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This article concerns the problem of the estimation bound for tracking an extended target observed by a high resolution sensor. Two types of commonly used models for extended targets and the corresponding posterior Cramer-Rao lower bound (PCRLB) are discussed. The first type is the equation-extension model which extends the state space to include parameters such as target size and shape. Thus, the extended state vector can be estimated through the measurements obtained by a high resolution sensor. The measurement vector is also an expansion of the conventional one, and the additional measurements such as target extent can provide extra information for the estimation. The second model is based on multiple target measurements, each of which is an independent random draw from a spatial probability distribution. As the number of measurements per frame is unknown and random, the general form of the measurement contribution to the Fisher information matrix (FIM) conditional on the number of measurements is presented, and an extended information reduction factor (EIRF) approach is proposed to calculate the overall FIM and, therefore, the PCRLB. The bound of the second extended target model is also less than that of the point model, on condition that the average number of measurements is greater than one. Illustrative simulation examples of the two models are discussed and demonstrated.

In a conventional target tracking framework, it is usually assumed that the sensor obtains one measurement of a single target (if detected) at each time step, which is referred to as the point target model. However, high resolution sensors have recently become more widely used and are able to resolve multiple point features on a single extended target. The potential to make use of the multiple sensor measurements is referred to as extended target tracking. An extended target is usually seen as a rigid or semi-rigid body. In contrast to the conventional point target model, the measurements provided by high-resolution sensors can provide extra information to improve target identification and data association [

The calculation of the PCRLB for two different types of extended target tracking models is considered in this paper. The first type of extended target tracking model extends the state and measurement equations [

In the second type of extended target tracking approaches, the state space is the same as that of the point target model, and the measurement of the target is represented by a spatial probability distribution. The target states are estimated based on the multiple measurements, which come from a region of high spatial density [

The paper is organized as follows. Section 2 introduces the definition and recursive formulation of the PCRLB for the general nonlinear filtering problem. Section 3 introduces the calculation and theoretical development of the PCRLB for two types of extended target tracking models. In Section 4, illustrative simulation examples corresponding to the two types of models are presented and discussed. Conclusions are given in Section 6.

Let _{k}_{k}_{k}_{k}

Tichavsky _{k}_{k}_{k}_{k}_{k}_{k}_{k}_{φ}_{z}_{0} (the initial FIM is the inverse of the initial target distribution covariance,

If the dynamic and measurement noise are additive Gaussian, _{k}_{k}_{k}_{k}_{k}_{k}

Furthermore, if the target dynamics are linear (_{k}_{k}_{k}X_{k}_{z}_{k}

For the first type of extended target tracking models, the dynamic and measurement vectors are both extensions of the ones in the point model with additional states and measurements. The equations are thus also extensions of the point model and are still in the general form. The superscripts “^{n}^{p}

In the second type of extended target tracking framework, the state space and the dynamic equation of the extended target model are in the general form described by (2). However, a high resolution sensor can obtain multiple measurements at each time step. It is assumed that _{k}_{k}_{k}_{k}_{z}_{k}_{k}_{k}_{k}

For the spatial distribution model of the extended target, the number of sensor measurements per frame is unknown and random. Thus, the recursive calculation of the PCRLB cannot be applied directly. Referencing the ideas for calculating the PCRLB in the case of a single point target in a cluttered environment [_{k}

It is noticed that for each _{k}_{k}_{z}_{k}_{k}_{k}

The following expression is defined as the measurement contribution in the conventional case (where just one measurement originates from _{z}_{k}_{k}

Thus:

The conclusion above obeys the usual intuition that the measurement uncertainty is reduced by multiple _{k}

If the dynamic and measurement noise are additive Gaussian (described by (

_{k}_{k}_{k}_{k}

Because the probability distribution of _{k}_{k}

If the dynamic and measurement noise are additive Gaussian (described by (

Furthermore, if the target dynamics are linear (_{k}_{k}_{k}X_{k}_{k}_{z}

The following simulation examples are presented to illustrate the numerical results of PCRLBs for the two types of extended target tracking models.

The first example is tracking an extended target whose shape is modeled as a stick using an equation-extension model for extended target tracking, which is described in Section 3.1. As shown in _{0}, _{0}) on a 2-D plane, and the state vector of the target is ^{e}

The sensor obtains not only the conventional target position measurements, such as the distance and azimuth angle of the target centroid, but also the extended measurements that describe the target extent. The measurement model is:

The first four elements of the third and fourth row of the Jacobian

The last elements of the third and fourth row of the Jacobian

In this simulation example, the sensor is static and located at the origin of the coordinate system, _{0}, _{0}) = (0, 0). The target moves with initial velocity _{0} = 10 in the direction with the initial VLOS angle _{0} = 20°. The initial position of the target is (15,000, 10,000), and the initial length is _{0} = 50. The initial FIM is _{0} = {[^{2}}^{−1}, and the covariance of the state noise is _{k}_{ρ}_{θ}_{L}_{W}

Because the target dynamics are random (with non-zero dynamic noise) in the simulation scenario, the calculation of the measurement contribution using (_{k}_{+1}. A sampling scheme is used here. From the initial target state and the dynamic model, multiple target state sequences are generated, and the corresponding measurement contribution is computed. The overall measurement contribution is then computed as an average of the measurement contributions conditional on each state sequence. In this simulation, 10,000 state sequences were sampled to approximate the measurement contribution. The comparison of the bounds of target centroid dynamics (position and velocity) using both the extended and point target models is shown in

_{L}_{L}_{L}_{L}_{L}_{L}_{0} = 20°). The influence of the sensor-target geometry on the estimation bound is reported below.

Next, the impact of the VLOS angle on the PCRLB is analyzed. The effect of _{0} = 0°, 45°, and 90° for four combinations of the values of _{L}_{W}

In _{0} = 0° (180°), is dramatically smaller than that for _{0} = 90°. The superior performance bound is a result of the target orientation being along the direction with the best measurement accuracy.

A similar interpretation can be proposed for

The following simulation example is presented to illustrate the performance bounds for tracking an extended target that can be resolved as multiple point features at each sampling time by a high resolution sensor. The simulation scenario discussed below is similar to Example 1 discussed in Section 5.1. The observer is located at (_{0}, _{0}) on a 2-D plane. The target is moving with nearly constant velocity (NCV), and the dynamic model is of the conventional form:
^{T}_{k}_{k}_{k}_{T}_{k}_{k}

The method of calculating the mathematical expectation with respect to the state vector in (

In the simulation, the sensor is also static and at the origin of the coordinate system, while the target moves with initial velocity (_{x}_{0}, _{y}_{0}) = (10, 15) starting from the initial position (15000,10000). The initial FIM is _{0} = {[^{−2}}^{−1}. The covariance of the state noise is _{k}_{ρ}_{θ}_{T}_{T}_{T}

In this article, the calculation of the PCRLB for two types of extended target tracking models is reported. For the equation extension (first type) extended target model, the dynamic and measurement equations are extensions of those of the point target models and are still in the general nonlinear filtering form. The PCRLB is then calculated through the recursive formulation, and the bound of the target centroid dynamics estimation of the extended model is always smaller than that of the point model. For the spatial distribution (second type) extended target model, the general form of the measurement contribution for a specific number of measurements with no clutter is presented in the paper, and the EIRF approach is introduced to calculate the overall measurement contribution and therefore the PCRLB. Illustrative simulation examples for the two types of extended target tracking models are also presented to verify the theoretical development and demonstrate the influence of parameters on the PCRLB. The theoretical and numerical results suggest the superior performance bound for both the two types of extended target models.

This work was supported in part by the National Natural Science Foundation of China (No. 60901057) and in part by the National Basic Research Program of China (973 Program, No. 2010CB731901).

_{d}<1

The stick shaped extended target model.

The comparison of the

The comparison of the
_{L}

The impact of φ_{0} on the
_{L}_{W}_{L}_{W}_{L}_{W}_{L}_{W}_{L}_{W}

The

The PCRLB of the multi-measurement extended target tracking for different mean numbers of measurements for (a) the target position in the direction of the