Mark A. Davenport

Single-Pixel Imaging via Compressive Sampling

In this article, the authors present a new approach to building simpler, smaller, and cheaper digital cameras that can operate efficiently across a broader spectral range than conventional silicon-based cameras. The approach fuses a new camera architecture based on a digital micromirror device with the new mathematical theory and algorithms of compressive sampling.

Signal Processing With Compressive Measurements

The recently introduced theory of compressive sensing enables the recovery of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist-rate samples. Interestingly, it has been shown that random projections are a near-optimal measurement scheme. This has inspired the design of hardware systems that directly implement random measurement protocols. However, despite the intense focus of the community on signal recovery, many (if not most) signal processing problems do not require full signal recovery. In this paper, we take some first steps in the direction of solving inference problems-such as detection, classification, or estimation-and filtering problems using only compressive measurements and without ever reconstructing the signals involved. We provide theoretical bounds along with experimental results.

Analysis of Orthogonal Matching Pursuit Using the Restricted Isometry Property

Orthogonal matching pursuit (OMP) is the canonical greedy algorithm for sparse approximation. In this paper we demonstrate that the restricted isometry property (RIP) can be used for a very straightforward analysis of OMP. Our main conclusion is that the RIP of order K+1 (with isometry constant δ <; [ 1/( 3√K)]) is sufficient for OMP to exactly recover any K-sparse signal. The analysis relies on simple and intuitive observations about OMP and matrices which satisfy the RIP. For restricted classes of K-sparse signals (those that are highly compressible), a relaxed bound on the isometry constant is also established. A deeper understanding of OMP may benefit the analysis of greedy algorithms in general. To demonstrate this, we also briefly revisit the analysis of the regularized OMP (ROMP) algorithm.

Compressed Sensing: Introduction to compressed sensing

Compressed sensing (CS) is an exciting, rapidly growing, field that has attracted considerable attention in signal processing, statistics, and computer science, as well as the broader scientific community. Since its initial development only a few years ago, thousands of papers have appeared in this area, and hundreds of conferences, workshops, and special sessions have been dedicated to this growing research field. In this chapter, we provide an up-to-date review of the basics of the theory underlying CS. This chapter should serve as a review to practitioners wanting to join this emerging field, and as a reference for researchers. We focus primarily on the theory and algorithms for sparse recovery in finite dimensions. In subsequent chapters of the book, we will see how the fundamentals presented in this chapter are expanded and extended in many exciting directions, including new models for describing structure in both analog and discrete-time signals, new sensing design techniques, more advanced recovery results and powerful new recovery algorithms, and emerging applications of the basic theory and its extensions. Introduction We are in the midst of a digital revolution that is driving the development and deployment of new kinds of sensing systems with ever-increasing fidelity and resolution. The theoretical foundation of this revolution is the pioneering work of Kotelnikov, Nyquist, Shannon, and Whittaker on sampling continuous-time bandlimited signals [162, 195, 209, 247]. Their results demonstrate that signals, images, videos, and other data can be exactly recovered from a set of uniformly spaced samples taken at the so-called Nyquist rate of twice the highest frequency present in the signal of interest.

Sparse Signal Detection from Incoherent Projections

The recently introduced theory of compressed sensing (CS) enables the reconstruction or approximation of sparse or compressible signals from a small set of incoherent projections; often the number of projections can be much smaller than the number of Nyquist rate samples. In this paper, we show that the CS framework is information scalable to a wide range of statistical inference tasks. In particular, we demonstrate how CS principles can solve signal detection problems given incoherent measurements without ever reconstructing the signals involved. We specifically study the case of signal detection in strong inference and noise and propose an incoherent detection and estimation algorithm (IDEA) based on matching pursuit. The number of measurements and computations necessary for successful detection using IDEA is significantly lower than that necessary for successful reconstruction. Simulations show that IDEA is very resilient to strong interference, additive noise, and measurement quantization. When combined with random measurements, IDEA is applicable to a wide range of different signal classes

The smashed filter for compressive classification and target recognition

The theory of compressive sensing (CS) enables the reconstruction of a sparse or compressible image or signal from a small set of linear, non-adaptive (even random) projections. However, in many applications, including object and target recognition, we are ultimately interested in making a decision about an image rather than computing a reconstruction. We propose here a framework for compressive classification that operates directly on the compressive measurements without first reconstructing the image. We dub the resulting dimensionally reduced matched filter the smashed filter. The first part of the theory maps traditional maximum likelihood hypothesis testing into the compressive domain; we find that the number of measurements required for a given classification performance level does not depend on the sparsity or compressibility of the images but only on the noise level. The second part of the theory applies the generalized maximum likelihood method to deal with unknown transformations such as the translation, scale, or viewing angle of a target object. We exploit the fact the set of transformed images forms a low-dimensional, nonlinear manifold in the high-dimensional image space. We find that the number of measurements required for a given classification performance level grows linearly in the dimensionality of the manifold but only logarithmically in the number of pixels/samples and image classes. Using both simulations and measurements from a new single-pixel compressive camera, we demonstrate the effectiveness of the smashed filter for target classification using very few measurements.

The Pros and Cons of Compressive Sensing for Wideband Signal Acquisition: Noise Folding versus Dynamic Range

Compressive sensing (CS) exploits the sparsity present in many signals to reduce the number of measurements needed for digital acquisition. With this reduction would come, in theory, commensurate reductions in the size, weight, power consumption, and/or monetary cost of both signal sensors and any associated communication links. This paper examines the use of CS in the design of a wideband radio receiver in a noisy environment. We formulate the problem statement for such a receiver and establish a reasonable set of requirements that a receiver should meet to be practically useful. We then evaluate the performance of a CS-based receiver in two ways: via a theoretical analysis of its expected performance, with a particular emphasis on noise and dynamic range, and via simulations that compare the CS receiver against the performance expected from a conventional implementation. On the one hand, we show that CS-based systems that aim to reduce the number of acquired measurements are somewhat sensitive to signal noise, exhibiting a 3 dB SNR loss per octave of subsampling, which parallels the classic noise-folding phenomenon. On the other hand, we demonstrate that since they sample at a lower rate, CS-based systems can potentially attain a significantly larger dynamic range. Hence, we conclude that while a CS-based system has inherent limitations that do impose some restrictions on its potential applications, it also has attributes that make it highly desirable in a number of important practical settings.

Exact signal recovery from sparsely corrupted measurements through the Pursuit of Justice

Compressive sensing provides a framework for recovering sparse signals of length N from M ≪ N measurements. If the measurements contain noise bounded by ∈, then standard algorithms recover sparse signals with error at most C∈. 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.

On the Fundamental Limits of Adaptive Sensing

Suppose we can sequentially acquire arbitrary linear measurements of an n -dimensional vector x resulting in the linear model y = A x + z, where z represents measurement noise. If the signal is known to be sparse, one would expect the following folk theorem to be true: choosing an adaptive strategy which cleverly selects the next row of A based on what has been previously observed should do far better than a nonadaptive strategy which sets the rows of A ahead of time, thus not trying to learn anything about the signal in between observations. This paper shows that the folk theorem is false. We prove that the advantages offered by clever adaptive strategies and sophisticated estimation procedures-no matter how intractable-over classical compressed acquisition/recovery schemes are, in general, minimal.

Sparsity and Structure in Hyperspectral Imaging : Sensing, Reconstruction, and Target Detection

Hyperspectral imaging is a powerful technology for remotely inferring the material properties of the objects in a scene of interest. Hyperspectral images consist of spatial maps of light intensity variation across a large number of spectral bands or wavelengths; alternatively, they can be thought of as a measurement of the spectrum of light transmitted or reflected from each spatial location in a scene. Because chemical elements have unique spectral signatures, observing the spectra at a high spatial and spectral resolution provides information about the material properties of the scene with much more accuracy than is possible with conventional three-color images. As a result, hyperspectral imaging is used in a variety of important applications, including remote sensing, astronomical imaging, and fluorescence microscopy.