We focus on a multidimensional field with uncorrelated spectrum and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise. More specifically, we apply linear reconstruction techniques and take the mean-square error (MSE) of the field estimate as a metric to evaluate the signal reconstruction quality. We find that the MSE analysis could be carried out by using the closed-form expression of the eigenvalue distribution of the matrix representing the sampling system. Unfortunately, such distribution is still unknown. Thus, we first derive a closed-form expression of the distribution moments, and we find that the eigenvalue distribution tends to the Marcenko-Pastur distribution as the field dimension goes to infinity. Finally, by using our approach, we derive a tight approximation to the MSE of the reconstructed field.
In the context of sensor networks, efficient techniques for spatial sampling are proposed in @cite_28 @cite_30 . In particular, in @cite_30 , an adaptive sampling is described, which allows the central data-collector to vary the number of active sensors, i.e., samples, according to the desired resolution level. Data acquisition is also studied in @cite_36 , where the authors consider a unidimensional field, uniformly sampled at the Nyquist frequency by low precision sensors. The authors show that the number of samples can be traded-off with the precision of sensors. The problem of the reconstruction of a bandlimited signal from an irregular set of samples at unknown locations is addressed in @cite_5 . There, different solution methods are proposed, and the conditions for which there exist multiple solutions or a unique solution are discussed. Differently from @cite_5 , we assume that the sink can either acquire or estimate the sensor locations and that sensors are randomly deployed.
@cite_28 @cite_5 @cite_36 @cite_30
2133489575 2122735658 2131052843 2114129195 2159479457 2571527823 2061158652 2953338282 2105483458 2053334136 2964120646 2108634960 2098662489 1977252496 2147276092 1597674521 2576230203 2158643468 2163973643 2774751177
Distributed sampling and reconstruction of a physical field using an array of sensors is a problem of considerable interest in environmental monitoring applications of sensor networks. Our recent work has focused on the sampling of bandlimited sensor fields. However, sensor fields are not perfectly bandlimited but typically have rapidly decaying spectra. In a classical sampling set-up it is possible to precede the A D sampling operation with an appropriate analog anti-aliasing filter. However, in the case of sensor networks, this is infeasible since sampling must precede filtering. We show that even though the effects of aliasing on the reconstruction cannot be prevented due to the "filter-less" sampling constraint, they can be suitably controlled by oversampling and carefully reconstructing the field from the samples. We show using a dither-based scheme that it is possible to estimate non-bandlimited fields with a precision that depends on how fast the spectral content of the field decays. We develop a framework for analyzing non-bandlimited fields that lead to upper bounds on the maximum pointwise error for a spatial bit rate of R bits meter. We present results for fields with exponentially decaying spectra as an illustration. In particular, we show that for fields f(t) with exponential tails; i.e., F( spl omega ) < spl pi spl alpha sup - spl alpha | spl omega | , the maximum pointwise error decays as c2e sup -a sub 1 spl radic R+ c3 1 spl radic R (e sup -2a sub 1 spl radic R) with spatial bit rate R bits meter. Finally, we show that for fields with spectra that have a finite second moment, the distortion decreases as O((1 N) sup 2 3 ) as the density of sensors, N, scales up to infinity . We show that if D is the targeted non-zero distortion, then the required (finite) rate R scales as O (1 spl radic D log 1 D). A constructive solution of the irregular sampling problem for band- limited functions is given. We show how a band-limited function can be com- pletely reconstructed from any random sampling set whose density is higher than the Nyquist rate, and give precise estimates for the speed of convergence of this iteration method. Variations of this algorithm allow for irregular sampling with derivatives, reconstruction of band-limited functions from local averages, and irregular sampling of multivariate band-limited functions. In the irregular sampling problem one is asked whether and how a band- limited function can be completely reconstructed from its irregularly sam- pled values f(xi). This has many applications in signal and image processing, seismology, meteorology, medical imaging, etc. Finding constructive solutions of this problem has received considerable attention among mathematicians and engineers. The mathematical literature provides several uniqueness results (1, 2, 17, 18, 19). It is now part of the folklore that for stable sampling the sampling rate must be at least the Nyquist rate (18). These results, as deep as they are, have had little impact for the applied sciences, because they were not constructive. If the sampling set is just a perturbation of the regular oversampling, then a reconstruction method has been obtained in a seminal paper by Duffin and Schaeffer (6) (see also (29)): if for some L > 0, a > 0, and o > 0 the sampling points xk , k e Z , satisfy (a) - ok a, k ^ I, then the norm equivalence A iR (x) ) with w < n o. This norm equivalence implies that it is possible to reconstruct through an iterative procedure, the so-called frame method. Most of the later work on constructive methods consists of variations of this method (3, 21, 22, 26). The above conditions on the sampling set exclude random irregular sampling sets, e.g., sets with regions of higher sampling density. A partial, but undesirable remedy, to handle highly irregular sampling sets, would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration. This paper investigates data-adaptive path planning schemes for wireless networks of mobile sensor platforms. We focus on applications of environmental monitoring, in which the goal is to reconstruct a spatial map of environmental factors of interest. Traditional sampling theory deals with data collection processes that are completely independent of the target map to be estimated, aside from possible a priori specifications reflective of assumed properties of the target. We refer to such processes as passive learning methods. Alternatively, one can envision sequential, adaptive data collection procedures that use information gleaned from previous observations to guide the process. We refer to such feedback-driven processes as active learning methods. Active learning is naturally suited to mobile path planning, in which previous samples are used to guide the motion of the mobiles for further sampling. This paper presents some of the most encouraging theoretical results to date that support the effectiveness of active over passive learning, and focuses on new results regarding the capabilities of active learning methods for mobile sensing. Tradeoffs between latency, path lengths, and accuracy are carefully assessed using our theory. Adaptive path planning methods are developed to guide mobiles in order to focus attention in interesting regions of the sensing domain, thus conducting spatial surveys much more rapidly while maintaining the accuracy of the estimated map. The theory and methods are illustrated in the application of water current mapping in a freshwater lake. We address the problem of reconstructing a multiband signal from its sub-Nyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multi-coset sampling. To date, recovery methods for this sampling strategy ensure perfect reconstruction either when the band locations are known, or under strict restrictions on the possible spectral supports. In this paper, only the number of bands and their widths are assumed without any other limitations on the support. We describe how to choose the parameters of the multi-coset sampling so that a unique multiband signal matches the given samples. To recover the signal, the continuous reconstruction is replaced by a single finite-dimensional problem without the need for discretization. The resulting problem is studied within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate, and provides a first systematic study of compressed sensing in a truly analog setting. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate. While high resolution, regularly gridded observations are generally preferred in remote sensing, actual observations are often not evenly sampled and have lower-than-desired resolution. Hence, there is an interest in resolution enhancement and image reconstruction. This paper discusses a general theory and techniques for image reconstruction and creating enhanced resolution images from irregularly sampled data. Using irregular sampling theory, we consider how the frequency content in aperture function-attenuated sidelobes can be recovered from oversampled data using reconstruction techniques, thus taking advantage of the high frequency content of measurements made with nonideal aperture filters. We show that with minor modification, the algebraic reconstruction technique (ART) is functionally equivalent to Grochenig's (1992) irregular sampling reconstruction algorithm. Using simple Monte Carlo simulations, we compare and contrast the performance of additive ART, multiplicative ART, and the scatterometer image reconstruction (SIR) (a derivative of multiplicative ART) algorithms with and without noise. The reconstruction theory and techniques have applications with a variety of sensors and can enable enhanced resolution image production from many nonimaging sensors. The technique is illustrated with ERS-2 and SeaWinds scatterometer data. We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala [30], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of nonzero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate δ exceeds the (upper) Renyi information dimension of the signal, d(pX). More precisely, for a sequence of signals of diverging dimension n whose empirical distribution converges to pX, reconstruction is with high probability successful from d(pX) n+o(n) measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension n and k(n) nonzero entries, this implies reconstruction from k(n)+o(n) measurements. For “discrete” signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from o(n) measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal pX. We study the problem of optimal sub-Nyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support spl Fscr that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau's (1967) lower bound equal to the measure of spl Fscr . For signals with sparse spl Fscr , this rate can be much smaller than the Nyquist rate. Unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In a previous study, we derived bounds on the error due to mismodeling and sample additive noise. Adopting these bounds as performance measures, we consider the problems of optimizing the reconstruction sections of the system, choosing the optimal base sampling rate, and designing the nonuniform sampling pattern. We find that optimizing these parameters can improve system performance significantly. Furthermore, uniform sampling is optimal for signals with spl Fscr that tiles under translation. For signals with nontiling spl Fscr , which are not amenable to efficient uniform sampling, the results reveal increased error sensitivities with sub-Nyquist sampling. However, these can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate. We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math . In this paper, we consider the problem of sampling signals that are nonband-limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation. Streams of Diracs and piecewise polynomials are the examples of such signals, and thus are known as signals with finite rate of innovation (FRI). We know that the classical ("band-limited sine") sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band-limited. However, the recent results on FRI sampling suggest that it is possible to sample and perfectly reconstruct such nonband-limited signals using a rich class of kernels. In this paper, we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions). In fact, the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples. Using these moments and the annihilating filter method (Prony's method), the innovative part of the signal, and therefore, the signal itself is perfectly reconstructed. In particular, we present local (directional-derivatives-based) and global (complex-moments-based, Radon-transform-based) sampling schemes for classes of FRI signals such as sets of Diracs, bilevel, and planar polygons, quadrature domains (e.g., circles, ellipses, and cardioids), 2D polynomials with polygonal boundaries, and n-dimensional Diracs and convex polytopes. This work has been explored in a promising way in super-resolution algorithms and distributed compression, and might find its applications in photogrammetry, computer graphics, and machine vision. We consider wireless sensor networks whose nodes are randomly deployed and, thus, provide an irregular sampling of the sensed field. The field is assumed to be bandlimited; a sink node collects the data gathered by the sensors and reconstructs the field by using a technique based on linear filtering. By taking the mean square error (MSE) as performance metric, we evaluate the effect of quasi-equally spaced sensor layouts on the quality of the reconstructed signal. The MSE is derived through asymptotic analysis for different sensor spatial distributions, and for two of them we are able to obtain an approximate closed form expression. The case of uniformly distributed sensors is also considered for the sake of comparison. The validity of our asymptotic analysis is shown by comparison against numerical results and it is proven to hold even for a small number of nodes. Finally, with the help of a simple example, we show the key role that our results play in the deployment of sensor networks. In high-dimensional magnetic resonance imaging applications, time-consuming, sequential acquisition of data samples in the spatial frequency domain ( @math -space) can often be accelerated by accounting for dependencies in linear reconstruction, at the cost of noise amplification that depends on the sampling pattern. Common examples are support-constrained, parallel, and dynamic MRI, and @math -space sampling strategies are primarily driven by image-domain metrics that are expensive to compute for arbitrary sampling patterns. It remains challenging to provide systematic and computationally efficient automatic designs of arbitrary multidimensional Cartesian sampling patterns that mitigate noise amplification, given the subspace to which the object is confined. To address this problem, this paper introduces a theoretical framework that describes local geometric properties of the sampling pattern and relates them to the spread in the eigenvalues of the information matrix described by its first two spectral moments. This new criterion is then used for very efficient optimization of complex multidimensional sampling patterns that does not require reconstructing images or explicitly mapping noise amplification. Experiments with in vivo data show strong agreement between this criterion and traditional, comprehensive image-domain- and @math -space-based metrics, indicating the potential of the approach for computationally efficient (on-the-fly), automatic, and adaptive design of sampling patterns. The problem of sampling signals that are not admissible within the classical Shannon framework has received much attention in the recent past. Typically, these signals have a parametric representation with a finite number of degrees of freedom per time unit. It was shown that, by choosing suitable sampling kernels, the parameters can be computed by employing high-resolution spectral estimation techniques. In this letter, we propose a simple acquisition and reconstruction method within the framework of multichannel sampling. In the proposed approach, an infinite stream of nonuniformly-spaced Dirac impulses can be sampled and accurately reconstructed provided that there is at most one Dirac impulse per sampling period. The reconstruction algorithm has a low computational complexity, and the parameters are computed on the fly. The processing delay is minimal just the sampling period. We propose sampling circuits using inexpensive passive devices such as resistors and capacitors. We also show how the approach can be extended to sample piecewise-constant signals with a minimal change in the system configuration. We provide some simulation results to confirm the theoretical findings. Recently, it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals, namely certain signals of finite rate of innovation. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time and can be reconstructed from a finite number of uniform samples. In order to prove sampling theorems, considered the case of deterministic, noiseless signals and developed algebraic methods that lead to perfect reconstruction. However, when noise is present, many of those schemes can become ill-conditioned. In this paper, we revisit the problem of sampling and reconstruction of signals with finite rate of innovation and propose improved, more robust methods that have better numerical conditioning in the presence of noise and yield more accurate reconstruction. We analyze, in detail, a signal made up of a stream of Diracs and develop algorithmic tools that will be used as a basis in all constructions. While some of the techniques have been already encountered in the spectral estimation framework, we further explore preconditioning methods that lead to improved resolution performance in the case when the signal contains closely spaced components. For classes of periodic signals, such as piecewise polynomials and nonuniform splines, we propose novel algebraic approaches that solve the sampling problem in the Laplace domain, after appropriate windowing. Building on the results for periodic signals, we extend our analysis to finite-length signals and develop schemes based on a Gaussian kernel, which avoid the problem of ill-conditioning by proper weighting of the data matrix. Our methods use structured linear systems and robust algorithmic solutions, which we show through simulation results. We study the notion of compressed sensing (CS) as put forward by Donoho, Candes, Tao and others. The notion proposes a signal or image, unknown but supposed to be compressible by a known transform, (e.g. wavelet or Fourier), can be subjected to fewer measurements than the nominal number of data points, and yet be accurately reconstructed. The samples are nonadaptive and measure 'random' linear combinations of the transform coefficients. Approximate reconstruction is obtained by solving for the transform coefficients consistent with measured data and having the smallest possible l1 norm.We present initial 'proof-of-concept' examples in the favorable case where the vast majority of the transform coefficients are zero. We continue with a series of numerical experiments, for the setting of lp-sparsity, in which the object has all coefficients nonzero, but the coefficients obey an lp bound, for some p ∈ (0, 1]. The reconstruction errors obey the inequalities paralleling the theory, seemingly with well-behaved constants.We report that several workable families of 'random' linear combinations all behave equivalently, including random spherical, random signs, partial Fourier and partial Hadamard.We next consider how these ideas can be used to model problems in spectroscopy and image processing, and in synthetic examples see that the reconstructions from CS are often visually "noisy". To suppress this noise we postprocess using translation-invariant denoising, and find the visual appearance considerably improved.We also consider a multiscale deployment of compressed sensing, in which various scales are segregated and CS applied separately to each; this gives much better quality reconstructions than a literal deployment of the CS methodology.These results show that, when appropriately deployed in a favorable setting, the CS framework is able to save significantly over traditional sampling, and there are many useful extensions of the basic idea. Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper, we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modeled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a block-sparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed lscr2 lscr1 program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP, we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently. We address the problem of deterministic oversampling of bandlimited sensor fields in a distributed communication-constrained processing environment, where it is desired for a central intelligent unit to reconstruct the sensor field to maximum pointwise accuracy.We show, using a dither-based sampling scheme, that is is possible to accomplish this using minimal inter-sensor communication with the aid of a multitude of low-precision sensors. Furthermore, we show the feasibility of having a flexible tradeoff between the average oversampling rate and the Analog to Digital (A D) quantization precision per sensor sample with respect to achieving exponential accuracy in the number of bits per Nyquist-period, thereby exposing a key underpinning "conservation of bits" principle. That is, we can distribute the bit budget per Nyquist-period along the amplitude-axis (precision of A D converter) and space (or time or space-time) using oversampling in an almost arbitrary discrete-valued manner, while retaining the same reconstruction error decay profile. Interestingly this oversampling is possible in a highly localized communication setting, with only nearest-neighbor communication, making it very attractive for dense sensor networks operating under stringent inter-node communication constraints. Finally we show how our scheme incorporates security as a by-product due to the presence of an underlying dither signal which can be used as a natural encryption device for security. The choice of the dither function enhances the security of the network. The design of conventional sensors is based primarily on the Shannon?Nyquist sampling theorem, which states that a signal of bandwidth W Hz is fully determined by its discrete time samples provided the sampling rate exceeds 2 W samples per second. For discrete time signals, the Shannon?Nyquist theorem has a very simple interpretation: the number of data samples must be at least as large as the dimensionality of the signal being sampled and recovered. This important result enables signal processing in the discrete time domain without any loss of information. However, in an increasing number of applications, the Shannon-Nyquist sampling theorem dictates an unnecessary and often prohibitively high sampling rate (see lWhat Is the Nyquist Rate of a Video Signal?r). As a motivating example, the high resolution of the image sensor hardware in modern cameras reflects the large amount of data sensed to capture an image. A 10-megapixel camera, in effect, takes 10 million measurements of the scene. Yet, almost immediately after acquisition, redundancies in the image are exploited to compress the acquired data significantly, often at compression ratios of 100:1 for visualization and even higher for detection and classification tasks. This example suggests immense wastage in the overall design of conventional cameras. This paper develops an optimal decentralized algorithm for sparse signal recovery and demonstrates its application in monitoring localized phenomena using energy-constrained large-scale wireless sensor networks. Capitalizing on the spatial sparsity of localized phenomena, compressive data collection is enforced by turning off a fraction of sensors using a simple random node sleeping strategy, which conserves sensing energy and prolongs network lifetime. In the absence of a fusion center, sparse signal recovery via decentralized in-network processing is developed, based on a consensus optimization formulation and the alternating direction method of multipliers. In the proposed algorithm, each active sensor monitors and recovers its local region only, collaborates with its neighboring active sensors through low-power one-hop communication, and iteratively improves the local estimates until reaching the global optimum. Because each sensor monitors the local region rather than the entire large field, the iterative algorithm converges fast, in addition to being scalable in terms of transmission and computation costs. Further, through collaboration, the sensing performance is globally optimal and attains a high spatial resolution commensurate with the node density of the original network containing both active and inactive sensors. Simulations demonstrate the performance of the proposed approach. Recent compressive sensing results show that it is possible to accurately reconstruct certain compressible signals from relatively few linear measurements via solving nonsmooth convex optimization problems. In this paper, we propose the use of the alternating direction method - a classic approach for optimization problems with separable variables (D. Gabay and B. Mercier, ?A dual algorithm for the solution of nonlinear variational problems via finite-element approximations,? Computer and Mathematics with Applications, vol. 2, pp. 17-40, 1976; R. Glowinski and A. Marrocco, ?Sur lapproximation par elements finis dordre un, et la resolution par penalisation-dualite dune classe de problemes de Dirichlet nonlineaires,? Rev. Francaise dAut. Inf. Rech. Oper., vol. R-2, pp. 41-76, 1975) - for signal reconstruction from partial Fourier (i.e., incomplete frequency) measurements. Signals are reconstructed as minimizers of the sum of three terms corresponding to total variation, ?1-norm of a certain transform, and least squares data fitting. Our algorithm, called RecPF and published online, runs very fast (typically in a few seconds on a laptop) because it requires a small number of iterations, each involving simple shrinkages and two fast Fourier transforms (or alternatively discrete cosine transforms when measurements are in the corresponding domain). RecPF was compared with two state-of-the-art algorithms on recovering magnetic resonance images, and the results show that it is highly efficient, stable, and robust. A novel on-demand, adaptive compressed sensing approach is proposed and discussed for spatial monitoring of correlated massive data in dense wireless sensor field. The spatial monitoring is based on collecting the sensor observations of a subset of sensors that their reading is in Δ margin of a set of contour levels. The contour level set is adaptively updated in order to reduce the mean square error of reconstruction of the spatial distribution. The reconstruction error and the number of involved sensing nodes as cost are used to compare the proposed approach with spatial monitoring using equally spaced contour levels.