Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit: The Gaussian Case
01 Aug 2007-
TL;DR: In this paper, a greedy algorithm called Orthogonal Matching Pursuit (OMP) was proposed to recover a signal with m nonzero entries in dimension 1 given O(m n d) random linear measurements of that signal.
Abstract: This report demonstrates theoretically and empirically that a greedy algorithm called
Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension
d given O(mln d) random linear measurements of that signal. This is a massive improvement
over previous results, which require O(m2) measurements. The new results for OMP are comparable
with recent results for another approach called Basis Pursuit (BP). In some settings, the
OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal
recovery problems.
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TL;DR: A 2D CS sparse image reconstruction algorithm based on iterative gradient projection is proposed that is more efficient and robust, not only yielding higher peak-signal-to-noise ratio but also reconstructing images of better subjective visual quality.
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07 Jun 2015
TL;DR: A novel reweighted Laplace prior based hyperspectral compressive sensing method that outperforms several state-of-the-art hyperspectrals CS methods on the reconstruction accuracy and unifies signal recovery, prior learning and noise estimation into a variational framework to infer the parameters automatically.
Abstract: Compressive sensing(CS) has been exploited for hype-spectral image(HSI) compression in recent years. Though it can greatly reduce the costs of computation and storage, the reconstruction of HSI from a few linear measurements is challenging. The underlying sparsity of HSI is crucial to improve the reconstruction accuracy. However, the sparsity of HSI is unknown in reality and varied with different noise, which makes the sparsity estimation difficult. To address this problem, a novel reweighted Laplace prior based hyperspectral compressive sensing method is proposed in this study. First, the reweighted Laplace prior is proposed to model the distribution of sparsity in HSI. Second, the latent variable Bayes model is employed to learn the optimal configuration of the reweighted Laplace prior from the measurements. The model unifies signal recovery, prior learning and noise estimation into a variational framework to infer the parameters automatically. The learned sparsity prior can represent the underlying structure of the sparse signal very well and is adaptive to the unknown noise, which improves the reconstruction accuracy of HSI. The experimental results on three hyperspectral datasets demonstrate the proposed method outperforms serveral state-of-the-art hyperspectral CS methods on the reconstruction accuracy.
40 citations
01 Nov 2015
TL;DR: Numerical results indicate that the proposed training-free detectors offer improved detection performance over covariance matrix based detectors when the latter have a moderate amount of training signals.
Abstract: This paper examines moving target detection (MTD) in distributed multi-input multi-output (MIMO) radar with sensors placed on moving platforms. Unlike previous works which were focused on the case of stationary platforms, we consider explicitly the effects of platform motion, which exacerbate the location-induced clutter non-homogeneity inherent in such systems and thus make the MTD problem significantly more challenging. We propose a sparsity based detector which, by exploiting a sparse representation of the clutter in the Doppler domain, adaptively estimates from the test signal the clutter subspace, which is allowed to be distinct for different transmit/receive (Tx/Rx) pairs and, moreover, may spread over the entire Doppler bandwidth. The proposed detector requires no range training signals that are indispensable for conventional detectors but are difficult to obtain in practice. Numerical results indicate that the proposed range training-free detector offers improved detection performance over covariance matrix based detectors when the latter are provided with a moderate amount of training signals.
40 citations
01 Jan 2006
40 citations
References
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TL;DR: It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients, and a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing.
Abstract: Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0
18,609 citations
TL;DR: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions.
Abstract: The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries --- stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB).
Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising.
BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.
9,950 citations
TL;DR: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions, chosen in order to best match the signal structures.
Abstract: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive time-frequency transform. They derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. They compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser see (IEEE Trans. Informat. Theory, vol. 38, Mar. 1992). >
9,380 citations
Stanford University1, Cleveland Clinic2, University of Toronto3, Centre national de la recherche scientifique4, Université Paris-Saclay5, University of Paris-Sud6, Avaya7, Rutgers University8, RAND Corporation9, IBM10, University of Pennsylvania11, University of Western Australia12, University of Minnesota13
TL;DR: A publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates is described.
Abstract: The purpose of model selection algorithms such as All Subsets, Forward Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. Three main properties are derived: (1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods. (2) A different LARS modification efficiently implements Forward Stagewise linear regression, another promising new model selection method; this connection explains the similar numerical results previously observed for the Lasso and Stagewise, and helps us understand the properties of both methods, which are seen as constrained versions of the simpler LARS algorithm. (3) A simple approximation for the degrees of freedom of a LARS estimate is available, from which we derive a Cp estimate of prediction error; this allows a principled choice among the range of possible LARS estimates. LARS and its variants are computationally efficient: the paper describes a publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates.
7,828 citations