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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 64 (13 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 30 (2 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 25 (8 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.
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.
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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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
TVMIN AND GREEDY PURSUIT FOR CONSTRAINED JOINT SPARSITY AND APPLICATION TO INVERSE SCATTERING
"... Abstract. This paper proposes an appealing framework for analyzing total variation minimization (TVmin) and extends Candès, Romberg and Tao’s proof of exact recovery of piecewise constant objects with noiseless incomplete Fourier data to the case of noisy data. The approach is based on reformulatio ..."
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Abstract. This paper proposes an appealing framework for analyzing total variation minimization (TVmin) and extends Candès, Romberg and Tao’s proof of exact recovery of piecewise constant objects with noiseless incomplete Fourier data to the case of noisy data. The approach is based on reformulation of TVmin as compressed sensing of constrained joint sparsity (CJS). TV and 2norm error bounds, independent of the ambient dimension, are derived for Basis Pursuit and for Orthogonal Matching Pursuit for CJS. 1.
Remote sensing via ℓ1minimization
"... We consider the problem of detecting the locations of targets in the far field by sending probing signals from an antenna array and recording the reflected echoes. Drawing on key concepts from the area of compressive sensing, we use an ℓ1based regularization approach to solve this, in general illp ..."
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We consider the problem of detecting the locations of targets in the far field by sending probing signals from an antenna array and recording the reflected echoes. Drawing on key concepts from the area of compressive sensing, we use an ℓ1based regularization approach to solve this, in general illposed, inverse scattering problem. As common in compressive sensing, we exploit randomness, which in this context comes from choosing the antenna locations at random. With n antennas we obtain n 2 measurements of a vector x ∈ C N representing the target locations and reflectivities on a discretized grid. It is common to assume that the scene x is sparse due to a limited number of targets. Under a natural condition on the mesh size of the grid, we show that an ssparse scene can be recovered via ℓ1minimization with high probability if n 2 ≥ Cs log 2 (N). The reconstruction is stable under noise and under passing from sparse to approximately sparse vectors. Our theoretical findings are confirmed by numerical simulations.
Vol. xx, pp. x c ○ xxxx Society for Industrial and Applied Mathematics x–x CoherencePatternGuided Compressive Sensing with Unresolved Grids
"... 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), Subspace Pursuit
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"... For the last one and half years, my research has been focused on developing compressive imaging methods. Compressive imaging is the idea of imaging sparse objects with comparably sparse measurements. The approach is based the compressed sensing (CS) theory originating in information theory and stati ..."
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For the last one and half years, my research has been focused on developing compressive imaging methods. Compressive imaging is the idea of imaging sparse objects with comparably sparse measurements. The approach is based the compressed sensing (CS) theory originating in information theory and statistics. In the standard theory, problems of imaging are typically formulated in the continuum setting and results are usually qualitative. For example, the standard inverse scattering theory asserts the uniqueness of scatterer given the entirety of scattering data (i.e. all incident and scattering angles). But in reality one has only a finite (sometimes small) number of scattering data, let alone the complete continuum set of scattering data! From this perspective, CS is a natural, alternative route to inverse scattering and, more generally, imaging. As such, compressive imaging has seen a flurry of activities in the last few years. Few, though, really address the heart of the problem. A simple fact is that the striking results of CS demand equally striking assumptions on sensing matrices and objects of interest. Moreover, for problems of imaging physical constraints often result in