Results 11  20
of
296
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
Lower Bounds for Sparse Recovery
"... We consider the following ksparse recovery problem: design an m × n matrix A, such that for any signal x, given Ax we can efficiently recover ˆx satisfying ..."
Abstract

Cited by 60 (23 self)
 Add to MetaCart
We consider the following ksparse recovery problem: design an m × n matrix A, such that for any signal x, given Ax we can efficiently recover ˆx satisfying
Sparse recovery using sparse random matrices
, 2008
"... We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a highdimensional vector x from its lowerdimensional sketch Ax. A popular way of performing this recovery is by finding x # such that Ax = Ax # , and �x # �1 is minimal. It is known that this approach ..."
Abstract

Cited by 59 (4 self)
 Add to MetaCart
We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a highdimensional vector x from its lowerdimensional sketch Ax. A popular way of performing this recovery is by finding x # such that Ax = Ax # , and �x # �1 is minimal. It is known that this approach “works” if A is a random dense matrix, chosen from a proper distribution. In this paper, we investigate this procedure for the case where A is binary and very sparse. We show that, both in theory and in practice, sparse matrices are essentially as “good” as the dense ones. At the same time, sparse binary matrices provide additional benefits, such as reduced encoding and decoding time.
Democracy in Action: Quantization, Saturation, and Compressive Sensing
"... Recent theoretical developments in the area of compressive sensing (CS) have the potential to significantly extend the capabilities of digital data acquisition systems such as analogtodigital converters and digital imagers in certain applications. A key hallmark of CS is that it enables subNyquis ..."
Abstract

Cited by 59 (22 self)
 Add to MetaCart
Recent theoretical developments in the area of compressive sensing (CS) have the potential to significantly extend the capabilities of digital data acquisition systems such as analogtodigital converters and digital imagers in certain applications. A key hallmark of CS is that it enables subNyquist sampling for signals, images, and other data. In this paper, we explore and exploit another heretofore relatively unexplored hallmark, the fact that certain CS measurement systems are democractic, which means that each measurement carries roughly the same amount of information about the signal being acquired. Using the democracy property, we rethink how to quantize the compressive measurements in practical CS systems. If we were to apply the conventional wisdom gained from conventional ShannonNyquist uniform sampling, then we would scale down the analog signal amplitude (and therefore increase the quantization error) to avoid the gross saturation errors that occur when the signal amplitude exceeds the quantizer’s dynamic range. In stark contrast, we demonstrate that a CS system achieves the best performance when it operates at a significantly nonzero saturation rate. We develop two methods to recover signals from saturated CS measurements. The first directly exploits the democracy property by simply discarding the saturated measurements. The second integrates saturated measurements as constraints into standard linear programming and greedy recovery techniques. Finally, we develop a simple automatic gain control system that uses the saturation rate to optimize the input gain.
Compressed sensing: how sharp is the restricted isometry property?
, 2009
"... Compressed sensing is a recent technique by which signals can be measured at a rate proportional to their information content, combining the important task of compression directly into the measurement process. Since its introduction in 2004 there have been hundreds of manuscripts on compressed sens ..."
Abstract

Cited by 51 (7 self)
 Add to MetaCart
(Show Context)
Compressed sensing is a recent technique by which signals can be measured at a rate proportional to their information content, combining the important task of compression directly into the measurement process. Since its introduction in 2004 there have been hundreds of manuscripts on compressed sensing, a large fraction of which have focused on the design and analysis of algorithms to recover a signal from its compressed measurements. The Restricted Isometry Property (RIP) has become a ubiquitous property assumed in their analysis. We present the best known bounds on the RIP, and in the process illustrate the way in which the combinatorial nature of compressed sensing is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners.
Stable image reconstruction using total variation minimization
 SIAM Journal on Imaging Sciences
, 2013
"... This article presents nearoptimal guarantees for accurate and robust image recovery from undersampled noisy measurements using total variation minimization, and our results may be the first of this kind. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be ..."
Abstract

Cited by 50 (2 self)
 Add to MetaCart
(Show Context)
This article presents nearoptimal guarantees for accurate and robust image recovery from undersampled noisy measurements using total variation minimization, and our results may be the first of this kind. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be reconstructed to within the best sterm approximation of its gradient, up to a logarithmic factor. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of a suitably incoherent matrix. 1
Exact Signal Recovery from Sparsely Corrupted Measurements through the Pursuit of Justice
"... Abstract—Compressive sensing provides a framework for recovering sparse signals of length N from M ≪ N measurements. If the measurements contain noise bounded by ɛ, then standard algorithms recover sparse signals with error at most Cɛ. However, these algorithms perform suboptimally when the measurem ..."
Abstract

Cited by 40 (2 self)
 Add to MetaCart
(Show Context)
Abstract—Compressive sensing provides a framework for recovering sparse signals of length N from M ≪ N measurements. If the measurements contain noise bounded by ɛ, then standard algorithms recover sparse signals with error at most Cɛ. However, these algorithms perform suboptimally when the measurement noise is also sparse. This can occur in practice due to shot noise, malfunctioning hardware, transmission errors, or narrowband interference. We demonstrate that a simple algorithm, which we dub Justice Pursuit (JP), can achieve exact recovery from measurements corrupted with sparse noise. The algorithm handles unbounded errors, has no input parameters, and is easily implemented via standard recovery techniques. I.
P2c2: Programmable pixel compressive camera for high speed imaging
 Computer Vision and Pattern Recognition, IEEE Computer Society Conference on
, 2011
"... We describe an imaging architecture for compressive video sensing termed programmable pixel compressive camera (P2C2). P2C2 allows us to capture fast phenomena at frame rates higher than the camera sensor. In P2C2, each pixel has an independent shutter that is modulated at a rate higher than the cam ..."
Abstract

Cited by 37 (6 self)
 Add to MetaCart
(Show Context)
We describe an imaging architecture for compressive video sensing termed programmable pixel compressive camera (P2C2). P2C2 allows us to capture fast phenomena at frame rates higher than the camera sensor. In P2C2, each pixel has an independent shutter that is modulated at a rate higher than the camera framerate. The observed intensity at a pixel is an integration of the incoming light modulated by its specific shutter. We propose a reconstruction algorithm that uses the data from P2C2 along with additional priors about videos to perform temporal superresolution. We model the spatial redundancy of videos using sparse representations and the temporal redundancy using brightness constancy constraints inferred via optical flow. We show that by modeling such spatiotemporal redundancies in a video volume, one can faithfully recover the underlying highspeed video frames from the observed low speed coded video. The imaging architecture and the reconstruction algorithm allows us to achieve temporal superresolution without loss in spatial resolution. We implement a prototype of P2C2 using an LCOS modulator and recover several videos at 200 fps using a 25 fps camera. 1.
Compressive Acquisition of Dynamic Scenes
"... Compressive sensing (CS) is a new approach for the acquisition and recovery of sparse signals and images that enables 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 acquis ..."
Abstract

Cited by 37 (10 self)
 Add to MetaCart
(Show Context)
Compressive sensing (CS) is a new approach for the acquisition and recovery of sparse signals and images that enables 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 infeasible. 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, from which the image frames are then reconstructed. 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 dynamic part of the scene at each instant and accumulates measurements over time to estimate the static parameters. This enables us to considerably lower the compressive measurement rate considerably. We validate our approach with a range of experiments including classification experiments that highlight the effectiveness of the proposed approach.
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