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Compressive Sensing and Structured Random Matrices
 RADON SERIES COMP. APPL. MATH XX, 1–95 © DE GRUYTER 20YY
"... These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to ..."
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Cited by 162 (19 self)
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These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using ℓ1minimization.
Sensitivity to basis mismatch of compressed sensing,” preprint
, 2009
"... Abstract—The theory of compressed sensing suggests that successful inversion of an image of the physical world (e.g., a radar/sonar return or a sensor array snapshot vector) for the source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the c ..."
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Cited by 80 (8 self)
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Abstract—The theory of compressed sensing suggests that successful inversion of an image of the physical world (e.g., a radar/sonar return or a sensor array snapshot vector) for the source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in passive and active radar and sonar, this basis is usually taken to be a DFT basis. The compressed sensing measurements are then inverted using an ℓ1minimization principle (basis pursuit) for the nonzero source amplitudes. This seems to make compressed sensing an ideal image inversion principle for high resolution modal analysis. However, in reality no physical field is sparse in the DFT basis or in an apriori known basis. In fact the main goal in image inversion is to identify the modal structure. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and there is always mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of basis pursuit to mismatch between the assumed and the actual sparsity bases and compare the performance of basis pursuit with that of classical image inversion. Our mathematical analysis and numerical examples show that the performance of basis pursuit degrades considerably in the presence of mismatch, and they suggest that the use of compressed sensing as a modal analysis principle requires more consideration and refinement, at least for the problem sizes common to radar/sonar. I.
Compressive Sensing
, 2010
"... Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many poten ..."
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Cited by 50 (13 self)
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Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.
SAMPLING AND RECONSTRUCTING DIFFUSION FIELDS WITH LOCALIZED SOURCES
"... We study the spatiotemporal sampling of a diffusion field generated by K point sources, aiming to fully reconstruct the unknown initial field distribution from the sample measurements. The sampling operator in our problem can be described by a matrix derived from the diffusion model. We analyze the ..."
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We study the spatiotemporal sampling of a diffusion field generated by K point sources, aiming to fully reconstruct the unknown initial field distribution from the sample measurements. The sampling operator in our problem can be described by a matrix derived from the diffusion model. We analyze the important properties of the sampling matrices, leading to precise bounds on the spatial and temporal sampling densities under which perfect field reconstruction is feasible. Moreover, our analysis indicates that it is possible to compensate linearly for insufficient spatial sampling densities by oversampling in time. Numerical simulations on initial field reconstruction under different spatiotemporal sampling densities confirm our theoretical results. Index Terms — Diffusion equation, initial inverse problems, spatiotemporal sampling, point sources localization, compressed sensing 1.
Beyond incoherence: stable and robust sampling strategies for compressive imaging,” preprint
, 2012
"... In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained ..."
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Cited by 12 (1 self)
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In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because loworder wavelets and loworder frequencies are correlated, so compressed sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper we turn to a more refined notion of coherence – the socalled local coherence – measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled, so for matrices comprised of frequencies sampled from suitable powerlaw densities, we can prove the restricted isometry property with nearoptimal embedding dimensions. Consequently, the variabledensity sampling strategies we provide — which are independent of the ambient dimension up to logarithmic factors — allow for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by ℓ1minimization and by total variation minimization. 1
Coherencepattern guided compressive sensing with unresolved grids
 SIAM J. Imaging Sci
"... Abstract. Highly coherent sensing matrices arise in discretization of continuum imaging problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold. Algorithms based on techniques of band exclusion (BE) and local optimization (LO) are proposed to deal with such c ..."
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Abstract. Highly coherent sensing matrices arise in discretization of continuum imaging problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold. Algorithms based on techniques of band exclusion (BE) and local optimization (LO) are proposed to deal with such coherent sensing matrices. These techniques are embedded in the existing compressed sensing algorithms such as Orthogonal Matching Pursuit (OMP),
Stable and robust sampling strategies for compressive imaging
, 2013
"... In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained ..."
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Cited by 8 (1 self)
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In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because loworder wavelets and loworder frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper we turn to a more refined notion of coherence – the socalled local coherence – measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square powerlaw density, we can prove the restricted isometry property with nearoptimal embedding dimensions. Consequently, the variabledensity sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by ℓ1minimization and by total variation minimization. The local coherence framework developed in this paper should be of independent interest in sparse recovery problems more generally, as it implies that for optimal sparse recovery results, it suffices to have bounded average coherence from sensing basis to sparsity basis – as opposed to bounded maximal coherence – as long as the sampling strategy is adapted accordingly. 1
EXACT LOCALIZATION AND SUPERRESOLUTION WITH NOISY DATA AND RANDOM ILLUMINATION
, 1008
"... Abstract. This paper studies the problem of exact localization of multiple objects with noisy data. The crux of the proposed approach consists of random illumination. Two recovery methods are analyzed: the Lasso and the OneStep Thresholding (OST). For independent random probes, it is shown that bot ..."
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Abstract. This paper studies the problem of exact localization of multiple objects with noisy data. The crux of the proposed approach consists of random illumination. Two recovery methods are analyzed: the Lasso and the OneStep Thresholding (OST). For independent random probes, it is shown that both recovery methods can localize exactly s = O(m), up to a logarithmic factor, objects where m is the number of data. Moreover, when the number of random probes is large the Lasso with random illumination has a performance guarantee for superresolution, beating the Rayleigh resolution limit. Numerical evidence confirms the predictions and indicates that the performance of the Lasso is superior to that of the OST for the proposed setup with random illumination. 1.
Compressive matchedfield processing
 The Journal of the Acoustical Society of America
"... Source localization by matchedfield processing (MFP) generally involves solving a number of computationally intensive partial differential equations. This paper introduces a technique that mitigates this computational workload by “compressing ” these computations. Drawing on key concepts from the ..."
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Cited by 5 (1 self)
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Source localization by matchedfield processing (MFP) generally involves solving a number of computationally intensive partial differential equations. This paper introduces a technique that mitigates this computational workload by “compressing ” these computations. Drawing on key concepts from the recently developed field of compressed sensing, it shows how a lowdimensional proxy for the Green’s function can be constructed by backpropagating a small set of random receiver vectors. Then, the source can be located by performing a number of “short ” correlations between this proxy and the projection of the recorded acoustic data in the compressed space. Numerical experiments in a Pekeris ocean waveguide are presented which demonstrate that this compressed version of MFP is as effective as traditional MFP even when the compression is significant. The results are particularly promising in the broadband regime where using as few as two random backpropagations per frequency performs almost as well as the traditional broadband MFP, but with the added benefit of generic applicability. That is, the computationally intensive backpropagations may be computed offline independently from the received signals, and may be reused to locate any source within the search grid area.
MUSIC for SingleSnapshot Spectral Estimation: Stability and Superresolution
, 2014
"... This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The singlesnapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The M ..."
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This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The singlesnapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noisespace correlation function and identifying the s smallest local minima of the noisespace correlation as the frequency set. In the noisefree case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noisespace correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh length, the stability of the noisespace correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLOOMP and SDP (TVmin). While BLOOMP is the stablest algorithm for frequencies separated above 4RL, MUSIC becomes the best performing one for frequencies separated between 2RL and 3RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to one RL and below when all other methods fail. Indeed, the resolution of MUSIC apparently decreases to zero as noise decreases to zero.