Algorithm 2. CRPF algorithm for target tracking problem

Initialization

1.    $x_0^{\left(i\right)}\sim p_0\left(x_0\right)$ $, {\mathcal{C}}_0^{\left(i\right)}=0$ $, {\mathbf{\sigma }}_0^{2,\left(i\right)}$ $, i=1,\dots ,N_s$ $, \mathrm{the} \mathrm{weighted}\text{-}\mathrm{particle} \mathrm{set} {\mathsf{\Xi }}_0={\left\{x_0^{\left(i\right)},{\mathcal{C}}_0^{\left(i\right)}\right\}}_{i=1}^{N_s}$

PMF Update

2.    ${\mathcal{R}}_k^{\left(i\right)}=\lambda {\mathcal{C}}_{k-1}^{\left(i\right)}+{\Vert {\mathbf{z}}_k-f_y\left(f_x\left(x_{k-1}^{\left(i\right)}\right)\right)\Vert }^q$ $, q=1,2$ $, {\widehat{\mathbf{\pi }}}_k^{\left(i\right)}\propto \mu \left({\mathcal{R}}_k^{\left(i\right)}\right)=\frac{1}{{\left({\mathcal{R}}_k^{\left(i\right)}-\min {\left\{{\mathcal{R}}_k^{\left(i\right)}\right\}}_{i=1}^{N_s}+\delta \right)}^{\beta }}$ $, i=1,\dots ,N_s.$

3.    ${\left\{{\widehat{x}}_{k-1}^{\left(i\right)},{\widehat{\mathcal{C}}}_{k-1}^{\left(i\right)}\right\}}_{i=1}^{N_s}=\mathrm{RESAMPLE}\left[{\left\{{\widehat{\mathbf{\pi }}}_k^{\left(i\right)}\right\}}_{i=1}^{N_s}\right]$

Particle Propagation and Variance Update

4.    $E_{p_k\left(x_k^{}|{\widehat{x}}_{k-1}^{\left(i\right)}\right)}\left[x_k^{}\right]=f_x\left({\widehat{x}}_{k-1}^{\left(i\right)}\right)$ $, Co{v}_{p_k\left(x_k^{}|{\widehat{x}}_{k-1}^{\left(i\right)}\right)}\left[x_k^{}\right]={\mathbf{\sigma }}_k^{2,\left(i\right)}{\mathit{I}}_{\left[x\right]}$ $, x_k^{\left(i\right)}\sim p_k\left(x_k^{}|{\widehat{x}}_{k-1}^{\left(i\right)}\right).$

5.    ${\mathbf{\sigma }}_k^{2,\left(i\right)}=\{\begin{array}{ll}{\sigma }_{k-1}^{2,\left(i\right)}& \mathrm{t}\le 10\\ \frac{k-1}{k}{\sigma }_{k-1}^{2,\left(i\right)}+\frac{{\Vert x_k^{\left(i\right)}-g\left({\widehat{x}}_{k-1}^{\left(i\right)}\right)\Vert }^2}{k\times \dim \left[x\right]}& \mathrm{t}>10\end{array}$ $, i=1,\dots ,N_s$

State Estimation

6.    ${\mathcal{C}}_k^{\left(i\right)}=\lambda {\mathcal{C}}_{k-1}^{\left(i\right)}+{\Vert {\mathbf{z}}_k-f_y\left(x_k^{\left(i\right)}\right)\Vert }^q$ $, {\widehat{\mathbf{\pi }}}_k^{\left(i\right)}\propto {\mu }_2\left({\mathcal{C}}_k^{\left(i\right)}\right)=\frac{1}{{\left({\mathcal{C}}_k^{\left(i\right)}-\min {\left\{{\mathcal{C}}_k^{\left(i\right)}\right\}}_{i=1}^{N_s}+\delta \right)}^{\beta }}$ $, \delta ,\beta >0$ $, \mathrm{normalized} \mathrm{to} {\mathbf{\pi }}_k^{\left(i\right)},$

7.    ${\widehat{x}}_k=x_k^{mean}={\displaystyle \sum _{i=1}^{N_s}{\mathbf{\pi }}_k^{\left(i\right)}{x}_k^{\left(i\right)}}$ $, i=1,\dots ,N_s$