Results 1  10
of
111
Compressive Sensing and Structured Random Matrices
 RADON SERIES COMP. APPL. MATH XX, 1–95 © DE GRUYTER 20YY
, 2011
"... These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to ..."
Abstract

Cited by 157 (18 self)
 Add to MetaCart
These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using ℓ1minimization.
Sensitivity to basis mismatch of compressed sensing,” preprint
, 2009
"... Abstract—The theory of compressed sensing suggests that successful inversion of an image of the physical world (e.g., a radar/sonar return or a sensor array snapshot vector) for the source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the c ..."
Abstract

Cited by 86 (8 self)
 Add to MetaCart
Abstract—The theory of compressed sensing suggests that successful inversion of an image of the physical world (e.g., a radar/sonar return or a sensor array snapshot vector) for the source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in passive and active radar and sonar, this basis is usually taken to be a DFT basis. The compressed sensing measurements are then inverted using an ℓ1minimization principle (basis pursuit) for the nonzero source amplitudes. This seems to make compressed sensing an ideal image inversion principle for high resolution modal analysis. However, in reality no physical field is sparse in the DFT basis or in an apriori known basis. In fact the main goal in image inversion is to identify the modal structure. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and there is always mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of basis pursuit to mismatch between the assumed and the actual sparsity bases and compare the performance of basis pursuit with that of classical image inversion. Our mathematical analysis and numerical examples show that the performance of basis pursuit degrades considerably in the presence of mismatch, and they suggest that the use of compressed sensing as a modal analysis principle requires more consideration and refinement, at least for the problem sizes common to radar/sonar. I.
Learning to Sense Sparse Signals: Simultaneous Sensing Matrix and Sparsifying Dictionary Optimization
, 2008
"... Abstract Sparse signals representation, analysis, and sensing, has received a lot of attention in recent years from the signal processing, optimization, and learning communities. On one hand, the learning of overcomplete dictionaries that facilitate a sparse representation of the image as a liner c ..."
Abstract

Cited by 69 (5 self)
 Add to MetaCart
(Show Context)
Abstract Sparse signals representation, analysis, and sensing, has received a lot of attention in recent years from the signal processing, optimization, and learning communities. On one hand, the learning of overcomplete dictionaries that facilitate a sparse representation of the image as a liner combination of a few atoms from such dictionary, leads to stateoftheart results in image and video restoration and image classification. On the other hand, the framework of compressed sensing (CS) has shown that sparse signals can be recovered from far less samples than those required by the classical ShannonNyquist Theorem. The goal of this paper is to present a framework that unifies the learning of overcomplete dictionaries for sparse image representation with the concepts of signal recovery from very few samples put forward by the CS theory. The samples used in CS correspond to linear projections defined by a sampling projection matrix. It has been shown that, for example, a nonadaptive random sampling matrix satisfies the fundamental theoretical requirements of CS, enjoying the additional benefit of universality. On the other hand, a projection sensing matrix that is optimally designed for a certain signal class can further improve the reconstruction accuracy or further reduce the necessary number of samples. In this work we introduce a framework for the joint design and optimization, from a set of training images, of the
Compressive Sensing
, 2010
"... Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many poten ..."
Abstract

Cited by 50 (12 self)
 Add to MetaCart
Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.
On the Interplay Between Routing and Signal Representation for Compressive Sensing in Wireless Sensor Networks
, 2009
"... Compressive Sensing (CS) shows high promise for fully distributed compression in wireless sensor networks (WSNs). In theory, CS allows the approximation of the readings from a sensor field with excellent accuracy, while collecting only a small fraction of them at a data gathering point. However, th ..."
Abstract

Cited by 42 (6 self)
 Add to MetaCart
Compressive Sensing (CS) shows high promise for fully distributed compression in wireless sensor networks (WSNs). In theory, CS allows the approximation of the readings from a sensor field with excellent accuracy, while collecting only a small fraction of them at a data gathering point. However, the conditions under which CS performs well are not necessarily met in practice. CS requires a suitable transformation that makes the signal sparse in its domain. Also, the transformation of the data given by the routing protocol and network topology and the sparse representation of the signal have to be incoherent, which is not straightforward to achieve in real networks. In this work we address the data gathering problem in WSNs, where routing is used in conjunction with CS to transport random projections of the data. We analyze synthetic and real data sets and compare the results against those of random sampling. In doing so, we consider a number of popular transformations and we find that, with real data sets, none of them are able to sparsify the data while being at the same time incoherent with respect to the routing matrix. The obtained performance is thus not as good as expected and finding a suitable transformation with good sparsification and incoherence properties remains an open problem for data gathering in static WSNs.
Compressive Imaging using Approximate Message Passing and a Markovtree Prior
 PROC. ASILOMAR CONF. ON SIGNALS, SYSTEMS, AND COMPUTERS
, 2010
"... We propose a novel algorithm for compressive imaging that exploits both the sparsity and persistence across scales found in the 2D wavelet transform coefficients of natural images. Like other recent works, we model wavelet structure using a hidden Markov tree (HMT) but, unlike other works, ours is b ..."
Abstract

Cited by 41 (6 self)
 Add to MetaCart
We propose a novel algorithm for compressive imaging that exploits both the sparsity and persistence across scales found in the 2D wavelet transform coefficients of natural images. Like other recent works, we model wavelet structure using a hidden Markov tree (HMT) but, unlike other works, ours is based on loopy belief propagation (LBP). For LBP, we adopt a recently proposed “turbo ” message passing schedule that alternates between exploitation of HMT structure and exploitation of compressivemeasurement structure. For the latter, we leverage Donoho, Maleki, and Montanari’s recently proposed approximate message passing (AMP) algorithm. Experiments on a large image database show that our turbo LBP approach maintains stateoftheart reconstruction performance at half the complexity.
DISTRIBUTED COMPRESSIVE VIDEO SENSING
"... Lowcomplexity video encoding has been applicable to several emerging applications. Recently, distributed video coding (DVC) has been proposed to reduce encoding complexity to the order of that for still image encoding. In addition, compressive sensing (CS) has been applicable to directly capture co ..."
Abstract

Cited by 35 (4 self)
 Add to MetaCart
(Show Context)
Lowcomplexity video encoding has been applicable to several emerging applications. Recently, distributed video coding (DVC) has been proposed to reduce encoding complexity to the order of that for still image encoding. In addition, compressive sensing (CS) has been applicable to directly capture compressed image data efficiently. In this paper, by integrating the respective characteristics of DVC and CS, a distributed compressive video sensing (DCVS) framework is proposed to simultaneously capture and compress video data, where almost all computation burdens can be shifted to the decoder, resulting in a very lowcomplexity encoder. At the decoder, compressed video can be efficiently reconstructed using the modified GPSR (gradient projection for sparse reconstruction) algorithm. With the assistance of the proposed initialization and stopping criteria for GRSR, derived from statistical dependencies among successive video frames, our modified GPSR algorithm can terminate faster and reconstruct better video quality. The performance of our DCVS method is demonstrated via simulations to outperform three known CS reconstruction algorithms. Index Terms—compressive video sensing, (distributed) compressive sampling/sensing, distributed video coding
Various thresholds for ℓ1optimization in compressed sensing
, 2009
"... Recently, [14, 28] theoretically analyzed the success of a polynomial ℓ1optimization algorithm in solving an underdetermined system of linear equations. In a large dimensional and statistical context [14, 28] proved that if the number of equations (measurements in the compressed sensing terminolog ..."
Abstract

Cited by 33 (17 self)
 Add to MetaCart
Recently, [14, 28] theoretically analyzed the success of a polynomial ℓ1optimization algorithm in solving an underdetermined system of linear equations. In a large dimensional and statistical context [14, 28] proved that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of nonzero elements of the unknown vector) also proportional to the length of the unknown vector such that ℓ1optimization succeeds in solving the system. In this paper, we provide an alternative performance analysis of ℓ1optimization and obtain the proportionality constants that in certain cases match or improve on the best currently known ones from [28, 29].
Prasanna “Compressedsensingenabled Video Streaming for Wireless Multimedia Sensor Networks
 IEEE Transactions on Mobile Computing
"... Abstract—This paper presents the design of a networked system for joint compression, rate control and error correction of video over resourceconstrained embedded devices based on the theory of Compressed Sensing (CS). The objective of this work is to design a crosslayer system that jointly control ..."
Abstract

Cited by 24 (7 self)
 Add to MetaCart
(Show Context)
Abstract—This paper presents the design of a networked system for joint compression, rate control and error correction of video over resourceconstrained embedded devices based on the theory of Compressed Sensing (CS). The objective of this work is to design a crosslayer system that jointly controls the video encoding rate, the transmission rate, and the channel coding rate to maximize the received video quality. First, compressed sensingbased video encoding for transmission over Wireless Multimedia Sensor Networks (WMSNs) is studied. It is shown that compressed sensing can overcome many of the current problems of video over WMSNs, primarily encoder complexity and low resiliency to channel errors. A rate controller is then developed with the objective of maintaining fairness among different videos while maximizing the received video quality. It is shown that the rate of Compressed Sensed Video (CSV) can be predictably controlled by varying only the compressed sensing sampling rate. It is then shown that the developed rate controller can be interpreted as the iterative solution to a convex optimization problem representing the optimization of the rate allocation across the network. The error resiliency properties of compressed sensed images and videos are then studied, and an optimal error detection and correction scheme is presented for video transmission over lossy channels. Finally, the entire system is evaluated through simulation and test bed evaluation. The rate controller is shown to outperform existing TCPfriendly rate control schemes in terms of both fairness and received video quality. The test bed results show that the rates converge to stable values in real channels. Index Terms—Compressed sensing, optimization, multimedia content, congestion control, sensor networks. Ç