# Universal Behavior of the Image Resolution for Different Scanning Trajectories
## 2 Scanning Methods  
### Scanning Trajectory Pattern and Density  
To comprehend the relationship between image resolution and scanning trajectory, it is necessary to first define the pixel size of any scanning trajectory. Pixel size can be defined as the half of the shortest distance between two non-overlapping scanning lines that are usually parallel to each other [25]. Let us first identify and utilize the coordinate position of crossing sites by which two scanning lines overlap as reference points; these  
Figure 1: Scanning trajectories with nearest (red dashed lines)/furthest (green dashed lines) nearest neighbor points and maximum perpendicular distance \(H_{max}\) for \(N_{P}=12\). (**a**) Bidirectional Cartesian. (**b**) Triangular Lissajous. (**c**) Sinusoidal Lissajous. (**d**) Radial Lissajous. points are represented by the blue dots in Figure 1. Depending on the scanning patterns, these crossing points divide the image into either a uniform or non-uniform grid. The patterns created by Cartesian (Figure 1a) and non-Cartesian (Figure 1b-d) scanning paths and their overlapping sites are shown in Figure 1. It can be seen here that for each crossing point that is not on the boundary, the closest six neighboring points are those that set the minimum (\(D_{min}\)) and maximum (\(D_{max}\)) distances between this crossing point and those belonging to its next adjacent lines (see Figure A1 in Appendix A for a zoomed in version of all trajectories). Distances between each crossing point and its six nearest neighbors in the FOV are first calculated and sorted to obtain the \(D_{min}\) (shown by the red dashed line) and \(D_{max}\) (indicated by the green dashed line) values [24]. The pixel size of the scanning pattern should at least be established using the value with the greatest minimum distance because each crossing point has its own \(D_{min}\) and \(D_{max}\) values. In general, each crossing point should have eight nearest neighbors; however, the final two farthest next nearest neighbor points are not required for calculating the image resolution. This is because the shortest distance between the two parallel nearest neighbor lines will always be used to determine the pixel size, and, hence, image resolution. Despite the fact that \(D_{min}\) and \(D_{max}\) are straightforward definitions of distances that nearly approximate the pixel size, they do not precisely depict the true pixel size of the scanning pattern. In reality, a considerably more precise representation that is usually employed to identify the pixel size is the largest/longest perpendicular distance between two contiguous scanning lines in the FOV, represented by \(H_{max}\) in Figure 1[25]. To calculate \(H_{max}\) at each crossing point, the largest perpendicular distance from the focus/center crossing point to all nearest adjacent scanning lines are calculated. \(D_{min}\) and \(D_{max}\) are required to calculate \(H_{max}\). Two to four \(H_{max}\) were acquired depending on the irregularity of the scanning trajectory, and they were sorted to achieve the highest value of \(H_{max}\). In other words, the values of \(H_{max}\) depend on the scanning trajectory and can be either four distinct values or two couples of values. The maximum value of \(H_{max}\), \(D_{min}\), and \(D_{max}\) for all points in the FOV are recorded because these values closely represent the pixel size, especially the \(H_{max}\). This method of calculating the pixel size eliminates the risk of underestimating or overestimating the image resolution, which would occur if the average values of all estimated distances within the FOV were used. A theoretical formulation for \(H_{max}\), \(D_{min}\), and \(D_{max}\) does not exist for the majority of scanning trajectories.