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17
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
On stable reconstructions from nonuniform Fourier measurements
 SIAM J. Imaging Sci
, 2014
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Fundamental performance limits for ideal decoders in highdimensional linear inverse problems. arXiv:1311.6239
, 2013
"... The primary challenge in linear inverse problems is to design stable and robust “decoders” to reconstruct highdimensional vectors from a lowdimensional observation through a linear operator. Sparsity, lowrank, and related assumptions are typically exploited to design decoders which performance is ..."
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The primary challenge in linear inverse problems is to design stable and robust “decoders” to reconstruct highdimensional vectors from a lowdimensional observation through a linear operator. Sparsity, lowrank, and related assumptions are typically exploited to design decoders which performance is then bounded based on some measure of deviation from the idealized model, typically using a norm. This paper focuses on characterizing the fundamental performance limits that can be expected from an ideal decoder given a general model, i.e., a general subset of “simple ” vectors of interest. First, we extend the socalled notion of instance optimality of a decoder to settings where one only wishes to reconstruct some part of the original high dimensional vector from a lowdimensional observation. This covers practical settings such as medical imaging of a region of interest, or audio source separation when one is only interested in estimating the contribution of a specific instrument to a musical recording. We define instance optimality relatively to a model much beyond the traditional framework of sparse recovery, and characterize the existence of an instance optimal decoder in terms of joint properties of the model and the considered linear operator. Noiseless and noiserobust settings are both considered. We show somewhat surprisingly that the existence of noiseaware instance optimal decoders for all noise levels implies the existence of a noiseblind decoder. A consequence of our results is that for models that are rich enough to contain an orthonormal basis, the existence of an `2/`2 instance optimal decoder is only possible when the linear operator is not substantially dimensionreducing. This covers wellknown cases (sparse vectors, lowrank matrices) as well as a number of seemingly new situations (structured sparsity and sparse inverse covariance matrices for instance). We exhibit an operatordependent norm which, under a modelspecific generalization of the Restricted Isometry Property (RIP), always yields a feasible instance optimality property. This norm can be upper bounded by an atomic norm relative to the considered model. 1
The quest for optimal sampling: Computationally efficient, structureexploiting sampling strategies for compressed sensing. Compressed Sensing and Its Applications
, 2014
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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
Computationally efficient, structureexploiting
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STRUCTURED RANDOM MEASUREMENTS IN SIGNAL PROCESSING
"... ABSTRACT. Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first show ..."
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ABSTRACT. Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (unstructured) Gaussian random measurement matrices, applications require certain structure of the measurements leading to structured random measurement matrices. Near optimal recovery guarantees for such structured measurements have been developed over the past years in a variety of contexts. This article surveys the theory in three scenarios: compressed sensing (sparse recovery), low rank matrix recovery, and phaseless estimation. The random measurement matrices to be considered include random partial Fourier matrices, partial random circulant matrices (subsampled convolutions), matrix completion, and phase estimation from magnitudes of Fourier type measurements. The article concludes with a brief discussion of the mathematical techniques for the analysis of such structured random measurements. 1.
On Asymptotic Incoherence and its Implications for Compressed Sensing of Inverse Problems
"... Recently, it has been shown that incoherence is an unrealistic assumption for compressed sensing when applied to infinitedimensional inverse problems. Instead, the key property that permits efficient recovery in such problems is socalled asymptotic incoherence. The purpose of this paper is to stud ..."
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Recently, it has been shown that incoherence is an unrealistic assumption for compressed sensing when applied to infinitedimensional inverse problems. Instead, the key property that permits efficient recovery in such problems is socalled asymptotic incoherence. The purpose of this paper is to study this new concept, and its implications towards the design of optimal sampling strategies. We determine how fast the asymptotic incoherence can decay in general for isometries. Furthermore it is shown that Fourier sampling and wavelet sparsity, whilst globally coherent, yield optimal asymptotic incoherence as a power law up to a constant factor. Sharp bounds on the asymptotic incoherence for Fourier sampling with polynomial bases are also provided. A numerical experiment is also presented to demonstrate the role of asymptotic incoherence in finding good subsampling strategies. 1
compressed sensing and the
"... the absence of the RIP in realworld applications of ..."
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