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41
Compressed Sensing: Theory and Applications
, 2012
"... Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that highdimensional signals, which allow a sparse representati ..."
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Cited by 120 (30 self)
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Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that highdimensional signals, which allow a sparse representation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements by using efficient algorithms. This article shall serve as an introduction to and a survey about compressed sensing. Key Words. Dimension reduction. Frames. Greedy algorithms. Illposed inverse problems. `1 minimization. Random matrices. Sparse approximation. Sparse recovery.
CHiLasso: A collaborative hierarchical sparse modeling framework
, 2010
"... Sparse modeling is a powerful framework for data analysis and processing. Traditionally, encoding in this framework is performed by solving an ℓ1regularized linear regression problem, commonly referred to as Lasso or basis pursuit. In this work we combine the sparsityinducing property of the Lasso ..."
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Cited by 36 (6 self)
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Sparse modeling is a powerful framework for data analysis and processing. Traditionally, encoding in this framework is performed by solving an ℓ1regularized linear regression problem, commonly referred to as Lasso or basis pursuit. In this work we combine the sparsityinducing property of the Lasso model at the individual feature level, with the blocksparsity property of the Group Lasso model, where sparse groups of features are jointly encoded, obtaining a sparsity pattern hierarchically structured. This results in the Hierarchical Lasso (HiLasso), which shows important practical modeling advantages. We then extend this approach to the collaborative case, where a set of simultaneously coded signals share the same sparsity pattern at the higher (group) level, but not necessarily at the lower (inside the group) level, obtaining the collaborative HiLasso model (CHiLasso). Such signals then share the same active groups, or classes, but not necessarily the same active set. This model is very well suited for applications such as source identification and separation. An efficient optimization procedure, which guarantees convergence to the global optimum, is developed for these new models. The underlying presentation of the new framework and optimization approach is complemented with experimental examples and theoretical results regarding recovery guarantees for the proposed models.
Sparse fusion frames: Existence and construction,
 Adv. Comput. Math.
, 2011
"... Abstract. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame can be regarded as a framelike collection of subspaces in a Hilbert space, and thereby generalizes the concept of a frame for signal repres ..."
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Cited by 23 (12 self)
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Abstract. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame can be regarded as a framelike collection of subspaces in a Hilbert space, and thereby generalizes the concept of a frame for signal representation. However, when the signal and/or subspace dimensions are large, the decomposition of the signal into its fusion frame measurements through subspace projections typically requires a large number of additions and multiplications, and this makes the decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a 'uniform basis' over all subspaces, thereby enabling lowcomplexity fusion frame decompositions. We study the existence and construction of sparse fusion frames, but our focus is on developing simple algorithmic constructions that can easily be adopted in practice to produce sparse fusion frames with desired (given) operators. By a desired (or given) operator we simply mean one that has a desired (or given) set of eigenvalues for the fusion frame operator. We start by presenting a complete characterization of Parseval fusion frames in terms of the existence of special isometries defined on an encompassing Hilbert space. We then introduce two general methodologies to generate new fusion frames from existing ones, namely the Spatial Complement Method and the Naimark Complement Method, and analyze the relationship between the parameters of the original and the new fusion frame. We proceed by establishing existence conditions for 2sparse fusion frames for any given fusion frame operator, for which the eigenvalues are greater than or equal to two. We then provide an easily implementable algorithm for computing such 2sparse fusion frames.
BlockSparse Recovery via Convex Optimization
, 2012
"... Given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks, we study the problem of finding a blocksparse representation of the signal, i.e., a representation that uses the minimum number of blocks. Motivated by signal/image processing and co ..."
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Cited by 19 (1 self)
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Given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks, we study the problem of finding a blocksparse representation of the signal, i.e., a representation that uses the minimum number of blocks. Motivated by signal/image processing and computer vision applications, such as face recognition, we consider the blocksparse recovery problem in the case where the number of atoms in each block is arbitrary, possibly much larger than the dimension of the underlying subspace. To find a blocksparse representation of a signal, we propose two classes of nonconvex optimization programs, which aim to minimize the number of nonzero coefficient blocks and the number of nonzero reconstructed vectors from the blocks, respectively. Since both classes of problems are NPhard, we propose convex relaxations and derive conditions under which each class of the convex programs is equivalent to the original nonconvex formulation. Our conditions depend on the notions of mutual and cumulative subspace coherence of a dictionary, which are natural generalizations of existing notions of mutual and cumulative coherence. We evaluate the performance of the proposed convex programs through simulations as well as real experiments on face recognition. We show that treating the face recognition problem as a blocksparse recovery problem improves the stateoftheart results by 10 % with only 25 % of the training data.
STREAMING COMPRESSIVE SENSING FOR HIGHSPEED PERIODIC VIDEOS
"... The ability of Compressive Sensing (CS) to recover sparse signals from limited measurements has been recently exploited in computational imaging to acquire highspeed periodic and nearperiodic videos using only a lowspeed camera with coded exposure and intensive offline processing. Each lowspeed ..."
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Cited by 11 (5 self)
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The ability of Compressive Sensing (CS) to recover sparse signals from limited measurements has been recently exploited in computational imaging to acquire highspeed periodic and nearperiodic videos using only a lowspeed camera with coded exposure and intensive offline processing. Each lowspeed frame integrates a coded sequence of highspeed frames during its exposure time. The highspeed video can be reconstructed from the lowspeed coded frames using a sparse recovery algorithm. This paper presents a new streaming CS algorithm specifically tailored to this application. Our streaming approach allows causal online acquisition and reconstruction of the video, with a small, controllable, and guaranteed buffer delay and low computational cost. The algorithm adapts to changes in the signal structure and, thus, outperforms the offline algorithm in realistic signals. 1.
PERTURBATIONS OF MEASUREMENT MATRICES AND DICTIONARIES IN COMPRESSED SENSING
"... Abstract. The compressed sensing problem for redundant dictionaries aims to use a small number of linear measurements to represent signals that are sparse with respect to a general dictionary. Under an appropriate restricted isometry property for a dictionary, reconstruction methods based on ℓ q min ..."
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Cited by 7 (2 self)
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Abstract. The compressed sensing problem for redundant dictionaries aims to use a small number of linear measurements to represent signals that are sparse with respect to a general dictionary. Under an appropriate restricted isometry property for a dictionary, reconstruction methods based on ℓ q minimization are known to provide an effective signal recovery tool in this setting. This note explores conditions under which ℓ q minimization is robust to measurement noise, and stable with respect to perturbations of the sensing matrix A and the dictionary D. We propose a new condition, the D null space property, which guarantees that ℓ q minimization produces solutions that are robust and stable against perturbations of A and D. We also show that ℓ q minimization is jointly stable with respect to imprecise knowledge of the measurement matrix A and the dictionary D when A satisfies the restricted isometry property. 1.
Random fusion frames are nearly equiangular and tight,” Linear algebra and its applications
, 2013
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A Review of Subspace Segmentation: Problem, Nonlinear Approximations, and Applications to Motion Segmentation
, 2013
"... The subspace segmentation problem is fundamental in many applications. The goal is to cluster data drawn from an unknown union of subspaces. In this paper we state the problem and describe its connection to other areas of mathematics and engineering. We then review the mathematical and algorithmic m ..."
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Cited by 4 (1 self)
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The subspace segmentation problem is fundamental in many applications. The goal is to cluster data drawn from an unknown union of subspaces. In this paper we state the problem and describe its connection to other areas of mathematics and engineering. We then review the mathematical and algorithmic methods created to solve this problem and some of its particular cases. We also describe the problem of motion tracking in videos and its connection to the subspace segmentation problem and compare the various techniques for solving it.