# Universal Behavior of the Image Resolution for Different Scanning Trajectories
## 3 Results and Discussion
### Relationship between Pixel Size and Scanning Parameters
Our results demonstrated that all of the defined distances \(H_{max}\), \(D_{min}\), and \(D_{max}\) for the four scanning trajectories examined in this work are proportional to \(1/N_{P}\). The findings showed that the inversely proportional relationship between the distance or pixel size and \(N_{P}\) is a common characteristic of all the analyzed trajectories, not just the sinusoidal Lissajous trajectories. Thus, setting the value of \(a_{1,2,3}\) to unity further simplified the formulas for \(H_{max}\), \(D_{min}\), and \(D_{max}\). Substituting this to (11) yields:  
\[ \begin{array}{l}H_{max}=\frac{b_{1}}{N_{P}}\\ D_{min}=\frac{b_{2}}{N_{P}}\\ D_{max}=\frac{b_{3}}{N_{P}}\\ \end{array}  \tag{12}\]  
Equation (12) states that \(N_{P}\) is inversely proportional to the pixel size, which is determined by a value from either \(H_{max}\), \(D_{min}\), or \(D_{max}\). This means that as \(N_{P}\) increases, the pixel size decreases, resulting in an improvement in image resolution or image quality. Table 4 shows the updated values of \(b_{1,2,3}\) that were determined by the fit when a = 1. It should be noted that new \(R^{2}\) values remain very close to 1 after setting a = 1, indicating a high-quality fit. Note that \(N_{P}\) is related to the scanning frequency \(f_{x}\) and the repetition time \(T_{R}\) via the following formula [8]:  
\[T_{R}=\frac{N_{P}}{f_{x}} \tag{13}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **B.C.** & **T.L.** & **S.L.** & **R.L.** \\ \hline \multirow{2}{*}{Max \(H_{max}\)} & \(b_{1}\) & 0.022 & 0.034 & 0.043 & 0.046 \\  & \(R_{1}^{2}\) & 0.999 & 0.999 & 0.999 & 0.999 \\ \multirow{2}{*}{Max \(D_{min}\)} & \(b_{2}\) & 0.022 & 0.031 & 0.040 & 0.058 \\  & \(R_{2}^{2}\) & 0.999 & 1.000 & 0.999 & 0.999 \\ \multirow{2}{*}{Max \(D_{max}\)} & \(b_{3}\) & 0.020 & 0.024 & 0.031 & 0.040 \\  & \(R_{3}^{2}\) & 0.999 & 0.999 & 0.999 & 0.998 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Constant \(b_{1,2,3}\) and their \(R^{2}\) values resulting from a hyperbola fit when \(a_{1,2,3}=1\), as shown in (12). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **B.C.** & **T.L.** & **S.L.** & **R.L.** \\ \hline \multirow{2}{*}{Max \(H_{max}\)} & \(a_{1}\) & 0.991 & 1.003 & 1.003 & 1.026 \\  & \(b_{1}\) & 0.021 & 0.034 & 0.044 & 0.052 \\ \multirow{2}{*}{Max \(D_{min}\)} & \(a_{2}\) & 0.992 & 1.000 & 0.998 & 0.991 \\  & \(b_{2}\) & 0.021 & 0.031 & 0.040 & 0.056 \\ \multirow{2}{*}{Max \(D_{max}\)} & \(a_{3}\) & 0.997 & 1.014 & 1.014 & 1.028 \\  & \(b_{3}\) & 0.020 & 0.026 & 0.033 & 0.045 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Constant \(a_{1,2,3}\) and \(b_{1,2,3}\) values resulting from a hyperbola fit, as shown in (11). Substituting (13) into (12) leads to the following equation:  
\[ \begin{split} H_{max}&=\frac{b_{1}}{Np}=\frac{b_{1}}{ \tau_{R}/\tau_{x}}\\ D_{min}&=\frac{b_{2}}{Np}=\frac{b_{2}}{\tau_{R}/ \tau_{x}}\\ D_{max}&=\frac{b_{2}}{Np}=\frac{b_{3}}{\tau_{R}/ \tau_{x}}\end{split}  \tag{14}\]  
Equation (13) shows that the scanning duration is directly related to \(N_{P}\), whereas the coefficient \(N_{P}\) is inversely proportional to the scanning trajectory's resolution. Combining these connections yields (14), which demonstrates that the frequency and repetition time are inversely related to the scan resolution. Such kinds of relationships between imaging parameters were not previously mentioned; thus, it is crucial and significantly important to be able to analyze the data.