## Compressed Sensing of Block-Sparse Signals: Uncertainty Relations and Efficient Recovery (2009)

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### BibTeX

@MISC{Eldar09compressedsensing,

author = {Yonina C. Eldar and Patrick Kuppinger and Helmut Bölcskei},

title = { Compressed Sensing of Block-Sparse Signals: Uncertainty Relations and Efficient Recovery},

year = {2009}

}

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### Abstract

We consider compressed sensing of block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block k-sparse signals in no more than k steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed ℓ2/ℓ1-optimization approach. This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.