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An analysis of block sampling strategies in compressed sensing
, 2013
"... Compressed sensing (CS) is a theory which guarantees the exact recovery of sparse signals from a few number of linear projections. The sampling schemes suggested by current CS theories are often of little relevance since they cannot be implemented on practical acquisition systems. In this paper, we ..."
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Compressed sensing (CS) is a theory which guarantees the exact recovery of sparse signals from a few number of linear projections. The sampling schemes suggested by current CS theories are often of little relevance since they cannot be implemented on practical acquisition systems. In this paper, we study a new random sampling approach that consists in selecting a set of blocks that are predefined by the application of interest. A typical example is the case where the blocks consist in horizontal lines in the 2D Fourier plane. We provide theoretical results on the number of blocks that are required for exact sparse signal reconstruction in a noise free setting. We illustrate this theory for various sensing matrices appearing in applications such as timefrequency bases. A typical result states that it is sufficient to acquire no more than O ( sln 2 (n) ) lines in the 2D Fourier domain for the perfect reconstruction of an ssparse image of size √ n × √ n. The proposed results have a large number of potential applications in systems such as magnetic resonance imaging, radiointerferometry or ultrasound imaging. Keywords: Compressed Sensing, blocks of measurements, sampling continuous trajectories, exact recovery, ℓ 1 minimization. 1
A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier measurements. arXiv
, 2014
"... This note complements the paper The quest for optimal sampling: Computationally efficient, structureexploiting measurements for compressed sensing [2]. Its purpose is to present a proof of a result stated therein concerning the recovery via compressed sensing of a signal that has structured sparsit ..."
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This note complements the paper The quest for optimal sampling: Computationally efficient, structureexploiting measurements for compressed sensing [2]. Its purpose is to present a proof of a result stated therein concerning the recovery via compressed sensing of a signal that has structured sparsity in a Haar wavelet basis when sampled using a multilevelsubsampled discrete Fourier transform. In doing so, it provides a simple exposition of the proof in the case of Haar wavelets and discrete Fourier samples of more general result recently provided in Breaking the coeherence barrier: A new theory for compressed sensing [1]. 1