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355
Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information
, 2006
"... This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set? A typical result of this pa ..."
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Cited by 2632 (50 self)
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This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set? A typical result of this paper is as follows. Suppose that is a superposition of spikes @ Aa @ A @ A obeying @�� � A I for some constant H. We do not know the locations of the spikes nor their amplitudes. Then with probability at least I @ A, can be reconstructed exactly as the solution to the I minimization problem I aH @ A s.t. ” @ Aa ” @ A for all
Compressive sampling
, 2006
"... Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired res ..."
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Cited by 1441 (15 self)
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Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired resolution of the image, i.e. the number of pixels in the image. This paper surveys an emerging theory which goes by the name of “compressive sampling” or “compressed sensing,” and which says that this conventional wisdom is inaccurate. Perhaps surprisingly, it is possible to reconstruct images or signals of scientific interest accurately and sometimes even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g. the number of pixels in the image. It is believed that compressive sampling has far reaching implications. For example, it suggests the possibility of new data acquisition protocols that translate analog information into digital form with fewer sensors than what was considered necessary. This new sampling theory may come to underlie procedures for sampling and compressing data simultaneously. In this short survey, we provide some of the key mathematical insights underlying this new theory, and explain some of the interactions between compressive sampling and other fields such as statistics, information theory, coding theory, and theoretical computer science.
Compressive sensing
 IEEE Signal Processing Mag
, 2007
"... The Shannon/Nyquist sampling theorem tells us that in order to not lose information when uniformly sampling a signal we must sample at least two times faster than its bandwidth. In many applications, including digital image and video cameras, the Nyquist rate can be so high that we end up with too m ..."
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Cited by 696 (62 self)
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The Shannon/Nyquist sampling theorem tells us that in order to not lose information when uniformly sampling a signal we must sample at least two times faster than its bandwidth. In many applications, including digital image and video cameras, the Nyquist rate can be so high that we end up with too many samples and must compress in order to store or transmit them. In other applications, including imaging systems (medical scanners, radars) and highspeed analogtodigital converters, increasing the sampling rate or density beyond the current stateoftheart is very expensive. In this lecture, we will learn about a new technique that tackles these issues using compressive sensing [1, 2]. We will replace the conventional sampling and reconstruction operations with a more general linear measurement scheme coupled with an optimization in order to acquire certain kinds of signals at a rate significantly below Nyquist. 2
Iterative hard thresholding for compressed sensing
 Appl. Comp. Harm. Anal
"... Compressed sensing is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper we present a theoretical analysis of the iterative hard thresholding algorithm when applied to the compressed sensing recovery probl ..."
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Cited by 329 (18 self)
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Compressed sensing is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper we present a theoretical analysis of the iterative hard thresholding algorithm when applied to the compressed sensing recovery problem. We show that the algorithm has the following properties (made more precise in the main text of the paper) • It gives nearoptimal error guarantees. • It is robust to observation noise. • It succeeds with a minimum number of observations. • It can be used with any sampling operator for which the operator and its adjoint can be computed. • The memory requirement is linear in the problem size. Preprint submitted to Elsevier 28 January 2009 • Its computational complexity per iteration is of the same order as the application of the measurement operator or its adjoint. • It requires a fixed number of iterations depending only on the logarithm of a form of signal to noise ratio of the signal. • Its performance guarantees are uniform in that they only depend on properties of the sampling operator and signal sparsity.
Matrix Completion with Noise
"... On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest ..."
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Cited by 255 (13 self)
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On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries, and comes up in many areas of science and engineering including collaborative filtering, machine learning, control, remote sensing, and computer vision to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown lowrank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclearnorm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n × n matrix of low rank r from just about nr log 2 n noisy samples with an error which is proportional to the noise level. We present numerical results which complement our quantitative analysis and show that, in practice, nuclear norm minimization accurately fills in the many missing entries of large lowrank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout.
Signal reconstruction from noisy random projections
 IEEE Trans. Inform. Theory
, 2006
"... Recent results show that a relatively small number of random projections of a signal can contain most of its salient information. It follows that if a signal is compressible in some orthonormal basis, then a very accurate reconstruction can be obtained from random projections. We extend this type of ..."
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Cited by 239 (26 self)
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Recent results show that a relatively small number of random projections of a signal can contain most of its salient information. It follows that if a signal is compressible in some orthonormal basis, then a very accurate reconstruction can be obtained from random projections. We extend this type of result to show that compressible signals can be accurately recovered from random projections contaminated with noise. We also propose a practical iterative algorithm for signal reconstruction, and briefly discuss potential applications to coding, A/D conversion, and remote wireless sensing. Index Terms sampling, signal reconstruction, random projections, denoising, wireless sensor networks
Robust Recovery of Signals From a Structured Union of Subspaces
, 2008
"... Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structu ..."
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Cited by 221 (47 self)
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Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a blocksparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed ℓ2/ℓ1 program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals
, 2009
"... Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, alt ..."
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Cited by 158 (18 self)
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Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system’s performance that supports the empirical observations.
Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets StrangFix
, 2006
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Sensing by Random Convolution
 IEEE Int. Work. on Comp. Adv. MultiSensor Adaptive Proc., CAMPSAP
, 2007
"... Abstract. This paper outlines a new framework for compressive sensing: convolution with a random waveform followed by random time domain subsampling. We show that sensing by random convolution is a universally efficient data acquisition strategy in that an ndimensional signal which is S sparse in a ..."
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Cited by 112 (7 self)
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Abstract. This paper outlines a new framework for compressive sensing: convolution with a random waveform followed by random time domain subsampling. We show that sensing by random convolution is a universally efficient data acquisition strategy in that an ndimensional signal which is S sparse in any fixed representation can be recovered from m � S log n measurements. We discuss two imaging scenarios — radar and Fourier optics — where convolution with a random pulse allows us to seemingly superresolve finescale features, allowing us to recover highresolution signals from lowresolution measurements. 1. Introduction. The new field of compressive sensing (CS) has given us a fresh look at data acquisition, one of the fundamental tasks in signal processing. The message of this theory can be summarized succinctly [7, 8, 10, 15, 32]: the number of measurements we need to reconstruct a signal depends on its sparsity rather than its bandwidth. These measurements, however, are different than the samples that