# Bisector-Based Tracking of In Plane Subpixel Translations and Rotations
## 4 Results
### Experimental Results
#### 4.2.2 Video Processing and Object Tracking Calculation  
The video sequences were processed offline following our own algorithm implemented in Matlab as we explained in the Method section. * Rotation stage  
In Figure 7, we show images resulting from the main stages of the image processing algorithm used in the experiment of the rotation stage. The logo of our University was the object to be tracked,  
Figure 6: Scheme of the lab experiment to measure the torsion and bending of a cantilever steel beam. The units of distances drawn are millimeters. so, from each frame, we cropped a region where we expected to extract two nonparallel segments (Figure 7a). Then, we quantized the image using two quantization levels (Figure 7b) and extracted the edges (Figure 7c) using the Canny method [18]. Next, we applied a binary opening filter to remove objects smaller than 200 px, to avoid undesired interferences, and manually selected two regions of the processed image that contained the two nonparallel segments (\(s_{1}\) and \(s_{2}\)) (Figure 7d). There, we got the coordinates of both segments. The two masks were automatically transformed frame to frame following the relative movement. We computed a location vector for each frame of the sequence. Figure 7e shows a frame of the sequence with the selected segments (blue) and the computed intersection (red square). In Figure 8, we present the variation of the angle \(\beta\) of the computed location vector in time through the video. Since we were not able to compare the position with the theoretical value, we only evaluated the parameters of the rotation stage provided by the manufacturer. As it was said above, the trajectory was a rotation of 5 degrees, in accordance with the registered rotation. The start and the end of the curve correspond to the time intervals when the object is stopped. There, one can appreciate two extrema zones where the stage motor accelerates and decelerates, in concordance with the employed trapezoidal velocity profile. Between these zones, the stage theoretically reached a constant maximum angular velocity of 1 deg/s. We have approximately cropped these three zones in the curve and fitted them to uniformly accelerated and constant velocity movement Equations, and obtained significant correlations (\(p\)-value \(<\) 0.05). The first interval, which corresponds to the start of the engine until it reached constant velocity at around \(t=2\) s, was fitted assuming it was uniformly accelerated, i.e., \(\beta(t)=(\alpha_{1}/2)t^{2}\). The initial acceleration \(\alpha_{1}\) resulted in \(0.34\pm 0.02\) deg/s\({}^{2}\), with a Pearson coefficient \(R=0.89\). Next, approximately between \(t=2\) s and \(t=6\) s, the curve was fitted to the straight line \(\beta(t)=\omega_{m}(t-t_{1})\) with \(R=0.99\), thus assuming constant angular velocity movement. The slope resulted in \(0.978\pm 0.004\) deg/s, which deviates \(2\%\) from the maximum theoretical value, and \(t_{1}=1.29\pm 0.03\) s. At this instant, the angle in the uniformly accelerated phase would be \(\beta(t_{1})=0.28\pm 0.03\) degrees. Finally, we assumed that the rotation stage uniformly decelerated and fitted data from \(t=6\) s to the end to the curve \(\beta(t)=(1/2)\alpha_{2}(t-t_{2})^{2}+\omega(t-t_{2})+\beta(t_{2})\). We constrained the angular velocity to be \(\omega=\omega_{m}=1\) deg/s. We obtained \(\alpha_{2}=-0.41\pm 0.07\) deg/s\({}^{2}\), \(t_{2}=5.2\pm 0.3\) s, and \(\beta(t_{2})=3.8\pm 0.2\) degrees, with \(R=0.94\). Figure 7: The image of the object fixed to the rotation stage was (**a**) cropped, (**b**) quantized, (**c**) edge extracted, (**d**) and removed of small objects, successively. From the final step, we selected two areas from where the segments were extracted. (**e**) Frame of the recorded video (see Visualization 2) of the object in the rotation stage together with the selected segments in blue and the computed intersection.