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58
Relative Location Estimation in Wireless Sensor Networks
, 2003
"... Selfconfig uration in wireless sensor networks is ag eneral class of estimation problems which we study via the CramerRao bound (CRB).Specifically, we consider sensor location estimation when sensors measure received sig]P strengI (RSS) or timeofarrival (TOA) between themselves and neig boring s ..."
Abstract

Cited by 298 (16 self)
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Selfconfig uration in wireless sensor networks is ag eneral class of estimation problems which we study via the CramerRao bound (CRB).Specifically, we consider sensor location estimation when sensors measure received sig]P strengI (RSS) or timeofarrival (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 maximumlikelihood 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, timeofarrival, wireless sensor network testbed, selfconfig uration, CramerRao bound I.
Regularization for uniform spatial resolution properties in penalizedlikelihood image reconstruction
 IEEE Tr. Med. Im
, 2000
"... Traditional spaceinvariant regularization methods in tomographic image reconstruction using penalizedlikelihood estimators produce images with nonuniform spatial resolution properties. The local point spread functions that quantify the smoothing properties of such estimators are spacevariant, as ..."
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Cited by 59 (28 self)
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Traditional spaceinvariant regularization methods in tomographic image reconstruction using penalizedlikelihood estimators produce images with nonuniform spatial resolution properties. The local point spread functions that quantify the smoothing properties of such estimators are spacevariant, asymmetric, and objectdependent even for spaceinvariant imaging systems. We propose a new quadratic regularization scheme for tomographic imaging systems that yields increased spatial uniformity and is motivated by the leastsquares tting of a parameterized local impulse response to a desired global response. We have developed computationally e cient methods for PET systems with shiftinvariant geometric responses. We demonstrate the increased spatial uniformity of this new method versus conventional quadratic regularization schemes in simulated PET thorax scans.
A Theoretical Study of the Contrast Recovery and Variance of MAP Reconstructions From PET Data
 IEEE Trans. Med. Imag
, 1999
"... We examine the spatial resolution and variance properties of PET images reconstructed using maximum a posteriori (MAP) or penalizedlikelihood methods. Resolution is characterized by the contrast recovery coefficient (CRC) of the local impulse response. Simplified approximate expressions are derived ..."
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Cited by 44 (8 self)
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We examine the spatial resolution and variance properties of PET images reconstructed using maximum a posteriori (MAP) or penalizedlikelihood 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 CRCtonoise 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...
Statistical approaches in quantitative positron emission tomography
 Statistics and Computing
"... Positron emission tomography is a medical imaging modality for producing 3D images of the spatial distribution of biochemical tracers within the human body. The images are reconstructed from data formed through detection of radiation resulting from the emission of positrons from radioisotopes tagged ..."
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Cited by 38 (3 self)
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Positron emission tomography is a medical imaging modality for producing 3D images of the spatial distribution of biochemical tracers within the human body. The images are reconstructed from data formed through detection of radiation resulting from the emission of positrons from radioisotopes tagged onto the tracer of interest. These measurements are approximate line integrals from which the image can be reconstructed using analytical inversion formulae. However these direct methods do not allow accurate modeling either of the detector system or of the inherent statistical fluctuations in the data. Here we review recent progress in developing statistical approaches to image estimation that can overcome these limitations. We describe the various components of the physical model and review different formulations of the inverse problem. The wide range of numerical procedures for solving these problems are then reviewed. Finally, we describe recent work aimed at quantifying the quality of the resulting images, both in terms of classical measures of estimator bias and variance, and also using measures that are of more direct clinical relevance.
Statistical Image Reconstruction Methods for RandomsPrecorrected PET Scans
 Med. Im. Anal
, 1998
"... PET measurements are usually precorrected for accidental coincidence events by realtime 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 ex ..."
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Cited by 35 (16 self)
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PET measurements are usually precorrected for accidental coincidence events by realtime 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 loglikelihood of the precorrected measurements, one based on a "shifted Poisson" model, the other based on saddlepoint 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 dataweighted 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 saddlepoint method provides a more accurate approximation to the exact loglikelihood than the WLS, OP and shifted Poisson alternatives. However, the simpler shifted Poisson method yielded comparable biasvariance performance to the saddlepoint method in the simulations. The new methods offer improved image reconstruction in PET through more realistic statistical modeling, yet with negligible increase in computation over the conventional OP method.
Optimality analysis of sensortarget geometries in passive localization: Part 1  Bearingonly localization
 In ISSNIP’07
, 2007
"... In this paper we characterize the relative sensortarget geometry in R2 in terms of potential localization performance for timeofarrival based localization. Our aim is to characterize those relative sensortarget geometries which minimize the relative CramerRao lower bound. 1. ..."
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Cited by 25 (7 self)
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In this paper we characterize the relative sensortarget geometry in R2 in terms of potential localization performance for timeofarrival based localization. Our aim is to characterize those relative sensortarget geometries which minimize the relative CramerRao lower bound. 1.
New Statistical Models for RandomsPrecorrected PET Scans
 in Information Processing in Medical
, 2001
"... PET measurements are usually precorrected for accidental coincidence events by realtime 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 exa ..."
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Cited by 24 (19 self)
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PET measurements are usually precorrected for accidental coincidence events by realtime 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 loglikelihood of the precorrected measurements, one based on a "shifted Poisson" model, the other based on saddlepoint 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 dataweighted 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 saddlepoint method provides a more accurate approximation to the exact loglikelihood than the WLS, OP and shifted Poisson alternatives.
Resolution properties of regularized image reconstruction methods
 of EECS, Univ. of Michigan, Ann Arbor, MI
, 1995
"... This paper examines the spatial resolution properties of penalizedlikelihood image reconstruction methods by analyzing the local impulse response. The analysis shows that standard regularization penalties induce spacevariant local impulse response functions, even for spaceinvariant tomographic s ..."
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Cited by 20 (13 self)
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This paper examines the spatial resolution properties of penalizedlikelihood image reconstruction methods by analyzing the local impulse response. The analysis shows that standard regularization penalties induce spacevariant local impulse response functions, even for spaceinvariant tomographic systems. Paradoxically, for emission image reconstruction the local resolution is generally poorest in highcount 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 penalizedlikelihood methods.
Minimum variance in biased estimation: Bounds and asymptotically optimal estimators
 IEEE Trans. Signal Processing
, 2004
"... Abstract—We develop a uniform Cramér–Rao lower bound (UCRLB) on the total variance of any estimator of an unknown vector of parameters, with bias gradient matrix whose norm is bounded by a constant. We consider both the Frobenius norm and the spectral norm of the bias gradient matrix, leading to two ..."
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Cited by 19 (6 self)
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Abstract—We develop a uniform Cramér–Rao lower bound (UCRLB) on the total variance of any estimator of an unknown vector of parameters, with bias gradient matrix whose norm is bounded by a constant. We consider both the Frobenius norm and the spectral norm of the bias gradient matrix, leading to two corresponding lower bounds. We then develop optimal estimators that achieve these lower bounds. In the case in which the measurements are related to the unknown parameters through a linear Gaussian model, Tikhonov regularization is shown to achieve the UCRLB when the Frobenius norm is considered, and the shrunken estimator is shown to achieve the UCRLB when the spectral norm is considered. For more general models, the penalized maximum likelihood (PML) estimator with a suitable penalizing function is shown to asymptotically achieve the UCRLB. To establish the asymptotic optimality of the PML estimator, we first develop the asymptotic mean and variance of the PML estimator for any choice of penalizing function satisfying certain regularity constraints and then derive a general condition on the penalizing function under which the resulting PML estimator asymptotically achieves the UCRLB. This then implies that from all linear and nonlinear estimators with bias gradient whose norm is bounded by a constant, the proposed PML estimator asymptotically results in the smallest possible variance. Index Terms—Asymptotic optimality, biased estimation, bias gradient norm, Cramér–Rao lower bound, penalized maximum likelihood, Tikhonov regularization.
Linear Regression with Gaussian Model Uncertainty: Algorithms and Bounds
"... We consider the problem of estimating an unknown deterministic parameter vector in a linear regression model with random Gaussian uncertainty in the mixing matrix. We prove that the maximum likelihood (ML) estimator is a regularized least squares estimator and develop three alternative approaches f ..."
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Cited by 14 (3 self)
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We consider the problem of estimating an unknown deterministic parameter vector in a linear regression model with random Gaussian uncertainty in the mixing matrix. We prove that the maximum likelihood (ML) estimator is a regularized least squares estimator and develop three alternative approaches for finding the regularization parameter which maximizes the likelihood. We analyze the performance using the Cramér Rao bound (CRB) on the mean squared error, and show that the degradation in performance due the uncertainty is not as severe as may be expected. Next, we address the problem again assuming that the variances of the noise and the elements in the model matrix are unknown and derive the associated CRB and ML estimator. We compare our methods to known results on linear regression in the error in variables (EIV) model. We discuss the similarity between these two competing approaches, and provide a thorough comparison which sheds light on their theoretical and practical differences.