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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07 Jun 2015
TL;DR: Experimental results demonstrate that, for the first time to the authors' knowledge, the hyperspectral video frame rate reaches up to 100fps with decent quality, even when the incident light is not strong.
Abstract: We propose a novel dual-camera design to acquire 4D high-speed hyperspectral (HSHS) videos with high spatial and spectral resolution. Our work has two key technical contributions. First, we build a dual-camera system that simultaneously captures a panchromatic video at a high frame rate and a hyperspectral video at a low frame rate, which jointly provide reliable projections for the underlying HSHS video. Second, we exploit the panchromatic video to learn an over-complete 3D dictionary to represent each band-wise video sparsely, and a robust computational reconstruction is then employed to recover the HSHS video based on the joint videos and the self-learned dictionary. Experimental results demonstrate that, for the first time to our knowledge, the hyperspectral video frame rate reaches up to 100fps with decent quality, even when the incident light is not strong.
70 citations
TL;DR: A novel method of fast and efficient measurement matrices and random phase masks for color image encryption, in which Kronecker product (KP) is combined with chaotic map, based on two-dimension compressive sensing and fraction Fourier transform.
Abstract: This paper introduces a novel method of fast and efficient measurement matrices and random phase masks for color image encryption, in which Kronecker product (KP) is combined with chaotic map. The encryption scheme is based on two-dimension (2D) compressive sensing (CS) and fraction Fourier transform (FrFT). In this algorithm, the KP is employed to extend low dimension seed matrices to obtain high dimension measurement matrices and random phase masks. The low dimension seed matrices are generated by controlling chaotic map. The original image is simultaneously encrypted and compressed by the 2D CS, then re-encrypted with FrFT. The proposed encryption scheme fulfills high speed, low complexity and high security. Numerical simulation results demonstrate the excellent performance and security of the proposed scheme.
70 citations
TL;DR: A modified adaptive orthogonal matching pursuit algorithm which estimates the initial value of sparsity by matching test, and will decrease the number of subsequent iterations to improve recognition accuracy and efficiency comparing with other greedy algorithms.
Abstract: Aiming at the disadvantages of greedy algorithms in sparse solution, a modified adaptive orthogonal matching pursuit algorithm (MAOMP) is proposed in this paper. It is obviously improved to introduce sparsity and variable step size for the MAOMP. The algorithm estimates the initial value of sparsity by matching test, and will decrease the number of subsequent iterations. Finally, the step size is adjusted to select atoms and approximate the true sparsity at different stages. The simulation results show that the algorithm which has proposed improves the recognition accuracy and efficiency comparing with other greedy algorithms.
70 citations
TL;DR: A novel position-patch face super-resolution method based on Tikhonov regularized neighbor representation (TRNR) is developed, which achieves better performance in terms of reconstruction error and visual quality than existing state-of-the-art methods.
70 citations
TL;DR: Two novel compressed sensing based algorithms for joint channel estimation and impulsive noise mitigation in UA OFDM systems are proposed and the results show that the proposed approaches consistently improve the accuracy ofChannel estimation and the performance of impulsive Noise Mitigation in UA OfDM communication systems.
Abstract: Impulsive noise occurs frequently in underwater acoustic (UA) channels and can significantly degrade the performance of UA orthogonal frequency-division multiplexing (OFDM) systems. In this paper, we propose two novel compressed sensing based algorithms for joint channel estimation and impulsive noise mitigation in UA OFDM systems. The first algorithm jointly estimates the channel impulse response and the impulsive noise by utilizing pilot subcarriers. The estimated impulsive noise is then converted to the time domain and removed from the received signals. We show that this algorithm reduces the system bit-error-rate through improved channel estimation and impulsive noise mitigation. In the second proposed algorithm, a joint estimation of the channel impulse response and the impulsive noise is performed by exploiting the initially detected data. Then, the estimated impulsive noise is removed from the received signals. The proposed algorithms are evaluated and compared with existing methods through numerical simulations and on real data collected during a UA communication experiment conducted in the estuary of the Swan River, WA, Australia, during December 2015. The results show that the proposed approaches consistently improve the accuracy of channel estimation and the performance of impulsive noise mitigation in UA OFDM communication systems.
70 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