Sparse and Non-Negative BSS for Noisy Data

Non-negative blind source separation (BSS) has raised interest in various fields of research, as testified by the wide literature on the topic of non-negative matrix factorization (NMF). In this context, it is fundamental that the sources to be estimated present some diversity in order to be efficiently retrieved. Sparsity is known to enhance such contrast between the sources while producing very robust approaches, especially to noise. In this paper we introduce a new algorithm in order to tackle the blind separation of non-negative sparse sources from noisy measurements. We first show that sparsity and non-negativity constraints have to be carefully applied on the sought-after solution. In fact, improperly constrained solutions are unlikely to be stable and are therefore sub-optimal. The proposed algorithm, named nGMCA (non-negative Generalized Morphological Component Analysis), makes use of proximal calculus techniques to provide properly constrained solutions. The performance of nGMCA compared to other state-of-the-art algorithms is demonstrated by numerical experiments encompassing a wide variety of settings, with negligible parameter tuning. In particular, nGMCA is shown to provide robustness to noise and performs well on synthetic mixtures of real NMR spectra.Comment: 13 pages, 18 figures, to be published in IEEE Transactions on Signal Processin

Image Decomposition and Separation Using Sparse Representations: An Overview

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

Sparse BSS in the presence of outliers

submitted to SPARS15—While real-world data are often grossly corrupted, most techniques of blind source separation (BSS) give erroneous results in the presence of outliers. We propose a robust algorithm that jointly estimates the sparse sources and outliers without requiring any prior knowledge on the outliers. More precisely, it uses an alternative weighted scheme to weaken the influence of the estimated outliers. A preliminary experiment is presented and demonstrates the advantage of the proposed algorithm in comparison with state-of-the-art BSS methods. I. PROBLEM FORMULATION Suppose we are given m noisy observations {Xi} i=1..m of unknown linear mixtures of n ≤ m sparse sources {Sj} j=1..n with t > m samples. It is generally assumed that these data are corrupted by a Gaussian noise, accounting for instrumental or model imperfections. However in many applications, some entries are additionally corrupted by outliers, leading to the following model: X = AS + O + N, with X the observations, A the mixing matrix, S the sources, O the outliers, and N the Gaussian noise. In the presence of outliers, the key difficulty lies in separating the components O and AS. To this end, assuming that the term AS has low-rank, some strategies [4] suggest to pre-process the data to estimate and remove the outliers with RPCA [3]. However, besides the fact that low-rankness is generally restrictive for most BSS problems, the source separation is severely hampered if the outliers are not well estimated. Therefore, we introduce a method that estimates the sources in the presence of the outliers without pre-processing. For the best of our knowledge, it has only been studied in [5] by using the β-divergence. Unlike [5], we propose to estimate jointly the outliers and the sources by exploiting their sparsity. II. ALGORITH

Cosmic Dawn and Epoch of Reionization Foreground Removal with the SKA

The exceptional sensitivity of the SKA will allow observations of the Cosmic Dawn and Epoch of Reionization (CD/EoR) in unprecedented detail, both spectrally and spatially. This wealth of information is buried under Galactic and extragalactic foregrounds, which must be removed accurately and precisely in order to reveal the cosmological signal. This problem has been addressed already for the previous generation of radio telescopes, but the application to SKA is different in many aspects. In this chapter we summarise the contributions to the field of foreground removal in the context of high redshift and high sensitivity 21-cm measurements. We use a state-of-the-art simulation of the SKA Phase 1 observations complete with cosmological signal, foregrounds and frequency-dependent instrumental effects to test both parametric and non-parametric foreground removal methods. We compare the recovered cosmological signal using several different statistics and explore one of the most exciting possibilities with the SKA --- imaging of the ionized bubbles. We find that with current methods it is possible to remove the foregrounds with great accuracy and to get impressive power spectra and images of the cosmological signal. The frequency-dependent PSF of the instrument complicates this recovery, so we resort to splitting the observation bandwidth into smaller segments, each of a common resolution. If the foregrounds are allowed a random variation from the smooth power law along the line of sight, methods exploiting the smoothness of foregrounds or a parametrization of their behaviour are challenged much more than non-parametric ones. However, we show that correction techniques can be implemented to restore the performances of parametric approaches, as long as the first-order approximation of a power law stands.Comment: Accepted for publication in the SKA Science Book 'Advancing Astrophysics with the Square Kilometre Array', to appear in 201

Application of Non-negative Matrix Factorization to LC/MS data

International audienceLiquid Chromatography-Mass Spectrometry (LC/MS) provides large datasets from which one needs to extract the relevant information. Since these data are made of non-negative mixtures of non-negative mass spectra, non-negative matrix factorization (NMF) is well suited for its processing, but it has barely been used in LC/MS. Also, these data are very difficult to deal with since they are usually contaminated with non-Gaussian noise and the intensities vary on several orders of magnitude. In this article, we show the feasibility of the NMF approach on these data. We also propose an adaptation of one of the algorithms aiming at specifically dealing with LC/MS data. We finally perform experiments and compare standard NMF algorithms on both simulated data and an annotated LC/MS dataset. This lets us evaluate the influence of the noise model and the data model on the recovery of the sources

Morphological diversity and sparsity : new insights into multivariate data analysis

International audienceOver the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. From blind source separation (BSS) to multi/hyper-spectral data restoration, an extensive work has already been dedicated to multivariate data processing. Previous work has emphasized on the fundamental role played by sparsity and morphological diversity to enhance multichannel signal processing. Morphological diversity has been first introduced in the mono-channel case to deal with contour/texture extraction. The morphological diversity concept states that the data are the linear combination of several so-called morphological components which are sparse in different incoherent representations. In that setting, piecewise smooth features (contours) and oscillating components (textures) are separated based on their morphological differences assuming that contours (respectively textures) are sparse in the Curvelet representation (respectively Local Discrete Cosine representation). In the present paper, we define a multichannel-based framework for sparse multivariate data representation. We introduce an extension of morphological diversity to the multichannel case which boils down to assuming that each multichannel morphological component is diversely sparse spectrally and/or spatially. We propose the Generalized Morphological Component Analysis algorithm (GMCA) which aims at recovering the so-called multichannel morphological components. Hereafter, we apply the GMCA framework to two distinct multivariate inverse problems : blind source separation (BSS) and multichannel data restoration. In the two aforementioned applications, we show that GMCA provides new and essential insights into the use of morphological diversity and sparsity for multivariate data processing. Further details and numerical results in multivariate image and signal processing will be given illustrating the good performance of GMCA in those distinct applications

Synergic PDE3 and PDE4 control intracellular cAMP and cardiac excitation-contraction coupling in a porcine model

International audienc

Stacked Sparse Non-Linear Blind Source Separation

Linear Blind Source Separation (BSS) has known a tremendous success in fields ranging from biomedical imaging to astrophysics. In this work, we however depart from the usual linear setting and tackle the case in which the sources are mixed by an unknown non-linear function. We propose to use a sequential decomposition of the data enabling its approximation by a linear-by-part function. Beyond separating the sources, the introduced StackedAMCA can further empirically learn in some settings an approximation of the inverse of the unknown non-linear mixing, enabling to reconstruct the sources despite a severely ill-posed problem. The quality of the method is demonstrated experimentally, and a comparison is performed with state-of-the art non-linear BSS algorithms