This paper investigates the problem of localizing an unknown number of transient emitters using a network of passive sensors measuring angles of arrival in the presence of missed detections and false alarms. It is assumed that measurements within a certain time window of interest have to be associated before they can be fused to estimate the emitter locations. Two measurement models — either that any target can generate at most one measurement per sensor or that any target can generate several measurements per sensor — are possible within this time window. These two measurement models lead to two different problem formulations: one is an S-D assignment problem and the other is a cardinality selection problem. The S-D assignment problem can be solved by the Lagrangian relaxation algorithm efficiently with a high degree of accuracy when a small number of sensors are used. The sequential m-best 2-D assignment algorithm, which is resistant to the ghosting problem due to the estimation of the emitter signal’s emission time, is developed to solve the problem when the number of sensors becomes large. Simulation results show that the sequential m-best 2-D assignment algorithm is suitable for real time processing with reliable associations and estimates. The cardinality selection formulation models a list of measurements as a Poisson point process and is solved by applying the expectation-maximization (EM) algorithm and an information criterion. The convergence of the EM algorithm to the desired global maximum needs an initialization, which is close to the truth. Localization using passive sensors makes it difficult to obtain such an initial estimate. An assignment-based initialization approach is therefore presented. Simulation studies showed that the EM algorithm based on the assignment initialization is able to estimate the number of targets, target locations and directions with a high degree of accuracy.
KEYWORDS: Sensors, Satellites, Target detection, Space sensors, Missiles, Monte Carlo methods, Optical sensors, 3D acquisition, Optimization (mathematics), Statistical analysis
In this paper, an approach to bias estimation in the presence of measurement association uncertainty using
common targets of opportunity, is developed. Data association is carried out before the estimation of sensor angle
measurement biases. Consequently, the quality of data association is critical to the overall tracking performance.
Data association becomes especially challenging if the sensors are passive. Mathematically, the problem can
be formulated as a multidimensional optimization problem, where the objective is to maximize the generalized
likelihood that the associated measurements correspond to common targets, based on target locations and sensor
bias estimates. Applying gating techniques significantly reduces the size of this problem. The association
likelihoods are evaluated using an exhaustive search after which an acceptance test is applied to each solution
in order to obtain the optimal (correct) solution. We demonstrate the merits of this approach by applying it to
a simulated tracking system, which consists of two satellites tracking a ballistic target. We assume the sensors
are synchronized, their locations are known, and we estimate their orientation biases together with the unknown
target locations.
Integration of space based sensors into a Ballistic Missile Defense System (BMDS) allows for detection and tracking of threats over a larger area than ground based sensors [1]. This paper examines the effect of sensor bias error on the tracking quality of a Space Tracking and Surveillance System (STSS) for the highly non-linear problem of tracking a ballistic missile. The STSS constellation consists of two or more satellites (on known trajectories) for tracking ballistic targets. Each satellite is equipped with an IR sensor that provides azimuth and elevation to the target. The tracking problem is made more difficult due to a constant or slowly varying bias error present in each sensor's line of sight measurements. It is important to correct for these bias errors so that the multiple sensor measurements and/or tracks can be referenced as accurately as possible to a common tracking coordinate system. The measurements provided by these sensors are assumed time-coincident (synchronous) and perfectly associated. The line of sight (LOS) measurements from the sensors can be fused into measurements which are the Cartesian target position, i.e., linear in the target state. We evaluate the Cramér-Rao Lower Bound (CRLB) on the covariance of the bias estimates, which serves as a quantification of the available information about the biases. Statistical tests on the results of simulations show that this method is statistically efficient, even for small sample sizes (as few as two sensors and six points on the (unknown) trajectory of a single target of opportunity). We also show that the RMS position error is significantly improved with bias estimation compared with the target position estimation using the original biased measurements.
KEYWORDS: Sensors, Error analysis, Statistical analysis, Monte Carlo methods, Passive sensors, Optical sensors, Data fusion, Composites, 3D metrology, Detection and tracking algorithms
In order to carry out data fusion, registration error correction is crucial in multisensor systems. This requires estimation of the sensor measurement biases. It is important to correct for these bias errors so that the multiple sensor measurements and/or tracks can be referenced as accurately as possible to a common tracking coordinate system. This paper provides a solution for bias estimation for the minimum number of passive sensors (two), when only targets of opportunity are available. The sensor measurements are assumed time-coincident (synchronous) and perfectly associated. Since these sensors provide only line of sight (LOS) measurements, the formation of a single composite Cartesian measurement obtained from fusing the LOS measurements from different sensors is needed to avoid the need for nonlinear filtering. We evaluate the Cramer-Rao Lower Bound (CRLB) on the covariance of the bias estimate, i.e., the quantification of the available information about the biases. Statistical tests on the results of simulations show that this method is statistically efficient, even for small sample sizes (as few as two sensors and six points on the trajectory of a single target of opportunity). We also show that the
RMS position error is significantly improved with bias estimation compared with the target position estimation using the original biased measurements.
Pixel-size effects on the probability of detection and on measurement extraction accuracy (the position measurement noise variance) for point sources in the focal plane (FP) of an optical sensor are analyzed from a general target tracking perspective. The analysis uses the point spread function of the optical sensor, which causes distant targets | point sources | to appear as a spatially extended pattern in the FP. Measurement extraction, both via Maximum Likelihood Estimation (MLE) and pixel detection centroiding is examined. The impact of pixel size, target strength (SNR) and target motion uncertainty (“maneuvering index") on tracking accuracy are quantified theoretically and verified by simulations.
This work derives the Cramer-Rao lower bound (CRLB) for an acoustic target and sensor localization system
in which the noise characteristics depend on the location of the source. The system itself has been previously
examined, but without deriving the CRLB and showing the statistical efficiency of the estimator used. Two
different versions of the CRLB are derived, one in which direction of arrival (DOA) and range measurements
are available ("full-position CRLB"), and one in which only DOA measurements are available ("bearing-only
CRLB"). In both cases, the estimator is found to be statistically efficient; but, depending on the sensor-target
geometry, the range measurements may or may not significantly contribute to the accuracy of target localization.
Combining line-of-sight (LOS) measurements from passive sensors (e.g., satellite-based IR, ground-based cameras,
etc.), assumed to be synchronized, into a single composite Cartesian measurement (full position in 3D) via
maximum likelihood (ML) estimation, can circumvent the need for nonlinear filtering. This ML estimate is
shown to be statistically efficient, and as such, the covariance matrix obtainable from the Cramer-Rao lower
bound provides a consistent measurement noise covariance matrix for use in a target tracking filter.
Collision warning systems are used for commercial air traffic to provide pilots with an extra layer of situational
awareness (as well as avoidance actions). The recent surge in the use of unmanned aerial vehicles (UAV) means
the design of a collision warning system for UAVs is increasing in importance. This paper deals with the design for
consistency of a Passive Collision Warning System (PCWS) which utilizes low resolution infrared (IR) cameras
mounted to the airframe. The lack of range information, as well as the unknown measurement noise statistics,
make tracking and the decision for collision warning difficult. Of the utmost importance is the estimation of
the measurement noise variance of the sensor and the consistency of the resulting tracking filter. The proposed
PCWS adaptively estimates this noise variance. The resulting system was found to provide consistent tracking
filters as well as accurate estimates of the angular velocity of detected targets, verified through a number of test
flights with an aircraft passing both over a stationary camera as well as across its field of view. Subsequently,
the filter was modified for use on a light aircraft in conjunction with an inertial navigation system.
A number of methods exist to track a target's uncertain motion through space using inherently inaccurate sensor measurements.
A powerful method of adaptive estimation is the interacting multiple model (IMM) estimator. In order to carry
out state estimation from the noisy measurements of a sensor, however, the filter should have knowledge of the statistical
characteristics of the noise associated with that sensor. The statistical characteristics (accuracies) of real sensors, however,
are not always available, in particular for legacy sensors. This paper presents a method of determining the measurement
noise variances of a sensor by using multiple IMM estimators while tracking targets whose motion is not known-targets of opportunity. Combining techniques outlined in [1] and [3], the likelihood functions are obtained for a number of IMM
estimators, each with different assumptions on the measurement noise variances. Then a search is carried out to bracket
the variances of the sensor measurement noises. The end result consists of estimates of the measurement noise variances
of the sensor in question.
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