*2.1. PHD Filter*

The probability hypothesis density is defined as the first-order statistical moment of multiple-target posterior distribution. Similar to the constant-gain Kalman filter in single-target filtering, the PHD filter is the first-order moment approximation of the multi-target Bayes filter, which only recursively propagates first-order multi-target moments by time prediction and data-update steps. Suppose *Dk*−1|*k*−1(*x*) is the PHD at time *k* − 1, the predictor equation of the PHD filter can be expressed as

$$D\_{k|k-1}(\mathbf{x}) = b\_{k|k-1}(\mathbf{x}) + \int \left[ p\_S(\mathbf{x'}) \cdot f\_{k|k-1}(\mathbf{x}|\mathbf{x'}) + b\_{k|k-1}(\mathbf{x}|\mathbf{x'}) \right] \cdot D\_{k-1|k-1}(\mathbf{x'}) d\mathbf{x'},\tag{1}$$

where *fk*|*k*−1(*x* -- *x* ) is the single-target Markov transition density, *pS*(*x* ) is the probability that a target with state *x* at time *k* − 1 will survive at time *k*, *bk*|*k*−1(*x* - - *x* ) is the PHD of targets at time *k* spawned by a single target *x* at time *k* − 1, and *bk*|*k*−1(*x*) is the PHD of new targets entering the scene at time *k*.

At time *k* the sensor collects a new multi-target measurement set *Zk* = {*z*1, ··· , *zm*}, if we assume that the predicted multi-target distribution is approximately Poisson, the closed-form formula of corrector equation of the PHD filter can be derived as

$$D\_{k|k}(\mathbf{x}) \approx \left[1 - p\_D(\mathbf{x}) + p\_D(\mathbf{x}) \sum\_{z \in \mathbb{Z}\_k} \frac{L\_z(\mathbf{x})}{\lambda \cdot c(\mathbf{z}) + \int p\_D(\mathbf{x}) L\_z(\mathbf{x}) D\_{k|k-1}(\mathbf{x}) d\mathbf{x}} \right] \cdot D\_{k|k-1}(\mathbf{x}),\tag{2}$$

where *Lz*(*x*) is the single-target likelihood function, *pD*(*x*) is the probability that a target with state *x* at time *k* will be detected, λ is the average number of Poisson-distributed false alarms, the spatial distribution of which is governed by the probability density *c*(*z*).

The expected number of targets can be estimated by rounding the integral of the PHD over the entire state space, and then the state-estimates of the targets can be obtained from the local maxima of the PHD.
