Self-config uration in wireless sensor networks is ag eneral class of estimation problems which we study via the Cramer-Rao bound (CRB).Specifically, we consider sensor location estimation when sensors measure received sig]P strengI (RSS) or time-of-arrival (TOA) between themselves and neig boring sensors.A small fraction of sensors in the network have known location while 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 campaig in an indoor o#ce area illustrates MLE performance.Finally, relative location estimation alg orithms 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 errorsusing TOA, and 1 m to 2 m RMS location errors using RSS. Index Terms sensor position location estimation, radio channel measurement, sig nal streng h, time-ofarrival, wireless sensor network testbed, self-config uration, Cramer-Rao bound I.

Traditional space-invariant regularization methods in tomographic image reconstruction using penalized-likelihood estimators produce images with nonuniform spatial resolution properties. The local point spread functions that quantify the smoothing properties of such estimators are space-variant, asymmetric, and object-dependent even for space-invariant imaging systems. We propose a new quadratic regularization scheme for tomographic imaging systems that yields increased spatial uniformity and is motivated by the least-squares tting of a parameterized local impulse response to a desired global response. We have developed computationally e cient methods for PET systems with shift-invariant geometric responses. We demonstrate the increased spatial uniformity of this new method versus conventional quadratic regularization schemes in simulated PET thorax scans.

We examine the spatial resolution and variance properties of PET images reconstructed using maximum a posteriori (MAP) or penalized-likelihood methods. Resolution is characterized by the contrast recovery coefficient (CRC) of the local impulse response. Simplified approximate expressions are derived for the local impulse response CRCs and variances for each voxel. Using these results we propose a practical scheme for selecting spatially variant smoothing parameters to optimize lesion detectability through maximization of the local CRC-to-noise ratio in the reconstructed image. I. INTRODUCTION PET image reconstruction algorithms based on maximum likelihood (ML) or maximum a posteriori (MAP) principles can produce improved spatial resolution and noise properties in comparison to conventional filtered backprojection (FBP) methods. It is often important to be able to quantify this improvement in terms of the resolution (or bias) and variance of the resulting images. These measures can be...

PET measurements are usually precorrected for accidental coincidence events by real-time subtraction of the delayed window coincidences. Randoms subtraction compensates in mean for accidental coincidences but destroys the Poisson statistics. We propose and analyze two new approximations to the exact log-likelihood of the precorrected measurements, one based on a "shifted Poisson" model, the other based on saddle-point approximations to the measurement probability mass function (pmf). The methods apply to both emission and transmission tomography; however in this paper we focus on transmission tomography. We compare the new models to conventional data-weighted least squares (WLS) and conventional maximum likelihood (based on the ordinary Poisson (OP) model) using simulations and analytic approximations. The results demonstrate that the proposed methods avoid the systematic bias of the WLS method, and lead to significantly lower variance than the conventional OP method. The saddle-point method provides a more accurate approximation to the exact log-likelihood than the WLS, OP and shifted Poisson alternatives.

This paper examines the spatial resolution properties of penalized-likelihood image reconstruction methods by analyzing the local impulse response. The analysis shows that standard regularization penalties induce space-variant local impulse response functions, even for space-invariant tomographic systems. Paradoxically, for emission image reconstruction the local resolution is generally poorest in high-count regions. We show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections. This analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution. The modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalized-likelihood methods.

We have investigated the improvement in resolution and sensitivity for brain imaging which would result by the addition of a single stationary vertex view to the tomographic data. This method has the practical advantage of being relatively inexpensive and easy to implement. The uniform Cramer Rao bound is a plot of the minimum achievable standard deviation for estimating the pixel intensity as a function of the bias gradient length. Uniform CR bound analysis indicated an improvement in performance when the vertex detector is added, especially for centrally located pixels for which improvement is seen over the useful depth for brain imaging. Simulation experiments were done with a simple six slice phantom and with the Hoffman brain phantom. Visual inspection of the reconstructed images showed improved resolution and noise characteristics over reconstructed images without the vertex data. Quantitatively, substantial reduction in mean square error was observed for a plane close to the ver...

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Previously we introduced the Uniform Cram&-Rao (CR) Bound as a lower bound on the variance of biased estimators, along with the concept of the delta-sigma tradeoff curve. For an estimator whose variance lies on this curve, lower variance can only be achieved at the price of increased estimator bias gradient norm, and vice versa. However, for single pixel estimation, one can specify a variety of different estimator point response functions that have identical bias-gradient norm but with widely different resolution properties. This has lead to some counter-intuitive results and interpretation difficulties when using the Uniform CR Bound in performance studies of imaging systems. We now extend this tradeoff concept by introducing the 2nd-moment of the point response function as a measure of resolution for single-pixel estimation tasks. We derive an expression for the delta-gamma-sigma tradeoff surface. This surface specifies an "unachievable region " of estimator variance. For estimators that lie on this surface, lower variance can only be achieved at the price of increased bias gradient norm and/or decreased estimator resolution. We present a method for computing this surface for linear gaussian inverse problems. I.

This work investigates the lower bounds of wireless localization accuracy using signal strength on commodity hardware. Our work relies on trace-driven analysis using an extensive indoor experimental infrastructure. First, we report the best experimental accuracy, twice the best prior reported accuracy for any localization system. We experimentally show that adding more and more resources (e.g., training points or landmarks) beyond a certain limit, can degrade the localization performance for lateration-based algorithms, and that it could only be improved further by “cleaning ” the data. However, matching algorithms are more robust to poor quality RSS measurements. We next compare with a theoretical lower bound using standard Cramér Rao Bound (CRB) analysis for unbiased estimators, which is frequently used to provide bounds on localization precision. Because many localization algorithms are based on different mathematical foundations, we apply a diverse set of existing algorithms to our packet traces and found that the variance of the localization errors from these algorithms are smaller than the variance bound established by the CRB. Finally, we found that there exists a wide discrepancy from what freespace models predict in the signal to distance function even in an environment with limited shadowing and multipath, thereby imposing a fundamental limit on the achievable localization accuracy indoors.

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