**4. SMC Implementation of the SCM-JTC-CBMeMBer Filter**

In what follows, SMC implementation of the SCM-JTC-CBMeMBer filter recursion will be presented.

Supposing that the posterior multi-target density π*k*−<sup>1</sup> = %*r* (*i*) *k*−1 , *p* (*i*) *<sup>k</sup>*−1(*xk*−1|Z*k*−1) & *Mk*−<sup>1</sup> *i*=1 is given, and each component *p* (*i*) *<sup>k</sup>*−1(*xk*−1|Z*k*−1) is comprised of *<sup>n</sup><sup>i</sup> <sup>k</sup>*−<sup>1</sup> weighted particles *wi*,*<sup>j</sup> k*−1 , *x i*,*j k*−1 , *l i*,*j ni k*−1 *<sup>j</sup>*=<sup>1</sup> , that is

$$p\_{k-1}^{(i)}(\mathbf{x}\_{k-1}|\mathcal{Z}^{k-1}) = \sum\_{j=1}^{n\_{k-1}^{i}} w\_{k-1}^{i,j} \delta\_{\mathbf{x}\_{k-1}^{i,j}}(\mathbf{x}\_{k-1}) \tag{58}$$

Then the class probability *p* (*i*) *<sup>k</sup>*−1(*cm*|Z*k*−1) and the kinematic state distribution conditioned on classification *p* (*i*) *<sup>k</sup>*−1(*xk*−1|*cm*, <sup>Z</sup>*k*−1) can be obtained by

$$p\_{k-1}^{(i)}(c^m|\mathcal{Z}^{k-1}) = \sum\_{j=1}^{n\_{k-1}^l} w\_{k-1}^{i,j} \delta\_{l^{i,j}}(c^m) / \sum\_{j=1}^{n\_{k-1}^l} w\_{k-1}^{i,j} \tag{59}$$

$$w\_{k-1}^{(i)}(\mathbf{x}\_{k-1}|\mathbf{c}^m, \mathbf{\mathcal{Z}}^{k-1}) = \sum\_{j=1}^{n\_{k-1}^i} w\_{k-1}^{i,m,j} \delta\_{\mathbf{x}\_{k-1}^{i,m,j}}(\mathbf{x}\_{k-1}) \tag{60}$$

with

$$w\_{k-1}^{i,m,j} = w\_{k-1}^{i,j} \delta\_{\vec{r}^j}(c^m) / \sum\_{j=1}^{n\_{k-1}^i} w\_{k-1}^{i,j} \delta\_{\vec{r}^j}(c^m) \tag{61}$$

$$\mathbf{x}\_{k-1}^{i,m,j} = \mathbf{x}\_{k-1}^{i,j} \delta\_{l^{i,j}}(\boldsymbol{c}^m) \tag{62}$$

**Proposition 3.** *Given the importance density q* (*i*) *<sup>k</sup>* (·|*xk*−1, <sup>Z</sup>*k*) *of the posterior distribution and importance densities b* (*i*) *<sup>k</sup>* (·|*xk*−1, <sup>Z</sup>*k*) *of the birth targets, according to Proposition 1, if the prior distribution is multi-Bernoulli* π*k*−<sup>1</sup> = *r* (*i*) *k*−1 , \$*n*c *m*=1 *p* (*i*) *<sup>k</sup>*−1(*xk*−1|*cm*, <sup>Z</sup>*k*−1)*<sup>p</sup>* (*i*) *<sup>k</sup>*−1(*cm*|Z*k*−1) +*Mk*−<sup>1</sup> *i*=1 *and each* *p* (*i*) *<sup>k</sup>*−<sup>1</sup> <sup>=</sup> \$*n*<sup>c</sup> *m*=1 *p* (*i*) *<sup>k</sup>*−1(*xk*−1|*cm*, <sup>Z</sup>*k*−1)*<sup>p</sup>* (*i*) *<sup>k</sup>*−1(*cm*|Z*k*−1) *is comprised of <sup>n</sup><sup>i</sup> <sup>k</sup>*−<sup>1</sup> *weighted particles wi*,*<sup>j</sup> k*−1 , *x i*,*j k*−1 , *l i*,*j ni k*−1 *<sup>j</sup>*=<sup>1</sup> *, then the predicted multi-target density is also multi-Bernoulli and the SMC implementation is calculated as*

$$r\_{P,k|k-1}^{(i)} = r\_{k-1}^{(i)} \sum\_{j=1}^{n\_{k-1}^{i}} w\_{k-1}^{i,j} p\_{S,k} \tag{63}$$

$$p\_{P, \mathbb{k}|\mathbf{k}-1}^{(i)}(\mathbf{c}^{m}|\mathbf{Z}^{\mathbf{k}-1}) = \sum\_{j=1}^{n\_{k-1}^{i}} w\_{P, \mathbb{k}|\mathbf{k}-1}^{i,j} \delta\_{\mathbb{P}^{i,j}}(\mathbf{c}^{m}) / \sum\_{j=1}^{n\_{k-1}^{i}} w\_{P, \mathbb{k}|\mathbf{k}-1}^{i,j} \tag{64}$$

$$w\_{P,k|k-1}^{(i)}(\\\mathbf{x}\_k|\mathbf{c}^m, \mathbf{Z}^{k-1}) = \sum\_{j=1}^{n\_{k-1}^i} w\_{P,k-1}^{i,m,j} \delta\_{\mathbf{x}\_{P,k|k-1}^{i,m,j}}(\mathbf{x}\_k) \tag{65}$$

$$r^{(i)}\_{
 \Gamma \, \vec{\nu} \, \vec{k} - 1} = r^{(i)}\_{B, \vec{k} - 1} \tag{66}$$

$$p\_{\Gamma,k|k-1}^{(i)}(\mathbf{x}\_k|\mathbf{c}^m, \mathbf{Z}^{k-1}) = \sum\_{j=1}^{n\_{k-1}^i} w\_{B,k|k-1}^{i,m,j} \delta\_{\mathbf{x}\_{B,k|k-1}^{i,m,j}}(\mathbf{x}\_k) \tag{67}$$

$$w\_{\Gamma,k|k-1}^{(i)}(c^{m}|\mathcal{Z}^{k-1}) = \sum\_{j=1}^{n\_{k-1}^{t}} w\_{B,k|k-1}^{i,j} \delta\_{l;i}(c^{m}) / \sum\_{j=1}^{n\_{k-1}^{t}} w\_{B,k|k-1}^{i,j} \tag{68}$$

*with*

$$\mathbf{x}\_{P,k|k-1}^{i,j} \sim q\_k^{(i)}(\cdot|\mathbf{x}\_{k-1}, \mathcal{Z}\_k) \tag{69}$$

$$\mathbf{x}\_{B,k\mid k-1}^{i,j} \sim b\_k^{(i)}(\cdot|\mathbf{x}\_{k-1}, \mathcal{Z}\_k) \tag{70}$$

$$
\overline{w}\_{P,k\mathbb{jk}-1}^{i,j} = w\_{k-1}^{i,j} p\_{S,k} \tag{71}
$$

$$w\_{P,k-1}^{i,j} = \overleftarrow{w}\_{P,k\mathbb{k}-1}^{i,j} / \sum\_{j=1}^{n\_{k-1}^{i}} \overleftarrow{w}\_{P,k\mathbb{k}-1}^{i,j} \tag{72}$$

$$w\_{P,k|k-1}^{i,m,j} = w\_{P,k|k-1}^{i,j} \delta\_{l^{i,j}}(c^m) / \sum\_{j=1}^{n\_{k-1}^l} w\_{P,k|k-1}^{i,j} \delta\_{l^{i,j}}(c^m) \tag{73}$$

$$\mathbf{x}\_{P,k|k-1}^{i,m,j} = \mathbf{x}\_{P,k|k-1}^{i,j} \delta\_{l^i;i}(\mathbf{c}^m) \tag{74}$$

$$w\_{B,k|k-1}^{i,j} = w\_{B,k-1}^{i,j} \tag{75}$$

$$w\_{B,k|k-1}^{i,m,j} = w\_{B,k|k-1}^{i,j} \delta\_{i^{j}\boldsymbol{\ell}}(\boldsymbol{c}^{m}) / \sum\_{j=1}^{n\_{k-1}^{\boldsymbol{\ell}}} w\_{B,k|k-1}^{i,j} \delta\_{i^{j}\boldsymbol{\ell}}(\boldsymbol{c}^{m}) \tag{76}$$

$$\mathbf{x}\_{\mathsf{B},k|k-1}^{i,m,j} = \mathbf{x}\_{\mathsf{B},k|k-1}^{i,j} \delta\_{l^{i,j}}(\mathbf{c}^m) \tag{77}$$

The proof of Proposition 3 is given in Appendix C.

**Proposition 4.** *If the predicted multi-target density is the multi-Bernoulli* <sup>π</sup>*k*|*k*−<sup>1</sup> <sup>=</sup> *<sup>r</sup>* (*i*) *k*|*k*−1 , \$*n*c *m*=1 *p* (*i*) *<sup>k</sup>*|*k*−1(*xk*|*cm*, <sup>Z</sup>*k*−1)*<sup>p</sup>* (*i*) *<sup>k</sup>*|*k*−1(*cm*|Z*k*−1) +*Mk*|*k*−<sup>1</sup> *i*=1 *, then the updated multi-target density is also a multi-Bernoulli, and the SMC implementation can be computed by*

$$r\_{L,k}^{(i)} = r\_{k|k-1}^{(i)} \frac{1 - \rho\_{L,k}^i}{1 - r\_{k|k-1}^{(i)} \rho\_{L,k}^i} \tag{78}$$

$$p\_{L,k}^{(i)}(\boldsymbol{c}^{m}|\mathcal{Z}^{k}) = \sum\_{j=1}^{n\_{k-1}^{i}} w\_{k|k-1}^{i,j} \delta\_{l^{i,j}}(\boldsymbol{c}^{m}) / \sum\_{j=1}^{n\_{k-1}^{i}} w\_{k|k-1}^{i,j} \tag{79}$$

$$p\_{L,k}^{(i)}(\mathbf{x}\_k | c^m, \mathcal{Z}^k) = \sum\_{j=1}^{n\_{k-1}^i} w\_{L,k}^{i,m,j} \delta\_{\mathbf{x}\_{k|k-1}^{i,m,j}}(\mathbf{x}\_k) \tag{80}$$

$$r\_{lL,k}^\*(\overline{z}) = \frac{\sum\_{i=1}^{M\_{k|k-1}} \frac{r\_{k|k-1}^{(i)} (1 - r\_{k|k-1}^{(i)}) \rho\_{lL,k}^i}{\left(1 - r\_{k|k-1}^{(i)} \rho\_{l,k}^i\right)^2}}{\kappa(\overline{z}) + \sum\_{i=1}^{M\_{k|k-1}} \frac{r\_{k|k-1}^{(i)} \rho\_{lL,k}^i}{1 - r\_{k|k-1}^{(i)} \rho\_{l,k}^i}} \tag{81}$$

$$p\_{IJ,k}^{(i)}(\\\mathbf{x}\_k|\mathbf{c}^m, \mathbf{Z}^k) = \sum\_{i=1}^{M\_{k|k-1}} \sum\_{j=1}^{n\_{k-1}^i} w\_{IJ,k}^{i,m,j} \delta\_{\mathbf{x}\_{k|k-1}^{i,m,j}}(\mathbf{x}\_k) \tag{82}$$

$$p\_{IJ,k}^{(i)}(c^m|\mathcal{Z}^k) = \sum\_{i=1}^{M\_{kk}-1} \sum\_{j=1}^{n\_{k-1}^i} w\_{IJ,k}^{i,j} \delta\_{I^{i,j}}(c^m) / \sum\_{i=1}^{M\_{kk}-1} \sum\_{j=1}^{n\_{k-1}^i} w\_{IJ,k}^{i,j} \tag{83}$$

*with*

$$p\_{L,k}^{i} = \sum\_{j=1}^{n\_{k-1}^{i}} w\_{k|k-1}^{i,j} p\_{D,k} \tag{84}$$

$$\rho\_{l1k}^{j} = \sum\_{j=1}^{n\_{k-1}^{j}} w\_{k|k-1}^{i,j} g\_k^{\mathbf{k}}(\mathbf{x}\_k) g\_k^{\mathbf{c}}(\mathbf{x}\_k, c^m) p\_{D,k} \tag{85}$$

$$
\overrightarrow{w}\_{L,k}^{i,j} = (1 - p\_{D,k}) w\_{k|k-1}^{i,j} \tag{86}
$$

$$w\_{L,k}^{i,j} = \overline{w}\_{L,k}^{i,j} / \sum\_{j=1}^{n\_{k-1}^i} \overline{w}\_{L,k}^{i,j} \tag{87}$$

$$w\_{L,k}^{i,m,j} = w\_{L,k}^{i,j} \delta\_{l^{i,j}}(c^m) / \sum\_{j=1}^{n\_{k-1}^i} w\_{L,k}^{i,j} \delta\_{l^{i,j}}(c^m) \tag{88}$$

$$\mathbf{x}\_{L,k}^{i,m,j} = \mathbf{x}\_{L,k}^{i,j} \boldsymbol{\delta}\_{l^{ij}}(\boldsymbol{c}^m) \tag{89}$$

$$\begin{aligned} \; \; \; \#\_{\mathcal{U},k}^{i,j} &= \frac{r\_{k|k-1}^{(i)}}{1 - r\_{k|k-1}^{(i)}} p\_{\mathcal{D},k} w\_{k|k-1}^{i,j} g\_k^{\mathbf{k}}(\mathbf{x}\_k) g\_k^{\mathbf{c}}(\mathbf{x}\_{k\prime} c^{\text{nr}}) \end{aligned} \tag{90}$$

$$w\_{l\downarrow k}^{i,j} = \overline{w}\_{l\downarrow k}^{i,j} / \sum\_{j=1}^{n\_{k-1}^i} \overline{w}\_{l\downarrow k}^{i,j} \tag{91}$$

$$w\_{l1,k}^{i,m,j} = \frac{w\_{l1,k}^{i,j} \delta\_{l^{i,j}}(c^m)}{\sum\_{i=1}^{M\_{\&k-1}} \sum\_{j=1}^{n\_{\!
u^{i}}} w\_{l1,k}^{i,j} \delta\_{l^{i,j}}(c^m)}\tag{92}$$

The proof of Proposition 4 is detailed in Appendix D.

Particle resampling is also needed for SMC implementation of the SCM-JTC-CBMeMBer filter, and the same resampling strategy as introduced in Section 3.1 is used. Similar to the CBMeMBer filter, in the proposed SCM-JTC-CBMeMBer filter, the pruning and merge strategy (refer to [22] for the details) should be adopted to reduce the computational burden.
