Results 1  10
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62
Distributed compressed sensing
, 2005
"... Compressed sensing is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. In this paper we introduce a new theory for distributed compressed sensing (DCS) that enables new distributed coding algori ..."
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Cited by 84 (21 self)
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Compressed sensing is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. In this paper we introduce a new theory for distributed compressed sensing (DCS) that enables new distributed coding algorithms for multisignal ensembles that exploit both intra and intersignal correlation structures. The DCS theory rests on a new concept that we term the joint sparsity of a signal ensemble. We study in detail three simple models for jointly sparse signals, propose algorithms for joint recovery of multiple signals from incoherent projections, and characterize theoretically and empirically the number of measurements per sensor required for accurate reconstruction. We establish a parallel with the SlepianWolf theorem from information theory and establish upper and lower bounds on the measurement rates required for encoding jointly sparse signals. In two of our three models, the results are asymptotically bestpossible, meaning that both the upper and lower bounds match the performance of our practical algorithms. Moreover, simulations indicate that the asymptotics take effect with just a moderate number of signals. In some sense DCS is a framework for distributed compression of sources with memory, which has remained a challenging problem for some time. DCS is immediately applicable to a range of problems in sensor networks and arrays.
Random projections of smooth manifolds
 Foundations of Computational Mathematics
, 2006
"... We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N ..."
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Cited by 83 (23 self)
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We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N → R M, M < N, on a smooth wellconditioned Kdimensional submanifold M ⊂ R N. As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on M are wellpreserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in R N. Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifoldmodeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data.
A new compressive imaging camera architecture using opticaldomain compression
 in Proc. of Computational Imaging IV at SPIE Electronic Imaging
, 2006
"... Compressive Sensing is an emerging field based on the revelation that a small number of linear projections of a compressible signal contain enough information for reconstruction and processing. It has many promising implications and enables the design of new kinds of Compressive Imaging systems and ..."
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Cited by 69 (6 self)
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Compressive Sensing is an emerging field based on the revelation that a small number of linear projections of a compressible signal contain enough information for reconstruction and processing. It has many promising implications and enables the design of new kinds of Compressive Imaging systems and cameras. In this paper, we develop a new camera architecture that employs a digital micromirror array to perform optical calculations of linear projections of an image onto pseudorandom binary patterns. Its hallmarks include the ability to obtain an image with a single detection element while sampling the image fewer times than the number of pixels. Other attractive properties include its universality, robustness, scalability, progressivity, and computational asymmetry. The most intriguing feature of the system is that, since it relies on a single photon detector, it can be adapted to image at wavelengths that are currently impossible with conventional CCD and CMOS imagers.
Subspace pursuit for compressive sensing: Closing the gap between performance and complexity
, 2008
"... Abstract — We propose a new algorithm, termed subspace pursuit, for signal reconstruction of sparse and compressible signals with and without noisy perturbations. The algorithm has two important characteristics: low computational complexity, comparable to that of orthogonal matching pursuit techniqu ..."
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Cited by 64 (4 self)
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Abstract — We propose a new algorithm, termed subspace pursuit, for signal reconstruction of sparse and compressible signals with and without noisy perturbations. The algorithm has two important characteristics: low computational complexity, comparable to that of orthogonal matching pursuit techniques, and reconstruction capability of the same order as that of ℓ1norm optimization methods. The presented analysis shows that in the noiseless setting, the proposed algorithm is capable of exactly reconstructing an arbitrary sparse signals, provided that the linear measurements satisfy the restricted isometry property with a constant parameter which can be described in a closed form. In the noisy setting and the case where the signal is not exactly sparse, it can be shown that the mean squared error of the reconstruction is upper bounded by a constant multiple of the measurement and signal pertubation energy. Index Terms — Compressive sensing, orthogonal matching pursuit, reconstruction algorithms, restricted isometry property, sparse signal reconstruction I.
Combinatorial Algorithms for Compressed Sensing
 In Proc. of SIROCCO
, 2006
"... Abstract — In sparse approximation theory, the fundamental problem is to reconstruct a signal A ∈ R n from linear measurements 〈A, ψi 〉 with respect to a dictionary of ψi’s. Recently, there is focus on the novel direction of Compressed Sensing [1] where the reconstruction can be done with very few—O ..."
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Cited by 64 (1 self)
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Abstract — In sparse approximation theory, the fundamental problem is to reconstruct a signal A ∈ R n from linear measurements 〈A, ψi 〉 with respect to a dictionary of ψi’s. Recently, there is focus on the novel direction of Compressed Sensing [1] where the reconstruction can be done with very few—O(k log n)— linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [1], [2], [3] prove that there exists a single O(k log n) × n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications. In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a nonadaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, log n, 1/ε, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + ε and improves the reconstruction time from poly(n) to poly(k log n). Our second result is a randomized construction of O(k polylog(n)) measurements that work for each signal with high probability and gives perinstance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such perinstance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [4], [5], Streaming algorithms [6], [7], [8] and Complexity Theory [9] for this case. Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement. I.
An architecture for compressive imaging
 in IEEE International Conference on Image Processing (ICIP
, 2006
"... Compressive Sensing is an emerging field based on the revelation that a small group of nonadaptive linear projections of a compressible signal contains enough information for reconstruction and processing. In this paper, we propose algorithms and hardware to support a new theory of Compressive Imag ..."
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Cited by 53 (7 self)
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Compressive Sensing is an emerging field based on the revelation that a small group of nonadaptive linear projections of a compressible signal contains enough information for reconstruction and processing. In this paper, we propose algorithms and hardware to support a new theory of Compressive Imaging. Our approach is based on a new digital image/video camera that directly acquires random projections of the signal without first collecting the pixels/voxels. Our camera architecture employs a digital micromirror array to perform optical calculations of linear projections of an image onto pseudorandom binary patterns. Its hallmarks include the ability to obtain an image with a single detection element while measuring the image/video fewer times than the number of pixels — this can significantly reduce the computation required for video acquisition/encoding. Because our system relies on a single photon detector, it can also be adapted to image at wavelengths that are currently impossible with conventional CCD and CMOS imagers. We are currently testing a prototype design for the camera and include experimental results.
Bayesian Compressed Sensing via Belief Propagation
, 2010
"... Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can comple ..."
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Cited by 51 (12 self)
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Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can complement conventional CS methods based on linear programming or greedy algorithms. We perform asymptotically optimal Bayesian inference using belief propagation (BP) decoding, which represents the CS encoding matrix as a graphical model. Fast computation is obtained by reducing the size of the graphical model with sparse encoding matrices. To decode a length signal containing large coefficients, our CSBP decoding algorithm uses ( log ()) measurements and ( log 2 ()) computation. Finally, although we focus on a twostate mixture Gaussian model, CSBP is easily adapted to other signal models.
Compressive imaging for video representation and coding
 In Proceedings of Picture Coding Symposium (PCS
, 2006
"... Abstract. Compressive Sensing is an emerging field based on the revelation that a small group of nonadaptive linear projections of a compressible signal contains enough information for reconstruction and processing. In this paper, we propose algorithms and hardware to support a new theory of Compres ..."
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Cited by 43 (8 self)
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Abstract. Compressive Sensing is an emerging field based on the revelation that a small group of nonadaptive linear projections of a compressible signal contains enough information for reconstruction and processing. In this paper, we propose algorithms and hardware to support a new theory of Compressive Imaging. Our approach is based on a new digital image/video camera that directly acquires random projections of the light field without first collecting the pixels/voxels. Our camera architecture employs a digital micromirror array to perform optical calculations of linear projections of an image onto pseudorandom binary patterns. Its hallmarks include the ability to obtain an image with a single detection element while measuring the image/video fewer times than the number of pixels/voxels; this can significantly reduce the computation required for video acquisition/encoding. Since our system relies on a single photon detector, it can also be adapted to image at wavelengths that are currently impossible with conventional CCD and CMOS imagers. We are currently testing a prototype design for the camera and include experimental results. Index Terms: camera, compressive sensing, imaging, incoherent projections, linear programming, random
Compressed Sensing Reconstruction via Belief Propagation
, 2006
"... Compressed sensing is an emerging field that enables to reconstruct sparse or compressible signals from a small number of linear projections. We describe a specific measurement scheme using an LDPClike measurement matrix, which is a realvalued analogue to LDPC techniques over a finite alphabet. We ..."
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Cited by 39 (8 self)
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Compressed sensing is an emerging field that enables to reconstruct sparse or compressible signals from a small number of linear projections. We describe a specific measurement scheme using an LDPClike measurement matrix, which is a realvalued analogue to LDPC techniques over a finite alphabet. We then describe the reconstruction details for mixture Gaussian signals. The technique can be extended to additional compressible signal models. 1
Analogtoinformation conversion via random demodulation
 In Proc. IEEE Dallas Circuits and Systems Workshop (DCAS
, 2006
"... Abstract — Many problems in radar and communication signal processing involve radio frequency (RF) signals of very high bandwidth. This presents a serious challenge to systems that might attempt to use a highrate analogtodigital converter (ADC) to sample these signals, as prescribed by the Shanno ..."
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Cited by 37 (12 self)
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Abstract — Many problems in radar and communication signal processing involve radio frequency (RF) signals of very high bandwidth. This presents a serious challenge to systems that might attempt to use a highrate analogtodigital converter (ADC) to sample these signals, as prescribed by the Shannon/Nyquist sampling theorem. In these situations, however, the information level of the signal is often far lower than the actual bandwidth, which prompts the question of whether more efficient schemes can be developed for measuring such signals. In this paper we propose a system that uses modulation, filtering, and sampling to produce a lowrate set of digital measurements. Our “analogtoinformation converter ” (AIC) is inspired by the recent theory of Compressive Sensing (CS), which states that a discrete signal having a sparse representation in some dictionary can be recovered from a small number of linear projections of that signal. We generalize the CS theory to continuoustime sparse signals, explain our proposed AIC system in the CS context, and discuss practical issues regarding implementation. I.