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Single-pixel imaging via compressive sampling
- IEEE Signal Processing Magazine
"... Humans are visual animals, and imaging sensors that extend our reach – cameras – have improved dramatically in recent times thanks to the introduction of CCD and CMOS digital technology. Consumer digital cameras in the mega-pixel range are now ubiquitous thanks to the happy coincidence that the semi ..."
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Cited by 296 (19 self)
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Humans are visual animals, and imaging sensors that extend our reach – cameras – have improved dramatically in recent times thanks to the introduction of CCD and CMOS digital technology. Consumer digital cameras in the mega-pixel range are now ubiquitous thanks to the happy coincidence that the semiconductor material of choice for large-scale electronics integration (silicon) also happens to readily convert photons at visual wavelengths into electrons. On the contrary, imaging at wavelengths where silicon is blind is considerably more complicated, bulky, and expensive. Thus, for comparable resolution, a $500 digital camera for the visible becomes a $50,000 camera for the infrared. In this paper, we present a new approach to building simpler, smaller, and cheaper digital cameras that can operate efficiently across a much broader spectral range than conventional silicon-based cameras. Our approach fuses a new camera architecture based on a digital mi-cromirror device (DMD – see Sidebar: Spatial Light Modulators) with the new mathematical theory and algorithms of compressive sampling (CS – see Sidebar: Compressive Sampling in a Nutshell). CS combines sampling and compression into a single nonadaptive linear measurement process [1–4]. Rather than measuring pixel samples of the scene under view, we measure inner products
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 block-sparse vector whose non-zero 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.
A unified framework for high-dimensional analysis of M-estimators with decomposable regularizers
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Structured variable selection with sparsity-inducing norms
, 2011
"... We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual ℓ1-norm and the group ℓ1-norm by allowing the subsets to ov ..."
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Cited by 187 (27 self)
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We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual ℓ1-norm and the group ℓ1-norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problems. We first explore the relationship between the groups defining the norm and the resulting nonzero patterns, providing both forward and backward algorithms to go back and forth from groups to patterns. This allows the design of norms adapted to specific prior knowledge expressed in terms of nonzero patterns. We also present an efficient active set algorithm, and analyze the consistency of variable selection for least-squares linear regression in low and high-dimensional settings.
Block-sparse signals: Uncertainty relations and efficient recovery
- IEEE TRANS. SIGNAL PROCESS
, 2010
"... We consider efficient methods for the recovery of block-sparse signals — i.e., sparse signals that have nonzero entries occurring in clusters—from an underdetermined system of linear equations. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we ..."
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Cited by 161 (17 self)
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We consider efficient methods for the recovery of block-sparse signals — i.e., sparse signals that have nonzero entries occurring in clusters—from an underdetermined system of linear equations. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block k-sparse signals in no more than k steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed `2=`1-optimization approach. This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.
Iteratively reweighted least squares minimization for sparse recovery
- Comm. Pure Appl. Math
"... Under certain conditions (known as the Restricted Isometry Property or RIP) on the m ×Nmatrix Φ (where m < N), vectors x ∈ RN that are sparse (i.e. have most of their entries equal to zero) can be recovered exactly from y: = Φx even though Φ−1 (y) is typically an (N − m)dimensional hyperplane; in ..."
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Cited by 156 (5 self)
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Under certain conditions (known as the Restricted Isometry Property or RIP) on the m ×Nmatrix Φ (where m < N), vectors x ∈ RN that are sparse (i.e. have most of their entries equal to zero) can be recovered exactly from y: = Φx even though Φ−1 (y) is typically an (N − m)dimensional hyperplane; in addition x is then equal to the element in Φ−1 (y) of minimal ℓ1-norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an Iteratively Re-weighted Least Squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in Φ−1 (y) with smallest ℓ2(w)-norm. If x (n) is the solution at iteration step n, then the new weight w (n) is defined by w (n) i:=
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 136 (26 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 multi-signal ensembles that exploit both intra- and inter-signal 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 Slepian-Wolf 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 best-possible, 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.
Bayesian compressive sensing via belief propagation
- IEEE Trans. Signal Processing
, 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, sub-Nyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can comple ..."
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Cited by 125 (19 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, sub-Nyquist 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 approximate Bayesian inference using belief propagation (BP) decoding, which represents the CS encoding matrix as a graphical model. Fast encoding and decoding is provided using sparse encoding matrices, which also improve BP convergence by reducing the presence of loops in the graph. To decode a length-N signal containing K large coefficients, our CS-BP decoding algorithm uses O(K log(N)) measurements and O(N log 2 (N)) computation. Finally, sparse encoding matrices and the CS-BP decoding algorithm can be modified to support a variety of signal models and measurement noise. 1
Curvelet-Wavelet 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 Shannon-Nyquist theory requires. Many images ..."
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Cited by 119 (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 Shannon-Nyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent point-like features while curvelets represent line-like 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 sparsity-constrained 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 Douglas-Rachford Split algorithm. Numerical experiments for compressed sensing based Fourier-domain random imaging show good performances of the proposed curvelet-wavelet regularized split Bregman (CWSpB) methods,whereweparticularlyuseacombination of wavelet and curvelet coefficients as sparsity constraints.
