势平衡多目标多伯努利滤波器高斯混合实现的收敛性分析
Convergence analysis for the Gaussian mixture implementation of the CBMeMBer filter

研究了势平衡多目标多伯努利(cardinality balanced multi-target multi-Bernoulli, CBMeMBer) 滤波器高斯混 合(Gaussian mixture, GM)实现的收敛性问题. 证明在线性高斯条件下, 若GM–CBMeMBer 滤波器的高斯项足够多, 则它一致收敛于真实的CBMeMBer滤波器. 并且证明在弱非线性条件下, GM–CBMeMBer滤波器的扩展卡尔曼(extended Kalman, EK)滤波近似实现—– EK–GM–CBMeMBer滤波器, 若每个高斯项的协方差足够小, 也一致收敛于真 实的CBMeMBer 滤波器. 本文的研究目的是从理论上给出CBMeMBer 滤波器GM实现的收敛结果, 以完善CBMe- MBer 滤波器对多目标跟踪的理论研究.
The convergence for the Gaussian mixture (GM) implementation of the cardinality balanced multi-target multi-Bernoulli (CBMeMBer) filter is studied. This paper proves that the GM–CBMeMBer filter converges uniformly to the true CBMeMBer filter in the linear Gaussian model as the number of Gaussians in the mixture tends to infinity. In addition, this paper proves the extended Kalman (EK) filter approximations of the GM–CBMeMBer filter in weak nonlinear condition — EK–GM–CBMeMBer filter, converges uniformly to the true CBMeMBer filter as the covariance of each Gaussian term tends to zero. The purpose of this paper is to theoretically present the convergence results of the CBMeMBer filter’s GM implementation, perfecting the theoretical research of the CBMeMBer filter for the multi-target tracking problem.