# Universal Behavior of the Image Resolution for Different Scanning Trajectories
## 3 Results and Discussion
### Relationship between Pixel Size and Scanning Parameters  
To understand the relationship between pixel size and \(N_{P}\) for all four scanning trajectories, numerical calculations are performed to estimate values of \(H_{max}\), \(D_{min}\), and \(D_{max}\) for \(N_{P}\) values ranging from 10 to 400. Because the maximum values of \(H_{max}\) and \(D_{max}\) exhibit a hyperbola function of \(N_{P}\) for the SL trajectory, it is safe to assume that this trend will also hold for \(D_{min}\) and all remaining trajectories investigated in this work. Figure 3a-d display the trend of \(H_{max}\), \(D_{min}\), and \(D_{max}\) as a function of \(N_{P}\) for all four scanning techniques examined in this work. Here, we can see that \(D_{min}\) also follows the same trend as \(H_{max}\) and \(D_{max}\). The figure also \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Max \(H_{max}\)** & **Max \(D_{min}\)** & **Max \(D_{max}\)** \\ \hline B.C. & 0.502 & 0.502 & 0.503 \\ T.L. & 0.287 & 0.320 & 0.361 \\ S.L. & 0.225 & 0.251 & 0.283 \\ R.L. & 0.166 & 0.188 & 0.195 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum distance to nearest neighboring points in mm for different trajectories using the proposed method with \(N_{P}\) = 100. Figure 2: Theoretical and numerically calculated values of \(H_{max}\) (**a**) and \(D_{max}\) (**b**) for sinusoidal Lissajous over various \(N_{P}\) values. demonstrates that for all scanning trajectories, numerically computed values for \(H_{max}\), \(D_{min}\), and \(D_{max}\) fit well to a hyperbola, specified in (11). The obtained \(R^{2}\) values, which are almost unity, further support the quality of the fit. The percentage of variation in dependent variables that can be explained by the independent variable is shown statistically by the \(R^{2}\) coefficient, also known as the coefficient of determination. In other words, the result is a measurement of how well the data fit the model, and a value close to unity indicates a near perfect match. The variation in \(H_{max}\) values across all trajectories is depicted in Figure 3e. As can be seen at a particular value of \(N_{P}\), RL has the lowest \(H_{max}\) value, which is then followed by SL, TL, and then BC. \[ \begin{split}& H_{max}=\frac{b_{1}}{N_{P}a^{2}}\\ & D_{min}=\frac{b_{2}}{N_{P}a^{2}}\\ & D_{max}=\frac{b_{3}}{N_{P}a^{3}}\end{split}  \tag{11}\]  
where \(a_{1,2,3}\) and \(b_{1,2,3}\) are constants. Figure 3: The maximum values for \(H_{max}\), \(D_{min}\), and \(D_{max}\) over a range of \(N_{P}\) values were calculated numerically, along with the corresponding fitted curves for the following trajectories: (**a**) bidirectional Cartesian, (**b**) triangular Lissajous, (**c**) sinusoidal Lissajous, (**d**) radial Lissajous. Plots of the \(H_{max}\) values for each trajectory are shown in (**e**). It is important to note that the BC scanning trajectory formula that is currently in use is incomplete. This equation only performs well when \(N_{P}/2\) is even. For all \(N_{P}/2\) is odd cases, the scanning pattern is discontinuous at the haft-way point of the repetition time. This means that the BC trajectory formulas presented in (3) and (4) are only relevant in \(N_{P}/2\) even cases (Figure A2 in Appendix A shows trends for both even and odd \(N_{P}/2\) cases). Here, we can observe that the trend of distance value for odd cases is slightly larger than for even situations, showing that the formulation for odd and even circumstances is inconsistent. Only numerically generated values for \(N_{P}/2\) that are even are utilized in this work to ensure the consistency and accuracy of the computed data for the BC trajectory. The values of the constants \(a_{1,2,3}\) and \(b_{1,2,3}\) that were determined by fitting the data to (11) are listed in Table 3. It should be noted that the values for \(a_{1,2,3}\) are close to unity for all trajectories.