# A Particle PHD Filter for Dynamic Grid Map Building towards Indoor Environment
## 2 Particle Realization of PHD Filter under Grid Map  
### Multi-Object Bayesian Filter and PHD Filter Based on Random Finite Set  
Suppose that there are \(n_{k}\) target objects and \(m_{k}\) observations in the environment at time \(k\), which are represented as \(\left\{x_{k}^{(1)},\ldots,x_{k}^{(n_{k})}\right\}\) and \(\left\{z_{k}^{(1)},\ldots,z_{k}^{(m_{k})}\right\}\), respectively. The number of objects and the state of each object are random variables; then, the state of multiple targets of the environment at time \(k\) can be represented as a random finite set \(X_{k}=\left\{x_{k}^{(1)},\ldots,x_{k}^{(n_{k})}\right\}\in F(\varsigma)\), where \(x_{k}^{(n_{k})}\in\varsigma\) and \(\
abla(\varsigma)\) is the finite subset of \(\varsigma\). Similarly, the multi-target observations can be expressed by random finite sets \(Z_{k}=\left\{z_{k}^{(1)},\ldots,z_{k}^{(m_{k})}\right\}\in\
abla(Z)\), where \(z_{k}^{(m_{k})}\in Z\) and \(\
abla(Z)\) is the finite subset of \(Z\). The multi-objective Bayesian filter directly estimates the posterior probability density of multi-objective at \(k+1\) time through the observation until the time \(k+1\)[26, 27]. The iterative process of the algorithm can be divided into a prediction and update process. Assuming that the posterior probability density of multi-objective is \(\pi_{k}(X_{k}|Z_{1:k})\) at time \(k\), the predicted posterior probability density of multi-objective is:  
\[\pi_{k+1|k}(X_{k+1}|Z_{1:k})=\int f_{k+1|k}(X_{k+1}|X_{k})\pi_{k}(X_{k}|Z_{1:k })\delta X_{k} \tag{1}\]  
where \(f_{+}(X_{k+1}|X_{k})\) represents the multi-target transfer density, which can be approximated by the motion model of the object. When the observation \(Z_{k+1}\) is received at time \(k+1\), the updating equation of multi-target posterior probability density is as follows:  
\[\pi_{k+1|k+1}(X_{k+1}|Z_{1:k+1})=\frac{\psi_{k}(X_{k+1}|Z_{k+1})\pi_{k+1|k}(X_ {k+1}|Z_{1:k+1})}{\int\psi_{k+1}(X_{k+1}|Z_{k})\pi_{k+1|k}(X_{k+1}|Z_{1:k+1}) \delta X_{k+1}}. \tag{2}\]The integral over multi-object probability density function is set integral as:  
\[\int f(X)\delta X=f(\varnothing)+\sum_{i=1}^{\infty}\frac{1}{n!}\int f(x_{1}, \cdots x_{n})dx_{1}\cdots dx_{n}. \tag{3}\]  
The number of objects is also a variable that makes integration difficult at the same time. So, it can only be used to estimate the limit number of targets and usually adopts SMC (Sequential Monte Carlo) methods [7; 28]. In order to overcome this difficulty, [PERSON] proposed PHD filter [10; 11; 29], which operates directly on the first-order statistical moment, greatly simplifying the calculation. The standard PHD filter modeling process includes the prediction:  
\[D_{k+1|k}(x|z_{1:k})=\gamma_{k+1}(x)+\int\left(e_{k+1|k}(x_{k})f_{k+1|k}(x|x_{k })+b_{k+1|k}(x|x_{k})\right)D_{k}(x_{k}|z_{1:k})dx_{k} \tag{4}\]  
and update equation:  
\[ \begin{array}{c}D_{k+1|k+1}(x|z_{1:k+1})=(1-p_{D}(x))D_{k+1|k}(x)+\\ \sum\limits_{z\in\mathbb{Z}_{k+1}}\frac{p_{D}(x)g_{k+1}(z|x_{k+1}^{2})D_{k+1 |k}(x)}{\kappa_{k+1}(z)+\int p_{D}(\xi)g_{k+1}(z|\xi)D_{k+1|k}(\xi)d\xi}\end{array}  \tag{5}\]  
where \(b_{k+1|k}(x_{k+1}|x_{k})\) denotes the PHD of RFS \(B_{k+1|k}(x_{k})\) spawned by a target with previous state \(x_{k}\) and \(\gamma_{k+1}(x_{k+1})\) denotes the PHD of the spontaneous birth RFS \(\Gamma_{k+1}\) at time \(k+1\). \(f_{k+1|k}(\cdot|\cdot)\) denotes the transition probability density of individual targets. \(\kappa_{k+1}(z)=\lambda_{k+1}c(z)\) is the intensity of false alarms, \(\lambda_{k+1}\) is the average number of Poisson clutter points per scan, \(c(z)\) denotes clutter probability density. \(Z_{k+1}=\left\{z_{k+1}^{(1)},\ldots,z_{k+1}^{(n_{k+1})}\right\}\in\Psi(Z)\) is the time-sequence of observation sets, \(z\) is one of the measurements in \(z_{k+1}^{(n_{k+1})}\), \(p_{D}(x)\) is the state dependent probability of detection, and \(g_{k+1}(z|x_{k+1|k}^{r})\) is the likelihood of individual targets.