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31 July 2002 Point target-clusters and continuous-state multitarget statistics
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In conventional single-sensor, single-target statistics, many techniques depend on the ability to apply New- tonian 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 discontinuow 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 = {xi}, X = {x1, x2},. . . 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) of a fully continuous multitarget state variable X. 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 = (a, x). Here, * 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) that can potentially be used in developing practical computational techniques.
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Ronald P. S. Mahler "Point target-clusters and continuous-state multitarget statistics", Proc. SPIE 4729, Signal Processing, Sensor Fusion, and Target Recognition XI, (31 July 2002);

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