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11
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 218 (48 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.
Compressed Sensing of Analog Signals in ShiftInvariant Spaces
, 2009
"... A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worstcase scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that on ..."
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Cited by 74 (41 self)
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A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worstcase scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that only part of the bandwidth is used. In this paper, we develop methods for lowrate sampling of continuoustime sparse signals in shiftinvariant (SI) spaces, generated by m kernels with period T. We model sparsity by treating the case in which only k out of the m generators are active, however, we do not know which k are chosen. We show how to sample such signals at a rate much lower than m/T, which is the minimal sampling rate without exploiting sparsity. Our approach combines ideas from analog sampling in a subspace with a recently developed block diagram that converts an infinite set of sparse equations to a finite counterpart. Using these two components we formulate our problem within the framework of finite compressed sensing (CS) and then rely on algorithms developed in that context. The distinguishing feature of our results is that in contrast to standard CS, which treats finitelength vectors, we consider sampling of analog signals for which no underlying finitedimensional model exists. The proposed framework allows to extend much of the recent literature on CS to the analog domain.
Robust recovery of signals from a union of subspaces
 IEEE TRANS. INFORM. THEORY
, 2008
"... Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees a unique signal consistent with the given measurements is that x lies in a known subspace. Recently, there has been growing interest in non ..."
Abstract

Cited by 45 (14 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 a unique signal consistent with 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 is assumed to lie in a union of subspaces. An example is the case in which x is a finite length vector that is sparse in a given basis. 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 finite union of finite dimensional spaces and the samples are modelled as inner products with an arbitrary set of sampling functions. We first develop conditions under which unique and stable recovery of x is possible, albeit with algorithms that have combinatorial complexity. To derive an efficient and robust recovery algorithm, we then show that our problem can be formulated as that of recovering a block sparse vector, namely a vector whose nonzero elements appear in fixed blocks. To solve this problem, we suggest minimizing a mixed ℓ2/ℓ1 norm subject to the measurement equations. We then develop equivalence conditions under which the proposed convex algorithm is guaranteed to recover the original signal. These results rely 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. A special case of the proposed framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Specializing 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.
Block sparsity and sampling over a union of subspaces
 in Proc. Int. Conf. Digital Signal Process
"... Sparse signal representations have gained wide popularity in recent years. In many applications the data can be expressed using only a few nonzero elements in an appropriate expansion. In this paper, we study a blocksparse model, in which the nonzero coefficients are arranged in blocks. To exploi ..."
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Cited by 2 (0 self)
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Sparse signal representations have gained wide popularity in recent years. In many applications the data can be expressed using only a few nonzero elements in an appropriate expansion. In this paper, we study a blocksparse model, in which the nonzero coefficients are arranged in blocks. To exploit this structure, we redefine the standard (NPhard) sparse recovery problem, based on which we propose a convex relaxation in the form of a mixed `2/`1 program. Isometrybased analysis is used to prove equivalence of the solution to that of the optimal program, under certain mild conditions. We further establish the robustness of our algorithm to mismodeling and bounded noise. We then present theoretical arguments and numerical experiments demonstrating the improved recovery performance of our method in comparison with sparse reconstruction that does not incorporate a block structure. The results are then applied to two related problems. The first is that of simultaneous sparse approximation. Our results can be used to prove isometrybased equivalence properties for this setting. In addition, we propose an alternative approach to acquire the measurements, that leads to performance improvement over standard methods. Finally, we show how our results can be used to sample signals in a finite structured union of subspaces, leading to robust and efficient recovery algorithms. Index Terms — Block sparsity, compressed sensing, multiple measurement vectors (MMV), restricted isometry property, sparse
RECOVERING SIGNALS FROM LOWPASS DATA 1 Recovering Signals from Lowpass Data
, 907
"... Abstract — The problem of recovering a signal from its low frequency components occurs often in practical applications due to the lowpass behavior of many physical systems. Here we study in detail conditions under which a signal can be determined from its lowfrequency content. We focus on signals i ..."
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Abstract — The problem of recovering a signal from its low frequency components occurs often in practical applications due to the lowpass behavior of many physical systems. Here we study in detail conditions under which a signal can be determined from its lowfrequency content. We focus on signals in shiftinvariant spaces generated by multiple generators. For these signals, we derive necessary conditions on the cutoff frequency of the lowpass filter as well as necessary and sufficient conditions on the generators such that signal recovery is possible. When the lowpass content is not sufficient to determine the signal, we propose appropriate preprocessing that can improve the reconstruction ability. In particular, we show that modulating the signal with one or more mixing functions prior to lowpass filtering, can ensure the recovery of the signal in many cases, and reduces the necessary bandwidth of the filter. Index Terms — Sampling, shiftinvariant spaces, lowpass signals I.
ANALOG COMPRESSED SENSING
"... A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. Practical signals often posses a sparse structure so that a large part of the bandwidth is not exploited. In this paper, we consider a framework for utilizin ..."
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A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. Practical signals often posses a sparse structure so that a large part of the bandwidth is not exploited. In this paper, we consider a framework for utilizing this sparsity in order to sample such analog signals at a low rate. By relying on results developed in the context of compressed sensing (CS) of finitelength vectors, we develop a general framework for lowrate sampling of signals in shiftinvariant spaces. In contrast to the problems treated in the context of CS, here we explicitly consider sampling of analog signals for which no underlying finitedimensional model exists. Index terms – Sampling methods, Compressed sensing. 1.
THE VECTOR HYBRID WIENER FILTER: APPLICATION TO SUPERRESOLUTION
"... We address the problem of recovering a continuoustime (space) signal from several blurred and noisy sampled versions of it, a scenario commonly encountered in superresolution (SR) and arrayprocessing. We show that discretization, a key step in many SR algorithms, inevitably leads to inaccurate mod ..."
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We address the problem of recovering a continuoustime (space) signal from several blurred and noisy sampled versions of it, a scenario commonly encountered in superresolution (SR) and arrayprocessing. We show that discretization, a key step in many SR algorithms, inevitably leads to inaccurate modeling. Instead, we treat the problem entirely in the continuous domain by modeling the signal as a continuoustime random process and deriving its linear minimum meansquared error estimate given the samples. We also provide an efficient implementation scheme, valid for 1D applications. Simulation results on realworld data demonstrate the advantage of our approach with respect to SR techniques that rely on discretization. Index Terms — Superresolution, nonuniform interpolation. x(t) y1(t) s1(t) t = n u1(t) yK(t) sK(t) t = n uK(t) Fig. 1: Multichannel sampling scheme. c1[n] cK[n] 1.
Signal Recovery from Low Frequency Components
"... Abstract—In many applications only the low frequency components of a signal can be measured due to the lowpass behavior of many physical systems. Nevertheless, if additional information on the structure of the signal is known, it might still be possible to reconstruct the signal from its lowfrequen ..."
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Abstract—In many applications only the low frequency components of a signal can be measured due to the lowpass behavior of many physical systems. Nevertheless, if additional information on the structure of the signal is known, it might still be possible to reconstruct the signal from its lowfrequency content. This paper studies signals in shiftinvariant spaces with multiple generators and derives necessary conditions on the bandwith of the lowpass filter as well as sufficient conditions on the generators such that signal recovery is possible. If the signal can not be recovered from its low frequency components, an appropriate preprocessing of the signal is proposed which improves the reconstruction ability. In particular, it is shown that modulating the signal with one or more mixing functions prior to lowpass filtering can ensure the recovery of the signal in many cases. I.