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From Bernoulli-Gaussian deconvolution to sparse signal restoration

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Abstract

Formulated as a least-square problem under an $\ell_0$ constraint, sparse signal restoration is a discrete optimization problem, known to be NP complete. Classical algorithms include, by increasing cost and efficiency, Matching Pursuit (MP), Orthogonal Matching Pursuit (OMP), Orthogonal Least Squares (OLS) and the exhaustive search algorithm. In inverse problems involving highly correlated dictionaries, OMP and OLS are not guaranteed to find the optimal solution. It is of interest to develop slightly slower sub-optimal search algorithms yielding better approximations. We revisit the Single Most Likely Replacement (SMLR) algorithm, developed in the mid-80's for Bernoulli-Gaussian signal restoration. We show that the formulation of sparse signal restoration as a limit case of Bernoulli-Gaussian signal restoration leads to an $\ell_0$-penalized least-square minimization problem, to which SMLR can be straightforwardly adapted. The resulting algorithm, called Single Best Replacement (SBR), can be interpreted as a forward-backward extension of OLS. A fast and stable implementation is proposed. The approach is illustrated on a deconvolution problem with a Gaussian impulse response and on the joint detection of discontinuities at different orders in a signal.
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Dates and versions

hal-00443842 , version 1 (04-01-2010)
hal-00443842 , version 2 (28-01-2010)
hal-00443842 , version 3 (03-02-2010)
hal-00443842 , version 4 (17-06-2011)

Identifiers

  • HAL Id : hal-00443842 , version 2

Cite

Charles Soussen, Jérôme Idier, David Brie, Junbo Duan. From Bernoulli-Gaussian deconvolution to sparse signal restoration. 2010. ⟨hal-00443842v2⟩

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