## 3 Results
The main challenge in using triangle-similarity for tracking is that the motions that move the laser spot off its target and create a need for video analysis complicate its implementation because the perspective of the image is no longer the same as the perspective used in calibration. To a first approximation, we can allow for the change in perspective by calculating P\({}_{1}\) and P\({}_{2}\) in a model that corresponds to the analytical geometry of the perspective used in calibration. In this analytical geometry, the triangle formed by the intersection of the lines in the image--defined by the top edges of the air slot and the line of length P connecting points opposite each other across the top of the air slot--is isosceles. To test this model, we performed experiments in our lab with a vehicle carrying two cameras and the laser range finder, moved manually down the air-slot mockup provided by PNNL. A video frame taken from the rear camera in our experiments is shown in Figure 3. The channel has been carefully lined with graph paper so that points directly opposite each other on the top edge could be easily identified. Figure 3 shows four of these points connected by lines. The lengths of these lines in pixels, determined from the frame coordinates shown in Table 1, are 311 and 570 pixels for far and near lines, respectively. Using these results for P\({}_{1}\) and P\({}_{2}\) in the formula WF/P\({}_{1}\)-WF/P\({}_{2}\) with W = 73.9 mm, the measured width of the channel at its top, and F = 452 pixels, the experimentally determined focal length of the rear camera, triangle similarity predicts the lines are separated by 48.8 mm. This result differs from the expected 50 mm derived from marks on the graph paper by only 1.2 mm. The approach used in the laboratory to find P\({}_{1}\) and P\({}_{2}\) will not be available for analysis of the video footage taken in the air slots because the point that is directly opposite the target on the other side of the air slot will not be known. Edge-detection in video analysis can be used to determine the slopes of the intersections of an air slot with the tank bottom. Given these slopes and the coordinates of a feature in the air-slot-tank-bottom interface, the isosceles-triangle model can predict the width of the air slot in pixels at the targeted feature. We tested this approach on the images obtained in our laboratory experiments with the vehicle tilted left and right by comparison with the lengths of near and far lines, like those shown in Figure 3, calculated from the coordinates of their endpoints. Errors in the predictions of the isosceles-triangle model in 24 cases ranged between 3% and 18% of the true values. These results motivated us to develop a linear regression model of corrections to the isosceles-triangle model. Optimum slope and y-intercept obtained with a training set of 24 examples are 0.349055 and \(-\)0.220441, respectively. Predictions of the corrected model on six new cases that are midway between the near and far training examples are shown in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **x** & **y** \\ \hline p1 & 537 & 841 \\ p2 & 1099 & 748 \\ p3 & 697 & 885 \\ p4 & 1005 & 842 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coordinates of line end points in Figure 3. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Left Edge Slope**} & \multicolumn{2}{c}{**\% Difference from Actual Width**} & \multicolumn{1}{c}{**\% Difference after**} \\  & **Using Isosceles Method** & **Correction** \\ \hline
0.618892508 & 3.605\% & \(-\)4.047\% \\
0.980263158 & 9.758\% & 2.415\% \\
0.291066282 & \(-\)9.825\% & \(-\)2.059\% \\
0.605166052 & \(-\)1.622\% & 0.701\% \\
0.952380952 & 10.781\% & 0.418\% \\
0.282786885 & \(-\)15.158\% & 2.985\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance of corrected isosceles-triangle model on a test set of experimental data. Figure 3: View from rear camera showing graph-paper lined channel marked with points 50 mm apart on the top edges.