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BlockBased Compressed Sensing of Images and Video
 Foundations and Trends in Signal Processing
, 2012
"... A number of techniques for the compressed sensing of imagery are surveyed. Various imaging media are considered, including still images, motion video, as well as multiview image sets and multiview video. A particular emphasis is placed on blockbased compressed sensing due to its advantages in terms ..."
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Cited by 20 (4 self)
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A number of techniques for the compressed sensing of imagery are surveyed. Various imaging media are considered, including still images, motion video, as well as multiview image sets and multiview video. A particular emphasis is placed on blockbased compressed sensing due to its advantages in terms of both lightweight reconstruction complexity as well as a reduced memory burden for the randomprojection measurement operator. For multipleimage scenarios, including video and multiview imagery, motion and disparity compensation is employed to exploit frametoframe redundancies due to object motion and parallax, resulting in residual frames which are more compressible and thus more easily reconstructed from compressedsensing measurements. ExFoundations and Trends R ○ in Signal Processing, to appear, 2012. tensive experimental comparisons evaluate various prominent reconstruction algorithms for stillimage, motionvideo, and multiview scenarios in terms of both reconstruction quality as well as computational
Analyzing least squares and kalman filtered compressed sensing,” in long version, available at http://www.ece.iastate.edu/ ∼namrata/ AnalyzeKFCS.pdf
, 2008
"... In recent work, we studied the problem of causally reconstructing time sequences of spatially sparse signals, with unknown and slow timevarying sparsity patterns, from a limited number of linear “incoherent” measurements. We proposed a solution called Kalman Filtered Compressed Sensing (KFCS). The ..."
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Cited by 16 (9 self)
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In recent work, we studied the problem of causally reconstructing time sequences of spatially sparse signals, with unknown and slow timevarying sparsity patterns, from a limited number of linear “incoherent” measurements. We proposed a solution called Kalman Filtered Compressed Sensing (KFCS). The key idea is to run a reduced order KF only for the current signal’s estimated nonzero coefficients’ set, while performing CS on the Kalman filtering error to estimate new additions, if any, to the set. KF may be replaced by Least Squares (LS) estimation and we call the resulting algorithm LSCS. In this work, (a) we bound the error in performing CS on the LS error and (b) we obtain the conditions under which the KFCS (or LSCS) estimate converges to that of a genieaided KF (or LS), i.e. the KF (or LS) which knows the true nonzero sets.
LassoKalman smoother for tracking sparse signals
 in Asilomar Conf. on Signals, Systems and Computers 2009
, 2009
"... Abstract—Fixedinterval smoothing of timevarying vector processes is an estimation approach with welldocumented merits for tracking applications. The optimal performance in the linear GaussMarkov model is achieved by the Kalman smoother (KS), which also admits an efficient recursive implementatio ..."
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Cited by 12 (0 self)
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Abstract—Fixedinterval smoothing of timevarying vector processes is an estimation approach with welldocumented merits for tracking applications. The optimal performance in the linear GaussMarkov model is achieved by the Kalman smoother (KS), which also admits an efficient recursive implementation. The present paper deals with vector processes for which it is known a priori that many of their entries equal to zero. In this context, the process to be tracked is sparse, and the performance of sparsityagnostic KS schemes degrades considerably. On the other hand, it is shown here that a sparsityaware KS exhibits complexity which grows exponentially in the vector dimension. To obtain a tractable alternative, the KS cost is regularized with the sparsitypromoting ℓ1 norm of the vector process – a relaxation also used in linear regression problems to obtain the leastabsolute shrinkage and selection operator (Lasso). The Lasso (L)KS derived in this work is not only capable of tracking sparse timevarying vector processes, but can also afford an efficient recursive implementation based on the alternating direction method of multipliers (ADMoM). Finally, a weighted (W)LKS is also introduced to cope with the bias of the LKS, and simulations are provided to validate the performance of the novel algorithms. I.
SPARSE SIGNAL RECOVERY IN THE PRESENCE OF CORRELATED MULTIPLE MEASUREMENT VECTORS
"... Sparse signal recovery algorithms utilizing multiple measurement vectors (MMVs) are known to have better performance compared to the single measurement vector case. However, current work rarely consider the case when sources have temporal correlation, a likely situation in practice. In this work we ..."
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Cited by 12 (6 self)
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Sparse signal recovery algorithms utilizing multiple measurement vectors (MMVs) are known to have better performance compared to the single measurement vector case. However, current work rarely consider the case when sources have temporal correlation, a likely situation in practice. In this work we examine methods to account for temporal correlation and its impact on performance. We model sources as AR processes, and then incorporate such information into the framework of sparse Bayesian learning for sparse signal recovery. Experiments demonstrate the superiority of the proposed algorithms. They also show that the performance of existing algorithms are limited by temporal correlation, and that if such correlation can be fully exploited, as in our proposed algorithms, the limitation can be overcome.
Compressive acquisition of linear dynamical systems
, 2013
"... Compressive sensing (CS) enables the acquisition and recovery of sparse signals and images at sampling rates significantly below the classical Nyquist rate. Despite significant progress in the theory and methods of CS, little headway has been made in compressive video acquisition and recovery. Vid ..."
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Cited by 11 (5 self)
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Compressive sensing (CS) enables the acquisition and recovery of sparse signals and images at sampling rates significantly below the classical Nyquist rate. Despite significant progress in the theory and methods of CS, little headway has been made in compressive video acquisition and recovery. Video CS is complicated by the ephemeral nature of dynamic events, which makes direct extensions of standard CS imaging architectures and signal models difficult. In this paper, we develop a new framework for video CS for dynamic textured scenes that models the evolution of the scene as a linear dynamical system (LDS). This reduces the video recovery problem to first estimating the model parameters of the LDS from compressive measurements and then reconstructing the image frames. We exploit the lowdimensional dynamic parameters (the state sequence) and highdimensional static parameters (the observation matrix) of the LDS to devise a novel compressive measurement strategy that measures only the timevarying parameters at each instant and accumulates measurements over time to estimate the timeinvariant parameters. This enables us to lower the compressive measurement rate considerably. We validate our approach and demonstrate its effectiveness with a range of experiments involving video recovery and scene classification.
Dynamic Compressive Sensing of TimeVarying Signals via Approximate Message Passing
, 2013
"... In this work the dynamic compressive sensing (CS) problem of recovering sparse, correlated, timevarying signals from subNyquist, nonadaptive, linear measurements is explored from a Bayesian perspective. While there has been a handful of previously proposed Bayesian dynamic CS algorithms in the li ..."
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Cited by 10 (2 self)
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In this work the dynamic compressive sensing (CS) problem of recovering sparse, correlated, timevarying signals from subNyquist, nonadaptive, linear measurements is explored from a Bayesian perspective. While there has been a handful of previously proposed Bayesian dynamic CS algorithms in the literature, the ability to perform inference on highdimensional problems in a computationally efficient manner remains elusive. In response, we propose a probabilistic dynamic CS signal model that captures both amplitude and support correlation structure, and describe an approximate message passing algorithm that performs soft signal estimation and support detection with a computational complexity that is linear in all problem dimensions. The algorithm, DCSAMP, can perform either causal filtering or noncausal smoothing, and is capable of learning model parameters adaptively from the data through an expectationmaximization learning procedure. We provide numerical evidence that DCSAMP performs within 3 dB of oracle bounds on synthetic data under a variety of operating conditions. We further describe the result of applying DCSAMP to two real dynamic CS datasets, as well as a frequency estimation task, to bolster our claim that DCSAMP is capable of offering stateoftheart performance and speed on realworld highdimensional problems.
Realtime dynamic mr image reconstruction using kalman filtered compressed sensing
 in IEEE Intl. Conf. Acoustics, Speech, Sig. Proc. (ICASSP
, 2009
"... In recent work, Kalman Filtered Compressed Sensing (KFCS) was proposed to causally reconstruct time sequences of sparse signals, from a limited number of “incoherent ” measurements. In this work, we develop the KFCS idea for causal reconstruction of medical image sequences from MR data. This is th ..."
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Cited by 10 (3 self)
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In recent work, Kalman Filtered Compressed Sensing (KFCS) was proposed to causally reconstruct time sequences of sparse signals, from a limited number of “incoherent ” measurements. In this work, we develop the KFCS idea for causal reconstruction of medical image sequences from MR data. This is the first real application of KFCS and is considerably more difficult than simulation data for a number of reasons, for example, the measurement matrix for MR is not as “incoherent ” and the images are only compressible (not sparse). Greatly improved reconstruction results (as compared to CS and its recent modifications) on reconstructing cardiac and brain image sequences from dynamic MR data are shown.
Efficient highdimensional inference in the multiple measurement vector problem.” arXiv:1111.5272 [cs.IT
, 2011
"... Abstract—In this work, a Bayesian approximate message passing algorithm is proposed for solving the multiple measurement vector (MMV) problem in compressive sensing, in which a collection of sparse signal vectors that share a common support are recovered from undersampled noisy measurements. The alg ..."
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Cited by 10 (3 self)
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Abstract—In this work, a Bayesian approximate message passing algorithm is proposed for solving the multiple measurement vector (MMV) problem in compressive sensing, in which a collection of sparse signal vectors that share a common support are recovered from undersampled noisy measurements. The algorithm, AMPMMV, is capable of exploiting temporal correlations in the amplitudes of nonzero coefficients, and provides soft estimates of the signal vectors as well as the underlying support. Central to the proposed approach is an extension of recently developed approximate message passing techniques to the amplitudecorrelated MMV setting. Aided by these techniques, AMPMMV offers a computational complexity that is linear in all problem dimensions. In order to allow for automatic parameter tuning, an expectationmaximization algorithm that complements AMPMMV is described. Finally, a detailed numerical study demonstrates the power of the proposed approach and its particular suitability for application to highdimensional problems. Index Terms—Approximate message passing (AMP), belief propagation, compressed sensing, expectationmaximization algorithms, joint sparsity, Kalman filters, multiple measurement vector problem, statistical signal processing. I.
Regularized Modified BPDN for Noisy Sparse Reconstruction with Partial Erroneous Support and Signal Value Knowledge
"... We study the problem of sparse reconstruction from noisy undersampled measurements when the following two things are available. (1) We are given partial, and partly erroneous, knowledge of the signal’s support, denoted by T. (2) We are also given an erroneous estimate of the signal values on T, deno ..."
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Cited by 9 (5 self)
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We study the problem of sparse reconstruction from noisy undersampled measurements when the following two things are available. (1) We are given partial, and partly erroneous, knowledge of the signal’s support, denoted by T. (2) We are also given an erroneous estimate of the signal values on T, denoted by(ˆµ)T. In practice, both of these may be available from available prior knowledge. Alternatively, in recursive reconstruction applications, like realtime dynamic MRI, one can use the support estimate and the signal value estimate from the previous time instant as T and (ˆµ)T. In this work, we introduce regularized modifiedBPDN (regmodBPDN) to solve this problem and obtain computable bounds on its reconstruction error. RegmodBPDN tries to find the signal that is sparsest outside the set T, while being “close enough ” to (ˆµ)T on T and while satisfying the data constraint. Corresponding results for modifiedBPDN and BPDN follow as direct corollaries. A second key contribution is an approach to obtain computable error bounds that hold without any sufficient conditions. This makes it easy to compare the bounds for the various approaches. Empirical reconstruction error comparisons with many existing approaches are also provided. Index Terms compressive sensing, sparse reconstruction, modifiedCS, partially known support