A review of blind source separation in NMR spectroscopy

Toumi, Ichrak; Caldarelli, Stéfano; Torrésani, Bruno

doi:10.1016/j.pnmrs.2014.06.002

Cited by 30 publications

(35 citation statements)

References 107 publications

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“…Problems of the type (1) or (2) are ubiquitous in many applied scientific disciplines and in applications, see e.g [19], [46], [31], [38], [37], [25], [36], [41], [42], [13], [49]. Thus, there is a large body of works to solve different versions of these problems.…”

Section: A Related Workmentioning

confidence: 99%

Blind Deconvolution Meets Blind Demixing: Algorithms and Performance Bounds

Ling

Strohmer

2017

IEEE Trans. Inform. Theory

110

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Suppose that we have r sensors and each one intends to send a function g i (e.g. a signal or an image) to a receiver common to all r sensors. During transmission, each g i gets convolved with a function f i . The receiver records the function y, given by the sum of all these convolved signals. When and under which conditions is it possible to recover the individual signals g i and the blurring functions f i from just one received signal y? This challenging problem, which intertwines blind deconvolution with blind demixing, appears in a variety of applications, such as audio processing, image processing, neuroscience, spectroscopy, and astronomy. It is also expected to play a central role in connection with the future Internet-of-Things. We will prove that under reasonable and practical assumptions, it is possible to solve this otherwise highly ill-posed problem and recover the r transmitted functions g i and the impulse responses f i in a robust, reliable, and efficient manner from just one single received function y by solving a semidefinite program. We derive explicit bounds on the number of measurements needed for successful recovery and prove that our method is robust in the presence of noise. Our theory is actually sub-optimal, since numerical experiments demonstrate that, quite remarkably, recovery is still possible if the number of measurements is close to the number of degrees of freedom.

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Section: A Related Workmentioning

confidence: 99%

Blind Deconvolution Meets Blind Demixing: Algorithms and Performance Bounds

Ling

Strohmer

2017

IEEE Trans. Inform. Theory

110

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show abstract

“…It should however be emphasized that the novel proximity operators we derive could be applied in a variety of proximal algorithms, in the convex [34] or the non-convex case [35,36]. The latter case should be of particular interest in the context of blind signal restoration problems such as those encountered in [37,38] where the proposed hybrid penalties could be beneficial.…”

Section: Resultsmentioning

confidence: 99%

Proximity operators for a class of hybrid sparsity + entropy priors application to dosy NMR signal reconstruction

Cherni

Chouzenoux

Delsuc

2016

2016 International Symposium on Signal, Image, Video and Communications (ISIVC)

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Inverse problems arising from Laplace transform inversion are ill-posed, and require suitable regularization strategies. Although the maximum entropy regularization approach usually appears as an adequate strategy due to its ability to recover regular positive valued signals, it was observed to lead to poor reconstruction results when the sought signal contains narrow peaks. In that case, a sparsity promoting penalty such as the 1 norm, combined with a positivity constraint, is more suitable. In order to derive a flexible resolution method, hybrid approaches combining both entropy and sparsity regularization strategies should be envisaged. However, the choice of an efficient optimization algorithm remains a challenging task. Among available optimization techniques, proximal methods have shown their efficiency in solving large scale possibly nonsmooth problems. This paper provides an extensive list of new proximity operators for the sum of entropy and sparsity penalties. The applicability of these results is illustrated by means of experiments, in the context of DOSY NMR signal reconstruction.

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“…More precisely, they need to be sparsely represented, i.e., with few atoms, with respect to two incoherent dictionaries A f and A g . This method is usually called "morphological component analysis" (MCA), and was introduced by Starck et al in [69] (see [25,45,44,50,24,23,42,56,71,60] for related works and [55] for a survey on the topic. In particular, Donoho and Kutyniok [42,56] first provided a theoretical foundation of geometric image separation into point and curve singularities by using tools from sparsity methodologies.…”

Section: Introductionmentioning

confidence: 99%

Disjoint sparsity for signal separation and applications to hybrid inverse problems in medical imaging

Alberti

Ammari

2017

Applied and Computational Harmonic Analysis

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Abstract. The main focus of this work is the reconstruction of the signals f and g i , i = 1, . . . , N , from the knowledge of their sums h i = f + g i , under the assumption that f and the g i 's can be sparsely represented with respect to two different dictionaries A f and Ag. This generalizes the well-known "morphological component analysis" to a multi-measurement setting. The main result of the paper states that f and the g i 's can be uniquely and stably reconstructed by finding sparse representations of h i for every i with respect to the concatenated dictionary [A f , Ag], provided that enough incoherent measurements g i are available. The incoherence is measured in terms of their mutual disjoint sparsity.This method finds applications in the reconstruction procedures of several hybrid imaging inverse problems, where internal data are measured. These measurements usually consist of the main unknown multiplied by other unknown quantities, and so the disjoint sparsity approach can be directly applied. As an example, we show how to apply the method to the reconstruction in quantitative photoacoustic tomography, also in the case when the Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.

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A review of blind source separation in NMR spectroscopy

Cited by 30 publications

References 107 publications

Blind Deconvolution Meets Blind Demixing: Algorithms and Performance Bounds

Blind Deconvolution Meets Blind Demixing: Algorithms and Performance Bounds

Proximity operators for a class of hybrid sparsity + entropy priors application to dosy NMR signal reconstruction

Disjoint sparsity for signal separation and applications to hybrid inverse problems in medical imaging

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