Sparse Component Analysis and Blind Source Separation of Underdetermined Mixtures

Georgiev, Pando; Theis, Fabian J.; Cichocki, Andrzej

doi:10.1109/tnn.2005.849840

Cited by 308 publications

(280 citation statements)

References 9 publications

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“…The reason is that even when A is known solution of the linear system of equations (1) or (7) is not unique, because there are more unknowns (M) than equations (N). If pure components are (M-N+1)-sparse, a unique solution is obtained at the minimum of the 1  norm of s, [24][25][26][27][30][31][32][33][34]. We could formulate linear programming based solution in the time-scale basis (7).…”

Section: Linear Programming-based Solution Of the Underdetermined Sysmentioning

confidence: 99%

“…Thus, a BSS method capable to extract pure components from reduced number of mixtures (that is less than the number of pure components) appears to be of great importance. This leads to underdetermined BSS (uBSS) problem that is not solvable under standard ICA assumptions, [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

“…Here, we propose a sparse component analysis (SCA) approach to solve related uBSS problem by having at disposal two mixtures only, [24][25][26][27]. It combines geometric approach known as clustering in wavelet domain to estimate concentration or mixing matrix and linear programming in Fourier/frequency domain to estimate pure components.…”

Section: Introductionmentioning

confidence: 99%

“…When N<M the BSS problem is underdetermined because there are less measured mixtures N available than unknown pure components M. In order to solve the uBSS problem additional a priori information about the pure components must be available, [24][25][26][27]. A priori information that is used most often is sparseness of the source signals in suitably chosen basis, [24][25][26][27]. [8] by the fact that it maximally exploits redundancy of the linear data model (1) in the chosen basis.…”

mentioning

confidence: 99%

“…The SCA method in [8] solves BSS problem by finding de-mixing matrix W by minimizing cost function that measures sparseness of the sources, however it still requires N=M. On the other side the SCA approach used here, and referred in [24][25][26][27], breaks down BSS problem into two separate problems: estimation of the mixing or concentration matrix A using geometric concept known as data clustering [24][25][26][27][28][29], and estimation of the magnitude spectra of the pure components (based on estimated A) by solving resulting underdetermined system of linear equations through linear programming [24,25,30,31], 1  -regularized least square problem [32,33] or 2  -regularized linear problem, [34]. In the case of the NMR spectroscopy it is customary to assume that Fourier basis yields sparse representation, however wavelet basis with properly chosen wavelet function can yield even sparser representation.…”

mentioning

confidence: 99%

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Extraction of multiple pure component 1H and 13C NMR spectra from two mixtures: Novel solution obtained by sparse component analysis-based blind decomposition

Kopriva

Jerić

Smrečki

2009

Analytica Chimica Acta

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Sparse Component Analysis (SCA) is demonstrated for blind extraction of three pure component spectra from only two measured mixed spectra in 13 C and 1 H nuclear magnetic resonance (NMR) spectroscopy. This appears to be the first time to report such results and that is the first novelty of the paper. Presented concept is general and directly applicable to experimental scenarios that possibly would require use of more than two mixtures. However, it is important to emphasize that number of required mixtures is always less than number of components present in these mixtures. The second novelty is formulation of blind NMR spectra decomposition exploiting sparseness of the pure components in the wavelet basis defined by either Morlet or Mexican hat wavelet. This enabled accurate estimation of the concentration matrix and number of pure components by means of data clustering algorithm and pure components § Patent pending under the number PCT/HR2008/000037. 2 spectra by means of linear programming with constraints from both 1 H and 13 C NMR experimental data.The third novelty is capability of proposed method to estimate number of pure components in demanding underdetermined blind source separation (uBSS) scenario. This is in contrast to majority of the BSS algorithms that assume this information to be known in advance. Presented results are important for the NMR spectroscopy-associated data analysis in pharmaceutical industry, medicine diagnostics and natural products research.

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Section: Linear Programming-based Solution Of the Underdetermined Sysmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Extraction of multiple pure component 1H and 13C NMR spectra from two mixtures: Novel solution obtained by sparse component analysis-based blind decomposition

Kopriva

Jerić

Smrečki

2009

Analytica Chimica Acta

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2010

Wideband Beamforming

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Blind Source Separation

Bronstein

Zibulevsky

2006

Wiley Encyclopedia of Biomedical Engineering

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Over the last few years, the development of multi-channel 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-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 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 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 hyperspectral 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. Numerical 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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Sparse Component Analysis and Blind Source Separation of Underdetermined Mixtures

Cited by 308 publications

References 9 publications

Extraction of multiple pure component 1H and 13C NMR spectra from two mixtures: Novel solution obtained by sparse component analysis-based blind decomposition

Extraction of multiple pure component 1H and 13C NMR spectra from two mixtures: Novel solution obtained by sparse component analysis-based blind decomposition

Bibliography

Blind Source Separation

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