Maximum Likelihood H-PMHT

Abstract

The methods described in this book so far are all Bayesian formulations of the tracking problem. This means that we have treated the kinematic state as a random variable with a prior distribution $p(\mathbf {x}_0^t)$ and an evolution distribution $p(\mathbf {x}_t^m|\mathbf {x}_{t-1}^m)$ that is driven by noise. The tracker amounts to deriving the joint probability $p(\mathbb {X},\mathbb {Z})$ or a point estimate based on its moments. This idea of kinematic states as a random process is just a model: in practice the movements of objects in the scene will not be a realisation of the assumed motion model. A different way to model the kinematic state is to treat the estimation as a curve-fitting problem. For example, we can fit a straight line through point measurements. This is a deterministic model of the state evolution and is not Bayesian; instead of finding the posterior probability density of the state, the curve parameters are obtained by optimising a cost function. For example, we could choose parameters to maximise the measurement likelihood or minimise the mean squared error. For Gaussian noise, these are equivalent. An advantage of using a deterministic model is that it is simple to impose constraints and the resulting estimate is forced to come from a desired family of trajectories. A disadvantage is that we are forced to work with a batch of data.