Numerical Examples

For a given hypothesis $\mbox{\boldmath$\tau$ }_n^{i(K)}$ let J_m = T_m^i(K) + F_m^i(K), where T_m^i(K) is the number of measurement vectors at time m used in the Ktrack set $\mbox{\boldmath$\tau$ }_n^{i(K)}$ and F_m^i(K) is the number of false alarms. Here we assume that

$$p\left( \mbox{\boldmath$\tau$ }_n^{i(K)} \right) = \prod_{m=... ...m^{i(K)}} \; (1-P_d)^{K-T_m^{i(K)}} \; p( F_m^{i(K)} ) \; \; ,$$ (23)

where p( F_m^i(K) ) is the Poisson distributed probability of F_m^i(K) false alarms. Then the incremental a priori cost for (17) is

$$\lambda_m ( \mbox{\boldmath$\tau$ }_n^{i(K)} ) \ = \ F_m^{i(... ...-T_m^{i(K)} ) \ln (1-P_d ) \ - \ \ln (p(F_m^{i(K)} )) \; \; .$$ (24)

We have demonstrated [7] how, with proper selection of the list length L in conjunction with merging and pruning, the Viterbi MHT algorithm can recover form a diverged track after several missed detections. We also show that Viterbi MHT can generate lists of track-set estimates, each with a close to optimum cost, so that if other information becomes available the track-set that is most consistent can be selected. Here we focus on number-of-track estimation. Four sets of simulation results are presented here. For the first two, we assume the number of tracks is fixed over the processing. For the last two we use the Viterbi MHT algorithm which allows for variation in the number of tracks.

First we conducted a Monte Carlo simulation to compare the joint K/track estimator (referred to below as the MAP estimator) and the marginalization estimator. K=2 tracks were simulated which were linear in the ${\cal X}$ and ${\cal Y}$ position values values. N=500 trials were conducted, with n=6 measurement times per trial. Gaussian measurement noise was added to each target measurement. The number of ``false detect'' events was generated using a Poisson distribution with a false alarm rate of 2, and the false detections themselves were generated using a uniform distribution over the range [0,4] in both ${\cal X}$ and ${\cal Y}$. Probability of a missed detection was assumed to be 0.3. K_max=3 and L=16. Figure 2 shows estimator performance for varying maesurement and model noise variance. On the basis of these simulations, it can be concluded that estimation by marginalization can be advantageous over MAP estimation. Note, however, that this advantage is not significant for all scenarios. The advantage is more pronounced with limited data situations.

 
Second, Monte Carlo simulations were run to study K estimator performance for varying probability of detection P_d. N=100 trials were run per P_d value. K=2 linear tracks were simulated, with n=21 measurement times. Measurement noise variance was 0.01. A Poisson distribution with a false alarm rate of 1 was used for the number of false detections, and the false detections themselves were generated using a uniform distribution over the range [0,4] in both ${\cal X}$ and ${\cal Y}$. K_max = 3 and L=32 were used. Table 1 shows the percentages for different estimates of K vs. P_d.

 

Table 1: Number-of-track estimation results: estimated number of tracks vs. P_d.

 

${\bf\longrightarrow}$ K	0	1	2	3
$\downarrow$ Pd
1.00	0	7	93	0
0.95	0	0	96	4
0.90	0	0	91	9
0.85	0	3	85	12
0.80	1	6	72	21

 

For the third simulation two parabolic target tracks consisting of n=20 ${\cal X}$ and ${\cal Y}$ position values were generated, where ${\cal X}$ varied linearly over time from ${\cal X}= 0$ to ${\cal X} = 4$. The two targets were present over the entire processing interval. Gaussian measurement noise with a variance of 0.01 was added to the target tracks. The number of ``false detect'' events was generated using a Poisson distribution with a false alarm rate of 2, and the false detections themselves were generated using a uniform distribution over the range [0,4] in both ${\cal X}$ and ${\cal Y}$. Probability of a missed detection was assumed to be 0.2. K_max=3 and L=8. We partitioned the data into 4 blocks of T=5measurement times. Figure 3(a) shows the ${\cal X}$ and ${\cal Y}$ position values for the true tracks (solid lines), along with the noisy measurements (boxed and circled x's) and false alarms (x's) used as input to the algorithm. Two tracks were detected for each block (i.e. $\hat{K}_t = 2; \ t=1,2,3,4$). Figure 3(b) shows the resulting best 2-track set estimate.

 
For the fourth simulation two parabolic target tracks consisting of n=20 ${\cal X}$ and ${\cal Y}$ position values were again generated, where ${\cal X}$ varied linearly over time from ${\cal X}= 0$ to ${\cal X} = 4$. This time, one target was turned off over the middle 10 measurement times. Again, Gaussian measurement noise with a variance of 0.01 was added to the target tracks. The number of ``false detect'' events was generated using a Poisson distribution with a false alarm rate of 2, and the false detections themselves were generated using a uniform distribution over the range [0,4] in both ${\cal X}$ and ${\cal Y}$. Probability of a missed detection was assumed to be 0.2. K_max=3 and L=8. We partitioned the data into 4 blocks of T=5measurement times. Figure 4(a) shows the ${\cal X}$ and ${\cal Y}$ position values for the true tracks (solid lines), along with the noisy measurements (boxed and circled x's) and false alarms (x's) used as input to the algorithm. This time the algorithm correctly detected two tracks for the first and last block, and one track for the middle two (i.e. $\hat{K}_1 = \hat{K}_4 = 2$, $\hat{K}_2 = \hat{K}_3 = 1$). Figure 4(b) shows $\hat{K}$ vs. time.