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Item Detection and estimation with compressive measurements(2006-11-01) Baraniuk, Richard G.; Davenport, Mark A.; Wakin, Michael B.Show more The recently introduced theory of compressed sensing enables the reconstruction 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 satisfactory measurement scheme. This has inspired the design of physical systems that directly implement similar measurement schemes. However, despite the intense focus on the reconstruction of signals, many (if not most) signal processing problems do not require a full reconstruction of the signal { we are often interested only in solving some sort of detection problem or in the estimation of some function of the data. In this report, we show that the compressed sensing framework is useful for a wide range of statistical inference tasks. In particular, we demonstrate how to solve a variety of signal detection and estimation problems given the measurements without ever reconstructing the signals themselves. We provide theoretical bounds along with experimental results.Show more Item Method and apparatus for compressive imaging device(2014-09-30) Baraniuk, Richard G.; Baron, Dror Z.; Duarte, Marco F.; Kelly, Kevin F.; Lane, Courtney C.; Laska, Jason N.; Takhar, Dharmpal; Wakin, Michael B.; Rice University; United States Patent and Trademark OfficeShow more A new digital image/video camera that directly acquires random projections of the incident light field without first collecting the pixels/voxels. In one preferred embodiment, the camera employs a digital micromirror array to perform optical calculations of linear projections of an image onto pseudorandom binary patterns. Its hallmarks include the ability to obtain an image with only a single detection element while measuring the image/video fewer times than the number of pixels or voxels—this can significantly reduce the computation required for image/video acquisition/encoding. Since the system features a single photon detector, it can also be adapted to image at wavelengths that are currently impossible with conventional CCD and CMOS imagers.Show more Item Method and apparatus for compressive imaging device(2012-06-12) Baraniuk, Richard G.; Baron, Dror Z.; Duarte, Marco F.; Kelly, Kevin F.; Lane, Courtney C.; Laska, Jason N.; Takhar, Dharmpal; Wakin, Michael B.; Rice University; United States Patent and Trademark OfficeShow more Show more Item Method and apparatus for distributed compressed sensing(2009-03-31) Baraniuk, Richard G.; Baron, Dror Z.; Duarte, Marco F.; Sarvotham, Shriram; Wakin, Michael B.; Davenport, Mark A.; Rice University; United States Patent and Trademark OfficeShow more A method for approximating a plurality of digital signals or images using compressed sensing. In a scheme where a common component xc of said plurality of digital signals or images an innovative component xi of each of said plurality of digital signals each are represented as a vector with m entries, the method comprises the steps of making a measurement yc, where yc comprises a vector with only ni entries, where ni is less than m, making a measurement yi for each of said correlated digital signals, where yi comprises a vector with only ni entries, where ni is less than m, and from each said innovation components yi, producing an approximate reconstruction of each m-vector xi using said common component yc and said innovative component yi.Show more Item Method and apparatus for distributed compressed sensing(2007-09-18) Baraniuk, Richard G.; Baron, Dror Z.; Duarte, Marco F.; Sarvotham, Shriram; Wakin, Michael B.; Davenport, Mark A.; Rice University; United States Patent and Trademark OfficeShow more Show more Item Method and apparatus for on-line compressed sensing(2014-04-01) Baraniuk, Richard G.; Baron, Dror Z.; Duarte, Marco F.; Elnozahi, Mohamed; Wakin, Michael B.; Davenport, Mark A.; Laska, Jason N.; Tropp, Joel A.; Massoud, Yehia; Kirolos, Sami; Ragheb, Tamer; Rice University; United States Patent and Trademark OfficeShow more A typical data acquisition system takes periodic samples of a signal, image, or other data, often at the so-called Nyquist/Shannon sampling rate of two times the data bandwidth in order to ensure that no information is lost. In applications involving wideband signals, the Nyquist/Shannon sampling rate is very high, even though the signals may have a simple underlying structure. Recent developments in mathematics and signal processing have uncovered a solution to this Nyquist/Shannon sampling rate bottleneck for signals that are sparse or compressible in some representation. We demonstrate and reduce to practice methods to extract information directly from an analog or digital signal based on altering our notion of sampling to replace uniform time samples with more general linear functionals. One embodiment of our invention is a low-rate analog-to-information converter that can replace the high-rate analog-to-digital converter in certain applications involving wideband signals. Another embodiment is an encoding scheme for wideband discrete-time signals that condenses their information content.Show more Item Method and apparatus for signal detection- classification and estimation from compressive measurements(2013-07-09) Baraniuk, Richard G.; Duarte, Marco F.; Davenport, Mark A.; Wakin, Michael B.; Rice University; United States Patent and Trademark OfficeShow more The recently introduced theory of Compressive Sensing (CS) enables a new method for signal recovery from incomplete information (a reduced set of “compressive” linear measurements), based on the assumption that the signal is sparse in some dictionary. Such compressive measurement schemes are desirable in practice for reducing the costs of signal acquisition, storage, and processing. However, the current CS framework considers only a certain task (signal recovery) and only in a certain model setting (sparsity). We show that compressive measurements are in fact information scalable, allowing one to answer a broad spectrum of questions about a signal when provided only with a reduced set of compressive measurements. These questions range from complete signal recovery at one extreme down to a simple binary detection decision at the other. (Questions in between include, for example, estimation and classification.) We provide techniques such as a “compressive matched filter” for answering several of these questions given the available measurements, often without needing to first reconstruct the signal. In many cases, these techniques can succeed with far fewer measurements than would be required for full signal recovery, and such techniques can also be computationally more efficient. Based on additional mathematical insight, we discuss information scalable algorithms in several model settings, including sparsity (as in CS), but also in parametric or manifold-based settings and in model-free settings for generic statements of detection, classification, and estimation problems.Show more Item Multiscale random projections for compressive classification(2007-09-01) Duarte, Marco F.; Davenport, Mark A.; Wakin, Michael B.; Laska, Jason N.; Takhar, Dharmpal; Kelly, Kevin F.; Baraniuk, Richard G.Show more We propose a framework for exploiting dimension-reducing random projections in detection and classification problems. Our approach is based on the generalized likelihood ratio test; in the case of image classification, it exploits the fact that a set of images of a fixed scene under varying articulation parameters forms a low-dimensional, nonlinear manifold. Exploiting recent results showing that random projections stably embed a smooth manifold in a lower-dimensional space, we develop the multiscale smashed filter as a compressive analog of the familiar matched filter classifier. In a practical target classification problem using a single-pixel camera that directly acquires compressive image projections, we achieve high classification rates using many fewer measurements than the dimensionality of the images.Show more Item A simple proof of the restricted isometry property for random matrices(2007-01-18) Baraniuk, Richard G.; Davenport, Mark A.; DeVore, Ronald A.; Wakin, Michael B.Show more We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finite-dimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsity-inducing basis.Show more Item The smashed filter for compressive classification and target recognition(2007-01-01) Davenport, Mark A.; Duarte, Marco F.; Wakin, Michael B.; Laska, Jason N.; Takhar, Dharmpal; Kelly, Kevin F.; Baraniuk, Richard G.Show more 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.Show more Item Sparse Signal Detection from Incoherent Projections(2006-05-01) Davenport, Mark A.; Wakin, Michael B.; Duarte, Marco F.; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)Show more 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 dection 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.Show more Item The geometry of low-dimensional signal models(2007) Wakin, Michael B.; Baraniuk, Richard G.Show more Models in signal processing often deal with some notion of structure or conciseness suggesting that a signal really has "few degrees of freedom" relative to its actual size. Examples include: bandlimited signals, images containing low-dimensional geometric features, or collections of signals observed from multiple viewpoints in a camera or sensor network. In many cases, such signals can be expressed as sparse linear combinations of elements from some dictionary---the sparsity of the representation directly reflects the conciseness of the model and permits efficient algorithms for signal processing. Sparsity also forms the core of the emerging theory of Compressed Sensing (CS), which states that a sparse signal can be recovered from a small number of random linear measurements. In other cases, however, sparse representations may not suffice to truly capture the underlying structure of a signal. Instead, the conciseness of the signal model may in fact dictate that the signal class forms a low-dimensional manifold as a subset of the high-dimensional ambient signal space. To date, the importance and utility of manifolds for signal processing has been acknowledged largely through a research effort into "learning" manifold structure from a collection of data points. While these methods have proved effective for certain tasks (such as classification and recognition), they also tend to be quite generic and fail to consider the geometric nuances of specific signal classes. The purpose of this thesis is to develop new methods and understanding for signal processing based on low-dimensional signal models, with a particular focus on the role of geometry. Our key contributions include (i) new models for low-dimensional signal structure, including local parametric models for piecewise smooth signals and joint sparsity models for signal collections; (ii) multiscale representations for piecewise smooth signals designed to accommodate efficient processing; (iii) insight and analysis into the geometry of low-dimensional signal models, including the non-differentiability of certain articulated image manifolds and the behavior of signal manifolds under random low-dimensional projections, and (iv) dimensionality reduction algorithms for image approximation and compression, distributed (multi-signal) CS, parameter estimation, manifold learning, and manifold-based CS.Show more