Recovery Conditions of Sparse Representations in the Presence of Noise.

Fuchs, Jean–Jacques

doi:10.1109/icassp.2006.1660659

Cited by 4 publications

(3 citation statements)

References 11 publications

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“…The minimization problem in (83) has at least one solution by coercivity, and is non-convex. But, for fixed γ (resp.…”

Section: The Algorithmic Viewpointmentioning

confidence: 99%

“…γ) is convex. As solutions of problem (83) have no explicit formulation, we again propose solving it by means of a block-coordinate relaxation iterative algorithm by alternately minimizing with respect to γ holding ν fixed, and vice versa. Thus, by classical ideas in convex analysis, a necessary condition for (γ, ν) to be a minimizer is that the zero is an element of the subdifferential of the objective at (γ, ν).…”

Section: The Algorithmic Viewpointmentioning

confidence: 99%

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Blind Source Separation: The Sparsity Revolution

Bobin¹,

Starck²,

Moudden³

et al. 2008

Advances in Imaging and Electron Physics

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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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“…The minimization problem in (83) has at least one solution by coercivity, and is non-convex. But, for fixed γ (resp.…”

Section: The Algorithmic Viewpointmentioning

confidence: 99%

Section: The Algorithmic Viewpointmentioning

confidence: 99%

Blind Source Separation: The Sparsity Revolution

Bobin¹,

Starck²,

Moudden³

et al. 2008

Advances in Imaging and Electron Physics

View full text Add to dashboard Cite

show abstract

“…Sparse recovery and stability conditions have been studied in [41][42][43] in the monochannel case. More particularly, conditions are proved in [41] under which OMP verifies an Exact Selection Property in the presence of bounded noise Z < .…”

Section: Handling Bounded Noise With Mmcamentioning

confidence: 99%

Morphological Diversity and Sparsity for Multichannel Data Restoration

et al. 2008

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Over the last decade, overcomplete dictionaries and the very sparse signal representations they make possible, have raised an intense interest from signal processing theory. In a wide range of signal processing problems, sparsity has been a crucial property leading to high performance. As multichannel data are of growing interest, it seems essential to devise sparsity-based tools accounting for such specific multichannel data. Sparsity has proved its efficiency in a wide range of inverse problems. Hereafter, we address some multichannel inverse problems issues such as multichannel morphological component separation and inpainting from the perspective of sparse representation. In this paper, we introduce a new sparsity-based multichannel analysis tool coined multichannel Morphological Component Analysis (mMCA). This new framework focuses on multichannel morphological diversity to better represent multichannel data. This paper presents conditions under which the mMCA converges and recovers the sparse multichannel representation. Several experiments are presented to demonstrate the applicability of our approach on

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Sparsity and Morphological Diversity in Blind Source Separation

Bobin

Starck

Fadili

et al. 2007

IEEE Trans. on Image Process.

189

161

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Over the last few years, the development of multichannel 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-caIled 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 emergedas a novel and effective source of diversity for BSS. Here, we give some new and 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 introduces a new 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 BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.

show abstract

Recovery Conditions of Sparse Representations in the Presence of Noise.

Cited by 4 publications

References 11 publications

Blind Source Separation: The Sparsity Revolution

Blind Source Separation: The Sparsity Revolution

Morphological Diversity and Sparsity for Multichannel Data Restoration

Sparsity and Morphological Diversity in Blind Source Separation

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