A Note on Bounds for Target Tracking with pd>1

Recently several new results for Cramer-Rao lower bounds (CRLBs) in dynamical systems have been developed. Several different approaches and approximations have been presented. For the general case of target tracking with a detection probability smaller than one and possibly in the presence of false measurements, two main approaches have been presented. The first approach is the information reduction factor (IRF) approach. The second approach is the enumeration (ENUM) approach, also referred to as the conditioning approach. It has been found that the ENUM approach leads to a strictly larger covariance matrix than the IRF approach, however, still providing a lower bound on the attainable error covariance. Thus, the ENUM approach provides a strictly tighter bound on the attainable performance. It has been conjectured that these bounds converge to one another in the limit or equivalently after an initial transition stage. We demonstrate, using some recent results from the modified Riccati equation (MRE) and by means of counter examples, that this conjecture does not hold true in general. We also demonstrate that the conjecture does hold true in the special case of deterministic target motion, or equivalently in the absence of process noise. Furthermore, we show that the detection probability has an influence on the limiting behaviors of the bounds. Moreover, we show that the MRE approximation provides a very good and computationally efficient approximation of the ENUM bound. The various results are illustrated by means of representative examples.     

Interacting multiple model fixed-lag smoothing algorithm forMarkovian switching systems

We investigate a suboptimal approach to the fixed-lag smoothing problem for Markovian switching systems. A fixed-lag smoothing algorithm is developed by applying the basic Interacting Multiple Model (IMM) approach to a state-augmented system. The computational load is roughly d (the fixed lag) times beyond that of filtering for the original system. In addition, an algorithm that approximates the “fixed-lag” mode probabilities given measurements up to current time is proposed. The algorithm is illustrated via a target tracking simulation example where a significant improvement over the filtering algorithm is achieved at the cost of a time delay (i.e., data up to time k are used to produce the smoothed state estimate at time k-d where the fixed large d>0). the IMM fixed-lag smoothing performance for the given example is comparable to that of an existing IMM fixed-interval smoother. Compared with fixed-interval smoothers, the fixed-lag smoothers can be implemented in real-time with a small delay     

A Kalman Tracker with a Simple Input Estimator

Two Kalman filter based schemes are proposed for tracking maneuvering targets. Both schemes use least squares to estimate a target's acceleration input vector and to update the tracker by this estimate. The first scheme is simpler and by an approximation to its input estimator the computation can be considerably reduced with insignificant performance degradation. The second scheme requires two Kalman filters and hence is more complex. However, since one of its two filters assumes input noise, it may outperform the first scheme when input noise is indeed present. A detector that compares the weighted norm of the estimated input vector to a threshold is used in each scheme. Its function is to guard against false updating of the trackers and to keep the error covariance small during constant velocity tracks. Simulation results for various target profiles are included. They show that in terms of tracking performance, both schemes are comparable. However, because of its computation simplicity, the first scheme is far superior.     

Ballistic missile track initiation from satellite observations

An algorithm is presented to initiate tracks of a ballistic missile in the initial exoatmospheric phase, using line of sight (LOS) measurements from one or more moving platforms (typically satellites). The major feature of this problem is the poor target motion observability which results in a very ill-conditioned estimation problem. The Gauss-Newton iterative least squares minimization algorithm for estimating the state of a nonlinear deterministic system with nonlinear noisy measurements has been previously applied to the problem of angles-only orbit determination using more than three observations. A major shortcoming of this approach is that convergence of the algorithm depends strongly on the initial guess. By using the more sophisticated Levenberg-Marquardt method in place of the simpler Gauss-Newton algorithm and by developing robust new methods for obtaining the initial guess in both single and multiple satellite scenarios, the above mentioned difficulties have been overcome. In addition, an expression for the Cramer-Rao lower bound (CRLB) on the error covariance matrix of the estimate is derived. We also incorporate additional partial information as an extra pseudomeasurement and determine a modified maximum likelihood (ML) estimate of the target state and the associated bound on the covariance matrix. In most practical situations, probabilistic models of the target altitude and/or speed at the initial point constitute the most useful additional information. Monte Carlo simulation studies on some typical scenarios were performed, and the results indicate that the estimation errors are commensurate with the theoretical lower bounds, thus illustrating that the proposed estimators are efficient     

Stability of Parameter Estimates for a Gaussian Process

Maximum-likelihood estimates for the levels of the mean value function and the covariance function of a Gaussian random process are investigated. The stability of these estimates is examined as the actual covariance function of the process deviates from the form assumed in the estimators. It is found that the time-bandwidth product for stationary processes represents an upper bound on the number of estimator terms that can be safely used when estimating with uncertainty about the process covariance function. This result is consistent with other interpretations of the time-bandwidth product and tempers the conclusion that, in principle, an infinite number of estimator terms can be used to obtain a perfect estimate of the covariance level. In practice, the estimate of the level can never be perfect, and the accuracy of the estimate depends on the observation interval. Finally, conditions are established to ensure asymptotic stability of the estimates and physical interpretations are presented.     

Adaptive Phased-Array Tracking in ECM using Negative Information

Target tracking with adaptive phased-array radars in the presence of standoff jamming presents both challenges and opportunities to the track filter designer. A measurement likelihood function is derived for this situation which accounts for the effect of both positive and negative contact information. This likelihood function is approximated a? a weighted sum of Gaussian terms consisting of both positive and negative weights, accounting for the positive and negative contact information. Additionally, recent theoretical results have been reported which have derived an accurate measurement error covariance in the vicinity of the jammer when adaptive beamforming is used by the radar to null the effects of the jammer. We compare the impact of using a likelihood function that accounts for negative contact information and the corrected measurement error covariance by comparing five Kalman filter-based trackers in five different scenarios. We show that only those track filters which use both the negative contact information and the corrected measurement error covariance are effective in maintaining track on a maneuvering target as it passes through the jamming region. This approach can also be generalized to any target tracking problem where the sensor response is anisotropic.     

Efficient particle filtering for road-constrained target tracking

The variable-structure multiple-model particle filtering approach for state estimation of road-constrained targets is addressed. The multiple models are designed to account for target maneuvers including "move-stop-move" and motion ambiguity at an intersection; the time-varying active model sets are adaptively selected based on target state and local terrain condition. The hybrid state space is partitioned into the mode subspace and the target subspace. The mode state is estimated based on random sampling; the target state as well as the relevant likelihood function associated with a mode sample sequence is approximated as Gaussian distribution, of which the conditional mean and covariance are deterministically computed using a nonlinear Kalman filter which accounts for road constraints in its update. The importance function for the sampling of the mode state approximates the optimal importance function under the same Gaussian assumption of the target state.     

A novel target detection approach based on adaptive radar waveform design

To resolve problems of complicated clutter, fast-varying scenes, and low signal-clutterratio (SCR) in application of target detection on sea for space-based radar (SBR), a target detection approach based on adaptive waveform design is proposed in this paper. Firstly, complicated sea clutter is modeled as compound Gaussian process, and a target is modeled as some scatterers with Gaussian reflectivity. Secondly, every dwell duration of radar is divided into several sub-dwells. Regular linear frequency modulated pulses are transmitted at Sub-dwell 1, and the received signal at this sub-dwell is used to estimate clutter covariance matrices and pre-detection. Estimated matrices are updated at every following sub-dwell by multiple particle filtering to cope with fast-varying clutter scenes of SBR. Furthermore, waveform of every following sub-dwell is designed adaptively according to mean square optimization technique. Finally, principal component analysis and generalized likelihood ratio test is used for mitigation of colored interference and property of constant false alarm rate, respectively. Simulation results show that, considering configuration of SBR and condition of complicated clutter, 9 dB is reduced for SCR which reliable detection requires by this target detection approach. Therefore, the work in this paper can markedly improve radar detection performance for weak targets.     

Variational Bayesian Kalman filter using natural gradient

We propose a technique based on the natural gradient method for variational lower bound maximization for a variational Bayesian Kalman filter. The natural gradient approach is applied to the Kullback-Leibler divergence between the parameterized variational distribution and the posterior density of interest. Using a Gaussian assumption for the parametrized variational distribution, we obtain a closed-form iterative procedure for the Kullback-Leibler divergence minimization, producing estimates of...     

Recursive 3-D motion estimation from a monocular image sequence

Consideration is given to the design and application of a recursive algorithm to a sequence of images of a moving object to estimate both its structure and kinematics. The object is assumed to be rigid, and its motion is assumed to be smooth in the sense that it can be modeled by retaining an arbitrary number of terms in the appropriate Taylor series expansions. Translational motion involves a standard rectilinear model, while rotational motion is described with quaternions. Neglected terms of the Taylor series are modeled as process noise. A state-space model is constructed, incorporating both kinematic and structural states, and recursive techniques are used to estimate the state vector as a function of time. A set of object match points is assumed to be available. The problem is formulated as a parameter estimation and tracking problem which can use an arbitrarily large number of images in a sequence. The recursive estimation is done using an iterated extended Kalman filter (IEKF), initialized with the output of a batch algorithm run on the first few frames. Approximate Cramer-Rao lower bounds on the error covariance of the batch estimate are used as the initial state estimate error covariance of the IEKF. The performance of the recursive estimator is illustrated using both real and synthetic image sequences     

Covariance control for multisensor systems

As the profusion of different sensors improves the capabilities of tracking platforms, tracking objectives can move from simply trying to achieve the most with a limited sensor suite to developing the ability to achieve more specific tracking goals, such as reducing the uncertainty in a target estimate enough to accurately fire a weapon at a target or to ensure that a mobile robot does not collide with an obstacle. Multisensor manager systems that balance tracking performance with system resources have traditionally been ill-suited for achieving such specific control objectives. This work extends the methods developed in single-sensor management schemes to a multisensor application using an approach known as covariance control, which selects sensor combinations based on the difference between the desired covariance matrix and that of the predicted covariance of each target.     

CFAR detection of distributed targets in non-Gaussian disturbance

The subject of detection of spatially distributed targets in non-Gaussian noise with unknown statistics is addressed. At the design stage, in order to cope with the a priori uncertainty, we model noise returns as Gaussian vectors with the same structure of the covariance matrix, but possibly different power levels (heterogeneous environment). We also assume that a set of secondary data, free of signal components, is available to estimate the correlation properties of the disturbance The proposed detector assumes no a priori knowledge about the spatial distribution of the target scatterers and ensures the constant false alarm rate (CFAR) property with respect to both the structure of the covariance matrix and the power levels. Finally, the performance assessment, conducted modeling the disturbance as a spherically invariant random process (SIRP), confirms its validity to operate in real radar scenarios     

Multisensor deployment using PCRLBS, incorporating sensor deployment and motion uncertainties

Recently, a general framework for sensor resource deployment (Hernandez, et. al. 2004) has been shown to allow efficient and effective utilization of a multisensor system. The basis of this technique is to use the posterior Cramer-Rao lower bound (PCRLB) to quantify and control the optimal achievable accuracy of target state estimation. In the original formulation (Hernandez, et. al. 2004) it was assumed that the sensor locations were known without error. In the current paper, the authors extend this framework by addressing the issues of imperfect sensor placement and uncertain sensor movement (e.g., sensor drift). The crucial consideration is then how these two forms of uncertainty are factored into the sensor management strategy. If unaccounted for, these uncertainties will render the output of the resource manager inaccurate and overoptimistic. The authors adjust the PCRLB to account for sensor location uncertainty, and we also allow for measurement origin uncertainty due to missed detections and false alarms. The work is motivated by the problem of tracking a submarine by adaptively deploying sonobuoys from a helicopter. Simulation results are presented to show the advantages of accounting for sensor location uncertainty within this focal domain of antisubmarine warfare. The authors note that the generic nature of the technique allows it to be utilized within other problem domains, including tracking ground-based targets using unattended ground sensors (UGSs) or unmanned aerial vehicles (UAVs)     

Automatic target recognition using sequences of high resolutionradar range-profiles

In this paper, we address the problem of joint tracking and recognition of a target using a sequence of high resolution radar (HRR) range-profiles. The likelihood function for the scene configuration combines a dynamics-based prior on the sequence of target orientations with a likelihood for range-profiles given the target orientation. A deterministic model and a conditionally Gaussian model for range-profiles are introduced, and the likelihood functions under each model are compared. Simulations are presented demonstrating recognition of mobile aircraft and ground targets, and results showing performance of the algorithm are given in terms of the expected angular estimation error and the rate of correct recognition     

Cumulant-based blind optimum beamforming

Sensor response, location uncertainty, and use of sample statistics can severely degrade the performance of optimum beamformers. We propose blind estimation of the source steering vector in the presence of multiple, directional, correlated or coherent Gaussian interferers via higher order statistics. In this way, we employ the statistical characteristics of the desired signal to make the necessary discrimination, without any a-priori knowledge of array manifold and direction-of-arrival (DOA) information about the desired signal. We then improve our method to utilize the data in a more efficient manner. In any application, only sample statistics are available, so we propose a robust beamforming approach that employs the steering vector estimate obtained by cumulant-based signal processing. We further propose a method that employs both covariance and cumulant information to combat finite sample effects. We analyze the effects of multipath propagation on the reception of the desired signal. We show that even in the presence of coherence, cumulant-based beamformer still behaves as the optimum beamformer that maximizes the signal-to-interference-plus-noise ratio (SINR). Finally, we propose an adaptive version of our algorithm simulations demonstrate the excellent performance of our approach in a wide variety of situations     

均方意义下三维雷达无偏转换测量误差协方差阵的推导

在以前的研究中,无偏转测量误差协方差阵是基于当前测量值得到的.为了能利用所有历史数据以得到更精确的转换测量误差协方差阵估计,文中在均方意义下,推导了三维雷达的最优无偏转换测量误差协方差阵.     

Multisensor resource deployment using posterior Cramer-Rao bounds

The development of a general framework for the systematic management of multiple sensors in target tracking in the presence of clutter is described. The basis of the technique is to quantify, and subsequently control, the accuracy of target state estimation. The posterior Cramer-Rao lower bound (PCRLB) provides the means of achieving this aim by enabling us to determine a bound on the performance of all unbiased estimators of the unknown target state. The general approach is then to use optimization techniques to control the measurement process in order to achieve accurate target state estimation. We are concerned primarily with the deployment and utilization of limited sensor resources. We also allow for measurement origin uncertainty, with sensor measurements either target-generated or false alarms. An example in which the aim is to track a submarine by deploying a series of constant false-alarm rate passive sonobuoys is presented. We show that by making some standard assumptions, the effect of the measurement origin uncertainty can be expressed as a state-dependent information reduction factor which can be calculated off-line. This enables the Fisher information matrix (FIM) to be calculated quickly, allowing Cramer-Rao bounds to be utilized for real-time, dynamic sensor management. The sensor management framework is shown to determine deployment strategies that enable the target to be accurately localized, and at the same time efficiently utilize the limited sensor resources.     

Decorrelation of broadband signals by Toeplitz-Block-Toeplitzstructuring

We propose F-norm of the cross-correlation part of the array covariance matrix as a measure of correlation between the impinging signals and study the performance of different decorrelation methods in the broadband case using this measure. We first show that dimensionality of the composite signal subspace, defined as the number of significant eigenvectors of the source sample covariance matrix, collapses in the presence of multipath and the spatial smoothing recovers this dimensionality. Using an upper bound on the proposed measure, we then study the decorrelation of the broadband signals with spatial smoothing and the effect of spacing and directions of the sources on the rate of decorrelation with progressive smoothing. Next, we introduce a weighted smoothing method based on Toeplitz-block-Toeplitz (TBT) structuring of the data covariance matrix which decorrelates the signals much faster than the spatial smoothing. Computer simulations are included to demonstrate the performance of the two methods     

Performance complexity study of several approaches to automatictarget recognition from SAR images

A framework which allows for the direct comparison of alternate approaches to automatic target recognition (ATR) from synthetic aperture radar (SAR) images is described and applied to variants of several ATR algorithms. This framework allows comparisons to be made on an even footing while minimizing the impact of implementation details and accounts for variation in image sizes, in angular resolution, and in the sizes of orientation windows used for training. Alternate approaches to ATR are characterized in terms of the best achievable performance as a function of the complexity of the model parameter database. Several approaches to ATR from SAR images are described and the performance achievable by each for a range of database complexities is studied and compared. These approaches are based on a likelihood test under a conditionally Gaussian model, log-magnitude least squared error, and quarter power least squared error. All approaches are evaluated for a wide range of parameterizations and the dependence on these parameters of both the resulting performance and the resulting database complexity is explored. Databases for all of the approaches are trained using identical sets of images and their performance is assessed under identical testing scenarios in terms of probability of correct classification, confusion matrices, and orientation estimation error. The results indicate that the conditionally Gaussian approach outperforms the other two approaches on average for both target recognition and orientation estimation, that accounting for radar power fluctuation improves performance for all three methods, and that the conditionally Gaussian approach normalized for power delivers average performance that is equal or superior to all other considered approaches     

Implementation and performance analysis of accurate localization of short-range targets based on multisensor systems

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