**4. The Existing Spatial Clutter Measurement Density Estimator for Single Target Tracking**

The clutter measurement density is an important parameter to calculate the data association probability and the posterior target existence probability for track maintenance. In particular, when the LM approach is used in a situation where multiple targets are located in the vicinity, the merging and switching phenomena of the tracks are reduced by utilizing the modulated clutter density with moderate computational loads [10]. Therefore, it is crucial to estimate the clutter measurement density properly.

The clutter measurement density is defined as the average number of measurements that exist within a unit volume. For calculating the sparsity of **z***k*,*i*, the measurements are aligned in the ascending order of distance from **z***k*,*i*. If **Y***<sup>i</sup> <sup>k</sup>* denotes the set of the aligned measurements such as:

$$\mathbf{Y}\_k^i = \bigcup\_{l=1} \mathbf{z}\_{k,i'}^{(l)} \tag{30}$$

where **z** (*l*) *<sup>k</sup>*,*<sup>i</sup>* is the *l*th nearest measurement from **z***k*,*i*. Let *r* (*n*) *<sup>k</sup>*,*<sup>i</sup>* denote the distance from **z***k*,*<sup>i</sup>* to the *n*th nearest measurement, **z** (*n*) *<sup>k</sup>*,*<sup>i</sup>* in **<sup>Y</sup>***<sup>i</sup> <sup>k</sup>*. Then, *r* (*n*) *<sup>k</sup>*,*<sup>i</sup>* becomes:

$$r\_{k,i}^{(n)} = \left\| \mathbf{z}\_{k,i}^{(n)} - \mathbf{z}\_{k,i} \right\|. \tag{31}$$

The SCMDE estimates the sparsity of the measurements, which is the reciprocal of the clutter measurement density. The sparsity of **z***k*,*<sup>i</sup>* is obtained from:

$$
\hat{\rho}\_{k,i}^{(n)} = \frac{1}{\rho\_{k,i}^{(n)}} = \frac{V\left(r\_{k,i}^{(n)}\right)}{n},
\tag{32}
$$

where *n* and *V r* (*n*) *k*,*i* denote the sparsity order and the volume of the hyper-sphere with radius *r* (*n*) *k*,*i* for **z***k*,*i*, respectively. *V r* (*n*) *k*,*i* is expressed by:

$$V\left(r\_{k,i}^{(n)}\right) = \mathbb{C}\_{\mathbb{H}\_z}\left(r\_{k,i}^{(n)}\right)^{n\_z},\tag{33}$$

where *nz* represents the dimension of the measurement space and *Cn*<sup>1</sup> = 2, *Cn*<sup>2</sup> = *π*, and *Cn*<sup>3</sup> = <sup>4</sup>*<sup>π</sup>* 3 . Figure 1 schematically illustrates the hyper-spheres with various sparsity orders for **z***k*,*<sup>i</sup>* in a 2D measurement space.

**Figure 1.** Hyper-spheres for the existing spatial clutter measurement density estimator (SCMDE) in a 2D space.

In the process of deriving the sparsity, track information is not used. A measurement shared by two or more tracks has a unique clutter measurement density regardless of the track states.
