Article

# A Recursive Sparse Blind Source Separation Method and Its Application to Correlated Data in NMR Spectroscopy of Biofluids

Journal of Scientific Computing (Impact Factor: 1.7). 06/2011; 51(3):1-21. DOI: 10.1007/s10915-011-9528-9

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**ABSTRACT:**Given a set of mixtures, blind source separation attempts to retrieve the source signals without or with very little information of the mixing process. We present a geometric approach for blind separation of nonnegative linear mixtures termed {\em facet component analysis}. The approach is based on facet identification of the underlying cone structure of the data. Earlier works focus on recovering the cone by locating its vertices (vertex component analysis) based on a mutual sparsity condition which requires each source signal to possess a stand-alone peak in its spectrum. We formulate alternative conditions so that enough data points fall on the facets of a cone instead of accumulating around the vertices. To find a regime of unique solvability, we make use of both geometric and density properties of the data points, and develop an efficient facet identification method by combining data classification and linear regression. For noisy data, total variation technique may be employed. We show computational results on nuclear magnetic resonance spectroscopic data to substantiate our method.Signal Image and Video Processing 09/2014; · 1.02 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Fourier transform is the data processing naturally associated to most NMR experiments. Notable exceptions are Pulse Field Gradient and relaxation analysis, the structure of which is only partially suitable for FT. With the revamp of NMR of complex mixtures, fueled by analytical challenges such as metabolomics, alternative and more apt mathematical methods for data processing have been sought, with the aim of decomposing the NMR signal into simpler bits. Blind Source Separation is a very broad definition regrouping several classes of mathematical methods for complex signal decomposition that use no hypothesis on the form of the data. Developed outside NMR, these algorithms have been increasingly tested on spectra of mixtures. In this review, we shall provide an historical overview of the application of Blind Source Separation methodologies to NMR, including methods specifically designed for the specificity of this spectroscopy.Progress in Nuclear Magnetic Resonance Spectroscopy 08/2014; · 8.71 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we develop a novel blind source separation (BSS) method for nonnegative and correlated data, particularly for the nearly degenerate data. The motivation lies in nuclear magnetic resonance (NMR) spectroscopy, where a multiple mixture NMR spectra are recorded to identify chemical compounds with similar structures (degeneracy). There have been a number of successful approaches for solving BSS problems by exploiting the nature of source signals. For instance, independent component analysis (ICA) is used to separate statistically independent (orthogonal) source signals. However, signal orthogonality is not guaranteed in many real-world problems. This new BSS method developed here deals with nonorthogonal signals. The independence assumption is replaced by a condition which requires dominant interval(s) (DI) from each of source signals over others. Additionally, the mixing matrix is assumed to be nearly singular. The method first estimates the mixing matrix by exploiting geometry in data clustering. Due to the degeneracy of the data, a small deviation in the estimation may introduce errors (spurious peaks of negative values in most cases) in the output. To resolve this challenging problem and improve robustness of the separation, methods are developed in two aspects. One technique is to find a better estimation of the mixing matrix by allowing a constrained perturbation to the clustering output, and it can be achieved by a quadratic programming. The other is to seek sparse source signals by exploiting the DI condition, and it solves an $\ell_1$ optimization. We present numerical results of NMR data to show the performance and reliability of the method in the applications arising in NMR spectroscopy.10/2011;

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