6 research outputs found
Structured Compressed Sensing: From Theory to Applications
Compressed sensing (CS) is an emerging field that has attracted considerable
research interest over the past few years. Previous review articles in CS limit
their scope to standard discrete-to-discrete measurement architectures using
matrices of randomized nature and signal models based on standard sparsity. In
recent years, CS has worked its way into several new application areas. This,
in turn, necessitates a fresh look on many of the basics of CS. The random
matrix measurement operator must be replaced by more structured sensing
architectures that correspond to the characteristics of feasible acquisition
hardware. The standard sparsity prior has to be extended to include a much
richer class of signals and to encode broader data models, including
continuous-time signals. In our overview, the theme is exploiting signal and
measurement structure in compressive sensing. The prime focus is bridging
theory and practice; that is, to pinpoint the potential of structured CS
strategies to emerge from the math to the hardware. Our summary highlights new
directions as well as relations to more traditional CS, with the hope of
serving both as a review to practitioners wanting to join this emerging field,
and as a reference for researchers that attempts to put some of the existing
ideas in perspective of practical applications.Comment: To appear as an overview paper in IEEE Transactions on Signal
Processin
Measurement Bounds for Sparse Signal Ensembles via Graphical Models
In compressive sensing, a small collection of linear projections of a sparse
signal contains enough information to permit signal recovery. Distributed
compressive sensing (DCS) extends this framework by defining ensemble sparsity
models, allowing a correlated ensemble of sparse signals to be jointly
recovered from a collection of separately acquired compressive measurements. In
this paper, we introduce a framework for modeling sparse signal ensembles that
quantifies the intra- and inter-signal dependencies within and among the
signals. This framework is based on a novel bipartite graph representation that
links the sparse signal coefficients with the measurements obtained for each
signal. Using our framework, we provide fundamental bounds on the number of
noiseless measurements that each sensor must collect to ensure that the signals
are jointly recoverable.Comment: 11 pages, 2 figure
Model-Based Compressive Sensing
Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for
the acquisition of sparse or compressible signals that can be well approximated
by just K << N elements from an N-dimensional basis. Instead of taking periodic
samples, CS measures inner products with M < N random vectors and then recovers
the signal via a sparsity-seeking optimization or greedy algorithm. Standard CS
dictates that robust signal recovery is possible from M = O(K log(N/K))
measurements. It is possible to substantially decrease M without sacrificing
robustness by leveraging more realistic signal models that go beyond simple
sparsity and compressibility by including structural dependencies between the
values and locations of the signal coefficients. This paper introduces a
model-based CS theory that parallels the conventional theory and provides
concrete guidelines on how to create model-based recovery algorithms with
provable performance guarantees. A highlight is the introduction of a new class
of structured compressible signals along with a new sufficient condition for
robust structured compressible signal recovery that we dub the restricted
amplification property, which is the natural counterpart to the restricted
isometry property of conventional CS. Two examples integrate two relevant
signal models - wavelet trees and block sparsity - into two state-of-the-art CS
recovery algorithms and prove that they offer robust recovery from just M=O(K)
measurements. Extensive numerical simulations demonstrate the validity and
applicability of our new theory and algorithms.Comment: 20 pages, 10 figures. Typo corrected in grant number. To appear in
IEEE Transactions on Information Theor
Signal recovery in compressed sensing via universal priors. arXiv:1204.2611
Abstract We study the compressed sensing (CS) signal estimation problem where an input is measured via a linear matrix multiplication under additive noise. While this setup usually assumes sparsity or compressibility in the observed signal during recovery, the signal structure that can be leveraged is often not known a priori. In this paper, we consider universal CS recovery, where the statistics of a stationary ergodic signal source are estimated simultaneously with the signal itself. We focus on a maximum a posteriori (MAP) estimation framework that leverages universal priors such as Kolmogorov complexity and minimum description length. We provide theoretical results that support the algorithmic feasibility of universal MAP estimation through a Markov Chain Monte Carlo implementation. We also include simulation results that showcase the promise of universality in CS, particularly for low-complexity sources that do not exhibit standard sparsity or compressibility
Joint manifolds for data fusion
Abstract—The emergence of low-cost sensing architectures for diverse modalities has made it possible to deploy sensor networks that capture a single event from a large number of vantage points and using multiple modalities. In many scenarios, these networks acquire large amounts of very high-dimensional data. For example, even a relatively small network of cameras can generate massive amounts of high-dimensional image and video data. One way to cope with this data deluge is to exploit low-dimensional data models. Manifold models provide a particularly powerful theoretical and algorithmic framework for capturing the structure of data governed by a small number of parameters, as is often the case in a sensor network. However, these models do not typically take into account dependencies among multiple sensors. We thus propose a new joint manifold framework for data ensembles that exploits such dependencies. We show that joint manifold structure can lead to improved performance for a variety of signal processing algorithms for applications including classification and manifold learning. Additionally, recent results concerning random projections of manifolds enable us to formulate a scalable and universal dimensionality reduction scheme that efficiently fuses the data from all sensors. Index Terms—Camera networks, classification, data fusion, manifold learning, random projections, sensor networks