The Probability Generating Functional for Finite Point Processes, and Its Application to the Comparison of PHD and Intensity Filters

Many radar and sonar sensor systems generate several point measurements at every scan. Some measurements are due to targets and others are due to clutter, or scatterers, in the sensor field of view. The multitarget tracking problem is to estimate the number of targets and their states given the measurements. The multi-hypothesis tracking (MHT) method for solving this problem is based on two widely accepted assumptions: 1) targets are points; and 2) sensors generate at most one measurement per target per scan. The second is called the “at most one measurement per target” rule. It is the cause of the intrinsically high computational complexity of optimal MHT algorithms and, consequently, the reason so many diverse kinds of alternative suboptimal algorithms are widely studied. This paper concerns the class of multitarget tracking filters based on finite point process models for multiple target states and sensor measurement sets. Two specific kinds of filters are discussed–the PHD (probability hypothesis density) filter and the iFilter (intensity filter). Many of the differences between these filters are due to the different models of the measurement set. Contrasting these two filters in this way has the added benefit of revealing the fundamental importance of the classical methods of finite point processes for tracking applications. Section 2 provides background on finite point processes and reviews their application to multitarget tracking filters. The next two sections are largely didactic. Section 3 defines the probability generating functional (PGFL) of a single point process. Basic results related to the PGFL are derived there. PGFLs play a central role–they characterize the probability structures underpinning the filters. Section 4 defines the bivariate PGFL of two finite point processes. The general Bayes posterior point process is defined, and its PGFL is derived from the bivariate PGFL. Section 5 derives the PHD filter and iFilter as examples of the general Bayes posterior point process. The PHD filter uses a traditional clutter model, while the iFilter uses a scattering model. These modeling differences manifest themselves in the PGFLs of the filters, thus exposing the similarities and differences between them. Conclusions and concluding remarks are given in Section 6.