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THE RESTRICTED ISOMETRY PROPERTY FOR SIGNAL RECOVERY WITH COHERENT TIGHT FRAMES

  • FEN-GONG WU (a1) and DONG-HUI LI (a2)

Abstract

In this paper, we consider signal recovery via $l_{1}$ -analysis optimisation. The signals we consider are not sparse in an orthonormal basis or incoherent dictionary, but sparse or nearly sparse in terms of some tight frame $D$ . The analysis in this paper is based on the restricted isometry property adapted to a tight frame $D$ (abbreviated as $D$ -RIP), which is a natural extension of the standard restricted isometry property. Assuming that the measurement matrix $A\in \mathbb{R}^{m\times n}$ satisfies $D$ -RIP with constant ${\it\delta}_{tk}$ for integer $k$ and $t>1$ , we show that the condition ${\it\delta}_{tk}<\sqrt{(t-1)/t}$ guarantees stable recovery of signals through $l_{1}$ -analysis. This condition is sharp in the sense explained in the paper. The results improve those of Li and Lin [‘Compressed sensing with coherent tight frames via $l_{q}$ -minimization for $0<q\leq 1$ ’, Preprint, 2011, arXiv:1105.3299] and Baker [‘A note on sparsification by frames’, Preprint, 2013, arXiv:1308.5249].

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References

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