*5.2. MTT-SCMDE*

To estimate the clutter measurement density accurately for these multi-target tracking environments, we propose a method to calculate the probability that adjacent measurements are generated from clutter and use this probability to estimate the clutter measurement density.

To derive the clutter measurement probability, two events are defined.


Let **T***<sup>k</sup>* denote the set of the cluster targets at scan *k*, and *Hk* represent the event that **T***<sup>k</sup>* exists at scan *k* such as:

$$H\_k = \bigcup\_{\sigma \in \mathbf{T}\_k} \chi\_k^{\sigma}. \tag{36}$$

The probability that **z***k*,*<sup>j</sup>* is generated from clutter under *Hk*, the mutual exclusiveness of **z***k*,*<sup>j</sup>* sources, becomes:

$$P(\chi\_{k,j}^{0}|\mathbf{Z}^{k-1}, H\_k) = a\_k \prod\_{\sigma \in \mathbf{T}\_k} \left(1 - P\_{k,j}^{\sigma}\right),\tag{37}$$

where *P<sup>σ</sup> <sup>k</sup>*,*<sup>j</sup>* is the prior probability that **z***k*,*<sup>j</sup>* is generated from target *σ* introduced in (17) and where *α<sup>k</sup>* is a normalization constant.

The probability that **z***k*,*<sup>j</sup>* is generated from target *τ* ∈ **T***<sup>k</sup>* becomes:

$$P(\chi\_{k,j}^{\mathsf{T}}|\mathbf{Z}^{k-1}, H\_k) \quad = \quad \mathfrak{a}\_k P\_{k,j}^{\mathsf{T}} \prod\_{\substack{\eta \in \mathsf{T}\_k \\ \eta \neq \tau}} \left(1 - P\_{k,j}^{\eta}\right) \tag{38}$$

$$\alpha = \alpha\_k \frac{P\_{k,j}^{\tau}}{1 - P\_{k,j}^{\tau}} \prod\_{\eta \in \mathbf{T}\_k} \left( 1 - P\_{k,j}^{\eta} \right) . \tag{39}$$

From (37) and (39), *α<sup>k</sup>* can be obtained from the mutual exclusiveness of **z***k*,*<sup>j</sup>* sources such as:

$$P(\chi\_{k,j}^0|\mathbf{Z}^{k-1}, H\_k) + \sum\_{\mathbf{r} \in \mathbf{T}\_k} P(\chi\_{k,j}^\mathbf{r}|\mathbf{Z}^{k-1}, H\_k) = 1. \tag{40}$$

Then, *α<sup>k</sup>* is obtained as:

$$\mathfrak{a}\_{k} = \frac{1}{\prod\_{\sigma \in \mathbf{T}\_{k}} \left(1 - P\_{k,j}^{\sigma}\right) \left(1 + \sum\_{\eta \in \mathbf{T}\_{k}} \frac{P\_{k,j}^{\eta}}{1 - P\_{k,j}^{\eta}}\right)} \tag{41}$$

Therefore, the clutter measurement probability *P*(*χ*<sup>0</sup> *k*,*j* |**Z***k*−1, *Hk*) in (37) can be expressed as:

$$P(\chi\_{k,j}^0 | \mathbf{Z}^{k-1}, H\_k) = \frac{1}{1 + \sum\_{\eta \in \mathcal{T}\_k} \left( \frac{P\_{k,j}^{\eta}}{1 - P\_{k,j}^{\eta}} \right)}. \tag{42}$$

The proposed MTT-SCMDE utilizes the clutter measurement probabilities of the element of **Y***<sup>i</sup> k* defined in (30). Let *C*(*l*) *<sup>k</sup>*,*<sup>i</sup>* be the clutter measurement probability of **z** (*l*) *<sup>k</sup>*,*<sup>i</sup>* , the *l*th nearest measurement from **<sup>z</sup>***k*,*i*. If the cumulative sum *<sup>C</sup>*(*l*) *<sup>k</sup>*,*<sup>i</sup>* from *l* = 1 is bigger than the predetermined sparsity order *n*, which is the expected number of clutter measurements, the summation is stopped for the sparsity calculation. If:

$$\sum\_{l=1}^{m-1} \mathbb{C}\_{k,i}^{(l)} < n \le \sum\_{l=1}^{m} \mathbb{C}\_{k,i}^{(l)} \tag{43}$$

then the radius *r* (*n*) *<sup>k</sup>*,*<sup>i</sup>* for the hyper-sphere volume calculation becomes:

$$r\_{k,i}^{(n)} = \left\| \mathbf{z}\_{k,i}^{(m+1)} - \mathbf{z}\_{k,i} \right\|,\tag{44}$$

where **z** (*m*+1) *<sup>k</sup>*,*<sup>i</sup>* <sup>∈</sup> **<sup>Y</sup>***<sup>i</sup> k*.

In this paper, **z** (*n*+1) *<sup>k</sup>*,*<sup>i</sup>* is used to calculate *r* (*n*) *<sup>k</sup>*,*<sup>i</sup>* instead of **z** (*n*) *<sup>k</sup>*,*<sup>i</sup>* used in [23] for single target tracking to produce less biased estimates of the clutter measurement density for multi-target tracking environments.

The estimated sparsity of order *n* becomes:

$$\hat{\gamma}\_{k,i}^{(n)} = \frac{V\left(r\_{k,i}^{(n)}\right)}{\sum\_{l=1}^{m} \mathbb{C}\_{k,i}^{(l)}},\tag{45}$$

where *V r* (*n*) *k*,*i* is the volume of the hyper-sphere with radius *r* (*n*) *<sup>k</sup>*,*<sup>i</sup>* defined in (44).

When estimating the sparsity, the existing SCMDE utilizes the number of measurements in the hyper-sphere, while the proposed MTT-clutter measurement density estimation method utilizes the mean number of clutter measurements with the clutter measurement probability to reduce biases in the clutter measurement density estimates in multi-target tracking applications.

Figure 2 shows an expansion of the volume of the hyper-sphere of the MTT-SCMDE if it is used for data association in single target environments. Compared to Figure 1, the volume of the hyper-sphere for each sparsity order is increased.

**Figure 2.** Hyper-spheres for the multi-target tracking (MTT)-SCMDE used for single target tracking in a 2D space.
