Estimation with Random Finite Sets

Introduction

The previous chapter provided the motivation to adopt an RFS representation for the map in both FBRM and SLAM problems. The main advantage of the RFS formulation is that the dimensions of the measurement likelihood and the predicted FBRM or SLAM state do not have to be compatible in the application of Bayes theorem, for optimal state estimation. The implementation of Bayes theorem with RFSs (equation 2.15) is therefore the subject of this chapter. It should be noted that in any realistic implementation of the vector based Bayes filter, the recursion of equation 2.13 is, in general, intractable. Hence, the well known extended Kalman filter (EKFs), unscented Kalman filter (UKFs) and higher order filters are used to approximate multi-feature, vector based densities. Unfortunately, the general RFS recursion in equation 2.15 is also mathematically intractable, since multiple integrals on the space of features are required. This chapter therefore introduces principled approximations which propagate approximations of the full multi-feature posterior density, such as the expectation of the map. Techniques borrowed from recent research in point process theory known as the probability hypothesis density (PHD) filter, cardinalised probability hypothesis density (C-PHD) filter, and the multi-target, multi-Bernoulli (MeMBer) filter, all offer principled approximations to RFS densities. A discussion on Bayesian RFS estimators will be presented, with special attention given to one of the simplest of these, the PHD filter. In the remaining chapters, variants of this filter will be explained and implemented to execute both FBRM and SLAM with simulated and real data sets.

The notion of Bayes optimality is equally as important as the Bayesian recursion of equation 2.15 itself. The following section therefore discusses optimal feature map estimation in the case of RFS based FBRM and SLAM, and once again, for clarity, makes comparisons with vector based estimators. Issues with standard estimators are demonstrated, and optimal solutions presented.