*3.2. SCM-JTC-CBMeMBer Filter: Multi-Target Case*

For the JTC of multi-target, the available measurement set at time *<sup>k</sup>* is denoted as <sup>Z</sup>*<sup>k</sup>* <sup>=</sup> '*zk*,*l nk*,*<sup>z</sup> l*=0 and the measurement set up to time *<sup>k</sup>* is <sup>Z</sup>*<sup>k</sup>* = {Z*l*} *k <sup>l</sup>*=1. The posterior multi-target density at time *k* − 1 is modeled as a multi-Bernoulli

$$\begin{aligned} \pi\_{k-1} &= \left\{ \left( r\_{k-1}^{(i)}, p\_{k-1}^{(i)} (\mathbf{x}\_{k-1} | \mathcal{Z}^{k-1}) \right) \right\}\_{i=1}^{M\_{k-1}} \\ &= \left\{ \left( r\_{k-1'}^{(i)} \sum\_{m=1}^{\mathcal{U}\_{\mathbb{C}}} p\_{k-1}^{(i)} (\mathbf{x}\_{k-1} | c^{m}, \mathcal{Z}^{k-1}) p\_{k-1}^{(i)} (c^{m} | \mathcal{Z}^{k-1}) \right) \right\}\_{i=1}^{M\_{k-1}} \end{aligned} \tag{42}$$

In addition to Assumption 1 and Assumption 2, the following assumptions should also be followed to obtain the SCM-JTC-CBMeMBer filter.

**Assumption 3.** *Each target evolves motion and generates measurements independently.*

**Assumption 4.** *The clutter is modeled as Poisson RFS with Poisson average rate* λ*c, and it is independent of target-originated measurements. The spatial distribution of the clutter is a uniform distribution, denoted by* <sup>C</sup>('*z*)*. The clutter intensity function is* <sup>κ</sup>('*z*) = <sup>λ</sup>*c*C('*z*)*.*

**Assumption 5.** *The survival and detection probabilities are state-independent, i.e., pS*,*k*(*x*, *c*) = *pS*,*k, pD*,*k*(*x*, *c*) = *pD*,*k.*

**Assumption 6.** *The PDF of birth targets at time k* − 1 *is also a multi-Bernoulli, namely*

$$\begin{aligned} \pi\_{B,k-1} &= \left\{ \left( r\_{B,k-1}^{(i)}, p\_{B,k-1}^{(i)} (\mathbf{x}\_{k-1} | \mathbf{\mathcal{Z}}^{k-1}) \right) \right\}\_{i=1}^{M\_{B,k-1}} \\ &= \left\{ \left( r\_{B,k-1}^{(i)}, \sum\_{m=1}^{n\_c} p\_{B,k-1}^{(i)} (\mathbf{x}\_{k-1} | \mathbf{c}^m, \mathbf{\mathcal{Z}}^{k-1}) p\_{B,k-1}^{(i)} (c^m | \mathbf{\mathcal{Z}}^{k-1}) \right) \right\}\_{i=1}^{M\_{B,k-1}} \end{aligned} \tag{43}$$

**Proposition 1.** *If the posterior multi-target density at time k* − 1 *is a multi-Bernoulli, as shown in Equation (42), then the predicted multi-target density is also a multi-Bernoulli and is given by*

$$\begin{split} \mathbb{T}\_{\mathbb{M}\mathbb{R}-1} &= \left\{ \left( r\_{P,\mathbb{M}-1}^{(i)} p\_{P,\mathbb{M}-1}^{(i)} (\mathbf{x}\_{k} | \mathsf{T}^{k-1}) \right) \right\}\_{i=1}^{M\_{\mathbb{M}-1}} \cup \left\{ \left( r\_{\Gamma,\mathbb{M}-1}^{(i)} p\_{\Gamma,\mathbb{M}-1}^{(i)} (\mathbf{x}\_{k} | \mathsf{T}^{k-1}) \right) \right\}\_{i=1}^{M\_{\mathbb{M}}} \\ &= \left\{ \left( r\_{P,\mathbb{M}-1}^{(i)} \sum\_{m=1}^{n\_{\mathbb{M}}} p\_{P,\mathbb{M}|i-1}^{(i)} (\mathbf{x}\_{k} | \mathsf{C}^{m}, \mathsf{T}^{k-1}) p\_{P,\mathbb{M}|i-1}^{(i)} (\mathsf{C}^{m} | \mathsf{T}^{k-1}) \right) \right\}\_{i=1}^{M\_{\mathbb{M}}-1} \cup \left\{ \left( r\_{\Gamma,\mathbb{M}-1}^{(i)} p\_{\Gamma,\mathbb{M}|i-1}^{(i)} (\mathsf{c} \mathsf{A} | \mathsf{T}^{k-1}) \right) \right\}\_{i=1}^{M\_{\mathbb{M}}} \end{split} \tag{44}$$

*with*

$$r\_{P,k|k-1}^{(i)} = r\_{k-1}^{(i)} \sum\_{m=1}^{n\_c} p\_{k-1}^{(i)}(c^m | \mathcal{Z}^{k-1}) \Big\langle p\_{k-1}^{(i)}(\mathbf{x}\_{k-1} | c^m, \mathcal{Z}^{k-1}), p\_{S,k} \rangle \tag{45}$$

$$p\_{P\_k \mathbb{K} - 1}^{(i)}(\boldsymbol{c}^m | \mathcal{Z}^{k-1}) = p\_{k-1}^{(i)}(\boldsymbol{c}^m | \mathcal{Z}^{k-1}) \tag{46}$$

$$p\_{P, \text{kjk}-1}^{(i)}(\mathbf{x}|c^{m}, \mathcal{Z}^{k-1}) = \frac{\left\langle f\_{k|k-1}^{\mathbf{k}}(\mathbf{x}\_{k}|\mathbf{x}\_{k-1}), p\_{k-1}^{(i)}(\mathbf{x}\_{k-1}|c^{m}, \mathcal{Z}^{k-1})p\_{S,k} \right\rangle}{\sum\_{m=1}^{n\_{c}} p\_{k-1}^{(i)}(c^{m}|\mathcal{Z}^{k-1}) \left\langle p\_{k-1}^{(i)}(\mathbf{x}\_{k-1}|c^{m}, \mathcal{Z}^{k-1}), p\_{S,k} \right\rangle} \tag{47}$$

$$r\_{\Gamma,k|k-1}^{(i)} = r\_{B,k-1}^{(i)} \tag{48}$$

$$p\_{\Gamma,k|k-1}^{(i)}(\mathbf{x}|\mathcal{Z}^{k-1}) = \sum\_{m=1}^{n\_c} p\_{B,k-1}^{(i)}(\mathbf{x}\_{k-1}|c^m, \mathcal{Z}^{k-1}) p\_{B,k-1}^{(i)}(c^m|\mathcal{Z}^{k-1}) \tag{49}$$

The proof of Proposition 1 is given in Appendix A.

**Proposition 2.** *If the predicted multi-target density at time k is a multi-Bernoulli*

*r*

$$\begin{aligned} \pi\_{k|k-1} &= \left\{ \left( r\_{k|k-1}^{(i)}, p\_{k|k-1}^{(i)} (\mathbf{x}\_k | \mathbf{\mathcal{Z}}^{k-1}) \right) \right\}\_{i=1}^{M\_{k|k-1}} \\ &= \left\{ \left( r\_{k|k-1}^{(i)}, \sum\_{m=1}^{n\_k} p\_{k|k-1}^{(i)} (\mathbf{x}\_k | \mathbf{c}^m, \mathbf{\mathcal{Z}}^{k-1}) p\_{k|k-1}^{(i)} (\mathbf{c}^m | \mathbf{\mathcal{Z}}^{k-1}) \right) \right\}\_{i=1}^{M\_{k|k-1}} \end{aligned} \tag{50}$$

Then, the posterior multi-target density can be approximated by a multi-Bernoulli as

$$\begin{split} \pi\_{k} & \coloneqq \left\{ \left( r\_{L,k}^{(\boldsymbol{\ell})} p\_{L,k}^{(\boldsymbol{\ell})} (\mathbf{x}\_{k} | \boldsymbol{\mathcal{Z}}^{k}) \right) \right\}\_{i=1}^{M\_{\mathrm{l\!k}}} \cup \left\{ \left( r\_{L\boldsymbol{\lambda}}^{\*} (\boldsymbol{\Xi}), p\_{L\boldsymbol{\lambda}}^{\*} (\boldsymbol{\Xi}) \right) \right\}\_{\overline{\boldsymbol{z}} \boldsymbol{\varepsilon} \mathcal{L}\_{k}} \\ &= \left\{ \left( r\_{L,k}^{(\boldsymbol{\ell})} \sum\_{m=1}^{\frac{n\_{c}}{2}} p\_{L,k}^{(\boldsymbol{\ell})} (\mathbf{x}\_{k} | c^{m}, \boldsymbol{\mathcal{Z}}^{k}) p\_{L,k}^{(\boldsymbol{\ell})} (c^{m} | \boldsymbol{\mathcal{Z}}^{k}) \right) \right\}\_{i=1}^{M\_{\mathrm{l\!\!k}}} \cup \left\{ \left( r\_{L\boldsymbol{\lambda}}^{\*} (\boldsymbol{\Xi}), \sum\_{m=1}^{\frac{n\_{c}}{2}} p\_{L\boldsymbol{\lambda}}^{(\boldsymbol{\ell})} (\mathbf{x}\_{k} | c^{m}, \boldsymbol{\mathcal{Z}}^{k}) p\_{L\boldsymbol{\lambda}}^{(\boldsymbol{\ell})} (c^{m} | \boldsymbol{\mathcal{Z}}^{k}) \right) \right\}\_{\overline{\boldsymbol{z}} \boldsymbol{\varepsilon} \mathcal{L}\_{k}} \end{split} \tag{51}$$

with

$$r\_{L,k}^{(i)} = r\_{k|k-1}^{(i)} \frac{1 - \sum\_{m=1}^{n\_{\xi}} p\_{k|k-1}^{(i)}(c^m | \mathcal{Z}^{k-1}) \biglangle p\_{k|k-1}^{(i)}(\mathbf{x}\_k | \mathcal{Z}^m, \mathcal{Z}^{k-1}), p\_{D,k} \big\rangle}{1 - r\_{k|k-1}^{(i)} \sum\_{m=1}^{n\_{\xi}} p\_{k|k-1}^{(i)}(c^m | \mathcal{Z}^{k-1}) \big\langle p\_{k|k-1}^{(i)}(\mathbf{x}\_k | \mathcal{C}^m, \mathcal{Z}^{k-1}), p\_{D,k} \big\rangle} \tag{52}$$

$$p\_{L,k}^{(i)}(\mathbf{x}\_k|\boldsymbol{c}^m, \boldsymbol{\mathcal{Z}}^k) = \frac{(1 - p\_{D,k})p\_{k|k-1}^{(i)}(\mathbf{x}\_k|\boldsymbol{c}^m, \boldsymbol{\mathcal{Z}}^{k-1})}{1 - \sum\_{m=1}^{n\_c} p\_{k|k-1}^{(i)}(\boldsymbol{c}^m|\boldsymbol{\mathcal{Z}}^{k-1}) \Big\{p\_{k|k-1}^{(i)}(\mathbf{x}\_k|\boldsymbol{c}^m, \boldsymbol{\mathcal{Z}}^{k-1}), p\_{D,k}\Big\}}\tag{53}$$

$$p\_{L,k}^{(i)}(c^{\eta u}|\mathcal{Z}^k) = p\_{k|k-1}^{(i)}(c^{\eta u}|\mathcal{Z}^{k-1})\tag{54}$$

*r* ∗ *<sup>U</sup>*,*k*('*z*) = \$*Mk*|*k*−<sup>1</sup> *i*=1 *r* (*i*) *<sup>k</sup>*|*k*−1(1−*<sup>r</sup>* (*i*) *<sup>k</sup>*|*k*−1) *n* \$c *m*=1 *p* (*i*) *<sup>k</sup>*|*k*−1(*cm*|Z*k*−1) ! *p* (*i*) *<sup>k</sup>*|*k*−1(*xk*|*cm*,Z*k*−1),*g*<sup>k</sup> *<sup>k</sup>* (*xk*)*g*<sup>c</sup> *k* (*xk*,*cm*)*pD*,*<sup>k</sup>* " 1−*r* (*i*) *k*|*k*−1 *n* \$c *m*=1 *p* (*i*) *<sup>k</sup>*|*k*−1(*cm*|Z*k*−1) ! *p* (*i*) *<sup>k</sup>*|*k*−1(*xk*|*cm*,Z*k*−1),*pD*,*<sup>k</sup>* "2 <sup>κ</sup>('*z*) + \$*Mk*|*k*−<sup>1</sup> *i*=1 *r* (*i*) *k*|*k*−1 *n* \$c *m*=1 *p* (*i*) *<sup>k</sup>*|*k*−1(*cm*|Z*k*−1) ! *p* (*i*) *<sup>k</sup>*|*k*−1(*xk*|*cm*,Z*k*−1),*g*<sup>k</sup> *<sup>k</sup>* (*xk*)*g*<sup>c</sup> *k* (*xk*,*cm*)*pD*,*<sup>k</sup>* " 1−*r* (*i*) *k*|*k*−1 *n* \$c *m*=1 *p* (*i*) *<sup>k</sup>*|*k*−1(*cm*|Z*k*−1) ! *p* (*i*) *<sup>k</sup>*|*k*−1(*xk*|*cm*,Z*k*−1),*pD*,*<sup>k</sup>* " (55)

*p*

$$p\_{IJ,k}^{(i)}(\mathbf{x}\_{k}|\mathcal{C}^{m},\mathcal{Z}^{k}) = \frac{\sum\_{i=1}^{M\_{\text{light}}-1} \frac{r\_{ijkl-1}^{(i)}(\mathbf{x}\_{k}|\mathbf{x}^{m},\mathcal{Z}^{k-1})r\_{ijkl-1}^{(i)}(\mathbf{c}^{m}|\mathbf{Z}^{k-1})r\_{D\cup k}\mathbf{x}\_{k}^{k}(\mathbf{x}\_{k})\mathbf{x}\_{k}^{m}\mathbf{x}^{m}}{1-r\_{ikl-1}^{(i)}}}{\sum\_{i=1}^{M\_{\text{light}}-1} \frac{r\_{ikl-1}^{(i)}(\mathbf{x}\_{k}^{m}|\mathbf{C}^{m})r\_{k}^{(i)}(\mathbf{x}\_{k}^{m},\mathbf{Z}^{k-1})r\_{D\cup k}\mathbf{x}\_{k}^{k}(\mathbf{x}\_{k})\mathbf{y}\_{k}^{m}(\mathbf{x}\_{k},\mathcal{C}^{m})}{1-r\_{ikl-1}^{(i)}}}{1-r\_{ikl-1}^{(i)}}\tag{56}$$

$$p\_{IJ,k}^{(i)}(\mathbf{x}^{m}|\mathbf{Z}^{k}) = \frac{\sum\_{i=1}^{M\_{\text{light}}-1} \frac{r\_{ikl-1}^{(i)}r\_{ikl-1}^{(i)}(\mathbf{x}^{m}|\mathbf{Z}^{k-1})\left(p\_{ijkl-1}^{(i)}(\mathbf{x}\_{k}^{m},\mathbf{Z}^{k-1})r\_{D\cup k}\mathbf{x}\_{k}^{k}(\mathbf{x}\_{k})\mathbf{y}\_{k}^{c}(\mathbf{x}\_{k},\mathcal{C}^{m})\right)}{1-r\_{ikl-1}^{(i)}}}{1-r\_{ikl-1}^{(i)}}\tag{57}$$

The proof of Proposition 2 is shown in Appendix B.

The state extraction step is similar to the CBMeMBer filter, and the details can be found in [22].
