## 3 Experiments and Results
### Experiment 2  
This experiment was based on Set 1. The accuracy of the uniform model for radial measurement was evaluated using 10-fold cross-validation. To that end, Set 1 recordings were split into training and testing sets. The uniform model was trained using the largest enclosed circles of the training set. The trained uniform model was then evaluated for estimating mm radii of all circles from the testing set, in addition to smaller circles (those that were not used during the training process) of the training set. Figure 9 presents scatter plots of the magnitude of errors over all 10 folds relative to the radial length of the target circle and the working distance. Investigating scatter plots of Figure 9 reveals that the measurement error of the uniform model depends on the working distance and the length of the target object. However, the relationship seems to be non-linear. Additionally, our analysis showed that neither of the variables had a normal distribution. Therefore, both parametric and non-parametric tests were used to quantify the effect of working distance and length of the object on the magnitude of the error. Table 2 reports values of [PERSON]'s \(\tau\), [PERSON]'s \(\tau\), and [PERSON]'s \(\rho\). Based on Table 2, we see a moderate positive correlation between the magnitude of error and length of the target object and a strong positive correlation between the magnitude of the error and the working distance. Figure 8: Performance of estimating the working distance: (**A**) measurement accuracy of different laser points, (**B**) effect of working distance. Figure 9: Performance of uniform model for radial measurements: (**A**) effect of object length, (**B**) effect of working distance. Table 3 quantifies the effect of radial length of the object and working distance on the magnitude of error from the non-uniform model. Based on Table 3, we see the magnitude of error has very week associations with the working distance and length of the target object. Comparing the results of Tables 2 and 3 highlights a primary advantage of the non-uniform method over its uniform counterpart. Specifically, the non-uniform method has a stable and relatively constant error for a wide range of working distances and target lengths. Additionally, comparing Figures 9 and 10, we see the non-linear method reduces measurement error significantly. To better quantify this, the range of working distance was divided into separate intervals. Average and standard deviation of error, and magnitude (i.e., absolute value) of the error for both, uniform and non-uniform methods, were calculated in each interval. Table 4 presents the results. Based on this table we see another advantage of the non-uniform approach. The average value of error in the non-uniform method is almost zero; therefore, measurement error using the non-uniform approach has a random nature. Thus, multiple radial measurements can reduce the error significantly. Conversely, the average error in the uniform approach is not zero, indicating \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **[PERSON]’s** & \multicolumn{2}{c}{**[PERSON]’s**} & **[PERSON]’s** \\ \hline Stat parameter & r & P & \(\tau\) & P & \(\rho\) & P \\ \hline Radial Length & 0.59 & \(\varepsilon\) & 0.56 & \(\varepsilon\) & 0.69 & \(\varepsilon\) \\ \hline Working Distance & 0.76 & \(\varepsilon\) & 0.57 & \(\varepsilon\) & 0.74 & \(\varepsilon\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation coefficients of uniform model for radial measurement error. The symbol \(\varepsilon\) denotes a p < 0.0001. Figure 10: Performance of non-uniform model for radial measurements: (**A**) effect of object length, (**B**) effect of working distance.