Abstract
Over **the** last few years, **the** development of multi-channel sensors motivated inter- est 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-called **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 emerged as a novel and effective **source** of diversity for BSS. We give here some 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 overviews a **sparsity**-based BSS method coined General- ized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and **sparsity**, using recent sparse overcomplete or redun- dant signal representations. GMCA is a fast and efficient **blind** **source** **separation** method. In remote sensing applications, **the** specificity of hyperspectral data should be accounted for. We extend **the** proposed GMCA framework to deal with hyper- spectral data. In a general framework, GMCA provides a basis for multivariate data analysis in **the** scope of a wide range of classical multivariate data restorate. Nu- merical results are given in color image denoising and inpainting. Finally, GMCA is applied to **the** simulated ESA/Planck data. It is shown to give effective astrophysical component **separation**.

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Figure 10. Damage detection results using EPCA.
6 CONCLUSION
**The** philosophy pursued throughout this paper is to exploit experimental vibration measurements to ex- tract dynamic features of a system without resorting on modal identification results (i.e. natural frequen- cies and/or mode-shapes). To this purpose, tech- niques of **the** **Blind** **Source** **Separation** (BSS) family are considered and especially here, a variant of Prin- cipal Component Analysis based on **the** definition of Hankel matrices is used. In this method, **the** order (number of active principal components) is deter- mined by looking at **the** cumulated variance in **the** singular value diagram. Thus **the** problem of damage detection is tackled using **the** subspaces spanned by **the** active principal components. It consists in de- termining **the** angular coherence between subspaces obtained in current states with respect to a reference (healthy) state. **The** advantage of PCA over classical modal identification methods relies on its easiness of use. First results obtained on **the** Champangshiehl bridge are encouraging.

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partially correlated sources
J. Bobin, J. Rapin, A. Larue and J-L Starck
Abstract
**Blind** **source** **separation** (BSS) is a very popular technique to analyze multichannel data. In this context, **the** data are modeled as **the** linear combination of sources to be retrieved. For that purpose, standard BSS methods all rely on some discrimination principle, whether it is statistical independence or morphological diversity, to distinguish between **the** sources. However, dealing with real-world data reveals that such assumptions are rarely valid in practice: **the** signals of interest are more likely partially correlated, which generally hampers **the** performances of standard BSS methods. In this article, we introduce a novel **sparsity**-enforcing BSS method coined Adaptive Morphological Component Analysis (AMCA), which is designed to retrieve sparse and partially correlated sources. More precisely, it makes profit of an adaptive re-weighting scheme to favor/penalize samples based on their level of correlation. Extensive numerical experiments have been carried out which show that **the** proposed method is robust to **the** partial correlation of sources while standard BSS techniques fail. **The** AMCA algorithm is evaluated in **the** field of astrophysics for **the** **separation** of physical components from microwave data.

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Index Terms— **Blind** **Source** **Separation**; **Sparsity**; Inde- pendant Component Analysis; Optimization
1. INTRODUCTION
**The** instantaneous linear mixture model of BSS assumes that: x = As + e , (1) where x ∈ R M ×T and s ∈ R N ×T are **the** matrices of mixture channels and **source** signals respectively. A ∈ R M ×N is **the** mixing matrix and e ∈ R M ×T models **the** background noise. **The** ICA [1] methods are often applied when M ≥ N (over-determined case). These methods try to achieve sep- aration by minimizing an independence criterion between **the** components of **the** estimated sources. In **the** under- determined case (M < N ), two-steps methods based on **sparsity** are largely used [2]: **The** mixing system is first es- timated using clustering methods [3], then **the** sources are estimated thanks to optimization approaches [4].

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Starting with **blind** **separation** of toy mixtures in **the** mid 90’s, research has pro- gressed up to real-world scenarios today, with applications to speech enhancement and recognition, music editing, 3D sound rendering, and audio information retrieval, among others. This has mostly been made possible by **the** development of increasingly informed **separation** techniques incorporating knowledge about **the** sources and/or **the** mixtures at hand. For instance, speech **source** **separation** for remote conferencing can benefit from prior knowledge of **the** room geometry and/or **the** names of **the** speakers, while music remastering will exploit instrument characteristics and knowledge of sound engineers mixing habits.

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jalal.fadili@greyc.ensicaen.fr - GREYC CNRS UMR 6072, Image Processing Group, ENSICAEN 14050, Caen Cedex, France
Abstract. This paper describes a new **blind** **source** **separation** method for instantaneous linear mixtures. This new method coined GMCA (Gen- eralized Morphological Component Analysis) relies on morphological di- versity. It provides new insights on **the** use of **sparsity** for **blind** **source** **separation** in a noisy environment. GMCA takes advantage of **the** sparse representation of structured data in large overcomplete signal dictionar- ies to separate sources based on their morphology. In this paper, we define morphological diversity and focus on its ability to be a helpful **source** of diversity between **the** signals we wish to separate. We intro- duce **the** **blind** GMCA algorithm and we show that it leads to good re- sults in **the** overdetermined **blind** **source** **separation** problem from noisy mixtures. Both theoretical and algorithmic comparisons between mor- phological diversity and independence-based **separation** techniques are given. **The** effectiveness of **the** proposed scheme is confirmed in several numerical experiments.

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valeurs propres associées sont pour leur part égales, à une constante près, aux moments d’ordre quatre des sources. Deux ans plus tard nait la méthode FOOBI (Fourth Order Only **Blind** Identification) [26] qui sans blanchiment des observations permet d’identifier la matrice de mélange à partir du sous-espace signal d’ordre quatre. Cette méthode a récemment été améliorée par L. De Lathauwer et al. [49] en se servant notamment du procédé de diagonalisation conjointe. Les deux approches précédentes offrent l’avantage de permettre l’identification de mélanges sous-déterminés de sources (i.e. P > N ). En 1993 survient la méthode JADE (Joint Approximate Diagonalization of Eigen-matrices), au travers de laquelle J.-F. Cardoso et A. Souloumiac [30] présentent une solution algébrique à la maximization de leur contraste basé sur les cumulants d’ordre quatre. Ils étendent par la même occasion l’algorithme de Jacobi dans le but de diagonaliser conjointement un ensemble de matrices [31]. Ce dernier sera par la suite l’outil fard de nombreuses méthodes de SAS, et ce jusqu’à l’apogée des approches de décomposition tensorielle telle que PARAFAC [24]. En 1997, A. Ferréol et P. Chevalier [60] proposent une version de JADE, baptisée JADE cyclique, exploitant les éventuelles propriétés cyclostationnaires des signaux observés. Cette approche a pour intérêt d’être insensible à la présence d’un bruit de cohérence spatiale inconnue. Plus récemment, L. Albera et al. [1] mettent en oeuvre les méthodes ICAR (Independent Component Analysis using Redundancies) [3]–[5] et BIRTH (**Blind** Identification of mixtures of sources using Redundancies in **the** daTa Hexacovariance matrix) [2], [6], exploitant les redondances matricielles respectivement de la quadricovariance et de l’hexacovariance. Ces deux méthodes s’inscrivent au sein d’une même famille d’algorithmes baptisée BIOME (**Blind** Identification of Overcomplete Mixtures of sourcEs) [7]. En parallèle, A. Ferréol et al. étendent la méthode SOBI à l’ordre quatre sous le nom de FOBIUM (Fourth Order **Blind** Identification of Underdetermined Mixtures of sources) [58], [59]. Notons que les méthodes FOOBI, ICAR, BIRTH et FOBIUM ne nécessitent pas d’étape préalable de blanchiment et sont insensibles asymptotiquement à la présence d’un bruit de cohérence spatiale inconnue. En outre les algorithmes FOOBI, FOBIUM et BIRTH permettent de traiter des mélanges sous-déterminés de sources.

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IV. D ATASETS A. Generation of simulated data
**The** main goal of this subsection is to explain how we ob- tain synthetic but realistic data for comparing **the** above BSS methods in **the** particular context of epileptic rapid ictal dis- charges. **The** simulated EEG data are generated using a realis- tic head and **source** model as described in [11]. 32−Channels EEG data were simulated from a single distributed **source** of 5cm 2 , referred to as "patch" in **the** following, located in **the** left superior temporal gyrus. Rapid ictal-like activities gen- erated by a neural mass model are assigned to **the** patch. 50 realizations of rapid ictal discharges simulations were gen- erated. These signals corresponded to "clean data". In order to generate noisy EEG simulations, 50 epochs of EEG mus- cle activity were extracted from real 32−channel EEG data. Each trial of EEG muscle activity was then normalized with respect to **the** channel showing **the** maximal power. Then, dif- ferent levels of amplitude of noisy background and muscular activities were added to **the** simulated rapid ictal discharges to get noisy simulated signals with different SNR values. B. Real data

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matrices with large condition numbers yield two major bottlenecks: i) an increased noise level in **the** **source** domain, and ii) **the** mixtures are closer to co-linearity, which makes **the** sources harder to distinguish.
In these experiments, **the** noise level is fixed to 40 dB and **the** **sparsity** level is ρ = 0.1. **The** number of observations is set to 20, **the** number of sources to 5 and **the** number of samples per sources to 10000. Figure 3 shows **the** evolution of **the** mixing matrix criterion C A as a function of **the** mini-batch size t b for two values of **the** condition numbers: left panel 2.5 and right panel 7. As expected, **the** quality of **the** **separation** results of all methods decrease when **the** condition number increases. Similarly to **the** tests performed in **the** previous section, **the** dGMCA algorithm has better results for relatively small mini-batch sizes (but when **the** Fr´ echet mean is used, it eventually de- teriorates for t b < 25, cf. Fig. 3). **The** use of small batches along with **the** robust Fr´ echet mean leads to an improvement for t b < 25, which becomes more significant when **the** condition number increases up to a gain of about one order of magnitude. Similarly, when **the** mini-batch size decreases, **the** discrepancy between **the** two methods increases.

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1200 Montreal Road, Ottawa, Ontario, K1A 0R6, Canada (received June 7, 2010; accepted September 1, 2010 )
This paper proposes an improved method of solving **the** permutation problem inherent in frequency-domain of convolutive **blind** **source** **separation** (BSS). It com- bines a novel inter-frequency dependence measure: **the** power ratio of separated signals, and a simple but effective bin-wise permutation alignment scheme. **The** pro- posed method is easy to implement and surpasses **the** conventional ones. Simulations have shown that it can provide an almost ideal solution of **the** permutation problem for a case where two or three sources were mixed in a room with a reverberation time of 130 ms.

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Index Terms: **blind** **source** **separation**, beamforming, robot au- dition
1. Introduction
Robot audition consists in **the** aptitude of an humanoid to under- stand its acoustic environment, separate and localize sources, identify speakers and recognize their emotions. This complex task is one of **the** target points of **the** R OMEO project [1]. This project aims to build an humanoid (R OMEO ) to help aged peo- ple in their everyday lives. In this project, we focus on **blind** **source** **separation** (BSS) using a microphone array (more than 2 sensors). In a **blind** **source** **separation** task, **the** **separation** should be done from **the** received microphone signals without prior knowledge of **the** mixing process. **The** only knowledge is limited to **the** array geometry. **Source** **separation** is **the** most im- portant step for human-robot interaction: it allows latter tasks like speakers identification, speech and motion recognition and environmental sound analysis.

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A challenging problem of BSS occurs when there are more sources than sensors, and this is referred to as underdeter-
mined **blind** **source** **separation** (UBSS). A time-frequency based
UBSS algorithm has been recently proposed in [2, 3] to suc- cessfully separate speech sources using time-frequency dis- tributions (TFDs). This algorithm provides good **separation** performance when **the** sources are disjoint in **the** TF plane. It also provides **the** **separation** of TF quasi-disjoint sources, that is **the** sources are allowed to have a small degree of overlap- ping in **the** TF plane. However, **the** intersection points in **the** TF plane are not directly treated. More precisely, a point at **the** intersection of two sources is clustered “randomly” to be- long to one of **the** sources. As a result, **the** **source** that picks up this point now contains some information from **the** other **source** while **the** later **source** loses some information of its own. However, for **the** other **source**, there is an interference

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Fig. 16. Four first elution profiles and corresponding mass spectra estimated by SCSA-UNS, from HPLC-LTQ Orbitrap data
VIII. C ONCLUSION
In this paper, we propose a geometrical method for separat- ing non-negative sources. **The** proposal, denoted SCSA-UNS, estimates **the** mixing matrix and **the** sources, by first reducing **the** dimension of **the** mixed data, followed by fitting a Min- imum Aperture Simplicial Cone (MASC) to **the** scatter plot of **the** dimension reduced data. SCSA-UNS does not require **the** independence of sources, neither their local dominance, but **the** positive orthant must be **the** unique MASC containing **the** scatter plot of **the** sources, to ensure recovering **the** true mixing matrix and **the** true sources. In noisy case, **the** proposed method starts by discarding **the** points most corrupted by **the** noise, which can significantly expand **the** scatter plot of mixed data, before looking for **the** MASC containing **the** data. Simulation on synthetic data have showned that **the** proposed method performs good **separation** for both independent and mutually correlated sources. **The** proposal has also been suc- cessfully used to estimate **the** pharmacokinetic compartments of [18F]-FDG tracer on human brain (in particular to estimated **the** Arterial Input Function) and to separate **the** elementary mass spectra of differents chemical compounds, from **the** mass spectra measured at **the** output of a liquid chromatograph.

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In **the** future, we would like to further analyze **the** theoretical stability of **the** problem in order to improve **the** selection of
initial conditions for **the** algorithm. For **the** current version of **the** algorithm, **the** conditions are randomly selected and can sometimes lead to **the** algorithm not converging, in which case we restart **the** method until it succeeds. While this has not proven an issue for our current application (possibly due to **the** fact that **the** mixing coefficients are typically small), a more theoretical analysis would be important to avoid problems with more general mixing models, and possibly improve **the** accuracy and convergence speed of our proposed method.

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Z (f, k) = B (f ) X (f, k) (4) To design a fixed beamformer that will achieve **the** desired beam pattern (according to a desired direction response), **the** least-square (LS) technique is used [5] and thus **the** steering vectors are needed. In **the** case of robot audition, **the** microphones are often fixed in **the** head of **the** robot and it is generally hard to know exactly **the** ge- ometry of **the** microphone array (cf. figure 3). Besides, **the** phase and magnitude of **the** steering vectors do not take into account **the** influence of **the** head on **the** surrounding acoustic fields. So we pro- pose to use **the** Head Related Transfer Functions (HRTFs) as steering vectors {a (f, θ)} θ∈Θ , where Θ = {θ1 , . . . , θK} is a group of K a priori chosen steering directions (cf. figure 2). **The** HRTF charac- terizes how **the** signal emitted from a specific direction is received at a sensor fixed in a head. It takes into account **the** geometry of **the** head, and thus **the** geometry of **the** microphone array.

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Abstract—We consider **the** problem of **blind** **source** separa- tion for underdetermined convolutive mixtures. Based on **the** multiplicative narrowband approximation in **the** time-frequency domain with **the** help of Short-Time-Fourier-Transform (STFT) and **the** sparse representation of **the** **source** signals, we formulate **the** **separation** problem into an optimization framework. This framework is then generalized based on **the** recently investigated convolutive narrowband approximation and **the** statistics of **the** room impulse response. Algorithms with convergence proof are then employed to solve **the** proposed optimization problems. **The** evaluation of **the** proposed frameworks and algorithms for synthesized and live recorded mixtures are illustrated. **The** proposed approaches are also tested for mixtures with input noise. Numerical evaluations show **the** advantages of **the** proposed methods.

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VI. C ONCLUSION
**The** ICAR algorithm, exploiting **the** information contained in **the** data statistics at fourth order only, has been proposed in this paper. This algorithm allows to process overdetermined (including square) mixtures of sources, provided **the** latter have marginal FO cumulants with **the** same sign, which is generally **the** case in radio communications contexts. Three conclusions can be drawn: first, in **the** presence of a Gaussian noise spatially and temporally white, **the** proposed method yields satisfactory results. Second, contrary to most BSS algorithms, **the** ICAR method is not sensitive to a Gaussian colored noise whose spatial coherence is unknown. Last, **the** ICAR algorithm is robust with respect to an over estimation of **the** number of sources, which is not **the** case for some methods such as JADE. Forthcoming works include **the** search for a contrast criterion associated with ICAR in order to analyse accurately its performance.

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HAL author manuscript inserm-00186093, version 1

Received 14 January 2010; Accepted 1 June 2010 Academic Editor: Harvey Thornburg
Copyright © 2010 Lin Wang et al. This is an open access article distributed under **the** Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided **the** original work is properly cited. Frequency-domain **blind** **source** **separation** (BSS) performs poorly in high reverberation because **the** independence assumption collapses at each frequency bins when **the** number of bins increases. To improve **the** **separation** result, this paper proposes a method which combines two techniques by using beamforming as a preprocessor of **blind** **source** **separation**. With **the** sound **source** locations supposed to be known, **the** mixed signals are dereverberated and enhanced by beamforming; then **the** beamformed signals are further separated by **blind** **source** **separation**. To implement **the** proposed method, a superdirective fixed beamformer is designed for beamforming, and an interfrequency dependence-based permutation alignment scheme is presented for frequency- domain **blind** **source** **separation**. With beamforming shortening mixing filters and reducing noise before **blind** **source** **separation**, **the** combined method works better in reverberation. **The** performance of **the** proposed method is investigated by separating up to 4 sources in different environments with reverberation time from 100 ms to 700 ms. Simulation results verify **the** outperformance of **the** proposed method over using beamforming or **blind** **source** **separation** alone. Analysis demonstrates that **the** proposed method is computationally efficient and appropriate for real-time processing.

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