Second-order multi-object filtering with target interaction using determinantal point processes
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Second-order multi-object filtering with target interaction using determinantal point processes Nicolas Privault1
· Timothy Teoh1
Received: 15 June 2019 / Accepted: 3 November 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract The probability hypothesis density (PHD) filter, which is used for multi-target tracking based on sensor measurements, relies on the propagation of the first-order moment, or intensity function, of a point process. This algorithm assumes that targets behave independently, an hypothesis which may not hold in practice due to potential target interactions. In this paper, we construct a second-order PHD filter based on determinantal point processes which are able to model repulsion between targets. Such processes are characterized by their first- and second-order moments, which allows the algorithm to propagate variance and covariance information in addition to firstorder target count estimates. 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, and on suitable approximation formulas in the determinantal point process setting. The repulsive properties of determinantal point processes apply to the modeling of negative correlation between distinct measurement domains. Monte Carlo simulations with correlation estimates are provided. Keywords Probability hypothesis density (PHD) filter · Higher-order statistics · Correlation · Second-order moment · Determinantal point processes · Multi-object filtering · Multi-target tracking Mathematics Subject Classification 60G35 · 60G55 · 62M30 · 62L12
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Nicolas Privault [email protected] Timothy Teoh [email protected]
1
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
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Mathematics of Control, Signals, and Systems
1 Introduction Probability hypothesis density (PHD) filters have been introduced in Mahler [20] for multi-target tracking in cluttered environments. The construction of the prediction point process therein uses multiplicative point processes; see, e.g., Moyal [25], Moyal [24], by thinning and shifting a prior point process , and superposition with a birth point process. The posterior point process | is obtained by conditioning given a measurement point process of targets , also constructed by thinning, shifting and superposition. This step relies on Bayesian estimation with a Poisson point process prior; see, e.g., van Lieshout [33], Mori [23], Portenko et al. [27]. PHD filters have low complexity, and they allow for explicit update formulas; see, e.g., Clark et al. [3] for a review. While the PHD filter of Mahler [20] is based on Poisson point processes, several extensions of the PHD filter to non-Poisson prior distributions have been proposed. Cardinalized Probability hypothesis density (CPHD) filters have been introduced in Mahler [21] as a generalization in which
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