Generalized Morphological Component Analysis) is introduced to retrieve sparse sources in the presence of outliers. It explicitly estimates the sources, the mixing matrix, and the outliers. It also takes advantage of the estimation of the outliers to further implement a weighting scheme, which provides a highly robust separation procedure. Numerical experiments demonstrate the efficiency of rGMCA to estimate the mixing matrix in comparison with standard BSS techniques.

Blind source separation (BSS) is a very popular technique to analyze data which can be modeled as linear mixtures of elementary sources. Standard approaches generally make the assumption that such sources are statistically independent or at least uncorrelated. However, this is barely the case for real-world sources which are very often partially correlated. We present a new sparsity-enforcing BSS method coined Adaptive Morphological Component Analysis (AMCA) designed to retrieve sparse and partially correlated sources based on an adaptive weighting scheme. Numerical experiments have been carried out which show that the proposed method is robust to the partial correlation of the sources while standard BSS techniques fail. The performances of the proposed algorithm are further illustrated with simulations in the context of astrophysics.

Anomaly detection from multi-spectral images is a standard image processing problem in civilian and military applications. The major difficulty of this problem lies in the complex background modeling: not only the background is usually heterogeneous (contains multiple sources, e.g. vegetation, soil, etc) but also spectrally varying even for an homogeneous background. Unlike most existing methods that rely on a Gaussian background model, we do not consider any parametric statistical model for the background; the latter is modeled by a linear mixture of multiple sources estimated using recent sparse blind source separation methods. The detection procedure is then based on contrast measures derived from the estimated mixture parameters. We numerically show that our method is more relevant than two other methods of the state of the art.

Nonnegative blind source separation, which is also referred to as nonnegative matrix factorization (NMF), is a very active field in domains as different as astrophysics, audio processing, and biomedical signal processing. In this context, the efficient retrieval of the sources requires the use of signal priors such as sparsity. Although NMF has been well studied with sparse constraints in the direct domain, only very few algorithms can encompass nonnegativity together with sparsity in a transformed domain since simultaneously dealing with two priors in two different domains is challenging. In this paper, we show how a sparse NMF algorithm called nonnegative generalized morphological component analysis (nGMCA) can be extended to impose nonnegativity in the direct domain along with sparsity in a transformed domain, with both analysis and synthesis formulations. To the best of our knowledge, this work presents the first comparison of analysis and synthesis priors---as well as their reweighted versions---in the context of blind source separation. Comparisons with state-of-the-art NMF algorithms on realistic data show the efficiency as well as the robustness of the proposed algorithms.

After a series of successful full-sky CMB (Cosmic Microwave Background) experiments (COBE and WMAP to only name two), the latest spatial mission of the European Space Agency, Planck, has started observing the whole sky in mid-2009. This experiment is of premier importance for the cosmologists to study the birth of our universe via the analysis of the CMB. The latter astrophysical component, among others, is not directly observable in the Planck data but rather mixed up with other components. For the sake of scientific exploitation, accessing such precious physical information requires extracting several different astrophysi-cal components (CMB, Sunyaev-Zel'dovich clusters, galactic dust). Mathematically, this problem amounts to a component separation problem. In the field of CMB studies, a very large range of state-of-the art source separation methods have been applied. Most of these methods assume that the sought after components are mixed up in the data according to the standard global linear mixture model. Nevertheless, this model does not hold and more accurate models require modeling the mixtures locally rather than globally. The purpose of this paper is to introduce an extension of GMCA (Generalized Morphological Component Analysis) to handle a local mixture model. Preliminary results on simulated Planck data are presented.

Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. This paper applies a smoothing technique and an accelerated first-order algorithm, both from Nesterov [Math. Program. Ser. A, 103 (2005), pp. 127–152], and demonstrates that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems as (1) it is computationally efficient, (2) it is accurate and returns solutions with several correct digits, (3) it is flexible and amenable to many kinds of reconstruction problems, and (4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as total-variation minimization and convex programs seeking to minimize the $\ell_1$ norm of $Wx$ under constraints, in which W is not diagonal. The code is available online as a free package in the MATLAB language.

Generalized morphological component analysis (GMCA) is a recent algorithm for multichannel data analysis which was used successfully in a variety of applications including multichannel sparse decomposition, blind source separation (BSS), color image restoration and inpainting. Building on GMCA, the purpose of this contribution is to describe a new algorithm for BSS applications in hyperspectral data processing. It assumes the collected data is a mixture of components exhibiting sparse spectral signatures as well as sparse spatial morphologies, each in specified dictionaries of spectral and spatial waveforms. We report on numerical experiments with synthetic data and application to real observations which demonstrate the validity of the proposed method.

This paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by reviewing our recent work in this field. The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has far-reaching applications in science and technology. Starck , proposed a novel decomposition method-morphological component analysis (MCA)-based on sparse representation of signals. MCA assumes that each (monochannel) signal is the linear mixture of several layers, the so-called morphological components, that are morphologically distinct, e.g., sines and bumps. The success of this method relies on two tenets: sparsity and morphological diversity. That is, each morphological component is sparsely represented in a specific transform domain, and the latter is highly inefficient in representing the other content in the mixture. Once such transforms are identified, MCA is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will provide an overview the generalized MCA introduced by the authors in and as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation.

Wavelets have been used extensively for several years now in astronomy for many purposes, ranging from data filtering and deconvolution to star and galaxy detection or cosmic-ray removal. More recent sparse representations such as ridgelets or curvelets have also been proposed for the detection of anisotropic features such as cosmic strings in the cosmic microwave background. We review in this paper a range of methods based on sparsity that have been proposed for astronomical data analysis. We also discuss the impact of compressed sensing, the new sampling theory, in astronomy for collecting the data, transferring them to earth or reconstructing an image from incomplete measurements.

This paper introduces a new, fast and accurate algorithm for solving problems in the area of compressed sensing, and more generally, in the area of signal and image reconstruction from indirect measurements. This algorithm is inspired by recent progress in the development of novel first-order methods in convex optimization, most notably Nesterov's smoothing technique. In particular, there is a crucial property that makes these methods extremely efficient for solving compressed sensing problems. Numerical experiments show the promising performance of our method to solve problems which involve the recovery of signals spanning a large dynamic range.

Recently morphological diversity and sparsity have emerged as new and effective sources of diversity for blind source separation. Based on these new concepts, novel methods such as generalized morphological component analysis have been put forward. The latter takes advantage of the very sparse representation of structured data in large overcomplete dictionaries, to separate sources based on their morphology. Building on GMCA, the purpose of this contribution is to describe a new algorithm for hyperspectral data processing. Large-scale hyperspectral data refers to collected data that exhibit sparse spectral signatures in addition to sparse spatial morphologies, in specified dictionaries of spectral and spatial waveforms. Numerical experiments are reported which demonstrate the validity of the proposed extension for solving source separation problems involving hyperspectral data.

Over the last few years, the development of multichannel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-caIled blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emergedas a novel and effective source of diversity for BSS. Here, we give some new and essential insights into the use of sparsity in source separation, and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper introduces a new BSS method coined generalized morphological component analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.

In a recent paper, a method called morphological component analysis (MCA) has been proposed to separate the texture from the natural part in images. MCA relies on an iterative thresholding algorithm, using a threshold which decreases linearly towards zero along the iterations. This paper shows how the MCA convergence can be drastically improved using the mutual incoherence of the dictionaries associated to the different components. This modified MCA algorithm is then compared to basis pursuit, and experiments show that MCA and BP solutions are similar in terms of sparsity, as measured by the l1 norm, but MCA is much faster and gives us the possibility of handling large scale data sets.

This paper describes two extensions of the recent morphological component analysis (MCA) method to multichannel data. MCA takes advantage of the sparse representation of structured data in large overcomplete dictionaries to separate features in the data based on their morphology. It was shown to be an efficient technique in such problems as separating an image into texture and piecewise smooth parts or for inpainting applications. A first extension, MMCA, achieves a similar source separation objective based on morphological diversity. A second extension, GMMCA, takes advantage of the highly sparse representations of the sources that can be built using MCA. Indeed, sparsity is now generally recognized as a valuable property for blind source separation. The efficiency of MMCA and GMMCA is confirmed in numerical experiments

This letter describes a new method for blind source separation, adapted to the case of sources having different morphologies. We show that such morphological diversity leads to a new and very efficient separation method, even in the presence of noise. The algorithm, coined multichannel morphological component analysis (MMCA), is an extension of the morphological component analysis (MCA) method. The latter takes advantage of the sparse representation of structured data in large overcomplete dictionaries to separate features in the data based on their morphology. MCA has been shown to be an efficient technique in such problems as separating an image into texture and piecewise smooth parts or for inpainting applications. The proposed extension, MMCA, extends the above for multichannel data, achieving a better source separation in those circumstances. Furthermore, the new algorithm can efficiently achieve good separation in a noisy context where standard independent component analysis methods fail. The efficiency of the proposed scheme is confirmed in numerical experiments