## 4 Our Proposed GCI-IGGM-PMBM Scheme
### IGGM Implementation
Given that the intensity of Poisson process is in an IGGM form and the density of the \(ith\) Bernoulli component in the \(jth\) hypothesis is in a single IGGform, the predicted intensity of the Poisson and Bernoulli components of the MBM process are presented as described below. 1. PPP process \[ \begin{array}{c}D_{k|k-1}^{u}(X,\xi)=\sum\limits_{l=1}^{l_{k-1}^{u}}w_{k|k-1}^{ l_{k-1}^{u}}\mathcal{IGG}(X,\xi_{k;k};\mu_{k;k-1}^{l_{k-1}^{u}},p_{k;k-1}^{l_{k-1}^{u}}, \mu_{k;k-1}^{l_{k-1}^{u}},p_{k;k-1}^{l_{k-1}^{u}})\\ +\sum\limits_{l=1}^{l_{k-1}^{u}}w_{k|k-1}^{l_{k-1}^{u}}\mathcal{GAM}(\xi_{k; \varepsilon};\mu_{k;k-1}^{l_{k-1}^{u}},p_{k;k-1}^{l_{k-1}^{u}})\mathcal{IGG}(X, \xi_{k;\varepsilon};\mu_{k;k-1}^{l_{k-1}^{u}},p_{k;k-1}^{l_{k-1}^{u}},p_{k;k-1}^ {l_{k-1}^{u}})\\ \\ w_{k|k-1}^{l_{1},B}=p_{S,k|k-1}w_{k-1}^{l_{1},B},w_{k|k-1}^{l_{k-1}^{u}}=p_{S,k |k-1}w_{k-1}^{l_{1},u},\end{array} \] (63)
2. MBM process \[w_{k|k-1}^{jj}=w_{k-1}^{jj},\] (65) \[r_{k|k-1}^{jj}=p_{S,k}r_{k-1}^{j,i},\] (66) \[p_{k|k-1}^{j,i}(X,\xi)=\mathcal{GAM}(\xi_{k;\varepsilon};\mu_{k,k|k-1}^{j,i },\rho_{k,k|k-1}^{j,i})\mathcal{IGG}(X,\xi_{k;\varepsilon};\mu_{B,k|k-1}^{j,i },\rho_{B,k|k-1}^{j,i},m_{B,k|k-1}^{j,i},p_{B,k|k-1}^{j,i},m_{X,k|k-1}^{j,i},p_ {X,k|k-1}^{j,i}),\] (67) \[m_{B,k|k-1}^{j,i}=F_{k-1}m_{B,k-1}^{j,i},P_{B,k|k-1}^{j,i}=Q_{k-1}+F_{k-1 }p_{B,k-1}^{j,i}F_{k-1}^{T}.\] (68) Proposition 4 (Updating process). Given that the predicted intensity of the Poisson process is in an IGGM form, as shown in Equation (67), where \(J_{k|k-1}^{u}=J_{k-1}^{B}+J_{k-1}^{u}\), the predicted density of the MBM can be expressed as shown in Equation (69). \[D_{k|k-1}^{u}(X,\xi)=\sum\limits_{l=1}^{J_{k|k-1}^{u}}w_{k|k-1}^{l,u}\mathcal{ GAM}(\xi_{k;\varepsilon};\alpha_{k|k-1}^{l,u},\beta_{k|k-1}^{l,u})\mathcal{IGG}(X, \xi_{k;\varepsilon};\mu_{k|k-1}^{l,u},\beta_{k|k-1}^{l,u},m_{k|k-1}^{l,u},m_{k |k-1}^{l,u}),\] (69) \[p_{k|k-1}^{j,i}(X,\xi)=\mathcal{GAM}(\xi_{k;\varepsilon};\mu_{ \varepsilon k|k-1}^{j,i},\rho_{\varepsilon k|k-1}^{j,i})\mathcal{IGG}(X,\xi_{k ;\varepsilon};\mu_{B,k|k-1}^{j,i},P_{B,k|k-1}^{j,i},m_{B,k|k-1}^{j,i},p_{B,k|k-1 }^{j,i},m_{X,k|k-1}^{j,i},P_{X,k|k-1}^{j,i}).\] (70) Furthermore, the measurement \(M_{k}\), the update of the Poisson process, and the MBM process can be derived in four stages. 1. Update for undetected targets \[D_{k}^{u}(X,\xi)=\sum\limits_{l=1}^{l_{k|k-1}^{u}}w_{k|l}^{l,u}\mathcal{GAM}( \xi_{k;\varepsilon};\mu_{k|k}^{l,u},\rho_{k1}^{l,u})\mathcal{IGG}(X,\xi_{k; \varepsilon};\mu_{k|k}^{l,u},\rho_{k1}^{l,u},m_{k1}^{l,u},p_{k1}^{l,u}),\] (71) where \[\alpha_{k1}^{l,u}=\alpha_{k|k-1}^{l,u},\ \beta_{k1}^{l,u}=\beta_{k|k-1}^{l,u},\ m_{k1}^{l,u}=m_{k|k-1}^{l,u},\ P_{k1}^{l,u}=P_{k|k-1}^{l,u},\] and \[w_{k1}^{l,u}=w_{k|k-1}^{l,u}(\frac{\beta_{k|k-1}^{l,u}}{\rho_{k|k-1}^{l,u}+1}) ^{l_{k-1}^{l,u}}.\]
2. Update for potential targets detected for the first time \[r_{k,2}^{2}(M)=\frac{\sum\limits_{l=1}^{l_{k|k-1}^{l}}w_{k|k-1}^{l,u}\mathcal{ G}(M,\xi_{k;\varepsilon};H_{k}m_{k|k-1}^{l,u},H_{k}P_{k|k-1}^{l,u}H_{k}^{T}+R_{k})}{ \kappa_{C}(M_{k},\xi_{k})+\sum\limits_{l=1}^{l}w_{k|k-1}^{l,u}\mathcal{G}(M, \xi_{k;\varepsilon};H_{k}m_{k|k-1}^{l,u},H_{k}P_{k|k-1}^{l,u}H_{k}^{T}+R_{k})},\] (72) \[r_{k|k-1}^{jj}=p_{S,k}r_{k-1}^{j,i},\]0 (73) \[r_{k|k-1}^{jj}=p_{S,k}r_{k-1}^{j,i},\]1where \(\alpha_{k2}^{l,u}=\alpha_{k}^{l,u}\), \(\alpha_{k2}^{l,u}=\alpha_{k}^{l,u}\), \(m_{k2}^{l,u}=m_{k|k-1}^{l,u}+P_{k|k-1}^{l,u}H_{k}(H_{k}^{T}P_{k|k-1}^{l,u}H_{k}^{ T}+R_{k})^{-1}\), \(\left(M-H_{k}m_{k|k-1}^{l,u}\right)\), and \(P_{k2}^{l,u}=\left(I-P_{k|k-1}^{l,u}H_{k}^{T}(H_{k}P_{k|k-1}^{l,u}H_{k}^{T}+R_{ k})^{-1}H_{k}\right)P_{k|k-1}^{l,u}\). 3. Missed detection of MBM \[r_{k|k-1}^{jj}=p_{S,k}r_{k-1}^{j,i},\]2 (74) \[r_{k|k-1}^{jj}=p_{S,k}r_{k-1}^{j,i},\]3 (75) where \(\alpha_{k3}^{j,i}=\alpha_{k|k-1}^{j,i}\), \(\beta_{k3}^{j,i}=\beta_{k|k-1}^{j,i}+1\), \(m_{k3}^{j,i}=m_{k|k-1}^{j,i}\), and \(P_{k3}^{j,i}=P_{k|k-1}^{j,i}\). 4.