Second order multi-object filtering with target interaction using determinantal point processes

• Main Author: Teoh, Timothy Zhisheng
• Other Authors: Nicolas Privault
• Format: Theses and Dissertations
• Language: English
• Published: 2019
• Subjects: Science::Mathematics::Applied mathematics::Simulation and modeling; Engineering::Computer science and engineering::Mathematics of computing::Probability and statistics
• Online Access: https://hdl.handle.net/10356/90301; http://hdl.handle.net/10220/49907
• Institution: Nanyang Technological University

The Probability Hypothesis Density (PHD) filter is an algorithm that propagates the first-order moment of random finite sets for use in multi-target tracking based on sensor measurements. This algorithm usually assumes that targets behave independently, an hypothesis which may not hold in practice due to potential target interactions. We begin by discussing some general properties, functionals and branching dependency associated with spatial point processes that constitute the foundations in the construction of the filtering equations. Next, we show some extended computations incorporating Janossy densities in the derivation of closed-form solutions to the first and second order posterior factorial moment densities of the PHD filter. Implementation of the PHD filter using Sequential Monte Carlo (SMC) method on independent and interactive multi-target environments were explored, and the results obtained motivated the advancement to the Determinantal Point Processes (DPPs) setting. In the main contribution of this thesis, we constructed a second order PHD filter based on determinantal point processes, which are able to capture some level of interactions among configuration points in domain space. We derived the factorial moments and Janossy densities of determinantal point processes and branched determinantal point processes from its associated probability generating functionals (PGFls), which are described by the determinant of its kernel. Additionally, we derived combinatorial expressions of the marginal and joint Janossy densities of the general branched point processes. These results obtained are applied to derive closed-form first and second order factorial moment expressions of the determinantal PHD filtering equations. Our approach relies on posterior moment formulas for the estimation of a general hidden point process after a thinning operation and a superposition with a Poisson Point Process (PPP), and on suitable approximation formulas of the determinantal point process hypothesis. In our Determinantal PHD filter, the knowledge of first and second order moments of the prediction and posterior filtering equations suffices, in principle, to update the respective Janossy densities that characterize the underlying determinantal point process. The repulsive properties of determinantal point processes also apply to the modeling of negative correlation between distinct observation domains. An algorithm to implement the determinantal PHD filter using Sequential Monte Carlo (SMC) method is developed. Under specific scenarios, the numerical results obtained are shown to exhibit better performance when compared under the Poisson setting.