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: This paper provides an alternative performance analysis of $\ell_1$-optimization and obtains the proportionality constants that in certain cases match or improve on the best currently known ones from \cite{DonohoPol,DT}.
Abstract: Recently, \cite{CRT,DonohoPol} theoretically analyzed the success of a polynomial $\ell_1$-optimization algorithm in solving an under-determined system of linear equations. In a large dimensional and statistical context \cite{CRT,DonohoPol} proved that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that $\ell_1$-optimization succeeds in solving the system. In this paper, we provide an alternative performance analysis of $\ell_1$-optimization and obtain the proportionality constants that in certain cases match or improve on the best currently known ones from \cite{DonohoPol,DT}.
94 citations
TL;DR: This paper proposes a new SR framework that seamlessly integrates learning- and reconstruction-based methods for single image SR to avoid unexpected artifacts introduced by learning-based SR and restore the missing high-frequency details smoothed by reconstruction- based SR.
Abstract: It has been widely acknowledged that learning- and reconstruction-based super-resolution (SR) methods are effective to generate a high-resolution (HR) image from a single low-resolution (LR) input. However, learning-based methods are prone to introduce unexpected details into resultant HR images. Although reconstruction-based methods do not generate obvious artifacts, they tend to blur fine details and end up with unnatural results. In this paper, we propose a new SR framework that seamlessly integrates learning- and reconstruction-based methods for single image SR to: 1) avoid unexpected artifacts introduced by learning-based SR and 2) restore the missing high-frequency details smoothed by reconstruction-based SR. This integrated framework learns a single dictionary from the LR input instead of from external images to hallucinate details, embeds nonlocal means filter in the reconstruction-based SR to enhance edges and suppress artifacts, and gradually magnifies the LR input to the desired high-quality SR result. We demonstrate both visually and quantitatively that the proposed framework produces better results than previous methods from the literature.
94 citations
03 Aug 2010
TL;DR: This work presents a VLSI implementation of a computationally efficient algorithm named Orthogonal Matching Pursuit, and further optimize the algorithm to meet typical hardware constraints and describe the different block units of the design.
Abstract: Compressive Sampling reconstruction techniques require computationally intensive algorithms, often using L1 optimization to reconstruct a signal that was originally sampled at a sub-Nyquist rate. In this work we present a VLSI implementation of a computationally efficient algorithm named Orthogonal Matching Pursuit. We further optimize the algorithm to meet typical hardware constraints and describe the different block units of our design. We synthesize our design for the Xilinx Virtex 5 FPGA and give timing and area results. We summarize our work with a short discussion of the possible uses for our system.
94 citations
TL;DR: This work presents a review of the CS-based scene reconstruction techniques that address the unique challenges associated with fast and efficient imaging in urban operations, and focuses on ground-based imaging systems for indoor targets.
Abstract: Through-the-wall radar imaging (TWRI) is emerging as a viable technology for providing high-quality imagery of enclosed structures. TWRI makes use of electromagnetic waves to penetrate through building wall materials. Due to the “see” through ability, TWRI has attracted much attention in the last decade and has found a variety of important civilian and military applications. Signal processing algorithms have been devised to allow proper imaging and image recovery in the presence of high clutter, which is caused by front walls and multipath due to reflections from internal walls. Recently, research efforts have shifted toward effective and reliable imaging under constraints on aperture size, frequency, and acquisition time. In this respect, scene reconstructions are being pursued with reduced data volume and within the emerging compressive sensing (CS) framework. We present a review of the CS-based scene reconstruction techniques that address the unique challenges associated with fast and efficient imaging in urban operations. Specifically, we focus on ground-based imaging systems for indoor targets. We discuss CS-based wall mitigation, multipath exploitation, and change detection for imaging of stationary and moving targets inside enclosed structures.
94 citations
TL;DR: Both simulations and experiments show that the proposed techniques for sampling, representing, and reconstructing the space-time volume to overcome this trade-off to reconstruct a video from a single coded image while maintaining high spatial resolution.
Abstract: Cameras face a fundamental trade-off between spatial and temporal resolution. Digital still cameras can capture images with high spatial resolution, but most high-speed video cameras have relatively low spatial resolution. It is hard to overcome this trade-off without incurring a significant increase in hardware costs. In this paper, we propose techniques for sampling, representing, and reconstructing the space-time volume to overcome this trade-off. Our approach has two important distinctions compared to previous works: 1) We achieve sparse representation of videos by learning an overcomplete dictionary on video patches, and 2) we adhere to practical hardware constraints on sampling schemes imposed by architectures of current image sensors, which means that our sampling function can be implemented on CMOS image sensors with modified control units in the future. We evaluate components of our approach, sampling function and sparse representation, by comparing them to several existing approaches. We also implement a prototype imaging system with pixel-wise coded exposure control using a liquid crystal on silicon device. System characteristics such as field of view and modulation transfer function are evaluated for our imaging system. Both simulations and experiments on a wide range of scenes show that our method can effectively reconstruct a video from a single coded image while maintaining high spatial resolution.
93 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