Abstract
Error bounds for nonlinear filtering are very important for performance evaluation and sensor management. This paper presents a comparative study of three error bounds for tracking filtering, when the detection probability is less than unity. One of these bounds is the random finite set (RFS) bound, which is deduced within the framework of finite set statistics. The others, which are the information reduction factor (IRF) posterior Cramer-Rao lower bound (PCRLB) and enumeration method (ENUM) PCRLB are introduced within the framework of finite vector statistics. In this paper, we deduce two propositions and prove that the RFS bound is equal to the ENUM PCRLB, while it is tighter than the IRF PCRLB, when the target exists from the beginning to the end. Considering the disappearance of existing targets and the appearance of new targets, the RFS bound is tighter than both IRF PCRLB and ENUM PCRLB with time, by introducing the uncertainty of target existence. The theory is illustrated by two nonlinear tracking applications: ballistic object tracking and bearings-only tracking. The simulation studies confirm the theory and reveal the relationship among the three bounds.
Highlights
In the Bayesian framework, the complete posterior density of the state is necessary, in order to obtain the optimal recursive random state estimate for a classical nonlinear filtering problem by using various sensors [1], but this problem has no analytic closed-form solution
For a discrete-time nonlinear filtering problem, the target state is modeled as a random vector, and the state dynamic equation is given by: xk +1 = fk + w k where xk ∈ Rm is the target state at time step k, m is the dimensionality of the target state, fk is the state transition function, and wk is a zero-mean white Gaussian process noise, with covariance matrix Qk
The reason is that the random finite set (RFS) bound and enumeration method (ENUM) posterior Cramer-Rao lower bound (PCRLB) are calculated by different models
Summary
In the Bayesian framework, the complete posterior density of the state is necessary, in order to obtain the optimal recursive random state estimate for a classical nonlinear filtering problem by using various sensors [1], but this problem has no analytic closed-form solution. There are many tracking algorithms, such as [16] and [17], based on RFS statistics, and they generally have ignored the issue of data association Because both target states and estimates are modeled as RFS, traditional Euclidean distance could not be applied for calculating the error. The paper presents a comparative study of the RFS bound in [20] and the PCRLBs in the case where detection probability PD < 1, such as IRF PCRLB and ENUM PCRLB We discuss this problem in two cases, one is when the target exists from the beginning to the end, and the other is when new targets might appear and existing targets could disappear.
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