Richard W. Osborne

Statistical Efficiency of Composite Position Measurements from Passive Sensors

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-which involves, by necessity, approximations. This ML estimate is shown to be statistically efficient, even for small sample sizes (as few as two LOS measurements), and as such, the covariance matrix obtainable from the Cramer-Rao lower bound (CRLB) provides the correct measurement noise covariance matrix for use in a target tracking filter.

Radar measurement noise variance estimation with several targets of opportunity

A number of methods exist to track a targets 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. A method is presented of determining the measurement noise variances of a sensor, assumed to be constant, by using multiple IMM estimators while tracking targets whose motion is not known-targets of opportunity. Combining techniques outlined in (Bar-Shalom et al., 2001) and (Gauvrit, 1984), 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 over a varying grid of IMMs 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.

Design of an Adaptive Passive Collision Warning System for UAVs

Collision warning systems are used for commercial air traffic to provide pilots an extra layer of situational awareness (as well as avoidance actions), and are of increased importance due to the recent interest in shared-airspace deployment of unmanned aerial vehicles (UAVs). The design for consistency of a passive collision warning system (PCWS) that uses low- resolution infrared (IR) cameras mounted to the airframe is presented. 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 are estimation of the measurement noise variance of the sensor and consistency of the resulting tracks. The proposed PCWS adaptively estimates this noise variance. The resulting system was found to provide consistent errors as well as accurate estimates of the angular velocity of detected targets. The results were verified through a number of test flights that included an aircraft approaching and passing over a stationary camera, and also flying across its field of view at a nearly constant range. Subsequently, the filter was modified for use on a light aircraft in conjunction with an inertial navigation system (INS).

Statistical efficiency of composite position measurements from passive sensors

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.

CRLB for likelihood functions with parameter-dependent support

In this paper we discuss the regularity conditions required for the classical Craḿer-Rao lower bound (CRLB) for real-valued (nonrandom unknown) parameters to hold. It is shown that the commonly assumed requirement that the support of the likelihood function (LF) should be independent of the parameter to be estimated can be replaced by the much weaker requirement that the LF is continuous at the end points of its support. Parameter-dependent support of the LF arises when an unknown parameter is observed in the presence of additive measurement noise and the measurement noise probability density function (pdf) has a finite support. It is also pointed out that the commonly cited requirements of absolute integrability of the derivatives of the LF should be replaced by requirements on the log-LF (LLF). Some practical examples of finite-support measurement noises, which lead to parameter-dependent LF support, are discussed in light of the above. For the case where the LF is not continuous at the end points of its support, a new modified CRLB -designated as the Craḿer-Rao-Leibniz lower bound (CRLLB), since it relies on Leibniz integral rule - is presented and its use illustrated. The CRLLB is shown to provide valid bounds for a number of long-standing problems for which the CRLB was shown in the literature as not valid, in particular, for a uniformly distributed measurement noise.

Update to the Hybrid Conditional Averaging Performance Prediction of the IMM Algorithm

Traditionally the performance evaluation of a target tracking algorithm is accomplished via Monte Carlo simulations for each specific scenario of interest. For some applications, the time and computational resource requirements of performing the necessary simulations for algorithm design is excessive; so the need for performance prediction becomes paramount. One method of performance prediction developed during the early 1990s is the hybrid conditional averaging (HYCA) technique, which can be used to predict the performance of the interacting multiple model (IMM) algorithm. Applying the HYCA technique to the IMM algorithm as originally developed leads to poor performance prediction in certain situations. A new extension used in these circumstances is shown to lead to superior performance prediction without an increase in computational complexity compared with the originally developed algorithm for such situations.

CRLB for likelihood functions with parameter-dependent support and a new bound

In this paper we discuss the regularity conditions required for the classical Cramér-Rao Lower Bound (CRLB) for real-valued (nonrandom unknown) parameters to hold. It is shown that the commonly assumed requirement that the support of the likelihood function (LF) should be independent of the parameter to be estimated can be replaced by the much weaker requirement that the LF is continuous at the end points of its support. Parameter-dependent support of the LF arises when an unknown parameter is observed in the presence of additive measurement noise and the measurement noise pdf has a finite support. It is also pointed out that the commonly cited requirements of absolute integrability of the derivatives of the LF should be replaced by requirements on the log-LF. Some practical examples of finite-support measurement noises, which lead to parameter-dependent likelihood function support, are discussed in light of the above. For the case where the LF is not continuous at the end points of its support, a new modified CRLB - designated as the Cramér-Rao-Leibniz Lower Bound (CRLLB), since it relies on Leibniz integral rule - is presented and its use illustrated. The CRLLB is shown to provide valid bounds for a number of longstanding problems for which the CRLB was shown in the literature as not valid.

2D Location estimation of angle-only sensor arrays using targets of opportunity

Passive acoustic sensor arrays for tracking ground targets are becoming increasingly popular due to their low cost and ease of deployment. In this paper we present an algorithm for locating sensor arrays in two-dimensions in an acoustic network (or in any network where angle-only measurements are used) when external references, such as GPS or known-location targets, are unavailable. We consider sensor localization when angular measurements are taken from the sensor arrays to targets of opportunity when all sensors take measurements with respect to a common axis of unknown orientation and where the sensors can not “see” each other. The solutions provided consist of low-complexity (generally closed-form) methods of getting initial estimates with no prior information, followed by maximum likelihood (ML) optimization to refine the estimates. Simulation shows that the accuracy approaches the Cramér Rao Lower Bound (CRLB), something that similar algorithms from previous research have been unable to achieve.

A minimalist approach to bias estimation for passive sensor measurements with targets of opportunity

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.

Cramér-Rao-Leibniz Lower Bound — A new estimation bound for finite support measurement noise

In this paper we introduce a new bound on an estimators error, derived from the classical Cramér-Rao Lower Bound (CRLB), for cases where the support of the likelihood function (LF) exhibits parameter-dependence. Parameter-dependent support of the LF arises here when an unknown parameter is observed in the presence of additive measurement noise and the measurement noise pdf has a finite support. This new modified CRLB - designated as the Cramér-Rao-Leibniz Lower Bound (CRLLB), since it relies on Leibniz integral rule - is presented and its use illustrated. The CRLLB is shown to provide, for example, a valid bound for the problem of uniform measurement noise for which the CRLB was shown in the literature as not valid. Furthermore, it is demonstrated that, in light of the CRLLB, the ML estimator in the uniform measurement noise case is statistically efficient, i.e., the estimators variance is equal to the CRLLB.