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32
ModifiedCS: Modifying compressive sensing for problems with partially known support
 in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2009
"... Abstract—We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known, although the known part may contain some errors. The “known ” part of the support, denoted, may be available from prior knowledge. Alternatively, in a ..."
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Cited by 126 (33 self)
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Abstract—We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known, although the known part may contain some errors. The “known ” part of the support, denoted, may be available from prior knowledge. Alternatively, in a problem of recursively reconstructing time sequences of sparse spatial signals, one may use the support estimate from the previous time instant as the “known ” part. The idea of our proposed solution (modifiedCS) is to solve a convex relaxation of the following problem: find the signal that satisfies the data constraint and is sparsest outside of. We obtain sufficient conditions for exact reconstruction using modifiedCS. These are much weaker than those needed for compressive sensing (CS) when the sizes of the unknown part of the support and of errors in the known part are small compared to the support size. An important extension called regularized modifiedCS (RegModCS) is developed which also uses prior signal estimate knowledge. Simulation comparisons for both sparse and compressible signals are shown. Index Terms—Compressive sensing, modifiedCS, partially known support, prior knowledge, sparse reconstruction.
Recursive sparse recovery in large but correlated noise
 in Proc. 49th Allerton Conf. Commun. Control Comput
, 2011
"... Abstract—In this work, we focus on the problem of recursively recovering a time sequence of sparse signals, with timevarying sparsity patterns, from highly undersampled measurements corrupted by very large but correlated noise. It is assumed that the noise is correlated enough to have an approxima ..."
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Cited by 20 (13 self)
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Abstract—In this work, we focus on the problem of recursively recovering a time sequence of sparse signals, with timevarying sparsity patterns, from highly undersampled measurements corrupted by very large but correlated noise. It is assumed that the noise is correlated enough to have an approximately low rank covariance matrix that is either constant, or changes slowly, with time. We show how our recently introduced Recursive Projected CS (ReProCS) and modifiedReProCS ideas can be used to solve this problem very effectively. To the best of our knowledge, except for the recent work of dense error correction via ℓ1 minimization, which can handle another kind of large but “structured ” noise (the noise needs to be sparse), none of the other works in sparse recovery have studied the case of any other kind of large noise. 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
Reprocs: A missing link between recursive robust pca and recursive sparse recovery in large but correlated noise
 CoRR
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Modifiedcsresidual for recursive reconstruction of highly undersampled functional mri sequences
 in IEEE Intl. Conf. Image Proc. (ICIP
, 2011
"... In this work, we study the application of compressive sensing (CS) based approaches for blood oxygenation level dependent (BOLD) contrast functional MR imaging (fMRI). In particular, we show, via exhaustive experiments on actual MR scanner data for brain fMRI, that our recently proposed approach for ..."
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Cited by 8 (5 self)
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In this work, we study the application of compressive sensing (CS) based approaches for blood oxygenation level dependent (BOLD) contrast functional MR imaging (fMRI). In particular, we show, via exhaustive experiments on actual MR scanner data for brain fMRI, that our recently proposed approach for recursive reconstruction of sparse signal sequences, modifiedCSresidual, outperforms other existing CS based approaches. ModifiedCSresidual exploits the fact that the sparsity pattern of brain fMRI sequences and their signal values change slowly over time. It provides a fast, yet accurate, reconstruction approach that is able to accurately track the changes of the active pixels, while using only about 30 % measurements per frame. Significantly improved performance over existing work is shown in terms of practically relevant metrics such as active pixel time courses, activation maps and receiver operating characteristic (ROC) curves. Index Terms — Compressive Sensing, Functional MRI 1.
Exact Reconstruction Conditions for Regularized Modified Basis Pursuit
, 2010
"... is a continuous and unimodal function of 2, with the unique maximum 2 2 (p+1) 2 (p) achieved at = ( ) , see also (9a). We conclude that ( ) 0 ( 2) (p+1) must go to zero. The second claim of Theorem 1 follows. ..."
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Cited by 4 (2 self)
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is a continuous and unimodal function of 2, with the unique maximum 2 2 (p+1) 2 (p) achieved at = ( ) , see also (9a). We conclude that ( ) 0 ( 2) (p+1) must go to zero. The second claim of Theorem 1 follows.
Exploiting Correlation in Sparse Signal Recovery Problems: Multiple Measurement Vectors, Block Sparsity, and TimeVarying
"... A trend in compressed sensing (CS) is to exploit structure for improved reconstruction performance. In the basic CS model (i.e. the single measurement vector model), exploiting the clustering structure among ..."
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Cited by 4 (3 self)
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A trend in compressed sensing (CS) is to exploit structure for improved reconstruction performance. In the basic CS model (i.e. the single measurement vector model), exploiting the clustering structure among
Message passing approaches to compressive inference under structured signal priors
, 2013
"... Across numerous disciplines, the ability to generate highdimensional datasets is driving an enormous demand for increasingly efficient ways of both capturing and processing this data. A promising recent trend for addressing these needs has developed from the recognition that, despite living in hi ..."
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Cited by 3 (3 self)
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Across numerous disciplines, the ability to generate highdimensional datasets is driving an enormous demand for increasingly efficient ways of both capturing and processing this data. A promising recent trend for addressing these needs has developed from the recognition that, despite living in highdimensional ambient spaces, many datasets have vastly smaller intrinsic dimensionality. When capturing (sampling) such datasets, exploiting this realization permits one to dramatically reduce the number of samples that must be acquired without losing the salient features of the data. When processing such datasets, the reduced intrinsic dimensionality can be leveraged to allow reliable inferences to be made in scenarios where it is infeasible to collect the amount of data that would be required for inference using classical techniques. To date, most approaches for taking advantage of the low intrinsic dimensionality inherent in many datasets have focused on identifying succinct (i.e., sparse) representations of the data, seeking to represent the data using only a handful of “significant ” elements from an appropriately chosen dictionary. While powerful in
FROGS: A Serial Reversible Greedy Search Algorithm
"... Abstract—For compressed sensing, in the framework of greedy search reconstruction algorithms, we introduce the notion of initial supportset. The initial supportset is an estimate given to a reconstruction algorithm to improve the performance of the reconstruction. Furthermore, we classify existing ..."
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Cited by 3 (3 self)
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Abstract—For compressed sensing, in the framework of greedy search reconstruction algorithms, we introduce the notion of initial supportset. The initial supportset is an estimate given to a reconstruction algorithm to improve the performance of the reconstruction. Furthermore, we classify existing greedy search algorithms as being serial or parallel. Based on this classification and the goal of robustness to errors in the initial supportsets we develop a new greedy search algorithm called FROGS. We end the paper with careful numerical experiments concluding that FROGS perform well compared to existing algorithms (both in terms of performance and execution time) and that it is robust against errors in the initial supportset. Index Terms—Compressed sensing, greedy search, greedy pursuit, initial support. I.
TRACKING SPARSE SIGNAL SEQUENCES FROM NONLINEAR/NONGAUSSIAN MEASUREMENTS AND APPLICATIONS IN ILLUMINATIONMOTION TRACKING
"... In this work, we develop algorithms for tracking time sequences of sparse spatial signals with slowly changing sparsity patterns, and other unknown states, from a sequence of nonlinear observations corrupted by (possibly) nonGaussian noise. A key example of the above problem occurs in tracking movi ..."
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Cited by 1 (1 self)
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In this work, we develop algorithms for tracking time sequences of sparse spatial signals with slowly changing sparsity patterns, and other unknown states, from a sequence of nonlinear observations corrupted by (possibly) nonGaussian noise. A key example of the above problem occurs in tracking moving objects across spatially varying illumination changes, where motion is the small dimensional state while the illumination image is the sparse spatial signal satisfying the slowsparsitypatternchange property. Index Terms — particle filtering, compressed sensing, tracking 1.