Alfred O. Hero

Locating the nodes: cooperative localization in wireless sensor networks

Accurate and low-cost sensor localization is a critical requirement for the deployment of wireless sensor networks in a wide variety of applications. In cooperative localization, sensors work together in a peer-to-peer manner to make measurements and then forms a map of the network. Various application requirements influence the design of sensor localization systems. In this article, the authors describe the measurement-based statistical models useful to describe time-of-arrival (TOA), angle-of-arrival (AOA), and received-signal-strength (RSS) measurements in wireless sensor networks. Wideband and ultra-wideband (UWB) measurements, and RF and acoustic media are also discussed. Using the models, the authors have shown the calculation of a Cramer-Rao bound (CRB) on the location estimation precision possible for a given set of measurements. The article briefly surveys a large and growing body of sensor localization algorithms. This article is intended to emphasize the basic statistical signal processing background necessary to understand the state-of-the-art and to make progress in the new and largely open areas of sensor network localization research.

Relative location estimation in wireless sensor networks

Self-configuration in wireless sensor networks is a general class of estimation problems that we study via the Cramer-Rao bound (CRB). Specifically, we consider sensor location estimation when sensors measure received signal strength (RSS) or time-of-arrival (TOA) between themselves and neighboring sensors. A small fraction of sensors in the network have a known location, whereas the remaining locations must be estimated. We derive CRBs and maximum-likelihood estimators (MLEs) under Gaussian and log-normal models for the TOA and RSS measurements, respectively. An extensive TOA and RSS measurement campaign in an indoor office area illustrates MLE performance. Finally, relative location estimation algorithms are implemented in a wireless sensor network testbed and deployed in indoor and outdoor environments. The measurements and testbed experiments demonstrate 1-m RMS location errors using TOA, and 1- to 2-m RMS location errors using RSS.

Space-alternating generalized expectation-maximization algorithm

The expectation-maximization (EM) method can facilitate maximizing likelihood functions that arise in statistical estimation problems. In the classical EM paradigm, one iteratively maximizes the conditional log-likelihood of a single unobservable complete data space, rather than maximizing the intractable likelihood function for the measured or incomplete data. EM algorithms update all parameters simultaneously, which has two drawbacks: 1) slow convergence, and 2) difficult maximization steps due to coupling when smoothness penalties are used. The paper describes the space-alternating generalized EM (SAGE) method, which updates the parameters sequentially by alternating between several small hidden-data spaces defined by the algorithm designer. The authors prove that the sequence of estimates monotonically increases the penalized-likelihood objective, derive asymptotic convergence rates, and provide sufficient conditions for monotone convergence in norm. Two signal processing applications illustrate the method: estimation of superimposed signals in Gaussian noise, and image reconstruction from Poisson measurements. In both applications, the SAGE algorithms easily accommodate smoothness penalties and converge faster than the EM algorithms. >

Distributed weighted-multidimensional scaling for node localization in sensor networks

Accurate, distributed localization algorithms are needed for a wide variety of wireless sensor network applications. This article introduces a scalable, distributed weighted-multidimensional scaling (dwMDS) algorithm that adaptively emphasizes the most accurate range measurements and naturally accounts for communication constraints within the sensor network. Each node adaptively chooses a neighborhood of sensors, updates its position estimate by minimizing a local cost function and then passes this update to neighboring sensors. Derived bounds on communication requirements provide insight on the energy efficiency of the proposed distributed method versus a centralized approach. For received signal-strength (RSS) based range measurements, we demonstrate via simulation that location estimates are nearly unbiased with variance close to the Cramér-Rao lower bound. Further, RSS and time-of-arrival (TOA) channel measurements are used to demonstrate performance as good as the centralized maximum-likelihood estimator (MLE) in a real-world sensor network.

Sparse LMS for system identification

We propose a new approach to adaptive system identification when the system model is sparse. The approach applies ℓ1 relaxation, common in compressive sensing, to improve the performance of LMS-type adaptive methods. This results in two new algorithms, the zero-attracting LMS (ZA-LMS) and the reweighted zero-attracting LMS (RZA-LMS). The ZA-LMS is derived via combining a ℓ1 norm penalty on the coefficients into the quadratic LMS cost function, which generates a zero attractor in the LMS iteration. The zero attractor promotes sparsity in taps during the filtering process, and therefore accelerates convergence when identifying sparse systems. We prove that the ZA-LMS can achieve lower mean square error than the standard LMS. To further improve the filtering performance, the RZA-LMS is developed using a reweighted zero attractor. The performance of the RZA-LMS is superior to that of the ZA-LMS numerically. Experiments demonstrate the advantages of the proposed filters in both convergence rate and steady-state behavior under sparsity assumptions on the true coefficient vector. The RZA-LMS is also shown to be robust when the number of non-zero taps increases.

Using proximity and quantized RSS for sensor localization in wireless networks

For wireless sensor networks, received signal strength (RSS) and proximity (also known as connectivity) measurements have been proposed as simple and inexpensive means to estimate range between devices and sensor location. While RSS measurements are recognized to suffer from errors due to the random nature of the fading channel, proximity measurements, ie., knowing only whether or not two devices are in communication range, are often discussed without considering that they are affected by the same fading channel. Proximity measurements are actually just a binary quantization of RSS measurements. We use the Cramér-Rao bound (CRB) to compare the minimal attainable variances of unbiased sensor location estimators for the cases of RSS and proximity measurements. For completeness, we also present the CRB for sensor localization with systems using K-level quantized RSS (QRSS) measurements, of which proximity measurements are the special case: K=2. Examples are presented for the case of one unknown-location sensor, and for the case of a 5 by 5 grid of sensors. These examples show that lower bounds for standard deviation in proximity-based systems are, as a rule of thumb, about 50% higher than the bounds for RSS-based systems. Furthermore, results are presented which show how many bits of quantization are necessary for a QRSS-based system to nearly achieve the bounds of an unquantized RSS system.

A Fast Spectral Method for Active 3D Shape Reconstruction

Variational energy minimization techniques for surface reconstruction are implemented by evolving an active surface according to the solutions of a sequence of elliptic partial differential equations (PDEs). For these techniques, most current approaches to solving the elliptic PDE are iterative involving the implementation of costly finite element methods (FEM) or finite difference methods (FDM). The heavy computational cost of these methods makes practical application to 3D surface reconstruction burdensome. In this paper, we develop a fast spectral method which is applied to 3D active surface reconstruction of star-shaped surfaces parameterized in polar coordinates. For this parameterization the Euler-Lagrange equation is a Helmholtz-type PDE governing a diffusion on the unit sphere. After linearization, we implement a spectral non-iterative solution of the Helmholtz equation by representing the active surface as a double Fourier series over angles in spherical coordinates. We show how this approach can be extended to include region-based penalization. A number of 3D examples and simulation results are presented to illustrate the performance of our fast spectral active surface algorithms.Image sharpening in the presence of noise is formulated as a non-convex variational problem. The energy functional incorporates a gradient-dependent potential, a convex fidelity criterion and a high order convex regularizing term. The first term attains local minima at zero and some high gradient magnitude, thus forming a triple well-shaped potential (in the one-dimensional case). The energy minimization flow results in sharpening of the dominant edges, while most noisy fluctuations are filtered out.

Applications of entropic spanning graphs

This article presents applications of entropic spanning graphs to imaging and feature clustering applications. Entropic spanning graphs span a set of feature vectors in such a way that the normalized spanning length of the graph converges to the entropy of the feature distribution as the number of random feature vectors increases. This property makes these graphs naturally suited to applications where entropy and information divergence are used as discriminants: texture classification, feature clustering, image indexing, and image registration. Among other areas, these problems arise in geographical information systems, digital libraries, medical information processing, video indexing, multisensor fusion, and content-based retrieval.

Lower bounds for parametric estimation with constraints

A Chapman-Robbins form of the Barankin bound is used to derive a multiparameter Cramer-Rao (CR) type lower bound on estimator error covariance when the parameter theta in R/sup n/ is constrained to lie in a subset of the parameter space. A simple form for the constrained CR bound is obtained when the constraint set Theta /sub C/, can be expressed as a smooth functional inequality constraint. It is shown that the constrained CR bound is identical to the unconstrained CR bound at the regular points of Theta /sub C/, i.e. where no equality constraints are active. On the other hand, at those points theta in Theta /sub C/ where pure equality constraints are active the full-rank Fisher information matrix in the unconstrained CR bound must be replaced by a rank-reduced Fisher information matrix obtained as a projection of the full-rank Fisher matrix onto the tangent hyperplane of the full-rank Fisher matrix onto the tangent hyperplane of the constraint set at theta . A necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable, and examples for which the bound is achieved are presented. In addition to providing a useful generalization of the CR bound, the results permit analysis of the gain in achievable MSE performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization. >

Secure space-time communication

Network security is important for information protection in open, secure, or covert communications. One such requirement is to achieve high-rate communications between clients, e.g., terminals or sensors, in the network while hiding information about the transmitted symbols, signal activity, or other sensitive data from an unintended receiver, e.g., an eavesdropper. For wireless links, the single-user capacity advantages of deployment of multiple antennas at the transmitter is well known. One of the principal conclusions of this paper is that proper exploitation of space-time diversity at the transmitter can also enhance information security and information-hiding capabilities. In particular, we show that significant gains are achievable when the transmitter and the client receiver are both informed about their channel while the transmitter and eavesdropper receiver are uniformed about their channel. More generally, we compare capacity limits for both informed and uninformed transmitter and informed receiver scenarios subject to low probability of intercept (LPI) and low probability of detection (LPD) constraints. For several general cases, we can characterize the LPI- and LPD-optimal transmitted source distributions and compare them to the standard optimal source distribution satisfying a power constraint. We assume the standard quasi-static flat Rayleigh-fading channel model for the transmitter-receiver pairs. This paper is a step toward answering the fundamental question: what are the qualitative and quantitative differences between the information-carrying capabilities of open space-time channels versus secure space-time channels?