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Nearly LinearTime ModelBased Compressive Sensing
"... Compressive sensing is a method for recording a ksparse signal x ∈ Rn with (possibly noisy) linear measurements of the form y = Ax, where A ∈ Rm×n describes the measurement process. Seminal results in compressive sensing show that it is possible to recover the signal x from m = O(k log n k) measur ..."
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Compressive sensing is a method for recording a ksparse signal x ∈ Rn with (possibly noisy) linear measurements of the form y = Ax, where A ∈ Rm×n describes the measurement process. Seminal results in compressive sensing show that it is possible to recover the signal x from m = O(k log n k) measurements and that this is tight. The modelbased compressive sensing framework overcomes this lower bound and reduces the number of measurements further to m = O(k). This improvement is achieved by limiting the supports of x to a structured sparsity model, which is a subset of all
A NearlyLinear Time Framework for GraphStructured Sparsity
"... We introduce a framework for sparsity structures defined via graphs. Our approach is flexible and generalizes several previously studied sparsity models. Moreover, we provide efficient projection algorithms for our sparsity model that run in nearlylinear time. In the context of sparse recovery, ..."
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We introduce a framework for sparsity structures defined via graphs. Our approach is flexible and generalizes several previously studied sparsity models. Moreover, we provide efficient projection algorithms for our sparsity model that run in nearlylinear time. In the context of sparse recovery, we show that our framework achieves an informationtheoretically optimal sample complexity for a wide range of parameters. We complement our theoretical analysis with experiments demonstrating that our algorithms also improve on prior work in practice. 1.
Fast Algorithms for Structured Sparsity (ICALP 2015 Invited Tutorial)
"... Sparsity has become an important tool in many mathematical sciences such as statistics, machine learning, and signal processing. While sparsity is a good model for data in many applications, data often has additional structure that goes beyond the notion of “standard ” sparsity. In many cases, we ca ..."
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Sparsity has become an important tool in many mathematical sciences such as statistics, machine learning, and signal processing. While sparsity is a good model for data in many applications, data often has additional structure that goes beyond the notion of “standard ” sparsity. In many cases, we can represent this additional information in a structured sparsity model. Recent research has shown that structured sparsity can improve the sample complexity in several applications such as compressive sensing and sparse linear regression. However, these improvements come at a computational cost, as the data needs to be “fitted ” so it satisfies the constraints specified by the sparsity model. In this survey, we introduce the concept of structured sparsity, explain the relevant algorithmic challenges, and briefly describe the best known algorithms for two sparsity models. On the way, we demonstrate that structured sparsity models are inherently combinatorial structures, and employing structured sparsity often leads to interesting algorithmic problems with strong connections to combinatorial optimization and discrete algorithms. We also state several algorithmic open problems related to structured sparsity. 1
CODED APERTURE COMPRESSIVE 3D LIDAR
"... Continuous improvement in optical sensing components, as well as recent advances in signal acquisition theory provide a great opportunity to reduce the cost and enhance the capabilities of depth sensing systems. In this paper we propose a new depth sensing architecture that exploits a fixed coded ..."
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Continuous improvement in optical sensing components, as well as recent advances in signal acquisition theory provide a great opportunity to reduce the cost and enhance the capabilities of depth sensing systems. In this paper we propose a new depth sensing architecture that exploits a fixed coded aperture to significantly reduce the number of sensors compared to conventional systems. We further develop a modeling and reconstruction framework, based on modelbased compressed sensing, which characterizes a large variety of depth sensing systems. Our experiments demonstrate that it is possible to reduce the number of sensors by more than 85%, with negligible reduction on the sensing quality. Index Terms — 3D imaging, LIDAR, time of flight, compressed sensing, computational imaging.