Sequential Monte Carlo Smoothing with Finite Set Statistics

受 1960 年卡尔曼 (Kalman) 发现线性滤波问题解决方案的推动，过去半个世纪人们对随机滤波和平滑产生了相当大的兴趣。通过将这项工作纳入美国宇航局的阿波罗导航计算机进行轨迹估计，这项工作几乎立即得到了应用。卡尔曼的发现的重要性体现在它对控制论、概率论、金融数学和信号处理的影响。自20世纪60年代以来，在各种滤波问题的解决方案出现之后不久，就出现了相应的平滑问题的解决方案。最近关于非线性滤波和平滑的工作受到了顺序蒙特卡罗理论的启发。随机滤波、预测和平滑是动态系统估计理论中的基本概念。该系统是一个部分观测的物理对象，其随时间的行为由一组方程控制，这些方程对动力学以及观测与对象状态之间的关系进行建模。系统中的不确定性是由于问题的噪声性质造成的，要么来自系统的未知且不可预测的运动，要么来自通过噪声传感器观察系统测量的不准确性。过滤、预测和平滑是基于对系统行为随时间的噪声观察来估计系统状态的问题的精确数学描述：预测是根据截至当前时间的测量来预测系统在未来某个时间点的状态。过滤是根据到该点为止的所有测量来估计系统状态的每个时间点。通常，预测和过滤一起使用来形成一组递归方程，用于预测和更新系统状态的估计。平滑与预测和过滤的不同之处在于，可以根据一批测量来确定特定时间点的系统状态估计，其中一些测量可能晚于我们感兴趣的时间收集。这意味着，尽管由于可以获得有关系统的更多信息而可以获得更准确的估计，但在生成当时的状态估计时不可避免地存在延迟。尽管单对象过滤方面的研究很丰富，但从数学原理上将过滤概念推广到多对象系统是最近的发展，受航空航天应用中的多目标跟踪问题的启发，在有限集统计（FISST）框架中制定。多目标跟踪算法的目的是从一个或多个传感器提供的目标的噪声观测序列中检测、跟踪和识别目标。由于这些观测结果往往存在许多误报，并且目标可能并不总是引起观测结果，因此这个问题变得更加复杂。从单目标场景到多目标环境的扩展并非易事，因为目标的数量可能未知并且随时间变化，在未观察到目标的情况下会出现漏检，并且由于杂波，观察结果可能是误报。此外，可能需要知道目标的身份才能确定其轨迹。这项工作将开发用于平滑多对象系统的新方法。这里提出的这项工作旨在研究多对象平滑器，以便在可能存在许多误报且并不总是观察到目标的环境中联合估计对象的数量及其状态向量。这个问题的解决方案可能会导致使用顺序蒙特卡罗近似的实际实现。该问题的成功解决方案将直接适用于雷达、电光和声纳等许多传感器应用中的一系列工业多传感器多目标跟踪问题。

There has been considerable interest in stochastic filtering and smoothing in the last half-century, motivated by the discovery in 1960 of the solution to the linear filtering problemby Kalman. Applications of this work were found almost immediately through the incorporation into NASA's Apollo navigation computer for trajectory estimation. The importance of Kalman's discovery is illustrated by the the impact it has had in control theory, probability theory, financial mathematics, and signal processing. Since the 1960's, soon after the solutions to various filtering problems came corresponding solutions to the smoothing problems. More recent work on non-linear filtering and smoothing has been inspired by sequential Monte Carlo theory.Stochastic filtering, prediction, and smoothing are fundamental concepts in the theory of estimation of dynamic systems. The system is a partially observed physical object whose behaviour over time is governed by a set of equations modelling the dynamics and the relationship between the observations and the object state. Uncertainty in the system is due to the noisy nature of the problem, either from unknown and unpredictable motion of the system, or from inaccuracy in observing measurements of the system through a noisy sensor. Filtering, prediction and smoothing are precise mathematical descriptionsof the problem of estimating the state of the system based on noisy observations of its behaviour over time: Prediction is the forecasting of the state of the system at some future point in time based on measurements up to the current time. Filtering is the estimation at each point in time of the state of the system based on all of the measurements up to that point. Typically prediction and filtering are used together to form a set of recursive equations for predicting and updating the estimate of the state of a system. Smoothing differs from prediction and filtering in that the estimate of the state of the system at a specific point in time can be determined from a batch of measurements, some of which may be collected later than the time that we are interested in. This means that there is inevitably a delay in producing the estimate of the state at that time, though more accurate estimates can be obtained since more information is available about the system.Despite the wealth of research in single-object filtering, a mathematically principled generalisation of filtering concepts to multi-object systems is a recent development, formulated in the framework of Finite Set Statistics (FISST) motivated by the problem of multiple-target tracking in aerospace applications. The purpose of multiple target tracking algorithms is to detect, track and identify targets from sequences of noisy observations of the targets provided by one or more sensors. This problem is complicated by the fact that these observations tend to have many false alarms and targets may not always give rise to observations. The extension from a single-target scenario to a multiple-target environment is non-trivial since the number of targets may not be known and varies with time, there are missed detections where the target is not observed and observations may be false alarms due to clutter. In addition, the identities of the targets may need to be known to determine their trajectories. This work will develop new methodologies for smoothing of multi-object systems. This work proposed here aims to investigate multi-object smoothers for jointly estimating the number of objects and their state vectors in environments where there can be many false alarms and the targets are not always observed. Solutions to this problem could lead the way to practical implementations using sequential Monte Carlo approximations. The successful solution to this problem would be directly applicable to a range of industrial multi-sensor multi-target tracking problems in many sensor applications including radar, electro-optics and sonar.