4.1.3. Hybrid TBD Observation Model

In this simulation, we use the hybrid measurement model to track objects following a linear dynamic motion model. The surveillance region is 100 × 100 pixels with image cell size of 1, total time step of *K* = 100, and Δ = 1. The observation are the raw images, which are arrays of pixels. In particular, for a pixel *i* at the image coordinate (*a*(*i*), *b*(*i*)), the array value is given as follows [26,27]:

$$y^{(i)} = \left[ \sum\_{\mathbf{x} \in \mathcal{X}} \frac{I\_k}{2\pi\sigma\_{\hbar}} \exp\left( -\frac{(a^{(i)} - p\_{\mathbf{x}})^2 + (b^{(i)} - p\_{\mathbf{y}})^2}{2\sigma\_{\hbar}^2} \right) \right] + w^{(i)} \tag{27}$$

where *<sup>w</sup>*(*i*) <sup>N</sup> (0, *<sup>σ</sup>y*) is Gaussian noise. In this experiment, we set *<sup>σ</sup><sup>h</sup>* <sup>=</sup> 4 and *<sup>σ</sup><sup>y</sup>* <sup>=</sup> 1. We choose the value of *Ik* such that the signal to noise ratio (SNR) varies over the range 7 to 10 dB. For the observation model from the perspective of the filter, we fix its SNR value to 10 dB. From the raw images, we then use hard-shareholding to extract the points measurements at each frame.

**Figure 7.** OSPA2 error for nonlinear dynamic scenario.

**Figure 8.** Estimated cardinality for nonlinear dynamic scenario.

The dynamic model and standard observation model are similar to the ones in Section 4.1.1 with *σ<sup>v</sup>* = 1 pixel/s, *pS* = 0.98, and *σ* = 4 pixels with a clutter rate of 10. The expected new births states are *m*(1) *<sup>B</sup>* = [5, 0, 25, 0] *<sup>T</sup>*, *m*(2) *<sup>B</sup>* = [5, 0, 90, 0] *<sup>T</sup>* , *m*(3) *<sup>B</sup>* = [80, 0, 90, 0] *<sup>T</sup>* , *m*(4) *<sup>B</sup>* = [5, 0, 5, 0] *T*, and *m*(5) *<sup>B</sup>* = [90, 0, 30, 0] *<sup>T</sup>* with the covariance of *PB* = *diag*([3, 2, 3, 2]) and the probability *rB* of 0.03. The ground truth location of objects is shown in Figure 9 while Figure 10 shows samples of raw image observation along with points detection. The implementation of the filtering phase is as the same as in Reference [27]. The smoothing interval is set to the entire tracking time with the track pruning threshold of the smoother set to 3 time steps.

This experiment is run over 100 Monte Carlo trials. The means of OSPA error and OSPA2 error are shown respectively in Figures 11 and 12. The estimated cardinality is plotted in Figure 13.

**Figure 9.** Ground truth for a hybrid track-before-detect (TBD) scenario (circle: track start position, triangle: track end position).

**Figure 10.** Samples of raw images and point observations for a hybrid TBD scenario (red asterisk: ground truth position, green circle: point detection).

**Figure 12.** OSPA2 error for a hybrid TBD scenario.

**Figure 13.** Estimated cardinality for a hybrid TBD scenario.
