Point target-clusters and continuous-state multitarget statistics

In conventional single-sensor, single-target statistics, many techniques depend on the ability to apply Newtonian calculus techniques to functions of a continuous variable such as the posterior density, the sensor likelihood function, the Markov motion-transition density, etc. Unfortunately, such techniques cannot be directly generalized to multitarget situations, because conventional multitarget density functions f(X) are inherently discontinuous with respect to changes in target number. That is, the multitarget state variable X experiences discontinuous jumps in its number of elements: X = 0, X = {x(1)}, X = {x(1),x(2)},... In this paper we show that it is often possible to render a multitarget density function f(X) continuous and differentiable by extending it to a function f(X) over dot of a fully continuous multitarget state variable (X) over dot. This is accomplished by generalizing the concept of a point target, with state vector x, to that of a point target-cluster, with augmented state vector (x) over dot = (a, x). Here, (x) over dot is interpreted as multiple targets co-located at target-state x, whose expected number is a > 0. Consequently, it becomes possible to define a Newtonian differential calculus of multitarget functions f(X)over dot that can potentially be used in developing practical computational techniques.