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CurveletWavelet Regularized Split Bregman Iteration for Compressed Sensing
"... Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the ShannonNyquist theory requires. Many images ..."
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Cited by 118 (6 self)
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Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the ShannonNyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent pointlike features while curvelets represent linelike features well. For a suitable recovery of images, we propose models that contain weighted sparsity constraints in two different frames. Given the incomplete measurements f = Φu + ɛ with the measurement matrix Φ ∈ R K×N, K<<N, we consider a jointly sparsityconstrained optimization problem of the form argmin{‖ΛcΨcu‖1 + ‖ΛwΨwu‖1 + u 1 2‖f − Φu‖22}. Here Ψcand Ψw are the transform matrices corresponding to the two frames, and the diagonal matrices Λc, Λw contain the weights for the frame coefficients. We present efficient iteration methods to solve the optimization problem, based on Alternating Split Bregman algorithms. The convergence of the proposed iteration schemes will be proved by showing that they can be understood as special cases of the DouglasRachford Split algorithm. Numerical experiments for compressed sensing based Fourierdomain random imaging show good performances of the proposed curveletwavelet regularized split Bregman (CWSpB) methods,whereweparticularlyuseacombination of wavelet and curvelet coefficients as sparsity constraints.
Sampling and Recovery of Pulse Streams
"... Compressive Sensing (CS) is a new technique for the efficient acquisition of signals, images, and other data that have a sparse representation in some basis, frame, or dictionary. By sparse we mean that the Ndimensional basis representation has just K ≪ N significant coefficients; in this case, the ..."
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Cited by 13 (1 self)
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Compressive Sensing (CS) is a new technique for the efficient acquisition of signals, images, and other data that have a sparse representation in some basis, frame, or dictionary. By sparse we mean that the Ndimensional basis representation has just K ≪ N significant coefficients; in this case, the CS theory maintains that just M = O (K log N) random linear signal measurements will both preserve all of the signal information and enable robust signal reconstruction in polynomial time. In this paper, we extend the CS theory to pulse stream data, which correspond to Ssparse signals/images that are convolved with an unknown Fsparse pulse shape. Ignoring their convolutional structure, a pulse stream signal is K = SF sparse. Such signals figure prominently in a number of applications, from neuroscience to astronomy. Our specific contributions are threefold. First, we propose a pulse stream signal model and show that it is equivalent to an infinite union of subspaces. Second, we derive a lower bound on the number of measurements M required to preserve the essential information present in pulse streams. The bound is linear in the total number of degrees of freedom S + F, which is significantly smaller than the naive bound based on the total signal sparsity K = SF. Third, we develop an efficient signal recovery algorithm that infers both the shape of the impulse response as well as the locations and
Morphological Diversity and Sparsity for Multichannel Data Restoration
 J MATH IMAGING VIS
, 2008
"... Over the last decade, overcomplete dictionaries and the very sparse signal representations they make possible, have raised an intense interest from signal processing theory. In a wide range of signal processing problems, sparsity has been a crucial property leading to high performance. As multichann ..."
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Cited by 12 (4 self)
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Over the last decade, overcomplete dictionaries and the very sparse signal representations they make possible, have raised an intense interest from signal processing theory. In a wide range of signal processing problems, sparsity has been a crucial property leading to high performance. As multichannel data are of growing interest, it seems essential to devise sparsitybased tools accounting for such specific multichannel data. Sparsity has proved its efficiency in a wide range of inverse problems. Hereafter, we address some multichannel inverse problems issues such as multichannel morphological component separation and inpainting from the perspective of sparse representation. In this paper, we introduce a new sparsitybased multichannel analysis tool coined multichannel Morphological Component Analysis (mMCA). This new framework focuses on multichannel morphological diversity to better represent multichannel data. This paper presents conditions under which the mMCA converges and recovers the sparse multichannel representation. Several experiments are presented to demonstrate the applicability of our approach on
Compressed Sensing for Surface Characterization and Metrology
 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT
, 2009
"... Surface metrology is the science of measuring smallscale features on surfaces. In this paper, a novel compressed sensing (CS) theory is introduced for the surface metrology to reduce data acquisition. We first describe that the CS is naturally fit to surface measurement and analysis. Then, a geomet ..."
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Cited by 11 (2 self)
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Surface metrology is the science of measuring smallscale features on surfaces. In this paper, a novel compressed sensing (CS) theory is introduced for the surface metrology to reduce data acquisition. We first describe that the CS is naturally fit to surface measurement and analysis. Then, a geometric waveletbased recovery algorithm is proposed for scratched and textural surfaces by solving a convex optimal problem with sparse constrained by curvelet transform and wave atom transform. In the framework of compressed measurement, one can stably recover compressible surfaces from incomplete and inaccurate random measurements by using the recovery algorithm. The necessary number of measurements is far fewer than those required by traditional methods that have to obey the Shannon sampling theorem. The compressed metrology essentially shifts online measurement cost to computational cost of offline nonlinear recovery. By combining the idea of sampling, sparsity, and compression, the proposed method indicates a new acquisition protocol and leads to building new measurement instruments. It is very significant for measurements limited by physical constraints, or is extremely expensive. Experiments on engineering and bioengineering surfaces demonstrate good performances of the proposed method.
Sparsity Averaging for Compressive Imaging
"... We propose a novel regularization method for sparse image reconstruction from compressive measurements. The approach relies on the conjecture that natural images exhibit strong average sparsity over multiple coherent frames. The associated reconstruction algorithm, based on an analysis prior and a ..."
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Cited by 7 (3 self)
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We propose a novel regularization method for sparse image reconstruction from compressive measurements. The approach relies on the conjecture that natural images exhibit strong average sparsity over multiple coherent frames. The associated reconstruction algorithm, based on an analysis prior and a reweighted ℓ1 scheme, is dubbed Sparsity Averaging Reweighted Analysis (SARA). We test our prior and the associated algorithm through extensive numerical simulations for spread spectrum and Gaussian acquisition schemes suggested by the recent theory of compressed sensing with coherent and redundant dictionaries. Our results show that average sparsity outperforms stateoftheart priors that promote sparsity in a single orthonormal basis or redundant frame, or that promote gradient sparsity. We also illustrate the performance of SARA in the context of Fourier imaging, for particular applications in astronomy and medicine.
Compressive Sensing of Streams of Pulses
"... as an enticing alternative to the traditional process of signal acquisition. For a lengthN signal with sparsity K, merely M = O (K log N) ≪ N random linear projections (measurements) can be used for robust reconstruction in polynomial time. Sparsity is a powerful and simple signal model; yet, rich ..."
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Cited by 5 (2 self)
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as an enticing alternative to the traditional process of signal acquisition. For a lengthN signal with sparsity K, merely M = O (K log N) ≪ N random linear projections (measurements) can be used for robust reconstruction in polynomial time. Sparsity is a powerful and simple signal model; yet, richer models that impose additional structure on the sparse nonzeros of a signal have been studied theoretically and empirically from the CS perspective. In this work, we introduce and study a sparse signal model for streams of pulses, i.e., Ssparse signals convolved with an unknown Fsparse impulse response. Our contributions are threefold: (i) we geometrically model this set of signals as an infinite union of subspaces; (ii) we derive a sufficient number of random measurements M required to preserve the metric information of this set. In particular this number is linear merely in the number of degrees of freedom of the signal S + F, and sublinear in the sparsity K = SF; (iii) we develop an algorithm that performs recovery of the signal from M measurements and analyze its performance under noise and model mismatch. Numerical experiments on synthetic and real data demonstrate the utility of our proposed theory and algorithm. Our method is amenable to diverse applications such as the highresolution sampling of neuronal recordings and ultrawideband (UWB) signals. I.
Compressed Sensing: Doppler Ultrasound Signal Recovery by Using Nonuniform Sampling
 Random Sampling, Proc. 28 th National Radio Science conference IEEE Catalog Number CFP11427PRT
, 2011
"... Several authors have shown that it is possible to reconstruct exactly a sparse signal from a fewer linear measurements, this method known as compressed sensing (CS). CS aim to reconstruct signals and images from significantly fewer measurements. With CS it‟s possible to make an accurate reconstructi ..."
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Cited by 4 (2 self)
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Several authors have shown that it is possible to reconstruct exactly a sparse signal from a fewer linear measurements, this method known as compressed sensing (CS). CS aim to reconstruct signals and images from significantly fewer measurements. With CS it‟s possible to make an accurate reconstruction from small number of samples (measurements). Doppler ultrasound is an important technique for noninvasively detecting and measuring the velocity of moving structure, and particularly blood, within the body. Doppler ultrasound signal has been reconstructed with CS by using random sampling and nonuniform sampling via ℓ1norm to generate Doppler sonogram. The result show that the recovered signals with nonuniform sampling are the same as the original signal, there is a loss of very small peaks, when random sampling used for recovering the signals, there is no significant different between the original signal and reconstructed one when we used more than 85 % of the data, when less than 85 % of the data used, the reconstructed signals and the original signal are different. The sonograms generated from the reconstructed signals with random and nonuniform sampling are same as the original one, but there are some losses in contrast. The error of the reconstructed images was calculated, the result shows that the error in the image decreased with increasing the number of samples.
Improved Iterative Curvelet Thresholding for Compressed Sensing
"... A new theory named compressed sensing for simultaneous sampling and compression of signals has been becoming popular in the communities of signal processing, imaging and applied mathematics. In this paper, we present improved/accelerated iterative curvelet thresholding methods for compressed sensing ..."
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Cited by 3 (1 self)
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A new theory named compressed sensing for simultaneous sampling and compression of signals has been becoming popular in the communities of signal processing, imaging and applied mathematics. In this paper, we present improved/accelerated iterative curvelet thresholding methods for compressed sensing reconstruction in the fields of remote sensing. Some recent strategies including BioucasDias and Figueiredo’s twostep iteration, Beck and Teboulle’s fast method, and Osher et al’s linearized Bregman iteration are applied to iterative curvelet thresholding in order to accelerate convergence. Advantages and disadvantages of the proposed methods are studied using the socalled pseudoPareto curve in the numerical experiments on singlepixel remote sensing and Fourierdomain random imaging.
EnergyAware Design for Compressed Sensing Systems for Wireless Sensors under Performance and Reliability Constraints,” to be published
, 2011
"... Abstract—This paper describes the system design of a compressed sensing (CS) based source encoding system for data compression in wireless sensor applications. We examine the tradeoff between the required transmission energy (compression performance) and desired recovered signal quality in the pres ..."
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Cited by 3 (0 self)
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Abstract—This paper describes the system design of a compressed sensing (CS) based source encoding system for data compression in wireless sensor applications. We examine the tradeoff between the required transmission energy (compression performance) and desired recovered signal quality in the presence of practical nonidealities such as quantization noise, input signal noise and channel errors. The endtoend system evaluation framework was designed to analyze CS performance under practical sensor settings. The evaluation shows that CS compression can enable over 10X in transmission energy savings while preserving the recovered signal quality to roughly 8 bits of precision. We further present low complexity error control schemestailoredtoCSthat further reduce the energy costs by 4X as well as diversity scheme to protect against burst errors. Results on a real electrocardiography (EKG) signal demonstrate 10X in energy reduction and corroborate the system analysis. Index Terms—Compressed sensing, error correction codes, source coding, wireless sensors, energy efficiency. I.