A practical implementation of joint multitarget probabilities

A Joint Multitarget Probability (JMP) is a posterior probability density p(T)(x(1),..., x(T) \ Z) that there are T targets (T an unknown number) with unknown locations specified by the multitarget state X = (x(1),..., x(T))(T) conditioned on a set of observations Z. This paper presents a numerical approximation for implementing JMP in detection, tracking and sensor management applications. A problem with direct implementation of JMP is that, if each x(1), t = 1,,..,T, is discretized on a grid of N elements, N-T variables are required to represent JMP on the T-target sector. This produces a large computational requirement even for small values of N and T. However, when the sensor easily separates targets, the resulting JMP factorizes and can be approximated by a product representation requiring only O((TN)-N-2) variables. Implementation of JMP for multitarget tracking requires a Bayes' rule step for measurement update and a Markov transition step for time update. If the measuring sensor is only influenced by the cell it observes, the JMP product representation is preserved under measurement update. However, the product form is not quite preserved by the Markov time update, but can be restored using a minimum discrimination approach. All steps for the approximation can be performed with O(N) effort. This notion is developed and demonstrated in numerical examples with at most two targets in a 1-dimensional surveillance region. In this case, numerical results for detection and tracking for the product approximation and the full JMP are very similar.