A Technique for Deriving Multitarget Intensity Filters Using Ordinary Derivatives

This paper shows that functional derivatives of the probability generating functional (PGFL) of a finite point process can be calculated using ordinary derivatives. The result is new, and it is potentially useful to the class of Bayesian multitarget tracking problems that is based on finite point process models for targets and measurements. In this class, the distribution of the Bayes posterior multitarget process is a ratio of functional derivatives of the joint measurement-target PGFL. In some problems evaluating the functional derivatives is only a tedious task, but in other problems the number of terms in the derivatives is prohibitively large and limits practical applications of the method. The proof is straightforward–we reduce the PGFL to an ordinary function that is conceptually straightforward to differentiate. This function is called a secular function to emphasize that it is an “ordinary” function and not a functional. Existing symbolic software packages can be used to differentiate the secular function, a fact that is potentially of practical importance since software for functional differentiation of the PGFL does not seem to be available. The methods of this paper use the established theory of PGFLs and their functional derivatives. The proposed methods are compatible with particle, or sequential Monte Carlo (SMC), filter implementations. The basic strategy is to embed symbolic differentiation software in the production code and evaluate the symbolic derivatives of the secular functions at the points of the particle filter. One of the purposes of this paper is to show that this is a theoretically feasible strategy. Its practical utility is outside the scope of the paper. Two tracking applications where functional differentiation causes serious difficulties are discussed. One is multisensor target tracking [9]. The other is extendedtarget tracking problems in which targets can produce more than one measurement [8, 11]. The secular functions for both problems are derived. Functional differentiation of the PGFL is the result of a double limit. A theoretical question naturally arises, “Can these limits be interchanged?” The answer is, “Yes, for the problems of interest here.” This result seems to be new. It gives a better understanding of PGFLs and their relationship to classical probability generating functions (PGFs). Section II speaks of the PGFL as an encoding of the multitarget tracking problem and functional differentiation of the PGFL as the decoding algorithm. Section III gives a simple example of the method we use to reduce PGFLs to secular functions. Section IV proves that for the class of PGFLs of interest in this paper, ordinary derivatives of secular functions are identical to functional derivatives of PGFLs. Section V gives several examples of secular functions, including those for multisensor and extended-target tracking problems. Section VI discusses finite differences and series expansion