On Selection of Search Space Dimension in Compressive Sampling Matching Pursuit

Compressive Sampling Matching Pursuit (CoSaMP) is one of the popular greedy methods in the emerging field of Compressed Sensing (CS). In addition to the appealing empirical performance, CoSaMP has also splendid theoretical guarantees for convergence. In this paper, we propose a modification in CoSaMP to adaptively choose the dimension of search space in each iteration, using a threshold based approach. Using Monte Carlo simulations, we show that this modification improves the reconstruction capability of the CoSaMP algorithm in clean as well as noisy measurement cases. From empirical observations, we also propose an optimum value for the threshold to use in applications.

Compressive video sensors using multichannel imagers.

We explore the possibilities of obtaining compression in video through modified sampling strategies using multichannel imaging systems. The redundancies in video streams are exploited through compressive sampling schemes to achieve low power and low complexity video sensors. The sampling strategies as well as the associated reconstruction algorithms are discussed. These compressive sampling schemes could be implemented in the focal plane readout hardware resulting in drastic reduction in data bandwidth and computational complexity.

Compressive holography.

Compressive sampling enables signal reconstruction using less than one measurement per reconstructed signal value. Compressive measurement is particularly useful in generating multidimensional images from lower dimensional data. We demonstrate single frame 3D tomography from 2D holographic data.

Design of universal multicoset sampling patterns for compressed sensing of multiband sparse signals

Many problems in digital communications involve wideband radio signals. As the most recent example, the impressive advances in Cognitive Radio systems make even more necessary the development of sampling schemes for wideband radio signals with spectral holes. This is equivalent to considering a sparse multiband signal in the framework of Compressive Sampling theory. Starting from previous results on multicoset sampling and recent advances in compressive sampling, we analyze the matrix involved in the corresponding reconstruction equation and define a new method for the design of universal multicoset codes, that is, codes guaranteeing perfect reconstruction of the sparse multiband signal.

Multiple transmitter localization and communication footprint identification using sparse reconstruction techniques

This paper considers the problem of identifying the footprints of communication of multiple transmitters in a given geographical area. To do this, a number of sensors are deployed at arbitrary but known locations in the area, and their individual decisions regarding the presence or absence of the transmitters' signal are combined at a fusion center to reconstruct the spatial spectral usage map. One straightforward scheme to construct this map is to query each of the sensors and cluster the sensors that detect the primary's signal. However, using the fact that a typical transmitter footprint map is a sparse image, two novel compressive sensing based schemes are proposed, which require significantly fewer number of transmissions compared to the querying scheme. A key feature of the proposed schemes is that the measurement matrix is constructed from a pseudo-random binary phase shift applied to the decision of each sensor prior to transmission. The measurement matrix is thus a binary ensemble which satisfies the restricted isometry property. The number of measurements needed for accurate footprint reconstruction is determined using compressive sampling theory. The three schemes are compared through simulations in terms of a performance measure that quantifies the accuracy of the reconstructed spatial spectral usage map. It is found that the proposed sparse reconstruction technique-based schemes significantly outperform the round-robin scheme.

Modified Greedy Pursuits for Improving Sparse Recovery

Compressive Sensing (CS) theory combines the signal sampling and compression for sparse signals resulting in reduction in sampling rate. In recent years, many recovery algorithms have been proposed to reconstruct the signal efficiently. Subspace Pursuit and Compressive Sampling Matching Pursuit are some of the popular greedy methods. Also, Fusion of Algorithms for Compressed Sensing is a recently proposed method where several CS reconstruction algorithms participate and the final estimate of the underlying sparse signal is determined by fusing the estimates obtained from the participating algorithms. All these methods involve solving a least squares problem which may be ill-conditioned, especially in the low dimension measurement regime. In this paper, we propose a step prior to least squares to ensure the well-conditioning of the least squares problem. Using Monte Carlo simulations, we show that in low dimension measurement scenario, this modification improves the reconstruction capability of the algorithm in clean as well as noisy measurement cases.

Compressed sensing based cyclic feature spectrum sensing for cognitive radios

Spectrum sensing is currently one of the most challenging design problems in cognitive radio. A robust spectrum sensing technique is important in allowing implementation of a practical dynamic spectrum access in noisy and interference uncertain environments. In addition, it is desired to minimize the sensing time, while meeting the stringent cognitive radio application requirements. To cope with this challenge, cyclic spectrum sensing techniques have been proposed. However, such techniques require very high sampling rates in the wideband regime and thus are costly in hardware implementation and power consumption. In this thesis the concept of compressed sensing is applied to circumvent this problem by utilizing the sparsity of the two-dimensional cyclic spectrum. Compressive sampling is used to reduce the sampling rate and a recovery method is developed for re- constructing the sparse cyclic spectrum from the compressed samples. The reconstruction solution used, exploits the sparsity structure in the two-dimensional cyclic spectrum do-main which is different from conventional compressed sensing techniques for vector-form sparse signals. The entire wideband cyclic spectrum is reconstructed from sub-Nyquist-rate samples for simultaneous detection of multiple signal sources. After the cyclic spectrum recovery two methods are proposed to make spectral occupancy decisions from the recovered cyclic spectrum: a band-by-band multi-cycle detector which works for all modulation schemes, and a fast and simple thresholding method that works for Binary Phase Shift Keying (BPSK) signals only. In addition a method for recovering the power spectrum of stationary signals is developed as a special case. Simulation results demonstrate that the proposed spectrum sensing algorithms can significantly reduce sampling rate without sacrifcing performance. The robustness of the algorithms to the noise uncertainty of the wireless channel is also shown.

Temporal Coding of Volumetric Imagery

'Image volumes' refer to realizations of images in other dimensions such as time, spectrum, and focus. Recent advances in scientific, medical, and consumer applications demand improvements in image volume capture. Though image volume acquisition continues to advance, it maintains the same sampling mechanisms that have been used for decades; every voxel must be scanned and is presumed independent of its neighbors. Under these conditions, improving performance comes at the cost of increased system complexity, data rates, and power consumption.

This dissertation explores systems and methods capable of efficiently improving sensitivity and performance for image volume cameras, and specifically proposes several sampling strategies that utilize temporal coding to improve imaging system performance and enhance our awareness for a variety of dynamic applications.

Video cameras and camcorders sample the video volume (x,y,t) at fixed intervals to gain understanding of the volume's temporal evolution. Conventionally, one must reduce the spatial resolution to increase the framerate of such cameras. Using temporal coding via physical translation of an optical element known as a coded aperture, the compressive temporal imaging (CACTI) camera emonstrates a method which which to embed the temporal dimension of the video volume into spatial (x,y) measurements, thereby greatly improving temporal resolution with minimal loss of spatial resolution. This technique, which is among a family of compressive sampling strategies developed at Duke University, temporally codes the exposure readout functions at the pixel level.

Since video cameras nominally integrate the remaining image volume dimensions (e.g. spectrum and focus) at capture time, spectral (x,y,t,\lambda) and focal (x,y,t,z) image volumes are traditionally captured via sequential changes to the spectral and focal state of the system, respectively. The CACTI camera's ability to embed video volumes into images leads to exploration of other information within that video; namely, focal and spectral information. The next part of the thesis demonstrates derivative works of CACTI: compressive extended depth of field and compressive spectral-temporal imaging. These works successfully show the technique's extension of temporal coding to improve sensing performance in these other dimensions.

Geometrical optics-related tradeoffs, such as the classic challenges of wide-field-of-view and high resolution photography, have motivated the development of mulitscale camera arrays. The advent of such designs less than a decade ago heralds a new era of research- and engineering-related challenges. One significant challenge is that of managing the focal volume (x,y,z) over wide fields of view and resolutions. The fourth chapter shows advances on focus and image quality assessment for a class of multiscale gigapixel cameras developed at Duke.

Along the same line of work, we have explored methods for dynamic and adaptive addressing of focus via point spread function engineering. We demonstrate another form of temporal coding in the form of physical translation of the image plane from its nominal focal position. We demonstrate this technique's capability to generate arbitrary point spread functions.

Coding Strategies for X-ray Tomography

This work focuses on the construction and application of coded apertures to compressive X-ray tomography. Coded apertures can be made in a number of ways, each method having an impact on system background and signal contrast. Methods of constructing coded apertures for structuring X-ray illumination and scatter are compared and analyzed. Apertures can create structured X-ray bundles that investigate specific sets of object voxels. The tailored bundles of rays form a code (or pattern) and are later estimated through computational inversion. Structured illumination can be used to subsample object voxels and make inversion feasible for low dose computed tomography (CT) systems, or it can be used to reduce background in limited angle CT systems.

On the detection side, coded apertures modulate X-ray scatter signals to determine the position and radiance of scatter points. By forming object dependent projections in measurement space, coded apertures multiplex modulated scatter signals onto a detector. The multiplexed signals can be inverted with knowledge of the code pattern and system geometry. This work shows two systems capable of determining object position and type in a 2D plane, by illuminating objects with an X-ray `fan beam,' using coded apertures and compressive measurements. Scatter tomography can help identify materials in security and medicine that may be ambiguous with transmission tomography alone.

DESIGN OF EFFICIENT IN-NETWORK DATA PROCESSING AND DISSEMINATION FOR VANETS

By providing vehicle-to-vehicle and vehicle-to-infrastructure wireless communications, vehicular ad hoc networks (VANETs), also known as the “networks on wheels”, can greatly enhance traffic safety, traffic efficiency and driving experience for intelligent transportation system (ITS). However, the unique features of VANETs, such as high mobility and uneven distribution of vehicular nodes, impose critical challenges of high efficiency and reliability for the implementation of VANETs. This dissertation is motivated by the great application potentials of VANETs in the design of efficient in-network data processing and dissemination. Considering the significance of message aggregation, data dissemination and data collection, this dissertation research targets at enhancing the traffic safety and traffic efficiency, as well as developing novel commercial applications, based on VANETs, following four aspects: 1) accurate and efficient message aggregation to detect on-road safety relevant events, 2) reliable data dissemination to reliably notify remote vehicles, 3) efficient and reliable spatial data collection from vehicular sensors, and 4) novel promising applications to exploit the commercial potentials of VANETs. Specifically, to enable cooperative detection of safety relevant events on the roads, the structure-less message aggregation (SLMA) scheme is proposed to improve communication efficiency and message accuracy. The scheme of relative position based message dissemination (RPB-MD) is proposed to reliably and efficiently disseminate messages to all intended vehicles in the zone-of-relevance in varying traffic density. Due to numerous vehicular sensor data available based on VANETs, the scheme of compressive sampling based data collection (CS-DC) is proposed to efficiently collect the spatial relevance data in a large scale, especially in the dense traffic. In addition, with novel and efficient solutions proposed for the application specific issues of data dissemination and data collection, several appealing value-added applications for VANETs are developed to exploit the commercial potentials of VANETs, namely general purpose automatic survey (GPAS), VANET-based ambient ad dissemination (VAAD) and VANET based vehicle performance monitoring and analysis (VehicleView). Thus, by improving the efficiency and reliability in in-network data processing and dissemination, including message aggregation, data dissemination and data collection, together with the development of novel promising applications, this dissertation will help push VANETs further to the stage of massive deployment.

Non-compressive wideband spectrum sensing with sub-Nyquist sampling rates

In this paper, we propose a low complexity and reliable wideband spectrum sensing technique that operates at sub-Nyquist sampling rates. Unlike the majority of other sub-Nyquist spectrum sensing algorithms that rely on the Compressive Sensing (CS) methodology, the introduced method does not entail solving an optimisation problem. It is characterised by simplicity and low computational complexity without compromising the system performance and yet delivers substantial reductions on the operational sampling rates. The reliability guidelines of the devised non-compressive sensing approach are provided and simulations are presented to illustrate its superior performance. © 2013 IEEE.

Nested Sampling and its Applications in Stable Compressive Covariance Estimation and Phase Retrieval with Near-Minimal Measurements

Compressed covariance sensing using quadratic samplers is gaining increasing interest in recent literature. Covariance matrix often plays the role of a sufficient statistic in many signal and information processing tasks. However, owing to the large dimension of the data, it may become necessary to obtain a compressed sketch of the high dimensional covariance matrix to reduce the associated storage and communication costs. Nested sampling has been proposed in the past as an efficient sub-Nyquist sampling strategy that enables perfect reconstruction of the autocorrelation sequence of Wide-Sense Stationary (WSS) signals, as though it was sampled at the Nyquist rate. The key idea behind nested sampling is to exploit properties of the difference set that naturally arises in quadratic measurement model associated with covariance compression. In this thesis, we will focus on developing novel versions of nested sampling for low rank Toeplitz covariance estimation, and phase retrieval, where the latter problem finds many applications in high resolution optical imaging, X-ray crystallography and molecular imaging. The problem of low rank compressive Toeplitz covariance estimation is first shown to be fundamentally related to that of line spectrum recovery. In absence if noise, this connection can be exploited to develop a particular kind of sampler called the Generalized Nested Sampler (GNS), that can achieve optimal compression rates. In presence of bounded noise, we develop a regularization-free algorithm that provably leads to stable recovery of the high dimensional Toeplitz matrix from its order-wise minimal sketch acquired using a GNS. Contrary to existing TV-norm and nuclear norm based reconstruction algorithms, our technique does not use any tuning parameters, which can be of great practical value. The idea of nested sampling idea also finds a surprising use in the problem of phase retrieval, which has been of great interest in recent times for its convex formulation via PhaseLift, By using another modified version of nested sampling, namely the Partial Nested Fourier Sampler (PNFS), we show that with probability one, it is possible to achieve a certain conjectured lower bound on the necessary measurement size. Moreover, for sparse data, an l1 minimization based algorithm is proposed that can lead to stable phase retrieval using order-wise minimal number of measurements.

Compressive sensing for music signals: comparison of transforms with coherent dictionaries

Compressive Sensing (CS) is a new sensing paradigm which permits sampling of a signal at its intrinsic information rate which could be much lower than Nyquist rate, while guaranteeing good quality reconstruction for signals sparse in a linear transform domain. We explore the application of CS formulation to music signals. Since music signals comprise of both tonal and transient nature, we examine several transforms such as discrete cosine transform (DCT), discrete wavelet transform (DWT), Fourier basis and also non-orthogonal warped transforms to explore the effectiveness of CS theory and the reconstruction algorithms. We show that for a given sparsity level, DCT, overcomplete, and warped Fourier dictionaries result in better reconstruction, and warped Fourier dictionary gives perceptually better reconstruction. “MUSHRA” test results show that a moderate quality reconstruction is possible with about half the Nyquist sampling.

Event-triggered Sampling and Reconstruction of Sparse Real-valued Trigonometric Polynomials

We propose data acquisition from continuous-time signals belonging to the class of real-valued trigonometric polynomials using an event-triggered sampling paradigm. The sampling schemes proposed are: level crossing (LC), close to extrema LC, and extrema sampling. Analysis of robustness of these schemes to jitter, and bandpass additive gaussian noise is presented. In general these sampling schemes will result in non-uniformly spaced sample instants. We address the issue of signal reconstruction from the acquired data-set by imposing structure of sparsity on the signal model to circumvent the problem of gap and density constraints. The recovery performance is contrasted amongst the various schemes and with random sampling scheme. In the proposed approach, both sampling and reconstruction are non-linear operations, and in contrast to random sampling methodologies proposed in compressive sensing these techniques may be implemented in practice with low-power circuitry.

New directions in sparse sampling and estimation for underdetermined systems

A central objective in signal processing is to infer meaningful information from a set of measurements or data. While most signal models have an overdetermined structure (the number of unknowns less than the number of equations), traditionally very few statistical estimation problems have considered a data model which is underdetermined (number of unknowns more than the number of equations). However, in recent times, an explosion of theoretical and computational methods have been developed primarily to study underdetermined systems by imposing sparsity on the unknown variables. This is motivated by the observation that inspite of the huge volume of data that arises in sensor networks, genomics, imaging, particle physics, web search etc., their information content is often much smaller compared to the number of raw measurements. This has given rise to the possibility of reducing the number of measurements by down sampling the data, which automatically gives rise to underdetermined systems.

In this thesis, we provide new directions for estimation in an underdetermined system, both for a class of parameter estimation problems and also for the problem of sparse recovery in compressive sensing. There are two main contributions of the thesis: design of new sampling and statistical estimation algorithms for array processing, and development of improved guarantees for sparse reconstruction by introducing a statistical framework to the recovery problem.

We consider underdetermined observation models in array processing where the number of unknown sources simultaneously received by the array can be considerably larger than the number of physical sensors. We study new sparse spatial sampling schemes (array geometries) as well as propose new recovery algorithms that can exploit priors on the unknown signals and unambiguously identify all the sources. The proposed sampling structure is generic enough to be extended to multiple dimensions as well as to exploit different kinds of priors in the model such as correlation, higher order moments, etc.

Recognizing the role of correlation priors and suitable sampling schemes for underdetermined estimation in array processing, we introduce a correlation aware framework for recovering sparse support in compressive sensing. We show that it is possible to strictly increase the size of the recoverable sparse support using this framework provided the measurement matrix is suitably designed. The proposed nested and coprime arrays are shown to be appropriate candidates in this regard. We also provide new guarantees for convex and greedy formulations of the support recovery problem and demonstrate that it is possible to strictly improve upon existing guarantees.

This new paradigm of underdetermined estimation that explicitly establishes the fundamental interplay between sampling, statistical priors and the underlying sparsity, leads to exciting future research directions in a variety of application areas, and also gives rise to new questions that can lead to stand-alone theoretical results in their own right.