# Research Trends in Collaborative Drones
## 4 Target Recognition
### Approach and Methodology  
Consider a drone flying over a certain highway represented as a line segment \(L\) of length \(\ell=|L|\). During the flight, the drone retains a given altitude \(x\). The speed \(v(x)\) of the drone may be decided by prevailing regulations on the drone's flying territory and is a function of \(x\) (see Figure 3, left picture).  
The drone's camera has a field of view (or visibility angle) \(\alpha\). When flying at height \(x\), the drone's visibility angle \(\alpha\) subtends a subsegment of the line of length \(\tau(x)\); this is the portion of the line \(\ell\) on which the drone's camera can detect objects. Since the field of view of the camera is \(\alpha\) and the drone is flying at altitude \(x\), we must have that \(\tau(x)=2x\tan(\alpha/2)\) degrees.  
First, we determine the target object density as a fraction of the total length \(\ell\) of the segment \(L\). As the drone is moving, it is detecting objects located on the line segment.  
Figure 3: Drone \(M\) is flying over a line segment \(L\) of length \(\ell\) and a certain object of length \(\sigma\). At height \(x\), the drone flies with speed \(v(x)\) which depends on the altitude \(x\). The picture in the righthand side indicates that the drone’s camera has a sliding window effect of width \(\tau(x)\) within which it can detect an object of length \(\sigma\).  
The drone's camera at altitude \(x\) has a sliding window of width \(\tau(x)\). This is moving with speed \(v(x)\), thus yielding a time of possible observation equal to \(\tau(x)/v(x)\).  
The following analysis is inspired from [38; 40]. An object of length \(\sigma\) is placed on the line \(L\). As indicated in Figure 3 (right picture), assuming it is moving left to right, the drone starts observing the object at its leftmost endpoint \(A\). It stops observing when at the point \(B\), thus yielding a total distance of observation equal to \(\sigma+\tau(x)\). The total duration of observation is equal to \(\frac{\sigma+\tau(x)}{v(x)}\). It follows that the drone is observing the target object a fraction \(f(x)\), where  
\[f(x)=\frac{(\sigma+\tau(x))/v(x)}{(\ell-\tau(x))/v(x)}=\frac{\sigma+\tau(x)}{ \ell-\tau(x)}, \tag{1}\]  
of its total flying time over the segment of length \(\ell\), assuming that the rightmost point of the target object is at a distance at most \(\tau(x)\) from the rightmost point of the segment of length \(\ell\). In a way, the righthand side of Equation (1) represents the target object density.  
Next, we look at distributions for the probability that the object is detected by the drone. Using distance sampling statistics, one could use the exponential distribution  
\[p(x)=\frac{1}{\ln(1+1/k)}\cdot\frac{e^{-x}}{k+e^{-x}}, \tag{2}\]  
as the probability that a drone flying at altitude \(x\geq 0\) detects the target object when its camera is directed toward it.  
The parameter \(k\) appearing in Equation (2) is chosen so as to indicate the probability that an autonomous ground vehicle detects the target when traveling at ground level (altitude \(x=0\)), e.g., \(p(0)=\frac{1}{k\ln(1+1/k)}\). The term \(\frac{1}{\ln(1+1/k)}\) is the normalizing coefficient and is derived as follows. Observe that \(\int\frac{e^{-x}}{e^{-x}+k}dx=x-\ln(ke^{x}+1)\) and for \(x\geq 0\), this yields the formula  
\[\int_{0}^{\infty}\frac{e^{-x}}{e^{-x}+k}dx=\ln(1+\frac{1}{k})\]  
Note that the (exponential) probability distribution above is one of several potential choices; others may be considered depending on the situation.  
Using the target object density calculations in the previous subsection, we conclude that the probability \(P_{\sigma}(x)\) that the drone detects a target object of length \(\sigma\) is equal to  
\[P_{\sigma}(x):=p(x)\cdot f(x)=\frac{1}{\ln(1+k)}\cdot\frac{e^{-x}}{k+e^{-x}} \cdot\frac{\sigma+\tau(x)}{\ell-\tau(x)},\]  
where \(f(x)\) is the fraction of its total flying time that the drone will be observing a fixed object of length \(\sigma\) over the segment of length \(\ell\).