Random finite set filters for superpositional sensors

Abstract

The multi–object filtering problem is a generalization of the well–known single– object filtering problem. In essence, multi–object filtering is concerned with the joint estimation of the unknown and time–varying number of objects and the state of each of these objects. The filtering problem becomes particular challenging when the number of objects cannot be inferred from the collected observations and when no association between an observation and an object is possible. A rather new and promising approach to multi–object filtering is based on the principles of finite set statistics (FISST). FISST is a methodology, originally proposed by R. Mahler, that allows the formulation of the multi–object filtering problem in a mathematical rigorous way. One of the main building blocks of this methodology are random finite sets (RFSs), which are essentially finite set (FS) – valued random variables (RVs). Hence, a RFS is a RV which is not only random in the values of each element but also random in the number of elements of the FS. Under the premise that the observations are generated by detection–type sensors, many practical and efficient multi–object filters have been proposed. In general, detection–type sensors are assumed to generate observations that either originate from a single object or are false alarms. While this is a reasonable assumption in many multi–object filtering scenarios, this is not always the case. Central to this thesis is another type of sensors, the superposition (SPS)–type sensors. Those types of sensors are assumed to generate only one single observation that encapsulates the information about all the objects in the monitored area. More specifically, a single SPS observation is comprised out of the additive contribution of all the observations which would be generated by each object individually. In this thesis multi–object filters for SPS–type sensors are derived in a formal mathematical manner using the methodology of FISST. The first key contribution is a formulation of a SPS sensor model that, alongside errors like sensor noise, accounts for the fact that an object might not be visible to a sensor due to being outside of the sensor’s restricted field of view (FOV) or because it is occluded by obstacles. The second key contribution is the derivation of multi–object Bayes filter for SPS sensors that incorporates the aforementioned SPS sensor model. The third key contribution is the formulation of a filter variant that incorporates a multi–object multi–Bernoulli distribution as underlying multi–object state distribution, thus providing a multi–object multi–Bernoulli (MeMBer) filter variant for SPS–type sensors. As the stated variant turns out not to be conjugate, two approximations to the exact solution are given. The fourth key contribution is the derivation of computationally tractable implementations of the SPS MeMBer filters.