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35
Trust, but verify: Fast and accurate signal recovery from 1bit compressive measurements
, 2010
"... Abstract—The recently emerged compressive sensing (CS) framework aims to acquire signals at reduced sample rates compared to the classical ShannonNyquist rate. To date, the CS theory has assumed primarily realvalued measurements; it has recently been demonstrated that accurate and stable signal ac ..."
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Cited by 29 (2 self)
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Abstract—The recently emerged compressive sensing (CS) framework aims to acquire signals at reduced sample rates compared to the classical ShannonNyquist rate. To date, the CS theory has assumed primarily realvalued measurements; it has recently been demonstrated that accurate and stable signal acquisition is still possible even when each measurement is quantized to just a single bit. This property enables the design of simplified CS acquisition hardware based around a simple sign comparator rather than a more complex analogtodigital converter; moreover, it ensures robustness to gross nonlinearities applied to the measurements. In this paper we introduce a new algorithm — restrictedstep shrinkage (RSS) — to recover sparse signals from 1bit CS measurements. In contrast to previous algorithms for 1bit CS, RSS has provable convergence guarantees, is about an order of magnitude faster, and achieves higher average recovery signaltonoise ratio. RSS is similar in spirit to trustregion methods for nonconvex optimization on the unit sphere, which are relatively unexplored in signal processing and hence of independent interest. Index Terms—1bit compressive sensing, quantization, consistent reconstruction, trustregion algorithms I.
Recursive sparse recovery in large but correlated noise
 in Proc. 49th Allerton Conf. Commun. Control Comput
, 2011
"... Abstract—In this work, we focus on the problem of recursively recovering a time sequence of sparse signals, with timevarying sparsity patterns, from highly undersampled measurements corrupted by very large but correlated noise. It is assumed that the noise is correlated enough to have an approxima ..."
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Cited by 21 (13 self)
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Abstract—In this work, we focus on the problem of recursively recovering a time sequence of sparse signals, with timevarying sparsity patterns, from highly undersampled measurements corrupted by very large but correlated noise. It is assumed that the noise is correlated enough to have an approximately low rank covariance matrix that is either constant, or changes slowly, with time. We show how our recently introduced Recursive Projected CS (ReProCS) and modifiedReProCS ideas can be used to solve this problem very effectively. To the best of our knowledge, except for the recent work of dense error correction via ℓ1 minimization, which can handle another kind of large but “structured ” noise (the noise needs to be sparse), none of the other works in sparse recovery have studied the case of any other kind of large noise. I.
1 Compressive Sensing Based High Resolution Channel Estimation for OFDM System
"... Abstract — Orthogonal frequency division multiplexing (OFDM) is a technique that will prevail in the next generation wireless communication. Channel estimation is one of the key challenges in OFDM, since highresolution channel estimation can significantly improve the equalization at the receiver an ..."
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Cited by 7 (3 self)
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Abstract — Orthogonal frequency division multiplexing (OFDM) is a technique that will prevail in the next generation wireless communication. Channel estimation is one of the key challenges in OFDM, since highresolution channel estimation can significantly improve the equalization at the receiver and consequently enhance the communication performances. In this paper, we propose a system with an asymmetric DAC/ADC pair and formulate OFDM channel estimation as a compressive sensing problem. By skillfully designing pilots and taking advantages of the sparsity of the channel impulse response, the proposed system realizes high resolution channel estimation at a low cost. The pilot design, the use of a highspeed DAC and a regularspeed ADC, and the estimation algorithm tailored for channel estimation distinguish the proposed approach from the existing estimation approaches. We theoretically show that in the proposed system, a Nresolution channel can be faithfully obtained with an ADC speed at M = O(S 2 log(N/S)), where N is also the DAC speed and S is the channel impulse response sparsity. Since S is small and increasing the DAC speed to N> M is relatively cheap, we obtain a highresolution channel at a low cost. We also present a novel estimator that is both faster and more accurate than the typical ℓ1 minimization. In the numerical experiments, we simulated various numbers of multipaths and different SNRs and let the transmitter DAC run at 16 times the speed of the receiver ADC for estimating channels at the 16x resolution. While there is no similar approaches (for asymmetric DAC/ADC pairs) to compare with, we derive the CramérRao lower bound. I.
An adaptive inverse scale space method for compressed sensing
, 2011
"... In this paper we introduce a novel adaptive approach for solving ℓ 1minimization problems as frequently arising in compressed sensing, which is based on the recently introduced inverse scale space method. The scheme allows to efficiently compute minimizers by solving a sequence of lowdimensional n ..."
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Cited by 6 (2 self)
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In this paper we introduce a novel adaptive approach for solving ℓ 1minimization problems as frequently arising in compressed sensing, which is based on the recently introduced inverse scale space method. The scheme allows to efficiently compute minimizers by solving a sequence of lowdimensional nonnegative leastsquares problems. We provide a detailed convergence analysis in a general setup as well as refined results under special conditions. In addition we discuss experimental observations in several numerical examples.
Exact Reconstruction Conditions for Regularized Modified Basis Pursuit
, 2010
"... is a continuous and unimodal function of 2, with the unique maximum 2 2 (p+1) 2 (p) achieved at = ( ) , see also (9a). We conclude that ( ) 0 ( 2) (p+1) must go to zero. The second claim of Theorem 1 follows. ..."
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Cited by 4 (2 self)
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is a continuous and unimodal function of 2, with the unique maximum 2 2 (p+1) 2 (p) achieved at = ( ) , see also (9a). We conclude that ( ) 0 ( 2) (p+1) must go to zero. The second claim of Theorem 1 follows.
High resolution OFDM channel estimation with low speed ADC using compressive sensing
 in proceedings of IEEE International Conference on Communications
, 2011
"... Abstract — Orthogonal frequency division multiplexing (OFDM) is a technique that will prevail in the next generation wireless communication. Channel estimation is one of the key challenges in an OFDM system. In this paper, we formulate OFDM channel estimation as a compressive sensing problem, which ..."
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Cited by 3 (3 self)
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Abstract — Orthogonal frequency division multiplexing (OFDM) is a technique that will prevail in the next generation wireless communication. Channel estimation is one of the key challenges in an OFDM system. In this paper, we formulate OFDM channel estimation as a compressive sensing problem, which takes advantage of the sparsity of the channel impulse response and reduces the number of probing measurements, which in turn reduces the ADC speed needed for channel estimation. Specifically, we propose sending out pilots with random phases in order to “spread out ” the sparse taps in the impulse response over the uniformly downsampled measurements at the low speed receiver ADC, so that the impulse response can still be recovered by sparse optimization. This contribution leads to high resolution channel estimation with low speed ADCs, distinguishing this paper from the existing attempts of OFDM channel estimation. We also propose a novel estimator that performs better than the commonly used ℓ1 minimization. Specifically, it significantly reduces estimation error by combing ℓ1 minimization with iterative support detection and limitedsupport leastsquares. While letting the receiver ADC run at a speed as low as 1/16 of the speed of the transmitter DAC, we simulated various numbers of multipaths and different measurement SNRs. The proposed system has channel estimation resolution as high as the system equipped with the high speed ADCs, and the proposed algorithm provides additional 6 dB gain for signal to noise ratio. I.
Exact Recovery Conditions for Sparse Representations with Partial Support Information
, 2013
"... We address the exact recovery of a ksparse vector in the noiseless setting when some partial information on the support is available. This partial information takes the form of either a subset of the true support or an approximate subset including wrong atoms as well. We derive a new sufficient an ..."
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We address the exact recovery of a ksparse vector in the noiseless setting when some partial information on the support is available. This partial information takes the form of either a subset of the true support or an approximate subset including wrong atoms as well. We derive a new sufficient and worstcase necessary (in some sense) condition for the success of some procedures based on `prelaxation, Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS). Our result is based on the coherence Âµ of the dictionary and relaxes the wellknown condition Âµ < 1/(2k â 1) ensuring the recovery of any ksparse vector in the noninformed setup. It reads Âµ < 1/(2k â g+ b â 1) when the informed support is composed of g good atoms and b wrong atoms. We emphasize that our condition is complementary to some restrictedisometry based conditions by showing that none of them implies the other. Because this mutual coherence condition is common to all procedures, we carry out a finer analysis based on the Null Space Property (NSP) and the Exact Recovery Condition (ERC). Connections are established regarding the characterization of `prelaxation procedures and OMP in the informed setup. First, we emphasize that the truncated NSP enjoys an ordering property when p is decreased. Second, the partial ERC for OMP (ERCOMP) implies in turn the truncated NSP for the informed `1 problem, and the truncated NSP for p < 1.
Open Access
"... Refining transcriptional regulatory networks using network evolutionary models and gene histories ..."
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Refining transcriptional regulatory networks using network evolutionary models and gene histories
EDGE GUIDED RECONSTRUCTION FOR COMPRESSIVE IMAGING
"... Abstract. We propose EdgeCS — an edge guided compressive sensing reconstruction approach — to recover images of higher qualities from fewer measurements than the current stateoftheart methods. Edges are important images features that are used in various ways in image recovery, analysis, and under ..."
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Abstract. We propose EdgeCS — an edge guided compressive sensing reconstruction approach — to recover images of higher qualities from fewer measurements than the current stateoftheart methods. Edges are important images features that are used in various ways in image recovery, analysis, and understanding. In compressive sensing, the sparsity of image edges has been widely utilized to recover images. However, edge detectors have not been used on compressive sensing measurements to improve the edge recovery and thus the image recovery. This motivates us to propose EdgeCS, which alternatively performs edge detection and image reconstruction in a mutually beneficial way. The edge detector of EdgeCS is designed to faithfully return partial edges from intermediate image reconstructions even though these reconstructions may still have noise and artifacts. For complex–valued images, it incorporates joint sparsity between the real and imaginary components. EdgeCS has been implemented with both isotropic and anisotropic discretizations of total variation and tested on incomplete kspace (Fourier) samples. It applies to other types of measurements as well. Experimental results on largescale real/complexvalued phantom and magnetic resonance (MR) images show that EdgeCS is fast and returns highquality images. For example, it exactly recovers the 256by256 SheppLogan phantom from merely 7 radial lines (3.03 % kspace), which is impossible for most existing algorithms. It is able to accurately reconstruct a 512by512 MR image with 0.05 white noise from 20.87 % radial samples. On complexvalued MR images, it obtains recoveries with faithful phases,, which are important in many medical applications. Each of these tests took around 30 seconds on a standard PC. Finally, the algorithm is GPU friendly. Key words. Compressive sensing, edge detection, total variation, discrete Fourier transform, magnetic resonance imaging