## 2 Materials and Methods  
### Concept  
Cameras for plant detection are typically mounted on a boom of a sprayer or fertilizer spreader [10; 11; 13], as illustrated in Figure 1. They are oriented so that their optical axis is perpendicular to the field and captures top-view images of plants [33]. Depending on the distance between the camera lens and captured plane, lens properties, and sensor size, the frame width or height is equivalent to an actual linear distance. For simplicity, the linear length of the side of a field of view of a frame parallel to the direction of travel shall be denoted as \(S\), in meters per frame. To provide complete visual coverage of the traversed width of the boom, the maximum spacing between adjacent cameras is equal to the length of the side of a field of view of a frame perpendicular to the direction of travel, denoted as \(W\), in meters per frame [10]. During motion, the traverse distance between two consecutive frames of the camera (\(d_{f}\)), in meters, is equal to the product of travel velocity (\(\overrightarrow{v}\)), in m/s, and the time between the frames (\(1/fps\)), in seconds, as illustrated in Equation (1). \[d_{f}=\overrightarrow{v}\cdot\ \frac{1}{fps}=\frac{\overrightarrow{v}}{fps} \tag{1}\]  
The ratio of \(S\) to \(d_{f}\) is proposed as the overlapping rate (\(r_{o}\)), a dimensionless parameter, and is represented by Equation (2). \[r_{o}=\frac{S\times fps}{\overrightarrow{v}}=\frac{S}{d_{f}} \tag{2}\]  
With a single camera, the value of \(r_{o}\) describes whether there is an overlap or gap between frames. Depending on the value of \(r_{o}\), certain regions in the traversed field of the machine, with a single camera configuration, will be uniquely captured, captured in multiple frames, or completely missed, as shown in Figure 2 and illustrated by the following cases:  
* Case 1: \(r_{o}=1\). When \(S\) and \(d_{f}\) are equal, the extents of each consecutive processed frame are side by side. Hence, both gaps and overlaps are absent. This scenario is ideal since the vehicle velocity and camera frame rate match perfectly. * Case 2: \(r_{o}>1\). The vision system accounted for all regions in the traversed field, but there is an overlap between the frames. The length of the current frame that is already accounted for by the previous one is \(S-d_{f}\). The vehicle can run faster if the mechanical capacity allows. * Case 3: \(r_{o}<1\). Gaps will occur between each pair of consecutive frames, and the camera will miss certain plants. The length of each gap is \(d_{f}-S\). We can define a gap rate (\(r_{g}\)) as shown in Equation (3) to depict the significance of the gap. Figure 1: Camera mounting location and orientation in a boom (not drawn in scale). (3)  
Figure 2: Relative positions of two consecutive processed frames of a vision system: (**a**) \(r_{o}=1\) since \(d_{f}=S\), resulting in frames with unique bounded regions; (**b**) \(r_{o}>1\) since \(d_{f}<S\), resulting in overlap region (**b**); and (**c**) \(r_{o}<1\) since \(d_{f}>S\), resulting in missed region. 2.1.1 Theoretical Detection Rate  
The theoretical or maximum detection rate (\(r_{d,th}\)) can be defined as \(\min(1,\ r_{0})\), shown in Equation (4). The \(r_{d,th}\) was also a dimensionless parameter. \[r_{d,th}=\left\{ \begin{array}{ll}1,&r_{0}\geq 1\\ r_{o},&r_{o}<1\end{array} \right. \tag{4}\]  
2.1.2 Maximizing Travel Velocity  
Setting \(r_{o}=1\) in Equation (2) resulted in Equation (5), which is similar to the equation used by [PERSON] et al. (2018) [PERSON] et al. (2018) in calculating the maximum travel velocity of a sprayer. However, Equation (5) only describes the maximum forward velocity \(\overrightarrow{v}_{max}\) that a vision-equipped robot can operate to prevent gaps while traversing the field as a function of \(S\) and \(fps\). \[\overrightarrow{v}_{max}=S\times fps \tag{5}\]  
2.1.3 Increasing \(\overrightarrow{v}_{max}\)  
A consequence of Equation (5) was that increasing the length of the frame \(S\) at a constant \(fps\) shall increase \(\overrightarrow{v}_{max}\).