*2.3. The Multi-Object Observation Models*

In the RFS multi-object tracking framework, given a set of measurements *<sup>Z</sup>* = {*z*1:|*<sup>Z</sup>*|}, we have a standard observation model of the following form: [54]

$$\log(Z|\mathbf{X}) \propto \sum\_{\theta \in \Theta(\mathcal{L}(\mathbf{X}))} \prod\_{(\mathbf{x},l) \in \mathbf{X}} \psi\_Z^{(\theta(l))}(\mathbf{x}, l) \tag{8}$$

where

$$\psi\_{Z}^{(\theta(l))}(\mathbf{x},l) = \delta\_{0}(\theta(l))q\_{D}(\mathbf{x},l) + (1-\delta\_{0}(\theta(l)))\frac{p\_{D}(\mathbf{x},l)g(z\_{\theta(l)}|\mathbf{x},l)}{\kappa(z\_{\theta(l)})}$$

*κ*(·) is the clutter intensity, *pD*(·) and *qD*(·) are respectively the detection and miss-detection probabilities, *<sup>g</sup>*(*z*|*x*, *<sup>l</sup>*) is the likelihood that (*x*, *<sup>l</sup>*) generates measurement *<sup>z</sup>*, *<sup>θ</sup>* : <sup>L</sup> → {<sup>0</sup> : <sup>|</sup>*Z*|} is a positive 1-1 map, and Θ is the entire set of such mappings.

For image observation, with the assumption that object template *T*(·) is not overlapped, i.e., *T*(*x*1) = *T*(*x*2) given *x*<sup>1</sup> = *x*2, the separable likelihood is given by the following [26]:

$$\mathcal{S}(y|\mathcal{X}) = f\_B(y) \prod\_{\mathbf{x} \in \mathcal{X}} \mathcal{g}\_y(\mathbf{x}) \tag{9}$$

where *y* denotes the observed image, *fB* denotes the likelihood of the entire set of **X**, and *gy*(*x*) denotes the likelihood of a single object in the observed image. The designs of *fB* and *gy* vary according to the applications, characteristics of observed image, and object appearances.

First introduced in Reference [27], the concept of a hybrid TBD observation model takes advantage of both standard and separable likelihood models. Intuitively, while detected objects can be updated by the associated point measurements, the miss-detected objects can be updated directly from the image observation. This intuition can be described mathematically by defining the following:

$$\sigma\_T(T(y)|\mathbf{x}, l) \triangleq \frac{\mathcal{G}\_T(T(y)|\mathbf{x}, l)}{\mathcal{G}\_T(T(y)|\mathcal{Q})} \tag{10}$$

The hybrid likelihood can then be written as follows [27]:

$$\log(y|\mathbf{X}) \propto \sum\_{\theta \in \Theta(\mathcal{L}(\mathbf{X}))} \prod\_{(\mathbf{x},l) \in \mathbf{X}} \varphi\_y^{(\theta(l))}(\mathbf{x}, l) \tag{11}$$

where

$$\varrho\_y^{(\theta(l))}(\mathfrak{x},l) = \psi^{(\theta(l))}(\mathfrak{x},l|Z)[\sigma\_T(T(y)|\mathfrak{x},l)]^{\delta\_0\theta(l)}$$
