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Pré-Publication, Document De Travail Année : 2016

Sparse Spikes Super-resolution on Thin Grids II: the Continuous Basis Pursuit

Résumé

This article analyzes the performance of the Continuous Basis Pursuit (C-BP) method for sparse super-resolution. The C-BP has been recently proposed by Ekanadham, Tranchina and Simoncelli as a refined discretization scheme for the recovery of spikes in inverse problems regularization. One of the most well known discretization scheme, the Basis Pursuit (BP, also known as Lasso) makes use of a finite dimensional L1 norm on a grid. In contrast, the C-BP rather uses a linear interpolation of the spikes positions to enable the recovery of spikes between grid's points. When the sought-after solution is constrained to be positive, a remarkable feature of this approach is that it retains the convexity of the initial L1 problem. For deconvolution problems, it is well known that L1-type methods (including BP and C-BP) recover exactly the unknown sparse sum of positive Diracs when there is no noise in the measurements. Our main contribution identifies a non-degeneracy condition ensuring that the support of the solution enjoys some stability when noise is added. More precisely, we show that, in the small noise regime, when this non-degeneracy condition holds, the C-B method estimates a pair of Dirac masses around each spike of the input measure. We also derive the asymptotic of the recovery error when the grid size tends to zero. We show some numerical illustrations of this stability to noise for both the BP and C-BP methods, and evaluate numerically these non-degeneracy conditions.
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Dates et versions

hal-01389956 , version 1 (30-10-2016)
hal-01389956 , version 2 (01-06-2017)

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  • HAL Id : hal-01389956 , version 1

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Vincent Duval, Gabriel Peyré. Sparse Spikes Super-resolution on Thin Grids II: the Continuous Basis Pursuit. 2016. ⟨hal-01389956v1⟩
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