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THE RESTRICTED ISOMETRY PROPERTY FOR SIGNAL RECOVERY WITH COHERENT TIGHT FRAMES

Part of: Communication, information

Published online by Cambridge University Press:  19 August 2015

FEN-GONG WU  and
DONG-HUI LI
FEN-GONG WU*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China email wufengong@gdei.edu.cn
DONG-HUI LI
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China email dhli@scnu.edu.cn
*
✉wufengong@gdei.edu.cn
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Abstract

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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].

Keywords

compressed sensingsparse recoveryl1 -analysisD-restricted isometry propertytight frames

MSC classification

Primary: 94A12: Signal theory (characterization, reconstruction, filtering, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 3 , December 2015 , pp. 496 - 507
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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