*3.2. Adaptive to Unknown Backgrounds*

In practice, the both the clutter rate and detection profile are unknown and unpredictably vary with time. Prior knowledge of background models, therefore, are typically unavailable. Mismatch in background models results in degradation of tracker performance [4]. In this section, based on the suite of methods for tackling the unknown clutter rate and detection probability introduced in [4], the bootstrapping method will be proposed.

A technique that accommodates the jointly unknown clutter rate *λ* and the unknown probability of detection *pD* has been introduced in [29]. This technique considers clutter as an RFS of "generator targets" or "false targets", and incorporates the non-homogeneous and unknown detection probability into each target state. Each real target state *<sup>x</sup>* <sup>∈</sup> <sup>X</sup> is corresponded to an augmented state *xa* = (*x*, *<sup>a</sup>*), in which *<sup>a</sup>* <sup>∈</sup> <sup>X</sup>*<sup>d</sup>* = [0, 1] is the variable on the probability detecting *<sup>x</sup>*. The augmented multitarget state now can be described as follows

$$X\_a = (\mathbf{x}\_{a,1}, \dots, \mathbf{x}\_{a,n}) = \{ (\mathbf{x}\_1, a\_1), \dots, (\mathbf{x}\_n, a\_n) \} \tag{39}$$

Similarly, the augmented generator target state is *xc* = [*x*¯, *ac*] with *<sup>x</sup>*¯ <sup>∈</sup> <sup>X</sup>*<sup>c</sup>* be the generator target state, and *ac* <sup>∈</sup> <sup>X</sup>*<sup>d</sup>* = [0, 1]. The augmented generator multitarget state is

$$X\_{\mathcal{L}} = (\mathbf{x}\_{\mathcal{L},1}, \dots, \mathbf{x}\_{\mathcal{L},m}) = \{ (\mathbf{x}\_1, a\_{\mathcal{L}1}), \dots, (\mathbf{x}\_{n\prime} a\_{\mathcal{cm}}) \} \tag{40}$$

Then the probability of detection is replaced by *a* and *ac*, respectively.

$$p\_{D,a}^{(s)}(\mathbf{x}\_a) = p\_{D,a}^{(s)}(\mathbf{x}, a) \triangleq a \tag{41}$$

$$p\_{D,\mathcal{c}}^{(s)}(\mathbf{x}\_{\mathcal{c}}) = p\_{D,\mathcal{c}}^{(s)}(\vec{x}, a\_{\mathcal{c}}) \triangleq a\_{\mathcal{c}} \tag{42}$$

Assuming that the false and true targets are statistically independent, then each of the augmented generator targets can be modeled for their characteristics as appearances, disappearances, and transitions, together with likelihood, detection and missed detection. The multitarget state is then a combination of (augmented) actual targets and clutter generators. Meaning that, the augmented hybrid space X*<sup>h</sup>* involving the multitarget state can be defined as follows [29]

$$\mathbb{X}^{(q)} = \left(\mathbb{X} \times \mathbb{X}^{(q)}\right) \noplus \left(\mathbb{X}^{(c)} \times \mathbb{X}^{(q)}\right) \triangleq \mathbb{X}^{(q)}\_{(q)} \nplus \mathbb{X}^{c}\_{(q)}\tag{43}$$

where " " denotes the disjoint union, and " × " denotes the Cartesian product.

The multitarget state (4) and multitarget observation (8)) at time *k* now become the hybrid ones:

$$X\_h = X\_d \uplus X\_{\mathcal{L}} \tag{44}$$

$$Z\_h = Z\_u \uplus Z\_c \tag{45}$$

with *Za* and *Zc* be the augmented multitarget and augmented generator observations, respectively. The integral of a function *<sup>f</sup>* (*h*) : <sup>X</sup>(*h*) <sup>→</sup> <sup>R</sup> is given by [29]

$$\int\_{\mathcal{X}^{(h)}} f^{(h)}\left(\mathcal{X}\_{h}\right)d\mathbf{x}\_{h} = \int\_{\mathcal{X}\_{a}^{(d)}} f\_{a}^{(d)}\left(\mathcal{X}\_{a}\right)\delta\mathcal{X}\_{a} + \int\_{\mathcal{X}\_{a}^{\varepsilon}} f\_{a}^{(\varepsilon)}\left(\mathcal{X}\_{\varepsilon}\right)\delta\mathcal{X}\_{\varepsilon} \tag{46}$$

Here, the set integral (19) has been applied to both augmented multitarget state and augmented generator multitarget state terms, i.e, [4]

$$\int f\_a^{(d)}\left(\mathbf{X}\_a\right)\delta \mathbf{X}\_a = \sum\_{n\geq 0} \frac{1}{n!} f\_a\left(\left\{\mathbf{x}\_{a,1}, \dots, \mathbf{x}\_{a,n}\right\}\right) d\mathbf{x}\_{a,1}, \dots, d\mathbf{x}\_{a,n} \tag{47}$$

$$\int f\_c^{(d)}\left(\mathbf{X}\_{\mathbf{c}}\right)\delta\mathbf{X}\_{\mathbf{c}} = \sum\_{m\geq 0} \frac{1}{m!} f\_a^{\mathbf{c}}\left(\{\mathbf{x}\_{\mathbf{c},1\prime},\dots,\mathbf{x}\_{\mathbf{c},n}\}\right) d\mathbf{x}\_{\mathbf{c},1\prime}, \dots, d\mathbf{x}\_{\mathbf{c},m} \tag{48}$$

Noting that the the measurement likelihood is kept unchanged

$$\lg\left(\mathbf{x}\_{a}\right) = \lg\_{a}^{(s)}\left(\mathbf{x}, a\right) \triangleq \mathbf{g}^{(s)}\left(\mathbf{x}\right) \tag{49}$$

$$\mathcal{g}\left(\mathbf{x}\_{\mathcal{E}}\right) = \mathcal{g}\_{\mathcal{E}}^{(s)}\left(\bar{\mathbf{x}}, a\_{\mathcal{E}}\right) \triangleq \mathcal{g}^{(s)}\left(\bar{\mathbf{x}}\right) \tag{50}$$

While the method proposed in [29] results in good estimates of targets, it do not produce the trajectories of the targets. Moreover, although this method is a closed-form solution of the CPHD recursion with jointly unknown clutter rate and detection profile, it is proposed for single-sensor multiple targets estimation solely. In this paper, we propose a method of using the technique introduced in [29] to estimate the mentioned unknown parameters then bootstrapping them into the GLMB filter for tracking on-the-fly. The structure of the proposed method is given in Figure 1.

**Figure 1.** The proposed structure of the *B*-GLMB filter.
