*5.1. Drawbacks of the Existing SCMDE for Multi-Target Tracking*

It was introduced in [23] that the SCMDE for single target tracking environments yields an accurate clutter measurement density when the point of interest is the target detection. When the point of interest is a clutter detection, the SCMDE generates a biased and smaller sparsity than the actual value, which implies a bigger clutter measurement density. This phenomenon gives benefits to single target tracking as the bigger clutter measurement density decreases the data association probability for the clutter detection in the probabilistic data association (PDA) algorithm. It was also introduced in [23] that these benefits are reduced as the sparsity order increases. Therefore, the existing SCMDE improves target tracking performance for single target tracking.

In this subsection, the performance of the existing SCMDE is analyzed for multi-target tracking in homogeneous clutter environments. The detailed derivations are given in Appendix A of this paper. When the point of interest is a target detection for a two-target case, the average value of the sparsity estimate for the point of interest **z***k*,*<sup>i</sup>* becomes:

$$E\left\{\hat{\gamma}\_{k,i}^{(n)}\right\} = \begin{cases} \frac{1}{\rho} (1 - \frac{1 - e^{-\rho V^{(1)}}}{\rho V^{(1)}}), & n = 1\\ \frac{1}{\rho} (1 - \frac{1 - e^{-\rho V^{(2)}}}{\rho V^{(2)}} + \frac{e^{-\rho V^{(2)}}}{2}), & n = 2 \end{cases} \tag{34}$$

where *n* is the sparsity order, *ρ* is the clutter measurement density of the homogeneous clutter environment, and *V*(*n*) is the volume of the hyper-sphere used for the sparsity estimation. If (34) is compared to the true sparsity, <sup>1</sup> *<sup>ρ</sup>* , which can be obtained from the existing SCMDE for single target tracking as shown in (35) of [23], the sparsity estimates are smaller than the true ones and biased. The bias becomes reduced as *n* increases and *V*(*n*) becomes bigger. In contrast to single target tracking environments, the SCMDE generates bigger clutter measurement density estimates when the point of interest is a target detection, which results in a reduced data association probability for the target detection and deteriorated target tracking performance for multi-target tracking environments.

When the point of interest is a clutter detection for two-target cases, the average value of the sparsity estimates for the point of interest, **z***k*,*i*, becomes:

$$E\left\{\hat{\gamma}\_{k,i}^{(n)}\right\} = \begin{cases} \frac{1}{\rho} - \frac{1}{\rho^2 V^{(1)}} (1 - e^{-\rho V^{(1)}}), & n = 1\\ \frac{1}{\rho} - \frac{1}{2\rho^2 V^{(2)}} (1 - e^{-\rho V^{(2)}}), & n = 2 \end{cases} \tag{35}$$

The average sparsity estimates in (35) are smaller than the actual <sup>1</sup> *<sup>ρ</sup>* , and this fact results in bigger clutter measurement density estimates. The average sparsity estimate for *n* = 1 in (35) is the same as *n* = 1 for single target tracking environments specified in (23) of [23]. When *n* = 2, the average sparsity estimate becomes bigger than *n* = 1, and this indicates that the clutter measurement density estimates become less biased for *n* = 2. This indicates that more accurate clutter measurement density estimation is possible with the PDA algorithm as *n* increases. From the above analysis, the existing SCMDE has two incompatible aspects in tracking performance for multi-target tracking environments. One aspect is that tracking performance becomes deteriorated as it generates smaller data association probabilities than the actual ones for true target detections. Another aspect is that tracking performance is improved as it generates smaller data association probabilities than the actual one for clutter measurements. These incompatible aspects are due to the biased and reduced sparsity estimates described in (34) and (35).

In order to improve tracking performance for multi-target tracking environments, it is more important to have the improved data association results with less biased clutter measurement density estimates. This can be done by evaluating the clutter measurement probability of each validated measurement for counting only the number of clutter measurements (excluding the number of target measurements), inside the volume of the hyper-sphere *V*(*r* (*n*) *<sup>k</sup>*,*<sup>i</sup>* ) specified in (33). The clutter measurement probability is the probability that the measurement is a clutter detection not from a target. If the clutter measurement probability is used for the sparsity estimates, enhanced tracking is expected due to less biased clutter measurement density estimates. This has a more significant effect in performance improvement when the point of interest is a target detection rather than a clutter detection. From the analysis in this section, the magnitude of bias of the sparsity estimate of (34) and (35) becomes smaller as *n* increases and the volume of the hyper-sphere *V*(*n*) increases. In the next subsection, the adaptive SCMDE algorithm for multi-target tracking (MTT-SCMDE) is proposed to take into account the clutter measurement probability and increased hyper-sphere volume for each sparsity order *n* to achieve enhanced tracking performances.
