The Spline Probability Hypothesis Density Filter

The Probability Hypothesis Density (PHD) filter is a multitarget tracker that can alleviate the computational intractability of the optimal multitarget Bayes filter. The PHD filter recursively estimates the number of targets and their PHD from a set of observations and works well in scenarios with false alarms and missed detections. Two distinct PHD filter implementations are available in the literature: the Sequential Monte Carlo Probability Hypothesis Density (SMC-PHD) and the Gaussian Mixture Probability Hypothesis Density (GM-PHD) filters. While particle-based PHD implementations may suffer from degeneracy, GM-based methods may not be suitable for highly nonlinear non-Gaussian systems. This paper proposes a B-Spline based Spline Probability Hypothesis Density (SPHD) filter, which has the capability to better approximate any arbitrary probability density function. The resulting algorithm can handle linear, non-linear, Gaussian, and non-Gaussian models. The SPHD filter can provide continuous estimates of the probability density function of the system state and it is immune to the degeneracy problem. The SPHD filter can maintain highly accurate tracks by taking advantage of dynamic knot movement, but at the expense of higher computational complexity, which makes it suitable for tracking a few high-value targets under difficult conditions. The SPHD filter derivations and simulations are provided in this paper.